SCHOOLS MATHEMATICS LIBRARY THE UNIVERSITY OF ILLINOIS LIBRARY The Frank Hall collection of arithmetics, presented by Professor H. L. Rietz of the University of Iowa. OFO6H 513 W/6™m / 6 NOTICE: Return or renew all Library Materials! The Minimum Fee for each Lost Book is $50.00. The person charging this material is responsible for its return to the library from which it was withdrawn on or before the Latest Date stamped below. Theft, mutilation, and underlining of books are reasons for discipli- nary action and may result in dismissal from the University. To renew call Telephone Center, 333-8400 UNIVERSITY OF ILLINOIS LIBRARY AT URBANA-CHAMPAIGN AUG 2 4 1999 SEP 1 4 ECD L161—O-1096 pom Sen» ety ad eee MATHEMATICS FOR CoMMON SCHOOLS ARE eh GRAMMAR-SCHOOL ARITHMETIC INCLUDING EASY ALGEBRAIC EQUATIONS AND SIMPLE ) GEOMETRICAL PROBLEMS BY JOHN H. WALSH ASSOCIATE SUPERINTENDENT OF PUBLIC INSTRUCTION, BROOKLYN, N.Y. BOSTON, U.S.A. D. C. HEATH & CO., PUBLISHERS 1897 CoPpYRIGHT, 1895, By JOHN H. WALSH. Norwood J8ress : Jj. S. Cushing & Co. — Berwick & Smith. Boston, Mass., U.S.A. oy eg a cnet ie W \6-m marnennarics UA | es PREFACE. THE Primary ARITHMETIC, Part I., of the two-part edition of MATHE- MATICS FOR Common ScuHoots is designed to cover the work of the first four years, and contains those portions of the subject needed by all pupils of the common schools: addition, subtraction, multiplication, and division of whole numbers; simple fractions; and the most commonly used denomina- tions of compound numbers. Part II., the GramMMAR ScHoot ARITHMETIC, completes the ordinary grammar-school course in this subject, and contains, besides, two chapters on algebraic equations and one on elementary constructive geometry, with applications. The first algebra chapter should be taken up with the seventh year’s work in percentage and interest. The remaining one, Chapter XV., may be profitably studied where there is time to continue this subject. Although placed at the end of the book, it is intended that suitable portions of the geometry work of Chapter XVI. be taught from time to time during the last two years of the grammar school. The special features of the work are its division of the arithmetical portion into half-yearly chapters, instead of the ordinary arrangement by topics; the omission, as far as possible, of rules and definitions; the very great number and variety of the examples; the use of the equation in the solution of arithmetical problems, especially in those of percentage and interest; and the introduction of the elements of algebra and geometry. Believing that there is some foundation for the complaints frequently made by business men and high-school teachers that grammar-school gradu- _ ates are too often slow and inaccurate in ordinary computations, the author has furnished throughout the entire work systematic drills and reviews in the addition, subtraction, multiplication, and division of ordinary numbers and of fractions. 464193 lv ‘PREFACE. In this endeavor to enrich the grammar-school course in mathematics, the attempt has not been made to shorten it so much as some may desire. The intelligent teacher can and should do the remainder for himself, by rigorously omitting all such topics as he finds unnecessary. J. H. W. Brooktyy, N.Y., January, 1895. CONTENTS. — PART II. pte a CHAPTER VI. PAGE Mrxep NumBrers— FEDERAL Monry — Britis — DENoMINATE Num- BERS — DECIMALS — — MEASUREMENTS GUNN es AMG = staal oi, Lehane keen Mixep NuMBERS. . . oy Va Ang Ae NE irint TA Waly ene inh aaah NE Addition of Mixed Re PEC ERD ARR eR i uti No M8 ME UPMITOSPATIC TE CUTE trannies va ics Srt Wiel heat tyes Nk ake ht etna PDEPACuOt OL) WiIXOG) INUIMUETS Noh, Huretrees wr seh ihe tet al, on fea inal Reo PrestNman DE NUMEEATION( +) 4G Vict bestal ve Ble) lc avy et ep iehe ae Niinucanomot Mixed Numbers (5 save) ei oy ist) s hos «) em ete Division-of Mixed Numbers.) v2 ee a4 SES Ee ALON Ne MEET oedine Masitinge oie) hae Urar Uwat) 9tat aude unter Pracuorabe arts Ol SOU ET i". ue bse) edie Agi cn s,m aah Dire OO STA MONGY Cree and ccs Pine Mi) cat abuses oso veer entre Pere DECK IMALION A 4) Wall Vaio ils cet orn WAN Lowe, elt he iterabe at taal PENA TH ANU MEMES mre oy cums el | aiidt piece Gerd ey les yee aoe eee Ue VIDOES a Ree cls A i eee et OT Ai VR sage PRE NOC HSTIR Gum Aries Weta alts. | Miia ie Latics’ Eile ya. es sl) a oe engl PUR IECITSOAR LV CLOD Laie is ce Ye siGh, iei\ col Watt ve Mev lan Va aati os See a Rene Breer MN HGRULO Mme eater Myo Uw wr er Sud Pratl) ch atG Lega vats eee CATE VEE Bg VEE Gof pee a Be TOR ny ora ica St ec eae nm a) SENG og wake hy SST AM AR TTR Sy ce got ae SRE RIES DE a OR el My et tae ee SW) ot dae ian Ng, le Al WY cite MER POMRLROT CA TCS NUMIGL ALON 9 0. Jill’. sical es Rae Sac cay ble. lm. 1s ne eae AL) Pe RTCPRDAOT COR Ori Are oss Vale ag aed MEL Pate Hays kek del o'r Sete Subtraction of Decimals . .. . ROL Ppt ia OO) nanan Tee MANN PS Multiplication of a Decimal by an Tee! WON e SU sar, rap ete Pan Nel Gib? 8, Dimon. cis Decimal by an Integer... ew es [5 Ne eo) B44 Aan eT eee ee rae Sipe nS. CEN eh, tatty A aot went BAB al CONTENTS. — PART II. CHAPTER VII. Fractions — Decimats — Brnits — DENoMrINATE Numpers — MEas- pier UREMENTS: 5 ei ne ahh) peg ety, RCNP Og ADDITION OF FRACTIONS... 0. Una ewe ae ee SUBTRACTION OF IPRAOTIONS “4.0. ac a). 86 4 Ny Factors and Multiples’. ..(7 3) RAL 0%) oa ke Prime Namberp eo ss oa ake Gl eee Greatest Comnion Divisor .. 4) 4 6 Ga wh le ee Lowesb'Terms 20 4h awe ce le) ay cs Jieast’' Common Multiple’ 2. 5 2k ee 8 ADDITION AND SUBTRACTION OF FRACTIONS: (34/04), <9) oy oe Special Drills oy Sl he ey ce ee a Cancellation. ive ce. le ye ee eta MULTIPLICATION OF (EP RAOTIONS «oA. 6) isha as eee DIVISION OF PRAOTIONS «S/S 5, tees. ee ticey © ok od Fractional Parta of a: Dollar vi). 3.) ee Ve 8.) ee BrByS hoops ha) eo tary 8) aloe 561 ween el eG Ok 9 eee Short: Methods) (60 eg eee alle eg ol) a MOLTIPLIGATION OF DECIMALS 40: ss 5s ee et DIVISION: OF DEOIMAIS ie RP ge hay ek a tee ne Sight Approximations: ; (5 )...0!s) 3/5 0) ee DENOMINATE: NUMBERS «0.5/8 0k Gd ong’ Measure. hse bee er tt it ay a, MBASUREMENTS © 481 ae RS he ee Ol ke, nent CHAPTER VIII. DECIMALS — Brits — DENoMINATE NumBERS — MEASUREMENTS — PERCENTAGE — INTEREST’ =: 2)... of 3's eS Se DECIMAES) 0 Ry ae ns a gee ke: | bela a ata Reduction “0 eee oy a) oo Sys ea te Addition’ ai poe: ae i he len ge) Subtraction. 2) is) faa en) co al ce 293 Multiplication’: 2% 0.0.0) Whoa) et ns tice e) « Jira rtnnt re ee DiVIRIOD 6.556 ye SE, ee dag eee ae CONTENTS. — PART II. MEASUREMENTS Special Drills . Short Methods Approximations . DENOMINATE NUMBERS . Reduction Descending . Reduction Ascending Addition and Subtraction . Multiplication and Division . PERCENTAGE BILLs . INTEREST . la, AREAS OF Ricut-ANGLED TRIANGLES Short Methods CHAPTER IX. DENOMINATE NuMBERS— SURFACES AND VOLUMES — PERCENTAGE — INTEREST DENOMINATE NUMBERS . Senne Reduction Ascending and Descending Compound Addition Compound Subtraction . Compound Multiplication . Compound Division . Special Drills . Short Methods aetie Avoirdupois Weight (Long Ton) MEASUREMENTS Time between Dates PERCENTAGE INTEREST . Approximations . vil PAGE 297 298 303 305 307 311 311 312 314 316 319 320 322 323 326 326 326 329 331 332 333 337 340 341 342 345 349 349 303 Vill CONTENTS. — PART II. SURFACES Square Measure . VOLUMES : Approximations . Cubic Measure Troy Weight . SURO AER UF ANGLES, TRIANGLES, QUADRILATERALS . Areas CHAPTER X. ALGEBRAIC EQUATIONS. OnE UNKNOWN QUANTITY Clearing of Fractions Transposing CHAPTER. XI. PERCENTAGE — INTEREST — DiIscouNT— SURFACES AND VOLUMES . PERCENTAGE free ena To find the Base or the Rate Profit and Loss MEASUREMENTS INTEREST asthe Interest-bearing Notes . Special Drills . Approximations . Short Methods Bank Discount iE aE ee Discount of Interest-bearing Notes English Money . CoMMERCIAL DIscouNT . SURFACES AND VOLUMES . CHAPTER XII. SIMPLE AND CompounpD INTEREST — Discount — CAUSE AND EFFECT — PARTNERSHIP — Bonps AND Stocks — ExcHANGE — LONGITUDE AND TIME— SURFACES AND VOLUMES . PAGE 353 354 358 360 361 361 365 367 069 369 373 377 381 381 383 386 389 393 394 396 398 399 402 407 408 410 413 415 CONTENTS. — PART ITI. 1X SimPLE INTEREST. . . . LSBs iy HEM Mal ay) «| Cae EO To find Principal, Rate, or Time SOV ies h ceed Coe ReeMeae SAN ge Ley interes py Asiquot Farts 12 Teeny. Neer tee shee CATT DIE METER ML IRCOUNT. Poa Sets cs Leal, ele ieneatine Liduhielus \aelttern tes bak BANE DISCOUNT». <,' Sk eC edey e ea tel eeLe To find Face of Note, Bite ae Diaeoune or Time iteatt Fe) cia. NBS Ml ene ree SUS EU MAYTALS aR aL thea? ha re aad aC A ei a AT 0 BEES MOUNOCS Ree ete en Lah eR R ESR ies ocean UN ae MARR RSET ee, Gh al ao te Wage Mee Me RI RACY A TMT GROTON et iia ey Coe ih ay ce Macdkye Mra Satter) atinte sa uin ere ey CEL Pang TES 2 A PL PT Sg SO A POLAR es 2 Pe POEUN AON con Wine Mim eM re gh tes Meus nant ail ound Aen ng Pet nee PRR EPEC STOOLS) oh e Yshecte cme Ch cf SME ate (at ec a. caiiita ine ee ee ae Beem UMTS INTEREST eh. 5 fist tes) ls ai) co Mtlpenie es ES A Ne A ate 3 SBO EXCHANGE... POEL RES oy SUI aN Mate RIN Oat HAN VEE eae hue Domestic Sight vee RA Keok OHMS GI SNe Pela yenreane pa CHPCULAT? WLGABUNO ror) Sue cia ee roe hie Lah ape tar “aytanmieerL cmap OL) EGET IEALCAPET IIIS. chet eM ic) san te fauitica Is Welple W alata eer) woah y LoNGITUDE AND TIME. . . SERN eatin te ch cal ta A sep sat Aveda gm WML EDD Bills of Exchange Feathany BM Teme e Neh NC at! onl! Riera!) Sula es CHAPTER XIII. PARTIAL PAYMENTS—RATIO AND PROPORTION —SQuARE Root— RRDURFACES: AND) VOLUMES. Bee ppd eel Ghia 2 hs 458 eatTA PAY M ERTS UO ULE aes) Oe Pe Alay o) KAU 488 Presentaworou and True’ Discounts 67. 2) eee eo. 460 ene eee GVOLOMES Ie. Wt seh eh ay tn) RO wy 4OR Der H aTCO Teme terete isha Coy Nits 1) MOP eS ae UE Oa TRA TIO cha sene Mk, Pah ae ere ANI IG) eh Watts A RE EOS Special Drills EA a URS DRA RAB URS URC MCS RW DRL Be Yi PROPORTION. . . tga Ve Shae eR ly hie ne Mas es Ch wee Applications of ey EOOU MAEM tatu ge lie VR tut cumay Tenis Vas Stat heOo x CONTENTS. — PART II. MEASUREMENTS: 05's (solu SDR ce a Hixact Interest ecg i Picea eet ae PARTIAL PAYMENTS— MrRcHANTS RULE... ..... CHAPTER XIV. HQUATION OF PAYMENTS—MENSURATION oF SURFACES AND Vot- umMES — BoArD MEAsuRE— ANNUAL INTEREST — GOVERNMENT Lanps— Merrie System EQuaTION OF PAYMENTS MENSURATION OF PLANE SURFACES Special Drills . SURFACES OF SOLIDS PPISHIS ATIC ULI OOPS 5 one 7 i ae ae a ate Pyramids and Cones VAT NCIS 100) SET) ST Es SN ak ha ha es ca Lumber Measure Surface of Sphere CORE Ronis oyet bree ide ieee Nak est Galle) WO Ui an an Volume ofSphere)( i) Py 0). 4 ld eee ee ae ANNUAL LNTER EST Gir ct. (llc evil cunedine, wean Outen Can er Government Danda rye) Mle ese ate ene RMIETRIOUS YSTEMS 575. ho OUR ae oe AGED oe ee een CHAPTER XV. ALGEBRAIC Equations—Two Unknown QUANTITIES— THREE Un- KNOWN QUANTITIES — PURE QUADRATICS — AFFECTED QUAD- RATICS ADDITION oF ALGEBRAIC QUANTITIES SUBTRACTION OF ALGEBRAIC QUANTITIES Removing Parentheses . Two UNKNOWN QUANTITIES . THREE UNKNOWN QUANTITIES 2; . 3 wh 6 ue 499 499 503 505 511 511 512 513 514 518 519 521 523 524 525 530 530 532 534 538 544 CONTENTS. — PART II. Xl PAGE MULTIPLICATION OF ALGEBRAIC QUANTITIES . ........ 5A ROPER EGIL A UA TEC cee c! gle’ (ass etd MMOH MO UO eer a AC Gis POLO PUR PROTEINS AIGA DATION 9). Seu Wee ha ee) Leica AG Hat Py BBL CHAPTER XVI. ELEMENTARY GEOMETRY — PROBLEMS IN CONSTRUCTION — PRACTICAL _APPLICATIONS—CALCULATION OF HEIGHTS AND DIstTancES— BU TT O Mere fee ism bettie vise TAN Sieg eis Lo eat area EEN Pee CA LT BOM ET VG te ells SG eck be Wes Ugh gel Mgl Yene lhe Fe cayh OOy Eee EAU ITLA ETUC OID tins (cl eo an hls doled iar eae cede ahr ty ane PCE CC MI EUPUCHOT ear cae ee ee lat eno fa ieee EO Equal Triangles— Equivalent Triangles . ...... =. =. 584 MLC AUP lGM ew Aah lb ada a: Beams hirer tates “ky eh nat ee OOO PALQuuATION OF HEIGHTS AND DISTANCES... 8. 0. 6 6 ne 1087 Mee ere TOME OP EOUREAUES (0/200 sila (le. uids ah cel Meyiet oo ele le Dee Prams, Ovinderss Pyramids; Cones ah) 6) ss Ne sce, ee yn ia a OOS Frustum of Pyramid or Cone ..... . CLUE CNP SAAN it Sie, 8: Sirs meM MMP EN Laerir is cea (Ce Mt fal! all aioe se 4 tere AOS Stee ROM ASM Sac esi eh taf al Ls she) aie, oan, fer) ce ahve nel BO Breer UU OOTS ie Mrs eae hel at hd: tel fU pit wi Sin iin ell eh eig Mae Bye PDL STAT CML ORIN AEs Wig Tru'gic fs Wibeh Gea iier he lite) (#i-s)) ay Koay sh OMD Brmecama Of byramiag and Cones’. 0. a a Me week ot le ey LOOL ReeeVUL Ge Lips mens uae Ge wel e ag) ve tye alcUycay Tile) Tg tahh tet: baanl (<6 Liat IO Bere ms Era hte hah ho A Ant igh a Val a Winmteh S) # 7! ool CODE APPENDIX. iene ome W RIGHTS AND MEASUBES - lu pote ts ie 8 ei ce ees BOF PERC neo Lens Wee) RON) eve nin se hls Tarr Cre a BOO Days OF GRACE . . . . . . . . . . e . . . . e . . ° 609 Xi ‘CONTENTS. — PART II. PAGE RATES OF INTEREST °° ce 8)0) hi EA Es Ee eed ee PARTIAL Payments, Connecticut Rule. 2) o>. (cee Annual Interest: Notes)... oS ER i ee ee New: Hampshire Rule. .)5 ea ear. ote Vermont Rule!) \ oi apes Sree ele te eumen ee TAXES wel EE Woon) Met dM ial NS Tin eer tra ah Ae, Ar Vermont Method of Levying Taxes). 3 5 eee VaLuEs Or Forrian Gores) Woe Wie) ao eg area } B65). en PE PEERS UPAY RRR GR ie ne gS. GRAMMAR-SCHOOL ARITHMETIC. CHAPTER VI. MIXED NUMBERS. --FEDERAL MONEY. — BILLS. — DENOM- INATE NUMBERS. — DECIMALS. — MEASUREMENTS. MIXED NUMBERS. 447. Oral Exercises. How many halves in 1? How many fourths in 1? Six halves=? 12 fourths =? 6thirds=? 12 sixths=? 448. Slate Exercises. - Add: es 2. 58 3. 182 4, 32 5. 744 18 391 1502 4 34 274 17 572 272 1. 449. Oral Exercises, $=? $=? $=? §=? Ya? yo? 450. A mixed number is a whole number and a fraction. 451. Reduce to a whole number or to a mixed number: So re ee an a 199 200 ARITHMETIC. 452. Slate Exercises. Add: 6. 3h Tour 8. 9% 9. 3182 10. 875%, 95 291 481 53 172 2544 734 351 527 353, 74 61 34 14 694 453. Oral Exercises. How many quarts in a gallon? What part of a gallon is a quart ? 4 gallon = how many quarts? += how many fourths? How many quarts in a peck? What part of a peck is one quart? One-half peck is how many quarts? One-half peck = how many eighths? + peck is how many quarts? +=—how many eighths? 2= how many eighths? 3= how many eighths? coleo at = eal ete fa 1 al Ale ne? 12 12 12 12 12 12 12 12 1z 12 12 454. Draw a line one foot long. Draw a second line of the same length; divide it into halves. Divide a third line of the same length into three equal parts. Divide three other lines, one into fourths, one into sixths, and one into twelfths. : : MIXED NUMBERS. 201 How many inches in a foot? What part of a foot is one inch? 4 foot = how many inches? 4= how many twelfths? = how many twelfths? #—how many twelfths? Change 1 to twelfths. Change 3, ? ia twelfths. How many twelfths = 4? 22 82 42 22 8? ee eer PETES Sef 4! sy ames spears ye deci aris Rewer s et? Steet? Sie eS Peri Pes Siege Daehn Grea by ones 10 =? pe EN Tele PE IAG | oer oO PE Pe ares | wert Bina - How many inches in 3 ft.+ 4 ft.+ 4 ft. +2 ft. +7, ft.? How many feet and inches? How many 12thsin3+4+41+1+4,45? Change to a mixed number. Change the fractional part to a different fraction hav- ing the same value. What fraction of a dime is 1 cent? 4 dime = how many cents? += 75. 4 dime=how many cents? 4,5. Change #to tenths. 3. 4. 2. Add 4 dime, + dime, and ;4; dime. How many cents? How many tenths = 4++4-+44,? Can you change the answer to a different fraction having the same value? 455. Slate Exercises, Add: sb a BS 1262) EG 8 ie os ees os Beige aa Bs 2704 53h 61 164 1834 31 9535 172 O54 O1 16 " eae u uae: 4.56. Oral Exercises. Show by a diagram that 4 is the same as 2. How do we add 4 andi? Show by a diagram. How many hours ina day? Indday? Iniday? In} day? Iniday? Iniday? In +, day? Change 1 to oa -fourths. 4. 4 4. 4. 4. Reduce 2, $, 8, 2, 3, z5, $, 44 to 24ths. 202 ARITHMETIC, 457. Slate Exercises. Add: 16. 94 17. 273 18. 33,5; 19. 634 20. 871 8} of 675 af 53g 268 1003 924 or 9,1, 21. 46,5, 22.36 23. 2751 24. 932 25.. 238 52 743 54,3, 64 654 1 91, 274 742 234 2074 812 64 87 94 458. Oral Problems. 1. I spent 4 of a dollar fora ball and +4 of a dollar for a bat. What part of a dollar did I spend for both? 2. += how many fourths? 3. What will be the cost of a pen-knife at 3 of a dollar, and a book at 4 of a dollar? 4. Write 4% with smaller numerator and denominator. 5. I need 4 of a yard of ribbon for one hat and 4 of a yard for another. How much ribbon must I buy? 6. Write a fraction equal to 5% with the smallest numbers youcan. (This is called reducing a fraction to lowest terms.) 7. Sold 2 of a pound of tea to one customer and { to another. How much was sold to both? 8. What quantity of oats must I buy to give $ of a peck to one horse and 3 to another ? 9. If I sell 3 of a dozen of oranges to one person and } of a dozen to another person, what part of a dozen do I sell? 10. What part of an hour is 45 minutes ? MIXED NUMBERS. 203 11. 2 of an hour is how many minutes? 12. I spent 4 of an hour reading and ,8, of an hour writing. What part of an hour did I spend at both? 13. A boy is carrying 64 lb. of flour, and 62 lb. of ham. What is the weight of his load ? 14. Reduce 43 to lowest terms. 15. 18 hours is what part of a day? 16. Reduce 4$ to lowest terms. ADDITION OF MIXED NUMBERS. 459. Add 124, 62, 84, 158, 4. In the fractions 4, 2, }, 3, 3, the numbers above the line, 1, 2, 1, 5, 3, are called nwmerators; the numbers below the line, 2, 3, 4, 6, 8, are called denomt- nators. To add fractions, they must have the same denominator. An inspection of the denominators, 2, 3, 4, 6, 8, shows that 48 or 24 will contain each without remainder. 24, which is the smallest number that will contain all, is called the least common denominator. Oo 121 | 12 62 | 16 81| 6 158 | 20 a| 9 Instead of writing the common denominator 24, with each fraction, we may place it above, and write only the new numerators. 4=332, =}, 1 = 6, etc. Write 12,16,6,20,9. The sum of these fractions is $3 = 2}$ = 23. 5 is placed under the fractions to be added, 2 is carried to the whole numbers, making 43. Nors. — The fractional parts of answers should be reduced to lowest terms. 204 ARITHMETIC. 460. Slate Exercises. O> Oe oA 19. 9921 + 888.4 778 20. 1058+ 322 + 472 Add: vast 2. 734 3. 932 4, 112 5. 181 634 8} 24 32 72 1% 393%5 4s 2055; 9.8, 8 yo 16} a 54 4 6. 125 + 355 + 27$ + 8} Te dont, SIERO 28 site ee Bib 7BR. cb OSE iat aioe 9. Ste + 388F + 233 + 1 10. 1003 + 754 + 9% + 494 11. 332 + 178 + 245+ 69); 12 Oly Oh Sea ee 13. 103 + 842 + 253 + 98 14,9188 0-2 3018s Sheen eons 15. 4444 +5188 + 37, + 952 is. 93%) 127aN- 18h pods 17. Sie 1 64S) 1268 Oe 18. 24 Ube aS ae eee -+- + CO ol MULTIPLES AND FACTORS. 461. A number that contains another number an exact num- ber of times is a multiple of that number. 24 is a multiple of 12; 36, 48, etc., are also multiples of 12. 30 is a multiple of 2, 3, 5, 6, 10, 15. MIXED NUMBERS. 205 462. Oral Exercises. 1. 95 is a multiple of what two numbers? 2. Give two factors of 51. 3. What number is a multiple of both 8 and 6? 4. Mention another number that is a multiple of both 8 and 6, 5. Find the smallest number that can be exactly divided by 8 and 12. 6. Give two factors of 91. 7. 571s a multiple of what two numbers ? 8. What is the smallest number that can be exactly divided by 4, 6, and 8. 463. The smallest number that is a multiple of two or more numbers is called the least common multiple of such numbers. 9. Find the least common multiple of 5, 10, 15. 10. Find the least common multiple of 2, 7, 14. 11. What part of a dollar is 70 cents? 12. 3—=how many 40ths? 13. Reduce 172 to thirds. 14. Change 7 to a mixed number. 15. Reduce 24 to lowest terms. 16. What part of a dollar is 15 cents? 17. Add 4 and 4. 18. From } take 4. 19. Reduce 3% to 60ths. 20. Which is largest, 2, #, or 3? 21. $o0f 84=—? 22. From 4} take s4. 23. From 4% take 3. 24. From 13 take +. 25. From 44 take 3%. 206 ARITHMETIC. SUBTRACTION OF MIXED NUMBERS. 464. Sight Exercises, Subtract : 1. 1614 2. 4919 3. 3828 4. 18,8 5. 275%, 1355 37438 2944 143, 16}; 6. 28,. 7 ATE 8. 3611 9. 2518 10. 321% 13% 295 18355 1944 187%, 465. Slate Exercises, 11. 35 12. 68h 13. 278 14. 553 15. 105,38; — 82 — 975 —174 — 253 — 81 16. 1208 17. 892.7 18) 188) 18s) OUR 20s — 847 — 884 — 155, — 214 — 594 21. 480 22. 375 23. 200 24. 873 25. 1,000 — 974 — 881 — 904 — 7572 — 9985 26s rev ou 27 et 1G 28. 999 29. 132 30. 23 — 2503 — 768 — 1382 — 46,3, —184 466. Oral Exercises. —1=? l}-}=? 10-1=? 10}-1=? 10h-1}=? 467. From 1972 take 682. 15 Reduce the fractions to a common denominator (15), as in 1972 9 addition of fractions. 42 being greater than ;%, we find the 682 | 10 difference between +2 and 1,%, or 24. Writing this difference, 12914 | 14 1$, we add 1 to 68 and subtract. Loe. & MIXED NUMBERS. 207 468. Subtract: 31. ~ 8} 32. 231 33. 344 34. 161 35. 211 5} 58 272 81 ga Bomcey ennsT Soh, 88. 77) 884) D014) 40. 998 293 14 594 244 881 41. 1008, 42. 25,8, 43. 93,2 44. 10199 45. 122 763 51 O44 983. 48 oe eee 48. 188%. 49. 161. 50... B7E loz5 3} 277% 348, 294 pimoeee 52.) 642 S08, 125 te) BA ATE! BB ys Tod 495 181 1008 88 504 Bos ale 957) 63, 58. Sy, 69. 252. 60. 1025, 273° 445 1} 174 864 469. Slate Problems. 1. From a piece of cloth containing 174 yards, 53 yards and 42 yards were sold. How many yards were left? 2. A boarding school uses 3 quarts of milk a day for 7 pupils. If there are 77 pupils in the school, how many gallons of milk will be needed per day? Per week? 3. A man pays $140.40 for 3 pieces of cloth. What is the length of each piece, if the cloth costs $1.80 per yard? 4. I buy 12% pounds coffee at 28 cents per pound, and twice as many pounds at 24 cents per pound. If I give the store- keeper $10, how much change will I receive? 5. A merchant pays $30 for 65 vases. He sells 17 of them at 40 cents each, 23 at 60 cents each, and receives 48 cents each for the others. What is his profit? . 208 ARITHMETIC, 6. Mrs. Jones buys 4 dozen chairs, a bureau for $17.40, and a mirror for $18.00. She pays for all $38.80. What do the chairs cost apiece ? 7. In selling 40 yards of velvet that cost me $1.40 per yard, I gained $24.00. What did I charge per yard? 8. How much heavier is a cheese weighing 403 pounds than one which weighs 26% pounds? 9. A farmer sold 364 dozen eggs to one storekeeper, 52 dozen to another, 17% dozen to a third, 83 dozen to a fourth, and 11,4 dozen to a fifth. How much did he receive for them at 12 cents per dozen? 10. A butcher going to market with $174.40, bought 15 sheep at $8 each. If he had had $25.60 more, he could have bought 4 hogs also. What were the hogs worth apiece ? 11. A scholar having to multiply a number by 30, mistook the 8 for a 5, and his answer was 600. What is the correct answer ? 12. Forty hats were bought for $104. At what price apiece should they be sold to make 40 cents on each hat? At what price per dozen ? 13. A farmer had 7 bushels of potatoes. He used 2 bushels and 3 pecks for seed. What would the remainder be worth at 20¢ per peck ? 14. A butcher buys an ox weighing 1,200 pounds alive, at 6 cents per lb. When killed and dressed, its weight is 2 of the live weight. What is the butcher’s profit, if he sells the meat at an average of 15 cents per lb.? 15. The weight of a tub of butter, including the weight of the tub, is 484 pounds. The tub weighs 94 lb. What is the butter worth at 24 cents per pound ? MIXED NUMBERS. 209 16. One boy had 15 marbles, another had 19, a third had 17, a fourth had 18. What was the average number of marbles for each boy? 17. A teacher divided 200 foreign postage stamps among the eight boys of his class. He gave one-fourth of them to the first boy, one-fifth of the remainder to the second boy, and then divided the rest equally among the other six boys.. How many did each receive? 18. A merchant sold 17% yards of muslin, 144 yards of silk, and as many yards of calico as of the other two together. How many yards did he sell in all? 19. A dealer mixed 23 pounds of black tea costing 32 cents per pound with 14 pounds of green tea costing 40 cents per pound. How much per pound does the mixed tea cost him? 20. A man can do 4 of a certain piece of work in a day; another man can do } of the same work in a day. What part of the work can both together do in a day? How long would it take both together to finish the work ? 470. Slate Exercises. Find answers: ha atte 2. 124 3. 282 4. A474 5. 28 x 7h x 192 x 8 x 12 x 48 ibe Aes 7. 848 St 975 69 102) Sl x 1062 fxd x 3h x 44 x 154 471. Divide: 11. 9383+3 15. 808 +5 19. 8252211 12. 564+4 16. 126126 20. 1,72812 +12 13. 985 +7 17. 450% + 9 21. 2,46012 + 10 14. 282-2 18. 3608 =8 22. 3,92613 + 13 210 ARITHMETIC. NOTATION AND NUMERATION. 472. The largest number that can be written with six figures is 999,999. 1,000,000 is called one million. Write in figures two million. Three million. Four million. Six million. Eight million. Ten million. 473. Read the following: 1. 1,234,567 6. 11,034,065 11. 30,100,021 2. 8,000,560 7. 14,602,500 12. 35,000,600 3. 5,009,008 8. 17,886,925 13. 401,023,160 4. 7,090,070 9. 20,007,316 14. 760,030,020 5. 9,843,000 10. 25,000,005 15. 980,750,000 474. Write in figures: 1. Seventy-eight million, one hundred eight thousand, ninety- SIX. 2. Three million, eight. 3. Fourteen million, seven thousand, five. — 4. Nine hundred. eighty-seven thousand, six hundred fifty- 5. Twenty million, thirty thousand, forty. 6. Three hundred seven million, nine hundred four thousand, SIX. 7. Nine hundred ninety-nine million, nine hundred ninety- nine thousand, nine hundred ninety-nine. 8. Four hundred seventy-six million, three hundred thou- sand. 9. Thirty-four thousand, eighteen. 10. Sixty-four million, thirty-two thousand, sixteen. Add the foregoing. REVIEW. 475. Review. Slate Exercises, Read the following numbers. 1. 27,088,549 116,908,070 3,006,005 20,080,070 1,647,893 206,045 73,000 180,059 2,316,045 54,006,000 2. 508,900,007 68,487,291 4,629,880 25,936,097 134,870,603 59,009,300 7,000,004 686,909 50,308 9,999 476. Add down and across: Lt 20 + 300 + 4,000 + 50,000 ++ 600,000 + 24 30 + 400 + 3,000 + 2, 40,000 + 30, 500,000-+ 400, 3+ 40 + 500 + 000 + 000 + 000 + 3. Add each column. 248 576,908 36,200,570 5,987,600 380,070 68,000 593,056 2,384,672 59,876,004 123,321,123 88,888,888 4=? 50 =? 600 =? 1,000 ==? 20,000 = ? 300,000 =? 7,000,000 + 8,000,000 -+ 6,000,000+ 5,000,000 = ? 30,000,000 + 40,000,000 -+ 50,000,000 + 60,000,000 = ? fmol -t, foe 4. (nd tts 212 477. Multiply across. 10x 2=? 10x 5=? 10x ?=? pipe gy agg TAs ordinal LOK —=9¢ Dia Wo CAM sr ¢ 123xX 3=? 123 x 20=? 123 x 100 =? 123x ?=? 478. Add: 1. $184,635.873 25,904.63 8,756.95 889.57 4.326.981 58,030.05 209,508.67 37,654.88 9,876,544 68,250.89 6,505.83 ARITHMETIC. 2. $263,005.95 38,462.77 159,076.50 50,318.92 36,485.73 9,860.44 76,035.00 8,900.56 4,056.02 26,465.54 7,826.98 Add multipliers and products. A a loo a IZWxX Ghee 50 x Wai? 50x 26==2 50 xe 560 x oi 4a 560 .) third inal How often is 1 cent contained in $1? 2 cents in a dollar? 4 cents in 2 dollars? 25 cents in 25 dollars? 518. Give answers at sight: PO wD $4 + 10¢ $5+ 5¢ Pee 12e 4 . $86+ 69 leozsin 2b? 4-02, anv lb es laouren 5. $638+ 3f 6: B74 25¢ 7. $20+3381¢ 8. $386+ 3¢ FEDERAL MONEY. 225 9. $40+50¢ 15. $16+16¢ 10. $9-+10¢ 16. $16 + 162¢ 11. $1+i3/ 17. $16+ 331¢ 12. $38+$1 18. $16-- 25¢ 13. $84-+ 50¢ 19. $16 + 50¢ 14. $1+ 163¢ 20. $12 + 20¢ 519. At 36 cents each, how many spellers can be bought for $27? 75 36)2700 $27 = 2,700 cents. Since 1 speller costs 36 cents, for 2,700 180 cents we can buy 239° spellers. Ans. 75 spellers. 0 520. Slate Problems. 1. At $2.75 per day, how long will it take a man to earn $110? 11,000 + 275 2. How many yards of muslin, at 12 cents per yard, can be bought for $126 ? 3. A farmer spent $140 for sheep at $5.60 each. How many did he buy? 4. A grocer pays $74.50 for tea at 4 of a dollar per pound. What is the weight of the tea? 5. When rye is worth 87 cents per bushel, how many bushels can be purchased for $261 ? 6. At 124 cents per pound, how many pounds of meat will cost $175.25 ? 7. If '75 spellers cost $27, what is the price of 1 speller? 8. A woman paid $24 for 36 yards of dress goods. What did she pay per yard? 226 ARITHMETIC. 9. At 6 for a dollar, how many rabbits can be bought for $87? 10. The cost of 13 houses was $36,887.50. What was the price of each ? 521. Sight Exercises. Approximate Answers, 1. What will be the cost of 389% lb. butter at 20¢ per lb.? (Nearly 40 lb. at 20%. The cost is nearly what ?) 2. Aman has 4,200 pounds of flour which he wishes to put into barrels containing 196 lb. each. About how many barrels will he need ? (Each barrel contains nearly how many pounds ?) 3. A merchant bought a hogshead of molasses, containing 472 gallons, at 50 cents per gallon. About how much did it cost? 4. How many lots at $1,975 each can be bought for $12,000? 5. Sold 3 pieces of cloth, 33 yd. to the piece, at $1.95 per yd. Give the approximate amount of the bill. 6. 2813 + 3748 = nearly what? 7. 175} + 24,9 = nearly what? 8. 18% x 92 = nearly what? 9. 87, — 4918 = nearly what ? 10. 43 x 48 x 4,9, = nearly what? 525. Learn the following tables: TIME. €0 seconds (sec.) 1 minute (min.) 60 minutes 1 hour (hr.) 24 hours 1 day (da.) 7 days 1 week (wk.) DENOMINATE NUMBERS. 27 Dry MEASURE. 2 pints (pt.) 1 quart (qt.) 8 quarts 1 peck (pk.) 4 pecks 1 bushel (bu.) AVoIRDUPOIS WEIGHT. 16 ounces (oz.) 1 pound (Ib.) 2000 pounds 1 ton (T.) The hundredweight (100 pounds) is written cwt. Liquip MEAsuRE. 2 pints (pt.) 1 quart (qt.) 4 quarts 1 gallon (gal.) A gill (gi.) is equal to one-fourth of a pint, It is very rarely used. DENOMINATE NUMBERS. 526. Slate Exercises, 1. How many hours in 74 days? 2. How many hours in 7 days 12 hours? 3. How many seconds in 2 hours? 4. A man buys 12 bu. and 3 pk. apples @ $1 per bu. What is the cost? 5. What will be the cost of 8 pk. 7 qt. chestnuts @ 8¢ per qt. ? 6. How many pints are there in 5 gallons of ice cream ? 7. How many half-pints are there in 10 gallons of ice cream ? 8. How many 4-ounce packages can be made from 5 pounds of pepper? Y 9. A boy pays $1.50 for 1 gallon and 2 quarts of ice cream. What is the price per quart? 228 10. people 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 16 yd. 29. ARITHMETIC, How many gallons of lemonade will be needed to give 96 + pt. each? How many seconds in 5 hours? How many minutes in 1 week ? Change 13 hours and 20 minutes to minutes. Change 15 bu. and 4 pk. to pecks. To quarts. How many ounces in 47 lb. 5 oz.? How many pounds and ounces in 237 ounces? Change 1,494 minutes to hours and minutes. Find the number of hours in 6 weeks. How many hours are there in January ? How many inches are there in 2 yd. 2 ft. and 2 in.? How many ounces in 4 tons? Reduce 5 cwt. and 80 lb. to oz. How many pounds in 3 of a ton? In ? of a ton how many cwt.? What will 400 lb. of coal cost at $5 per T.? What fraction of a ton is 1,500 lb. ? How many days and hours in 2 of a week? Find the number of yards in 3 pieces of cloth averaging 2 ft. each. When coal is $5.50 per ton, how much will I have to pay for 3,000 pounds ? 30. A store-keeper charged 70 cents for a roll of butter weigh- ing 2lb.30z. What was the price per ounce? What was the price per pound ? 31. 32. Change 60 lb. to the fraction of a hundredweight. How many pints are there in a barrel of oil that contains 434 gallons? REVIEW. SPECIAL DRILLS. 528. Give sums: 56 + 25 47 +47 22 + 68 39 + 31 529. Give remainders: 81 — 56 94 — 47 60 — 28 72 — 39 530. Give products : 310 x 9 420 x 4 630 x 3 740 x 2 531. Give quotients : 196 + 4 190 +5 192+6 196 + 7 532. Give answers: O14 1b +1 +1 +1 21+ 32+ 48 750 + 190 30 + 20+ 18 390 + 120 40+ 18+ 26 480 + 150 24 + 31+ 30 620 + 180 750 — 190 750 — 560 510 — 120 510 — 890 630 — 150 630 — 480 820 — 160 820 — 660 aoe 207 65 x 3 24x 8 49 x 4 APA oe AYE pada Leap a 196 + 49 450 + 25 190 + 38 375 + 25 192 + 32 225 +- 25 196 + 28 350 + 25 1i— 2 Zof 66 21 — 14 84x 2 bt — 24 4 of 100 41 — 31 186x 2 229 225 + 54 315 + 21 437 + 60 540 + 55 — Comb la ey On ise) oye ess | o> cto wmloo bole 230 ARITHMETIC. 533. Oral Problems. 1. Paid 59¢ for muslin and 25¢ for trimming. How much was paid in all? 2. A boy had 75%. How much had he after spending 25¢ for a knife and 15¢ for a ball? 3. If 8 lb. of raisins cost $1.04, what is the price per pound? 4. At $1.89 per yard of silk, what will be the cost of 1 foot? 5. If 382 lb. of flour cost 96 cents, how many pounds can be bought for 60 cents? 6. One girl has 16 cents, another has 24 cents, a third has 8 cents. How many dolls at 16 cents each can be bought with their money ? 7. What will be the weight of 3 bushels corn, weighing 56 pounds per bushel ? 8. How many ounces in 9 pounds? 9. How many pounds in 8 packages, each weighing 16 ounces ? 10. Find the cost of 3 lb. and 2 oz. of butter at 32 cents per lb. 11. Bought 4 pounds of 6-cent sugar and a pound of butter at 86 cents. How much change from $1? 12. Four boys have 144 marbles among them. If they were equally divided, how many would each have? 13. A man earns $100 per month, and spends $76. How much does he save? 14. If a man saves $32 per month, how many months will it take him to save $960? 15. Paid $27.90 for 9 jackets. What did they cost apiece? REVIEW. Zoos: 16. Mr. B's farm contains 520 acres. How many acres will he have left after selling 180 acres? 17. William’s kite string is 435 yards long, John’s is 62 yards longer. What is the length of John’s string? 18. A farmer raised 168 bushels of grain. He sold 4 of it. How many bushels did he sell ? 19. 64 yards of ribbon are cut into pieces a quarter of a yard long. How many pieces are there? 20. Ifit takes 18% yards of cloth to make 3 suits, how many yards does it take for 1 suit? 21. James has 150 marbles, Thomas has # as many. How many marbles have both? 22. A newsdealer received $6.36 for papers sold at 3 cents each. How many papers did he sell? 23. If it takes 44 days for one man to do a piece of work, how long will it take 2 men to do the same work? | 24. A farm is divided into 4 fields, each containing 49 acres. How many acres are there in the farm? 25. From a piece of cloth containing 104 yards, 53 yards are sold. How many yards are left? 26. Find the cost of 28 lb. coffee at 25% per lb. 27. How much does a farmer receive for 28 cows which he sells at $25 each? 28. Find the number of hours in a week. 29. How many pieces, each three-quarters of a yard long, can be cut from six yards of wire? 30. 3,600 seconds are equal to how many minutes? 31. If 25 yards of material are needed for a dress, how many yards will be required for 33 dresses? 32. At 7 for a cent what will 98 marbles cost ? 232 534. Sight Exercises, Divide : CL Wate 535. Give answers in whole numbers. 20. 960 + 240 780 + 260 960 + 480 720 + 180 1,170 + 130 960 + 241 779 + 260 959 + 480 720 +181 1,030 + 1380 536. Multiply: ol. 32. 33. 34. 35. 537. Perform indicated operations : 18 + (80 x 4) 7+(2x 8)—4 [((7 +2) x 8]—4 Lead) (6xP)+4 1 of 4 of 600 1,200 x 6 1,800 x 4 2.500 x 3 1,700 x 5 1,400 x 7 51. 52. 53. 54. 55. 56. 36. 37. 38. 39. 40. ARITHMETIC. 6. 8400+ 2,100 Ain ie 7. 8,600 + 4,300 ib} 8. 8,800 + 2,200 13. 9. 9,600 -- 3,200 14. 10. 9,900 + 8,300 15. 10,800 = 1,200 10,400 + 1,300 6,000 ~~ 1,500 5,700 -- 1,900 12,000 + 2,400 (Omit remainders.) 21. 8400+2,110 26. 10,800 +1,205 22. 89500+4,3800 27. 10,300~+ 1,800 23. 8800+2,199 28. 6,100+ 1,550 24. 9599-3199 29. 5,700+ 1,899 25. 10,000-+3,3380 30. 12,020+ 2,410 1,800 x 9 2,300 x 3 8,200 x 2 1,500 x 4 1,200 x 8 41. 42. 43. 44. 45. 2,100 x 4 1,400 x 6 4,100 x 2 1,600 x 5 2,200 x 3 46. 47. 48. 49. 50. 1,400 x 8 2,400 x 4 15300 Sra 1,200 x 9 6,300 x 2 57. 4 of (240+ 60) 58. (7+2)x(8—4) 59. 7+[2x(8—4)] 60, ate 61. 6X (4+4) 62. $x 12x23. 538. Place the following upon the board. The pupils write the answers and nothing else. 582 97 617 77 299 36 458 57 63. 64. 65. 66. 67. 68. 69. 70. 209 29 374 45 582 87 292 56 539. Slate Exercises. Add: 1. $3,947.25 14,816.00 956.83 2,469.98 95,783.04 9,005.79 6,598.86 1.58 39.99 463.27 85 4. $18,477.09 494.78 1,489.07 104.84 91.03 20,999.99 7,583.94 87.62 3.47 6,952.83 REVIEW. 233 will (Art. 385.) 907 524 Be Ol 1. — 5. — OF nes . 93 f 76 a 85 470 430 295 2. — 6. — ._—_ 67 u 86 oh 66 260 310 876 3. — ._— Poms ui 55 bd 54 : 95 400 865 573 74. — 8. — 82. — Tie keno mee ae Review, 2. $14.92 3. $1,094.07 3,120.50 789.14 18.72 9,870.00 79,841.24 4,009.89 3,972.87 484.78 104.99 9,741.96 19.90 419.74 19,877.46 4.58 387.24 999.10 91.85 23.46 901.09 98 5. $46.89 6. $48.34 VAY 875.39 3,538.39 82.76 468.438 9.87 56.19 538.49 3:37 835.47 786.49 3,457.96 5,898.39 85,473.89 65.40 4,938.78 808.34 453.48 —— - 234 ARITHMETIC. 540. Supply missing numbers : 7. 9,256,874 8. 348 863,052 2,967 24,635,998 36,847 7,007,007 243,837 ? 3,096,846 85,386,950 ee 6,875,634 183,634 3,987,456 986,246 30,068 8,216 705 586,237 139,049,086 6,000,000 10. 3,157,842 11. 749,809 re 980 1,308,215 9,876,543 930,084 1,234,567 17,521,938 468,208 743,150 63,593,065 9,807 82,389,659 420,985 1,293,714 73,612 460,045 9,708 i 65 15,813,477 35,986,210 293,352,032 541. Multiply: Lore. 20 (2xK15,014. 19. 14, 35,482 x 7982 20. 15. 5,290 x 6,075 21 16. 9,204 678% 22 LS 1 O,Uiax< 9 (395 23. 18. 68,431 x 9242 24 95 185 9. 7,293 82,538 786,324 ? 94,649 1,009,765 256,834 3,983,387 54,619 760,888 10,685,391 12. 8,852,465 37,947 40,897,654 390,784 ? 1,246,937 4,373,539 301,236 9,764,318 665,524 74,638 100,001,010 x 95x 95 SLU Pers 874 x 23x 386 . 706 x 804 x 509 482x 82x 74 . 0388 X 247 x 125 REVIEW. 250 542. Divide: 25. oec00= 1 810 $1. 68,703,705 + 12,345 26. 2,823,150~— 1,298 32. 861,420,135 + 56,789 27. Goo dae 11624 33. 70,870,088 + 25,986 28. 21,345,738 + 72,100 34. 4,510,940+ 4,900 29. 1,861,704= 3,510 35. 34,689,215 + 39,783 30. 20,857,884-+ 38,004 36. 12,845,678 + 57,095 543. Slate Problems. 1. Thesum of three numbers is 150. Two of the numbers are 8 and 48. What is the third? 68 + 43+ ?=150 2. The divisor is 24; the dividend is 264. Find the quotient. 38. The product is 228; the multiplicand is 19. What is the multiplier ? 19 «? = 228 4. The minuend is 175; the subtrahend is 87. What is the remainder ? 5. The remainder is 92; the subtrahend is 89. Find the minuend. MD RG I2E G0 6. The minuend is 176, and the remainder is 99. What is the subtrahend ? 7. The multiplier is 15; the multiplicand is 46. What is the product? 8. The multipler is 16; the product is 272. What is the anultiplicand ? 9. The dividend is 800; the divisoris 17. Find the remainder. 10. The quotient is 15; the remainder is 3; the divisor is 8. What is the dividend ? 8) 2? 153 236 ARITHMETIC. 11. The dividend is 273; the quotient is 21. What is the divisor ? DAE 21 12. The dividend is 267; the quotient is 138; the remainder is 7. What is the divisor? ?)267 132 544. Oral Problems. 1. What will be the cost of 8 pounds of meat at 15 cents per pound? 2. Paid $12.90 for 3 pieces of lace. How much did each cost ? 3. Gave $1 in payment for a 25-cent ball, and 4 ten-cent bats. How much change did I receive? 4. If 3 straw hats cost 63 cents, what will be the cost of 5? 5. At the rate of 3 oranges for 5 cents, what will be the cost of a dozen oranges? 6. A gross is 12 dozen. How many pens in } gross? 7. How many inches in 4 yards? 8. How many ounces in 62 pounds? 9. At 5 cents per pint, how much would be paid for a bushel of chestnuts ? 10. A person used 2 gal. and 3 qt. of milk in one week, and 8 gal. and 1 qt. the next week. How many gallons are used in the two weeks ? 11. I sold 33 yards of silk and 22 yards of velvet. How many yards in all did I sell? 12. A man had $64 dollars, and he spent $34. How much money had he left? REVIEW. 237 545. Slate Problems. 1. A man works 9 months, 26 days per month, and receives $702. What are his daily wages? 2. A merchant buys 136 vases for $272. If 86 are broken, how much must he charge apiece for the others to gain $28 on all? 3. On Monday, the receipts of a store are $180; on Tuesday, $30 less; on Wednesday, $110 less than the total of Monday and Tuesday. What are the receipts for the three days? 4. The yearly rent of a house is $480. What is the rent for 2 years 4 months? 5. A mechanic works 300 days per year, at $4 per day. It his daily expenses for 365 days average $3, how much money does he save each year? 6. A woman pays $5.20 for 3 lb. of tea and 56 lb. of sugar. What is the price per lb. of the sugar, if the tea costs 80 ¥ per lb.? 7. A man had $7,500. He paid 2 of it for a house, $575.60 for repairs, and $387.75 for furniture. How much money had he left? 8. How much hay will be required to feed 18 horses a year of 366 days, if each horse receives 15 pounds a day? 9. A person pays a debt of $576, giving 40 ten-dollar bills, 30 two-dollar bills, 6 one-dollar bills, and the remainder in five- dollar bills. How many of the last did he give? 10. A drover buys 64 sheep for $400. He sells +of them at $7 each, and the remainder at $8 each. What is his profit? 11. A merchant sells 56 yards of cloth for $84, gaining $14. What did it cost him per yard? 12. A package of coffee, costing 60 cents, was sold for 75 cents, the profit on each pound being 5 cents. What was the selling price per pound? 13. How many yards of cloth, at $1.75 per yard, can be bought for $105? 238 ARITHMETIC. 14. A tailor buys a piece of cloth for $50. From it he makes 4 pairs of trousers, which he sells at $7 per pair, and 4 coats, for each of which he receives $15. Thread, buttons, lining, etc., cost him $16. How much does he get for his labor ? 15. A man sold a certain number of papers for 50 cents. If he had sold 9 more, he would have received 95 cents. How many papers did he sell? BILLS. 546. Brooxuyrn, July 31, 1894. Mrs. H. J. SHoRtT Bought of ABRAHAM & STRAUS. 1} yd. Plaid $ 1.00 16 yd. Cambric .05 12 pr. Socks .20 1 Wrapper 1 | 98 4 yd. Silk .65 1 pr. Gloves 2 | 25 2 spools Silk .08 1. Copy the above, filling in the cost of each item and the total. ; 2. Robert J. Wildes buys of Caswell & Donaldson 64 |b. of sugar (@ 41¢; 28 lb. of lard @ 71; 24 lb. of coffee @ 25¢; 1 bbl. flour @ $5.75; and 12 gal. of molasses@ 251”. Make out the bill. 3. Make out a bill for 10 pairs of men’s shoes at $4.75; 4 pairs of boys’ shoes at $1.474; 6 pairs of slippers at $.871; 9 pairs of girls’ shoes at $2.43; 8 pairs of women’s shoes at $3.374. 4. Make outa bill for 84 lb. of ham at 14 per lb.; 34 Ib. of beefsteak at 24%; 9 lb. of corned beef at 12%; 101 Ib. of chicken at 30%; 12 lb. of roast beef at 18¥. DECIMALS. 239 5. Make out a bill for 14 doz. collars, at $1.50 per dozen; 6 doz. pairs of cutis, at $2.75 per dozen pairs; 4 doz. shirts, at $9.00 per dozen; 3 dozen ties, at $2.25 per dozen; 17 doz. pairs of socks, at $2.50 per dozen pairs. DECIMALS. 547. Oral Exercises. In the number 25, what does the 2 stand for? In the number 467, what does the 4 represent? The 6? The 7? In the number 383,333, give the value of the first-3 (commencing at the left). Of the second. Of the third. Of the fourth. Of the fifth. The last 8 is what part of the number represented by the fourth 3? The third 3 is what part of the second? Hach 3 is what part of the 3 to its left ? The value of each 3 in this number depends upon what? In the number XX XIII, what is the value of the first X? Of the second? Of the third? 548. When we write $784.56, the 7 stands for seven times how many dollars? The eight for 8 times how many dollars? The 4 for four times how many dollars? The 5 stands for five times what part of adollar? The 6 stands for six times what part of a dollar? Hundreds. Tens, Units, Decimal Point, Tenths. Hundredths. 7 8 4 5 This is read 784 and 56 hundredths. 37.5 1s read 37 and 5 tenths. 6.492 is read 6 and 492 thousandths. .0005 is read 5 ten-thousandths. 01234 is read 1,234 hundred-thousandths. © 56.000246 is read 56 and 246 millionths. $ 497.625 is read 497 dollars and 62 cents 5 mills. Norre.— Cent means hundredth. Mill means thousandth. 240 549. Nors. ARITHMETIC, NOTATION AND NUMERATION. In reading a number containing an integer and a decimal, the word and may be placed between the two, as is shown above. To avoid mistakes, the word units should be used after the integer in reading such numbers as 200.005, 3000.0075, etc. Say: Two hundred units and five thousandths, three thousand units and seventy-five ten-thousandths., 550. Read the following: LPC § Sone 2. 84.9 Oren a Oe Si) oO 10. OD 4 eo 11. 100.025 5. woreto 12} 125 6. 9.624 13. .99 Teo 14; .08 551. Express in decimals: 15. 005 16. 1.3848 17 oo.G 18. 100.25 19. 627.009 20. 099 5 ss 887 and 72 hundredths. 6,054 hundredths. 6,000 and 54 hundredths. 1. 7 tenths, 2. 86 and 47 thousandths. 3. One hundred twenty-five thousandths. 4. One hundred units and twenty-five thousandths. 5. 47 hundredths. 6. Four hundred units and six tenths. 7. Four hundred six thousandths. 8. 38 and 56 hundredths. 8. Twenty-eight. 15. 10. 65 tenths. 16. 345 tenths. 11. 6 and 5 tenths. shy es 12. 465 and 8 hundredths. 18. 13. 3875 hundredths. 19. 5 millionths. 14. 4,000 tenths. 20. 562 millionths. DECIMALS. 241 ADDITION OF DECIMALS. 554. Add: cs ey 2. 3.84 3. 28.978 4. 5.6 4.18 68.075 .28 387 005 oO 5.375 26.93 5.67 24.698 18.758 8.754 10.555 97.113 os © M 2 o 10. 7h 12. 13. 14. 15. 16. 17. 18. 19. 20. 027 + 1.89 + 48.6 + 72.978 234.96 + .675 +50.4-+ 6.02 + 1.001 8.047 + 54.79 + .097 + .76 + .862 8+ 38+ .479 + 27.87 + 875 445 + 34.75 + 306.973 + .004 + 48.56 81+ 12.654 + 234.79 + 8.6 -+ 603 + 42.96 45.78 + 237 + 6.987 + 18 + 372.003 + 87.5 4.745 + 36,58 + 725.894 + 9.87 + 75.357 + 86.74 59.8 + 83 -+9.64+4 48.565 + 6.98 + 8.795 + 963.826 13.387 + 72.563 -+ .7-+ .603 + 7.245 + .483 + 9.25 8.3 + 2.576 + 3.424 1.5 + 6.279 + .008 + 1.417 24-2.35641.144 24 4.96 + 3.2724 .7 + 3.54 4.7 4+.1.198 + .35 + 763.5 + 1,423 + 157.24 + .487-+9.5 7.369 + 1.72 + 32.948 + .429 + .74+ 3.14 + 695+.7.005 8.87 + 2.694 + .8 4251.47 +9.3-+ 1.916 + 41.5 + 751.006 87 + 6.3 + .008 + 9.63 + 96 + 47.82 + 637.46 + 1.923 242 ARITHMETIC. SUBTRACTION OF DECIMALS. 555. From 37. 182.01 Lakeovo.t 4.624 3434 TSG 556. Find answers: 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 1 — .057 1 — .245 6 — 3.324 4 — 2.491 3 — 1.568 7 — 4.736 3.587 — 1.34 91.3852 — 72.456 42.007 — 17.658 68.217 — 89.4 — 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 1 28.6 009 1,008 991 © 97.597 9.34 — 5.672 45.268 — 23.068 219.843 — 187.95 681.38 — 94.572 1000 — 465.874 30.053 — 18.7 2,568.91 — 1,925 1.234 — .825 478.5 — 298.572 57.088 — 44.95 MULTIPLICATION OF A DECIMAL BY AN INTEGER. 557. Three times 3 tenths equals how many tenths? Oo XO = what? Oe 4 Sie diy, 7.5 275 x8 x4 Gd De 39 xX 20 7.00 1G at 0036 x 110 8960 558. The products are 13.6, 30, 3.3, 7, and .396; the ciphers to the right of the decimals, having no value, are omitted in giving the answers, 359. Multiply : 560. 41. 42. 43. 44. 45. 46. 47. 86 x 8 ay Sai | 6.4 x 122 122 x 6.4 67 x 4 008 x 512 512 x .008 056 x 987 SP aa OS) . 9,430 x 8 Sight Exercises. Give products : 561. ahs 684 x 10 2. 68.4 x 10 3. 4 5 Bex LU. » O71 X-100 - 5.71 x-1,000 Give quotients : 11. 12. 13. 14. 15. 932 + 100 86 + 1,000 328 + 10 9 + 1,000 48 + 1,000 ‘DECIMALS. © 51. 52. 53. 54, 55. 56. BT. 58. 59. 60. © MW WD 2715 X 1.2 88.4 x 25 048 x 875 12.67 x 300 6.57 x 9 748 x .97 8.76 x 43 964 x .347 570 x 11 860 x .005 .961 x 100 57 X 1,000 » .09 x 1,000 .026 x 100 5.17 x 10 684 = 100 57.6 +10 24.3 + 100 8.75 + 10 . 932.5+ 100 243 244 ARITHMETIC. DIVISION OF A DECIMAL BY AN INTEGER. 562. Sight Exercises. 1. 8.64+2 6. .846+6 2. 48.244 7. .048+8 3. 465-3 8. 81+9 4. 840-5 9. 12+5 5. 84+5 10. .24+4 563. Where it is necessary, ciphers may be annexed to the right of the decimal in the dividend. 8) .12 15) .06 1.875 O15 .004 64) 120. 560 012 A418 480 125) 1.5 21)8.673 320 250 aT 0 0 63 564. Divide: 11. 25)1.00 21.0 L= 12. 4)21.80 22. i= 13. 8).2 PA ye ee 14. 13)3.913 24. tem 15. 12)48.12 25. = 16. 11) 70.07 26. += 17. 24)36.6 27. 1,00 — 18. 18).576 28. 180 — 19. 25) 11.1 29, 5990 — 20. 32) 62.000 $0: aks REVIEW. 245 565. Slate Problems. 31. A franc is 19.3 cents. Find the cost in United States money of goods amounting to 1,250 francs. 32. A merchant bought 1,800 meters of silk. How many yards did he buy, a meter being 39.37 inches? 33. A kilogram is 2.2046 pounds. What is the difference in weight between the English ton of 2,240 lb. and a French ton of 1,000 kilograms ? 34. A cubic foot of water weighs 1,000 ounces. How many pounds does a cubic foot of gold weigh, gold being 19.4 times as heavy as water ? 35. There are 128 cubic feet in a cord. How many tons of 2,000 lb. are there in a cord of pine wood, the latter being .66 times as heavy as water ? 36. A man buys three plots of ground containing 35.27, 17.8, and 40.375 acres, respectively. Find the total cost at $36 per acre. 37. How many pints are there in 2.375 gallons? 38. What decimal of a peck is a quart? 39. What will be the cost of carrying 468 tons of coal at $0.125 per ton? 40. A farmer sold one-eighth of his farm of 224.2 acres at $62.50 per acre. How much did he receive for it? 41. How long is a post which is 5% feet above water, one- half of its length in the water, and one-fourth of its length in the mud? (Diagram.) 42. Hight pounds of black tea costing 35% per Ib. are mixed with twelve pounds of green tea costing 50% per lb. What is the cost of a pound of the mixed tea? 43. How many bushels, pecks, and quarts, are there in 1,449 pounds of corn weighing 56 lb. per bushel? 246 ARITHMETIC. MEASUREMENTS. 569. Preliminary Exercises. Draw a square each side of which is one inch. This is called a square inch. Draw a rectangle two inches long, one inch wide. How many square inches will it contain ? Draw a rectangle three inches long, two inches wide. Divide it into one-inch squares. Count them. How many are there? How many square inches in the rectangle ? How many square inches in a rectangle 6 inches long, 3 inches wide? How many square inches in a rectangle 4 inches long, 4 inches wide ? About how long is your slate? About how wide? About how many square inches are there in its surface? How many square inches are there in a rectangle 12 inches long, 3 inches wide? Ina rectangle 1 foot long, 3 inches wide? In a rectangle 1 foot 1 inch long, 4 inches wide? 570. Slate Exercises, How many square inches in each of the following rectangles? . 18in. by 14in. 5. 2lin. by 19 in. 9. 18 in. by 22 in. .17in.by Qin. 6. 387in. by 14in. 10. 64 in. by 29 in. qleun. bye. ine 9 7513 1m. Dy 4240. 9 dele Lae eames 2010, by 15in, 8. 27 in by slin, 4 AZo itl in by rain fF WwW DO KF S71. Nore. Change each dimension to inches before multiplying. ASU th 1D yl an. 17a 2 it 5 Olin yt ihe a 14. 1 ft. by 1 ft. 18. 3 ft. Tin. by 2 ft. Qin. 15. 1 ft. 4 in. by 1 ft. 19. 4 ft. ll in. by 1 ft. 8 in. 16. 2 ft. 6 in. by 1 ft. 20. 5ft 3 in. by 2 ft. 11 in. MEASUREMENTS. 247 572. Oral Exercises. How many square feet in a rectangle 2 feet long, 1 foot wide ? 6 feet long by 5 feet wide? 9 feet long by 7 feet wide? 573. Slate Exercises. Find the square feet in the following: 1. 12 ft. by 14 ft. 6. 29 ft. by 12 ft. 2. 15 ft. by 17 ft. 7. 154 ft. by 12 ft. 3. 19 ft. by 11 ft. 8. 15 ft. 6 in. by 12 ft. 4. 23 ft. by 15 ft. 9, 183 ft. by 16 ft. 5. 18 ft. by 16 ft. 10. 18 ft. 9 in. by 16 ft. 574. Nors. Change the inches to fractions of a foot. 11, 232 ft. by 18 ft. 16. 36 ft. by 23 ft. 5 in. 12. 24 ft. 8 in. by 18 ft. 17. 18 ft. by 248 ft. 13. 19 ft. 3 in. by 16 ft. 18. 13 ft. 4 in. by 24 ft. 14, 24 ft. by 17 ft. 9 in. 19. 26 ft. 8 in. by 15 ft. 15. 24 ft. by 16 ft. 1 in. 20. 122 ft. by 12 ft. 575. Suggestive Examples. Measure the top of the desk, and calculate the surface in square inches. (Do not include fractions of an inch.) Measure the blackboard, and find how many square feet in its surface. (Do not include fractions of a foot.) Calculate the number of square inches in a pane of glass in the school-room window. Find the number of square feet in the floor of the class-room. Find the number of square feet in the class-room ceiling. Estimate the height of the class-room, and calculate the num- ber of square feet in the front wall. In the rear wall. In the right-hand wall. In the left-hand wall. 248 ARITHMETIC. 576. Slate Problems. Suacestion. When the surface is required in square inches, change each dimension to inches; when required in square feet, express each dimension in feet, or in feet and the fraction of a foot; when required in square yards, etc., express each dimension in yards, etc. 1. How many square feet are there in the surface of a field 125 feet long, 87.5 feet wide? | 2. A rug is 2 yards long, 13 yards wide. How many square yards does it contain ? 3. How many square yards are there in a strip of carpet 6 yards long, 27 inches ($ yd.) wide? 4. Find the number of square meters in a room 12 meters long, 9.75 meters wide. 5. At 50 cents per square yard, what will be the cost of the oil-cloth needed to cover a floor 18 feet (6 yd.) long, 15 feet (5 yd.) wide? 6. What will be the cost, at $1.50 per square yard, of car- peting a room 64 yards long, 15 feet wide? 7. At 3 cents a square foot, how much must be paid for 10 boards, each 16 feet long, 4 foot wide? 8. A field is 80 rods long and 24 rods wide. How many square rods will it contain after a strip 24 rods long and 2 rods wide is taken from it for a road ? 9. How many square yards of plastering will be required for a ceiling 18 feet long, 15 feet wide? 10. Ifa roll of wall paper is 24 feet long and 18 inches wide, ow many square yards does it contain? CHAPTER VII. FRACTIONS. — DECIMALS. — BILLS. DENOMINATE NUM- BERS. — MEASUREMENTS. ADDITION OF FRACTIONS. 577. Slate Exercises. 1. Find the sum of 102, 6%, 57. Since 9 is a multiple of 3, any multiple of 9 will be a multiple of 3. Omitting the latter number, we find the least common multiple of 8 and 9, 72, which is the least common denominator of the fractions. 2. Add 24, %, 52, 44 Omitting 4, which is a factor of 8, we find the least common multiple of 3, 7, 8. 3x7xX8=168 $. 6844478 49—7 L. C. M.=15~x 8 Omit 5 and 3. 4, 172-48 + 26% 8x 9x 7=504 epee SI -t O72 2b OF L. C. M. of 15 and 6 =? 6. 29,84 45 + 16.4, 8. 114534 13,5, 4+ 82+ 198 (Eo Sop tee tae 9. 885 -+37$ + 284 + 99% + 195 10. iy -+ 28d, + Slat + 15,2, 249 250 578. Find answers : 11: 1y-# 13. 14. 15. 579. 21. 22. 23. 24. 25. 580. Oral Exercises. Give two factors of : oO FP OW Ne ARITHMETIC, SUBTRACTION OF FRACTIONS. pee ee 23,5, — 16% 472. — 88 102% — 848 39,4, — 1941, REVIEW. 16. 17. 18. 19. 20. Perform the operations indicated : 16 x (25+ 1) (8h —8}) lf 18} — (8}—-1) (244 — 152) x 36 26. 27. 28. 29. 30. 39,9, — 23.7, 92,7, — 68,5, 8302, — 807% 12,345 — 6,08214 320-9, — 120%, (508 +5) + 122 100 — (4 x 188) (83% +172) x 24 881 + (172 x 24) UX Bae Ss 16x3 15 FACTORS AND MULTIPLES. 4 Ans. 2,2 6 Ans. 2,3 Gueeo Tlie 59 16. > 57 Tanne 12. 46 ysis. 8. 34 13. 49 18. 62 9. 35 it Bo tay k 19. 65 10. 38 15. 59 20. 69 PRIME NUMBERS. 581. Sight Exercises. Give three factors of : Whe te: 12 18 20 27 vee ereec mee oo nr & 28 30 42 44 45 11. 12. 13. 14. 15. 50 52 63 66 68 PRIME NUMBERS. 16. 17. 18. 19. 20. 70 75 18 98 99 582. A number that has no factors is a prime number. 1, 2, 3, 5, 7, etc., are prime numbers. 583. 1. Name the prime numbers between 10 and 20. 4. Between 50 and 70. 5. Between 70 and 100. 2. Between 20 and 380. 3. Between 30 and 50. 584. Sight Exercises. Give the prime factors, commencing with the smallest. 11. 12. 13. 14. 15. 1eLD 2. 16 3. 24 4. 32 5. 36 6. 7. 8. 9. 10. 40 585. Slate Exercises. 1. 86 2. 87 3.. 88 4. 90 6. 91 6. 7. 8. 9. 10. 92 93 94 95 96 Lis 12. 13. 14. 15. 64 72 74 76 {ire 100 120 210 240 360 16. 17. 18. 19. 20. 16. 17. 18. 19. 20. 80 81 82 84 85 576 840 1,152 1,728 2,016 251 252 ARITHMETIC. GREATEST COMMON DIVISOR. 586. A common faetor of two or more numbers is a number that will divide each of them without remainder. The largest number that is a factor of two or more numbers is called the greatest common divisor. 587. Sight Exercises. Find the greatest common divisor of : 1. 27 and 48 6. 34 and 51 Qo Andis 7. 382 and 48 3. 386 and 54 8. 45 and 75 4. 26 and 39 9. 40 and 65 5. 42 and 63 10. 54 and 69 588. Slate Exercises. Reduce the following fractions to lowest terms: 1. 14 5. 348 9. $14 2. 278 6. 328, 10. 324 3. 444 7. M1. $i 4, 132 8. 24 12. 43$ LOWEST TERMS. $89. How can you tell that a number is divisible by 2? By 5? A number is divisible by 3 when the sum of its digits (figures) is divisible by 3; it is divisible by 9, when the sum of its digits is divisible by 9. A number is divisible by 4, when the number expressed by its last two figures is divisible by 4. When is a number divisible by 25? 590. A fraction is reduced to lowest terms by dividing the numerator and the denominator by their greatest common divisor, LOWEST TERMS. 253 591. Reduce to its lowest terms 546%. In this example, it is not easy to ascertain by inspection any number that will divide both terms. In such cases, the greatest common divisor is found by dividing the denominator by the numerator. The remainder is divided into the numerator, and each subsequent remainder is divided into the corresponding divisor until there is no longer a remainder. This last divisor is the greatest common divisor of the two numbers. 5 169) 10011 156) 169° 13) 156. 12 13 is the greatest common divisor. IGG ee La) ae —_—_— = — lowest terms. POUT eels at. §92. In reducing fractions to lowest terms, the above method of finding the greatest common divisor should not be resorted to if it is possible to get along without it. 593. Reduce to lowest terms: 123 LL A look at both terms shows that 3 is a common factor. This reduces the fraction to 744. 41 is a prime number, and is not a factor of 100, so that 54), cannot be reduced to lower terms. 2. $34 44+3+4+2=9; 64+2+1=9 Since the sum of the digits of each term is divisible by 9, this number is a common factor, and reduces the fraction to €§, etc, 3. 44% 5 is clearly a common factor, etc. 4. 42 nr cam 12. 1 5. i 9. 3 13. 119 6. 4? 10. 5&5, 14, 115 7. 24 11. {45 15. 248 254 ARITHMETIC. LEAST COMMON MULTIPLE. 594. Sight Exercises. Give the least common multiple of : 1. 16 and 24 6 2B ORO RIO 2. 12iandds TaD BySSG) OAD Bee Oa 8: 8,6, 9/1254 18 An aso Lalo OF Dl las §2(2))5,450;6 10. '5,.10, 20° 26.400 595. Slate Exercises. Add 141, 732, 68, %, 232, 101.8, 584, 9,4. Here we have to find the least common multiple of 3, 9, 7, 14, 6, 14, 2, 12. Rejecting 3, 6, 2 because they are factors of 12; 7, a factor of 14; and one 14, we have to find the least common multiple of 9 14 12 Divide these numbers by a prime number that is exactly contained in any two of them, bringing down 2). he eh the numbers that are not multiples of the divisor. 3} 9 7 6 Taking 2 as a divisor, bring down 9, and write quo- 3 7 Je tients 7 and 6. 3 being a factor of two of the three numbers, 9, 7, 6, is taken as the next divisor. 3 is written as a quotient, 7 is brought down, 2 is a quotient. As there is no factor common to any two of the numbers, 3, 7, 2, we find the least common multiple by multiplying together the two divisors and these three numbers. 2%3 XK 3507-% Biss 252 LUG. Mi 252 Find the sum. FRACTIONS. 255 596. Find the least common multiple of the denominators of ; i Bee S28 oT the fractions 3, 34, 3, 44, 33, 7. 45 2| 20 30 45 12 Omit 4 and 6. = | 10 1 45 6 Strike out 15, a factor of 45. 5 A5 3 Strike out 5 and 3, factors of 45. L. C. M. = 2x 2 x 45 = 180 Add the fractions. 597. Find the L. C. M. of: #40,9, 0,0, 20. Strike out 4) 3.5: 9, 15, 15, 4, 4, 12, 25. Strike out one 15 and two 4’s. Poy, (, 0, 14. 10. 12° 94 000,070,910, 1), 16,80 20, 30, 40, 50 2, 3, 4, 6, 8, 12, 16, 24 24, 12, 5, 8; 10, 18 LPB Te Fe38 18, 5, 9, 40, 16 LOI DAT CO NAA Pw DY rey = ADDITION AND SUBTRACTION OF FRACTIONS. 598. In adding or subtracting fractions, they must be reduced to a common denominator. The least common denominator is the least common multiple of the denominators. In the following examples, determine the least common denomi- uator by wmspection, if possible. 599. Add: IMG Veneer 6 1. 8, 54, 34 3’ 5?) 10) 20° 25 1 32 474 OF 0? oF, ¢ bz. 77 453, 201, 83, OL B21, 198, 61, 812 O21 20, 88, 8, BL 81, 458, 217, 42 © © 2 op wo 1 3 3 4h 10. Liagoo 100) 635, 1000) 100 >) “ 56 ARITHMETIC. 600. Subtract: 11. 6544-575, 16. 251.8 — 2744 12. 1847 — 928 17. 75512 — 28325 13. 10043 — 1513 18. 10054, — 89343, 14. 102,8 — 2747 19. 12333 — 8018 15. 20814 — 12838 20. 671, — 587445 601. Perform the operations indicated : Dhe cabatitas 25+5 25 a2, 22 = 23. (873 — 112) — (28,5 — 1934) 24. 148 — 81 — 384 41 25. (Bf; + 68) — Bi — 69) 26. 48x 16 x 8% aT. (25 +54) + (t+ 24-484) 28. (8}+ 41) + (2k-+14) 29. (33; x 36) x 83 30. 42+ 31 — 68+171 — 94 602. Reduce to lowest terms: 31. 5 35. 375, 39. 875 32. x00 36. ify 40. 3125. 33. z00 87. 825 41. ;88, 34. slo00 38. ibty 42. 7625, REVIEW. 257 603. Oral Problems. 1. A person traveling from New York to Albany (140 miles apart) has gone 93 miles. How much farther has he to go? 2. There are 196 pounds of flour in a barrel. How many pounds in } bbl.? | 3. How many fourths in 24%? 4. Reduce 4% to lowest terms. 5. Change 19° to a mixed number. 6. Add 4, 4, and 4. 7. From a chest of tea containing 454 lb., 14 lb. are sold. How many pounds remain ? 8. From 4 of a dollar take 102 cents. 9. How many cents in++4- +3, of a dollar? 10. A farmer has 602 acres of pasture and 203 acres of wood- land. How many acres in both? 11. The circumference of a circle is about 3} times its diameter. If the diameter is 8 feet, about what is the circumference ? 12. Mary is 12 years and 7 months old; Jane is 3 years and 11 months older. How old is Jane? 13. Ina year anda half William will be 7 years 2 months old. How old is he now ? 14. What number multiplied by 3 equals 231? 15. What number between 7 and 12 is a prime number? 16. A boy received 9 marks in arithmetic, 8 in penmanship, and 7 in reading. What was his average mark ? 17. 4 of a class consists of boys. How many girls in the class if it contains 49 pupils? 18. If July 1 falls upon Tuesday, what will be the date of the third Tuesday of July ? 258 ARITHMETIC, 19. If July 1 falls upon Thursday, upon what day will the first of August fall ? 20. A man bought 204 Ib. of sugar; he sold 10% lb. at one time and 64 lb. at another. How much had he left? 21. If3 qt. 1 pt. oil cost 7 cents, what will 1 gal. 1 qt. cost? 22. How much will have to be paid for 7 cows at $50 each, and 4 horses at $150 each ? 23. %—how many hundredths? 24. What are the two factors of 87? 25. Find the G.C. D. of 36 and 54. 604. Slate Problems. 1. A merchant sold one firkin of butter at 193 per lb., a second at 203¢ per lb., a third at 16Z¢ per lb. What was the average price per lb., each firkin containing 56 lb. ? 2. If eggs are sold at the rate of 21 for 25 cents, how much will be paid for 54 dozen ? 3. Ifa train travels 45 miles per hour, how far will it go from half-past 9 in the morning to a quarter of 7 in the evening? 4. To the sum of 6% and 192 add their difference, and find how often the greater number is contained in this total. 5. A mechanic has deposited in the savings bank $35 per month for 11 months, and $20 the twelfth month. His expenses have averaged $3 each day of the year. What are his daily wages for the 300 days he has worked ? 6. A merchant sold 4 pieces of cloth containing 274 yd., 262 yd., 29% yd., and 281 yd., respectively. How much did he receive for the cloth at 96 cents per yard? 7. Reduce 18% to lowest terms. 8. A man has 8,5, bu. of peanuts. He puts them into bags holding @; bu. How many bags does he fill? REVIEW. 959 9. A 160-acre farm consists of 5 fields. The first contains 17.38 acres, the second 29.4 acres, the third 35.078 acres, the fourth 25.875 acres. How many acres are there in the fifth field? 10. How many seconds in 7 hours 15 minutes? 11. Find the total cost of 2 doz. rockets at $7.50 per gross, 3 dozen Roman candles at $9.60 per gross, and 24 doz. pin wheels at $1.35 per gross. (1 gross = 12 doz.) 12. From a piece of silk that contained 28 yd., there were sold 21 yd., 64 yd., and 133 yd. Find the value of the remainder at $1.20 per yd. : 13. Three pieces of cloth bought at $2 per yard cost $150. The first piece measures 234 yd., the second measures 302 yd. How many yards in the Hind piece? 14. Three lots of tea were sold for $330. The second con- tained twice as much as the first, and the third three times as much as the first. The third lot contained 330 pounds. Find the selling price of the tea per pound. 15. What part of a person’s income remains after he spends 4, =, and + of it? 16. A boy loses 4 of his marbles, and he gives away + of them. If he has 17 marbles left, how many had he at first ? 17. A cask of molasses contained 80 gallons. One-fourth of it leaked out. If the molasses cost 60 cents per gallon, what price must be charged for the remainder so that there will be no loss. 18. A dealer sells 13 gross, 3% gross, 441 gross, 974; gross, and 82 gross of lead anhiel at 36 cents per ose (she much does he receive for all? 19. There are four towns, A, B, 0, and D, on a certain rail- road running east and west. A is 414 miles west of C; D is 394 miles east of B; B is 221 miles west of C. How many miles from A to D? Make a Maree 20. If 121 dozen rockets cost $5.75, het will 15 dozen cost? 260 SPECIAL DRILLS. 605. Give sums: 59+75 22+34+18 67+83 19+47+430 94438 25+435+4 26 66+56 17+18+19 606. Give remainders : 134 — 75 750 — 290 150 — 83 510 — 220 132 — 94 630 —- 880 122 — 56 820 — 560 607. Give products: 49x 2 47x 3 48 x 4 43 x 5 123 x 3 431 x 2 122 x 4 don X38 608. Give quotients : 141+3 192 +4 215 +5 276 +6 111 + 37 192+ 48 215 + 48 276 + 46 609. Give answers: 213 +4 t+4 a+! wmloo celbo Fat pant | = oes ero osibo bole | celho bl a ARITHMETIC. 560 + 390 270 + 280 640 + 260 430 + 480 131 — 65 123 — 84 156 — 78 164 — 97 46 x 6 34 xX 6 38 X 7 49x 7 925 + 25 875 + 25 150 + 25 625 + 25 66 x 12 84 x 13 100 x 14 126 x 12 225 + 154 315+ 421 437 + 260 540 + 355 279 — 154 086 — 263 457 — 237 668 — 325 AT x 25 25 x 86 33 X 25 25 X 27 266 + 7 296-8 414+9 360-8 | alto ci imico cxleo celbo leo celbo bhme — oe) ex|bo Le) es|bo Ne) = SP oloo — s) olen | ae “aro fomcl =) bo orb REVIEW. 261 610. Oral Problems. 1. Find the cost of 1 lb. of tea at 75 cents, and a piece of ham at 56 cents. 2. A farmer sold 58 sheep from his flock of 121 sheep. How many remained ? 3. What will be paid for 8 lb. of coffee at 35¢ per lb.? 4. A laborer received $4.88 for four days’ work. How much did he earn per day? 5. At $40 each, how many cows can be purchased for $ 2,000? 6. Bought 20 lb. of sugar at 5¢ per Ib., and 22 lb. of butter at 80%. What was the amount of my bill ? 7. A piece of cloth measuring 311 yards was divided into 2 equal parts. Find the length of each. 8. How many weeks in a year of 366 days? 9. If I pay 25 cents for 3 pounds of cherries, how many pounds can I buy for $1.25? 10. Find the cost of a bushel and a peck of peanuts at the rate of 5 cents per quart. 11. A farmer had 164 acres of land. How much had he left after selling 87 acres? 12. Find the total number of pounds in 3 tubs of butter weighing respectively 25 pounds, 34 pounds, and 36 pounds. 13. At 60¥ per lb., how much tea can be bought for $5.85? 14. A drover paid $219 for oxen, at an average price of $73. How many did he buy ? 15. What will be the cost of 20 bu. of wheat at $1.044 per bu.? 16. At 24¢ per lb., how many ounces of butter can be bought for 18¢? 262 ARITHMETIC. 17. A woman pays $540 per year for a house. What is the rent per month ? ; 18. How many weeks in 294 days? 19. At 72 per yard, what will be the cost of 2 ft. 11 in. of lace? 20. How much does a grocer receive for a barrel of flour, 196 lb., retailed at 3 cents per lb.? 21. If 47 men can do a piece of work in 4 days, how long will it take 1 man to do the same work ? 22. Find the cost of 36 acres of land at $25 per acre. 23. If it takes 34 yards of cloth to make a coat, how many coats can be made from 244 yards? 24. How much will be paid for 84 yards of silk at $13 per yard? 25. If a certain quantity of provisions will last one man 215 days, how long will it last 43 men? 26. How many square yards are there in a rectangular field 36 yards long and 25 yards wide? 613. Sight Exercises, Add: Lethe 4. 11344} 7 88464 2. 41488 5. 78+ 941, 8. 153+ 81 3. 92478 6. 5+ 22 9. 98452 614. Subtract 84 from 10. (104 — 82) is how much greater than (10 — 84)? To subtract 84 from 10} mentally, we find the difference between 83 ana 10, which is 14, and to this add }. The answer is 13. 10. 1183—62=41438=? 11, 142-91 =4142=% CANCELLATION. 263 615. Subtract at sight: 12. 13. 14. 15. 16. 17. 232 — 1954 18. 141 —83 168 — 9% 19. 27, — 72 184) 38 2ONMaB Ee BE Ph 8s OI Lae OF yh Laarpes 22. 43.8, — 84 107;— 54 23. 502 — 48 616. Slate Exercises. Divide without writing products (Art. 385): 4,320 + 32 5387 +51 . 10,246 + 84 . 21,3821 + 97 1 2 3. 8,795 75 4 5 A514 36) 17837 343 197 iy: 6. 42,387+ 123 7. 73,690+ 345 8. 105,261+ 624 9. 423,958 + 1,008 10. 867,293 + 2,534 CANCELLATION. 617. Slate Problems. 1. If 17 horses cost $4,387, what will I pay for 51 at the same rate ? 618. Problems consisting of multiplication and division can be some- times shortened by indicating the operations, and then canceling. To solve the above, we indicate the cost of one horse, 17 3 BBR ie 1287 BL et and of 51 horses, Since 17 is contained in 51 three times, we cancel both numbers by drawing a line through them, and we write a 3 above the 51, The answer is $4,387 x 3 = $13,161. 264 ARITHMETIC. 2. If 15 eggs cost 25 cents, what will 10 dozen cost? 25 x 10 x 12 15 We indicate price of one egg by dividing by 15. Multiplying this by 10 times 12, we get the price of 10 dozen. In this case, 15 is not contained in any number ’ above the line. We divide 15 and 10 by 5, canceling OB y i %. 17 them and writing quotients 3 and 2 alongside. 3 is con- tained in 12 4 times; so we cancel 3 and 12. Our Ap answer now is 20 cents x 4 X 2 = 200 cents, or $2. p 3. Eighteen men can do a piece of work in 26 days. How long will it take 13 men to do the same work ? One man will take 26 days x 18. 4. Seventeen barrels of flour, 196 lb. each, were put into bags holding 49 pounds each. How many bags of flour were put up? 5. At the rate of 23 cents for 7 pounds, how much would be paid for 91 pounds of flour? 6. A bank pays $4 interest a year on every $100. How much interest will be paid for 3 years on $650? 7. At $45 per thousand for bricks, what must I pay for 63,200 bricks ? 8. If flour is $6 per barrel (196 lb.), what must be paid for a 49-pound bag? 9. A grocer buys 84 dozen eggs, and sells them at the rate of 21 eggs for 25 cents. What does he receive for them? 10. A miller buys 9,840 pounds of wheat at 90 cents per bushel of 60 pounds. How much does he pay for it? 11. What will be the cost of 64 sheep, if 18 cost $198? 12. If 18 men can do a piece of work in 42 days, how long will it take 21 men to do the same work? FRACTIONS. 265 13. What will be the cost of 66 doz. pens at 42 cents per gross of 12 doz. ? 14. A certain quantity of hay feeds 15 horses 56 days. How long will it feed 14 horses ? 15. A merchant bought 9 pieces of cloth, each containing 24 yards, for $189. What was the price per yard? MULTIPLICATION OF FRACTIONS. 619. Oral Exercises. What is 4 of 2 fifths? Of 4 sevenths? Of 6 elevenths? What ist of+? Of? Of4? Of? Show by a diagram. What is4 of 3? Of 8? Of? Of2? What is$ of 2? 2of4? of}? What ist of 8? 2o0f3? 3 of 2? What isd of 4? tof? BZofhZ? F of 3? What is the half of 14? Of 24? Of 3}? Of 4}? What is one-third of 12? 2o0f14? 4of2}3? 3 of 2}? 620. Multiply 2 by 4. This means to find 2 of 4. Since 4 of } = 44, 4 of $= x, and 2 of $= x. 2 4=,8; that is, the product of the numerators is divided by the product of the denominators. Norr. — Cancel when possible. 621. Multiply 2 by 3%. mi ie 3 tof =f Fo PAX To Show by a diagram that 2 x 3 tenths = 3 fifths. 266 ARITHMETIC. 622. Multiply 124 by 3,4. Reduce the mixed numbers to improper fractions. 35 49 119 o xa = 88 3 623. Slate Exercises. Multiply : 1. 2 by 96 16. 35 x 82 2. 128 by % 17. 3% by 124 3. § by F 18. 2x 412 4. ¢ by g 19. $ by 3 by 44 5. 2 by 2 20. 37 of 28 of # 6. 3,5, by 72 21. 14x$xe& 7. 242 by 18 22. 42 of 14 of 24 8. 698 by 32 23. 1 of 65% 9. 1114, by 28 24. % of 558 10. 67 by 16 25. 61x 73 11. 23 by 32 26. 41x dt 12. 9 x 22 27. % of 41 x 82° 13. 174 by 62 28. 3 of 34 x 4,1 14. 6LXF 29. 15 x 23 X 384 15. 41 by 84 30. 21x 21 x 21 624. Find answers: 31, Sx +64 - 6% 36. (82 x 21) — us 32. (8h—21) x 8 37. 54+ 62+ 73 33. 1 of (54 — 32) 38. 188 —32— 74 34. (244+ 161)+8 39. 2 of $ of (844 13) 35. (Bk +21) x (8k—2h) 40. (184 — 68) +11 FRACTIONS. 267 DIVISION OF FRACTIONS. 625. Oral Exercises. 3 fourths is contained in 4 fourths how many times? 4 fourths + 3 fourths =? 1 -- = how many thirds? 626. Give answers in improper fractions. 1S coho a: Cae 1+3=? Can you show by a diagram that 1+2=11=3? 627. Divide # by 3. We can divide by reducing both fractions to a common denominator: $+ $= 29+ 21 = 20+ 21 = 29 ad ae 27 The following is another method: 2 is contained in 1,3 times. In # of 1, it is contained + of 2 times, ris aT: 628. To divide by 3, we multiply by what fraction? 629. Divide ? by 5%. ys is contained in 1, 16 times. In 3, it is contained $ of 1¢ times. 4 Canceling, e we have 4=11. Ans. 3 630. To divide by 3%, we multiply by what? 631. Can you give a rule for dividing by a fraction? 632. N. B. Change mixed numbers to improper fractions, 268 ARITHMETIC. 633. Slate Exercises. Divide : 1. 5 83-4 16.45 Sar ae 2. 2+4 17. o3 + 3 8. 10+? 18. ++ 33 4. e+ 25 19. +6 5. = 8 + 10 20) Oe ore 6. gots Ah) oh ack 7 18+5 22. 4-3 8. 5+1% 23. 8+4 9840-91 DAN eet ae 10. 4,5-+17 25. 85+ 34 11. 241 20 26. 9% + 34 12. 68+9 27. 181-112 13. 4-74 28. 154. + 183 14. 11+ 4 29. 164-1381 nes aye 30. 231+ 6§ 634, Perform operations indicated : . (82 x 44) — 104 | (183-78) x3 , (20xH+4 - (20+4)xs 20+ ¥) | (2044) +3 - (143 x 7) —(9 x 10f) 42. dt X 13 X 3d X 64 22 X 44 X 31 REVIEW. 269 635. Oral Problems. Give analysis of each : 1. If base-balls are worth # of a dollar each, what will be the cost of 16 base-balls ? 2. Paid $12 for base-balls, at 3 of a dollar each. How many were bought? 3. What is the cost of 2 feet of ribbon at 30 cents per yard? 4. Find how much a yard of ribbon is worth, if 2 yard costs 20 cents. ' 5. If it takes $ yard of material to make one waist, how many can be made from a piece containing 24 yd.? 6. If 18 jackets require 24 yards of cloth, how much is needed for 1 jacket? 7. A man had 60 acres of land. How many acres had he left after selling ? of his land? 8. After spending # of his money, a person had $36 remain- ing. How much money had he at first ? 9. If tea is worth $ of a dollar per pound, how much can be bought for 4 of a dollar ? 10. When tea is $.50 per pound, how much can be bought for $.75? 11. When silk is selling at $.75 per yard, how much can be bought for one-fourth of a dollar? 12. Find the cost of a gallon of milk at the rate of 9 cents for 3 pints. 13. 3 of a gal. of milk costs 9%. What is the price per gal.? 14. 2 of what number is 12? 15. lyard and 1 foot of wire cost 16 cents. How much must be paid for a yard? 270 ARITHMETIC, 16. A man bought some cows at $35 each, and the same num- ber at $45 each. What was the average price ? 17. A girl received 100 per cent in three studies, and 80 per cent in the fourth. What was her average ? 18. A square floor contains 144 square feet. How many feet long and wide is it? 19. Mr. Brown mixed 3 pounds of black tea worth 40 cents a pound with 1 pound of 60-cent green tea. What is the mixed tea worth a pound? 640. Slate Problems. 1. A milliner sells 3 pieces of ribbon at 18 cents per yard. They measure 42 yd., 34 yd., and 535, yd., respectively. What is the amount of her bill? 2. How many feet and inches in 55 of a yd.? 3. To make powder, a man mixes 7+ lb. of saltpetre, 14% lb. of sulphur, and as much charcoal as sulphur. How many pounds of powder will there be? 4. Four men form a partnership; the first furnishes 4 of the capital, the second 8, and the third 53. What fraction of the capital is furnished by the fourth ? 5. I pay 15 cents more for a half pound of tea than I pay for a quarter pound of the same tea. What is its price per pound? 6. After doing # of a piece of work, a man requires 3 days more to finish it. How many hours does he take to do the whole work if he works 8 hours per day ? 7. If 1 lb. 7 oz. coffee cost 46 cents, what will 3 lb. 9 oz. cost ? j 8. Add 14 days 6 hours 50 minutes and 15 days 17 hours 10 minutes. 9. If a dozen pairs of gloves cost $15.25, what will be the cost of 60 pairs? Cancel. REVIEW. ee 10. 15 men doa piece of work in 102 days. How long would it take 5 men to do the same work? 11. To make a cloak 3 yd. of cloth 11 yards wide are re- quired. How much cloth # yd. wide would be required? 12. In3 years 4 months a gas company manufactures 4,200,000 cubic feet of gas. How many cubic feet are manfactured per year ? | 13. If 22 dozen hats cost $80, what will be the cost of 3 hats? 14. A boy hires a boat at 20 cents per hour. How much has he to pay if he uses it from 20 minutes before 9 a.m. until 10 minutes past 1 P.m.? 15. A and B kill an ox. A takes $8 and B the remainder. If B’s share weighs 3614 lb., what is the weight of the ox? 16. A grocer buys 30 dozen eggs at 18 cents per dozen. He sells them at the rate of 15 eggs for 25 cents. What is his profit? 17. How many cents in 55 of a dollar? 2 18. What fraction of 18% 1s 62? ~% =f 19. A farmer buys a horse for $140, and sells it at an advance of 8, of the cost. What is the selling price? 20. In 1893, A was 36 years old and B was 1$ times as old. In 1884, B was how many times as old as A? 21. From the sum of 18,4, and 252 take their difference. 22. If 23 acres of land cost $220, what will be the cost of 174 acres? Indicate the work, and cancel. 23. A can do a piece of work in 6 days, B can do it in 6 days, C can do it in 6 days. How long will it take all three working together ? 24. Find the value of ares 25. A man sold a horse for % of its cost, losing $40. What did the horse cost him ? OH ie ARITHMETIC. FRACTIONAL PARTS OF A DOLLAR. 641. Oral Problems. 1. How many 50-cent base-balls can be bought for $15? (15 +4) 2. How many 75-cent base-balls can be bought for $15? (15+ 4) 3. At 75¥ per lb., how much tea can be bought for $1? 4. How many hats at $1.25 each can be bought for $15? (15 + 14) 5. Paid $16 for coffee at 25% per lb. How many pounds were purchased ? 6. At 331% per lb., how many pounds of butter can be bought for $32? 7. Find the number of yards of ribbon, at 121 ¢ per yd., that will cost $45. 8. At 619 per bar, how many bars of soap will cost $11? 9. If 4 pieces of violet soap are svld for 25f, how many can be bought for $9? 10. $24 is paid for corn at 75% per bu. How many bushels? 11. I spent $30 for lace at 662 ¢ per yard. How many yards did I buy? 12. For $36 how many pairs of rubber shoes can be bought at 871 per pair ? 13. Oats are 621% per bu. How many bushels will $40 buy? 14. A farmer pays 873% per bu. for seed rye. If his bill amounted to $21, how many bushels did he buy? FEDERAL MONEY. pags: 15. A store-keeper sold $33 worth of collars, at 162% each. _ How many did he sell? 16. At the rate of 3 for 50%, how many collars can be bought for $25? 17. Corn is worth 20¢ per can. How many cans will cost $32? 18. Find the cost of 35 yards of cloth, at $1.25 per yard. 19. At $1.25 per yard, how many yards of cloth can be bought for $35? 20. How many pairs of gloves, at $1.75 per pair, will cost $28? 21. When coal is $5.25 per ton, how many tons can be bought for $42? 22. Cost of 16 pairs of shoes at $2.75? 23. 33 jackets at $3.334? 24. 18 yd. cloth at $2.162? 25. Paid $26 for cloth at $2.162 per yard. How many yards did I buy? 26. Find the cost of 16 pairs of skates at $1.874 per pair. 27. Ifsheep cost $3.124 each, how many can I get for $75? 28. How many 25-cent balls can be bought for $8.75? 29. Divide 775 by 25. 30. Divide $8.25 by 75 ¥. 31. How many square feet are there in a lot 96 ft. long 25 ft. wide? 32. Find the total cost of 82 head of cattle at $75 per head. 33. How much must be paid for 32 cows at $37.50 each? 34. If sheep are worth $3.75 each, how much will a farmer receive for 32 sheep? 35. Ifa train goes at the rate of 25 miles per hour, how many hours will it take to go 675 miles? 274 ARITHMETIC. BILLS. 642. New York, Oct. 1, 1894. Mrs. WILLIAM MARTIN, Bought of GRAY AND WINTER. 1894 Aug. | 13 | 44 yd. Carpet $ .90 15 | 3 Oak Chairs 1.75 1 Rocker 12 | — 19 | 18 yd. Oil Cloth .50 27 | 1 Parlor Suit 75 | — Sept. | 19 | 6 Kitchen Chairs 75 1 Table 4 | 50 26 | 36 yd. Matting 331 1. Copy the above. Supply the missing amounts. 2. John R. Schultz has bought the following goods of Arthur B. Rowe & Co. : Jan. 8, 1894, 50 lb. of sugar, at 54%; 4 Ib. of tea, at 622%. Jan. 4, 10 lb. of coffee, at 824%; 2 bbl. of flour, at $5.75. Jan. 9, 24 bars of soap, at 162 ¢. 42 |b. of starch, at 8¥. Make out a bill dated Feb. 1, 1894. 3. Make out a bill for the following articles bought during March and April. Supply the names of buyer and seller, also the dates. 231 yd. of silk, at 80%; 18 yd. of lace, at $2.40; 64 yd. of — muslin, at 62%; 8 spools of sewing silk, at 7%; 4 pr. of stockings, at 65%; 6 yd. of linen, at 871¢; 4 doz. collars, at $2.10. 4. Make out a bill for the following goods bought June 15. 3 cases of torpedoes, at $2.20; 12 boxes of fire-crackers, at $1.621; 3 gross pin wheels, at $1.35; 5 gross sky-rockets, at $3.25; 2 doz. balloons, at $2.25; 45 lanterns, at 9. ee stilted 647. Sight Exercises, 1 ie 2, 31 3. 12x — 4. 36 x 14 9 x 16 8 29 X 18 36 5. 8. REVIEW. 42x 28 ——— 9. pal 4. a X46 10. 23 67 Bo 11. 96 Bo Le ede ee 12, 99 648. Visitors to Prospect Park. Month. January . February March April . May June . July August September October . November December Carriages, 43,398 112,140 128,520 120,240 359,621 208,096 220,860 260,516 333,639 421,220 316,020 174,256 Equestrians. 1,953 8,032 12,027 8,827 15,805 14,687 5,575 9,578 11,926 16,246 15,324 12,157 Pedestrians. 408,230 485,990 526,270 655,925 1,944,353 1,233,873 1,443,173 1,640,651 1,704,611 1,699,851 1,268,101 719,569 13. 14. 15. 16. Sleighs, Totals. 6,680 16 ee, EEE ed en, Monthly average. . Daily average . ee Sf | | | [| In the foregoing table find the total number of visitors for each month, the number of visitors by carriage for the year, the number of equestrians and pedestrians, the number by sleighs, together with the grand total for the year. and the monthly average. Find also the daily 276 ~ ARITHMETIC. SHORT METHODS. 649. Oral Problems, 1. Multiply by 25: 16, 19, 21, 238, 25, 29, 33, 36, 42, 48. 2. How many square feet in a lot 84 ft. long, 25 ft. wide? 3. What is the weight of 25 bbl. of flour, each weighing 196 lb.? 4. Find the cost of 25 lb. of coffee at 82¢ per lb. 5. What will a woman have to pay for 25 yd. of silk at $1.60 per yd. ? 6. A man sold 25 cows at $44 each. How much did he receive for them? 7. Multiply 64 by 123. 8. Find the cost of 124 bu. of wheat at 96% per bushel. 9. At $12.50 per bbl., how much would I have to pay for 56 bbl. of pork ? 10. How many pens in 124 gross? (144 to gross.) 11. Find the cost of 124 lb. of tea at 56¢ per lb. 12. How many square yards in a field 96 yards long 75 yards wide ? 650. Blackboard Exercises. Write only the answers : 1. 887x 25 8. 25x 2,174 15. 124 x 1,084 Ree Oo axe EAD 9. 837 x 250 16. 123 x 2,196 3. 9384x 25 10. 763 x 250 17. 123 x 3,670 4. 508x 25 11. 864x 124 18. 123 x 6,281 5. 25x 686 1 A a 2 19. 864 x 125 Cee OU 13. 236xX 122 20. 776 xX 125 7. 25 x 1,089 14, 404 123 21. 125 x 1,020 —s FRACTIONS. 651. Add 152 and 83. Adding 2 and ? (or 22. 23. 24. 25. 26. 27. 28. 652. 36. 37. 38. 39. 40. 41. 42. 3414. 15,7, 4238 + 194 842 + 188 404 + 162 158 + 822 352 + 201 128 + 411 10 + 1 29. 30. 31. 32. 33. 34. 35. Write answers: 57} — 182 985 — 561 463 — 192 675 — 8h 745 — 401 572 — 18} 981 — 568 43. 44. 45. 46. 47. 48. 49. 653. Multiply 18% by 4. 8x4=3. 4 eights are 32, and 3 are 35 (put down 5). 4 ones are 4 and 3 are 7. 50. 51. 52. 53. 54. 55. 56. 274 x 10 334 X 12 168 x 8 172 x8 192 x 6 153 x 3 13% x 4 57. 58. 59. 60. 61. 62. 63. 5 *). we get ly. Write +4 and carry 1. O51 4+ 468 578 +178 29% + 844 68% +188 74,5, + 18% 871 + 60$ 138 +814 372 — 291 58,8, — 502 242 — 61 902; — 182 6511 — 94 872 — 292 O41 — 6 Ans. 75. 204 x 11 402 x 5 163 x 7 374 X 3 458 X 5 234 x 4 17i x 6 277 278 ARITHMETIC. 654. Do not change dividends to improper fractions. 64. 3)453 71. 6) 25} 65. 4)564 72. 7)10% 66. 12)361 73. 6)752 67. 5) 724 74. 7)97335 68. 11)833 75. 10)874 69. 8) 374 76. 4)662 70. 9)481 77. 8)941 MULTIPLICATION OF DECIMALS. 655. Oral Exercises. 3 and a decimal multiplied by 2 and a decimal gives about what product? 44.02 x 2.05 = about what? 656. Slate Exercises, Multiply : ; 1) O26 2,0 6 9.6 x 1.125 aoe Oi 7 34.9 x 2.34 8. 64x45 8. 5.625 x 8.4 4. 7.23.75 9. 1.875 x 128 By 12.8 x 5.7 10. 42.36 x 2.95 657. In multiplying 32 by 2.5, how many decimals are pointed off in the product? In multiplying 3.2 by 2.5, how many are pointed off? How many are pointed off in the product of 9.6 by 1.125? DECIMALS. 279 658. Can you tell the relation the number of decimal places in the product bears to the number in the multiplier and in the multiplicand ? qo Xqo=? ies tie chix hee abe el hiearg 11. 1.75 x 64 16. 18.4 « 20.25 12. 8.875 x 40 Le MEDIO A240 18. 24.5 18.2 18. 66.6 x 3.34 14. 96x12} 195 204 Sc TD 15. 7.48 x 3.6 20. 400.04 x 89.25 DIVISION OF DECIMALS. 659. Divide 42 by 2.1. Changing the decimal fraction in the divisor to a common fraction, we have = 4221 — 42 10 — 420 42 + 23, = cy MBean ROA: IG ‘ 42 + 2.1 = 420 + 2l. — 660. When we change the divisor 2.1 to 21, we have multiplied it by 10, and the same change must be made in the dividend. 661. In the following examples, make each divisor a whole number by removing the decimal point, and make a correspond- ing change in the dividend. 662. Divide: 21: 80 -- 2.5 30. 72.5 ae 8+2.5 31. 960+ .03 23. 840-12 32. .847+ .007 24. 36 + 3% 33. 27 + .002 25. 36 + .9 34. 10—18 26. 126-638 35. 1.263 + .03 27; 48-15 36. 196.8 + .018 28. 18.36 + .6 37. 19.68 + .013 29. 50 + .25 38. 1.963 + .013 280 ARITHMETIC. 663. Remove the decimal point in the divisor Ans. 15 100. three places to the right, and make a corresponding 6013.) 1966300. change in the dividend, adding two ciphers. ayy To show where the decimal point originally be- err longed, it may be enclosed in a small circle, instead —— of being erased. 00 When the divisor is thus made a whole number, the decimal point in the quotient will be placed under (or over) the new decimal point in the dividend. 1.736 + 16 17.36 + .16 017386 = 1.6 1085 Ans. 108.5 Ans. .01085 Ans, 16) 1.736 @16.) 17636. 1e6.) @0.1736 186 186 136 ~ 80 80 80 39. .504+ .024 47. 392 3.2 40. 504+ .24 48. 48 + 3,009 41. 504+24 49. 92 -- .23 42. 504+ 24 50. .875+125 43. 168+.7 51. 381.17+ 8.11 44. 86 +112 52. 624 -+ 9.75 45. .875+.25 53. 48.195 = 3.57 46. 123.6~ .01 54. 829.31 + .019 664. Divide 381.6 by 95.032. 4.015 + 950382.) 381.600. 147200 521680 The sign (+) after the last figure of the quotient indicates that there is a remainder. 665. Divide, carrying out the quotient to 3 places of decimals : 5D. 31+ 13 58. 7.049 + 1.6 56. 4.5+17 59. 81.22+ 3.275 57. 920.07 = 46 60. 246.3 + 93.473 Se eo ae 13. 14. 15. 16. 666. Sight Exercises. Write answers at sight : .042 x 200 13 x 800 014 x 50 8.1 x 60 5. 6 7. 8 DECIMALS. 40x.7 25 X .08 284 X .2 .13 X 30 667. Remember that 369 + 1,000 = .569; that 219 100 =2.19;: and that 6 + 100 — .06. 2,460 + 3,000 369 + 3,000 219 3800 48.6+ 60 189+ 90 668. Slate Exercises. ae 1S: 19. 20. 196 + 4, Gin 27.9 + 281 9. .121 x 4,000 10. .061 x 500 LE. 7 038:.2.000 LZ Ol Byrn 100 = 369. — 369 thousandths 1000 000 500 300 21. 4.68-= 20 22. 380.5+ 500 23. 188+ 200 24. 248+ 4,000 Cancel the ciphers in the divisor, and remove the decimal point in the dividend a corresponding number of places to the left, prefixing ciphers if necessary. 1. 1,728 + 1,200 1299) 17.286 1.44 Ans. 4. 2.486 + 3,000 5. 1386.5 + 1,300 6. 848+ 80 7. 100.1+ 700 8. 1+ 40 9. 22+ 50 2. 172.8 + 1,200 1200) 1.7268 144 Ans. 10. ul . 845.6 + 1,200 . 4004+ 110 3. 1.728 + 1,200 1200) 016728 .00144 Ans. 45~ 800 OF nat) 5.28 eto0 . 907.5 + 1,500 282 ARITHMETIC, SIGHT APPROXIMATIONS. 669. Give approximate answers. Whole numbers, 670. 671. ihe 2. 3. 4. 5. 6. 17,8, X 388; or, about 17 Xx about 4. 8 9 2075 + 24; or, about 25+ 4 nearly. 63 X 63 7. 79934 + 9918 3001, + 1138 8. Ta x Tp 863 x 4 Oo Ted tte 35% + 355 10. 64,8 Give answers in whole numbers: er 8.75 x 9.999; or, 8.75 x 10 nearly. 24.002 + .4999; or, 24 nearly 55, or 4. 25.125 x 11.834 Toh LO. ee 36.845 + 6.105 8. 7.999 X:7.999 86.4 = .983 9. 7.001 x 12.003 82.04 x 5.001 10. 64.001 + .249 Give the cost, approximately, of : 1. 49 horses at $199 each. ($200 x 49.) _ = 199 yd. 2 ft. 11 in. of cloth at $2.50 per yard. 3 lb. 15 oz. of butter at 25 ¢ per lb. 398 coats at $12 each. 7 bu. 3 pk. 7 qt. potatoes at $2 per bushel. 798 base-balls at 25, cents each. 19 gal. 3 qt. 1 pt. alcohol at $2.49 per gallon. 995 lb. tea at 59% cents per pound. 7 houses at $4,995 each. 597 pounds of hay at 99 cents per 100 pounds. SLATE PROBLEMS. 283 672. Slate Problems, 1. Find the cost of 18,756 ft. of lumber at $30 per 1,000 ft. 2. A field is 14.25 rods long by 7.4 rods wide. What is its area in square rods? 3. Arodis 16.5 feet. How many rods are there in 231 feet. ? 4. How many marks are there in $100? (A mark is equal to 23.8 cents.) 5. Add3and 4 tenths, 96 thousandths, 100 and 5 thousandths, 27 hundredths. 6. From 2,700 take 27 hundredths. 7. Multiply 8 and 4 tenths by 9 and 25 hundredths. 8. Divide 96 and 75 hundredths by 322 and 5 tenths. 9. A load of hay, at 75 cents per 100 pounds, cost $13.98. What was the weight of the hay? 10. The circumference of a circle is 3.1416 times the diameter. flow many inches in circumference is a circle whose diameter is 20 inches ? 11. Show by a diagram the number of pieces of wire ? yd. long that can be made from 4 yd. of wire. 12. Show by a diagram that three-fourths of one is equal to one-fourth of three. 13. If 2of a yd. of material will make an apron, how many half aprons can be made from a yard? Show by a diagram. 14. A boy paid 6 cents for three-eighths of a pie. What would be the cost of the whole pie at the same rate? Make a drawing. 15. Seven-eighths of an acre of land is sold for $140. What is the price of an acre? 284 ARITHMETIC, REVIEW. 676. In comparing two fractions, reduce both to a common denominator. Change denominate numbers to same denominate unit. 677. Oral Exercises. 1. _ = 11. 12. Oi eet Cle Ot wy Ot What part of 7% 1s 33? (What part of 7 is 3?) 4 is what part of <4? What part of 24 is 4? 3 pt. is what part of a gallon? (8 pt. is what part of 8 pt. ?) What part of a gallon is 1 qt. 1 pt.? Divide 2 by 3. (Divide 10 fifteenths by 9 fifteenths.) Divide 3 by 2. How many sq. ft. in a rectangle 12 ft. long, 13 ft. wide? 4 of a day is how many hours and minutes? 14 ounces is what part of 2 pounds? $ ft. is what part of a yard? A strip of tape 3 yards long is cut into four equal pieces. How many feet and inches in each piece? 13. At $30 per month, how much rent will I pay in 1 year, 8 months? 14. 24 months is what part of a year? 15. At #2 ofa dollar per lb., how much tea can I get for $1? 16. How many sq. yd. in a room 15 ft. long, 18 ft. wide? 17. A lot is 25 ft. by 100 ft. How many feet of fence will it take to enclose it ? i 18. 1 pk. 1 qt. is what part of a bushel? 19. 15 is what part of 4 dozen? 20. Reduce 2% to lowest terms. > —— LONG MEASURE. 285 DENOMINATE NUMBERS. 678. Slate Exercises. 1. Add 4 days 6 hours, 9 days 11 hours, 3 days 7 hours. 2. What part of a week is 1 day 18 hours? 3. Ifa man receives $60 interest per year, how much will he receive in 3 years 74 months? 4. Reduce 3 days 18 hours to minutes. ° 5. How many days and hours are there in 8,100 min.? 6. 34°) of a day is how many hours? 7. How many hours and minutes in .4 day? 8. A man receives $1,460 per year of 365 days. What is his salary per week ? 9. Find the cost of 1 bu. 1 pk. 1 qt. of potatoes at 8 cents per half-peck. 10. A piece of meat weighing 27 lb. 12 oz. is divided among 6 persons. How many pounds and ounces does each receive? 11. How many bu., pk., and qt., are there in 5 bags, each containing 1 bu. 1 pk. 1 qt.? 12. How many gallons, quarts, and pints of ice-cream will be needed to give a half-pint to each one of 67 persons? 13. Find the cost of 7 lb. 10 oz. of tea at 40 cents per lb. 14. From a pile of 20 bu. wheat there were sold 10 bu. 3 pk. 7 qt. How much remained? 679. Long Measure. 12 inches (in.) 1 foot (ft.) 3 feet 1 yard (yd.) 5} yards 1 rod (rd.) 320 rods 1 mile (mi.) 286 ARITHMETIC. 15. How many yards in a mile? How many feet? How © many inches? 16. A field is 16 rods long, 12 rods wide. How many square yards does it contain? How many rods of fence will be needed to enclose it? How many feet? 17. How many rails each 30 feet long will be needed for a single track road (two tracks) 40 miles long? 18. A boy steps $3 inches. How many steps will he take in going 2 miles? 19. Dec. 20 the sun rises at Boston at 7.26 a.m. and sets at 4.30 p.m. How long is it between sunrise and sunset? How much longer is the day at Charleston, 8. C., where the sun rises at 6.58 A.M. and sets at 4.57 p.m.? 20. On June 21 the sun rises at Boston at 4.23 a.m. and sets at 7.40 p.m. On the same day it rises at Charleston at 4.53 a.m. and sets at 7.11 P.m. What is the length of the day at each place? Change: 21. 17 lb. and 4 oz. to ounces. 22. 84 tons and 1,560 lb. to pounds. 23. 37 gal. and 3 qt. to quarts. 24. 45 gal. to pints. 25. 63 qt. and 1 pt. to pints. 26. 27 bu. and 3 pk. to pecks. 27. 48 pk. and 7 qt. to quarts. 28. 84 pk. to pints. 29. 27 mi. to yards. 30. 16 rd. and 3 yd. to yards. 31. 15,000 min. to days, etc. 32. 25,124 lb. to tons, etc, 33. 1,650 ft. to rods. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 34. 35. 36. 37. » 38. 39. 40. DENOMINATE NUMBERS. 287 876 pt. to gallons and quarts. 228 in. to yards and feet. 1,650 rods to miles and rods. 864 hours to weeks and days. 296 qt. to bushels and pecks. 315 oz. to pounds and ounces. 743 months to years and months, & ft. 6 in. + 9 ft. 5 in. + 12 ft. 3 in. 380 min. 15 sec. + 80 min. 18 sec. ++ 45 min. 24 sec. 9 yr. 35 mo. + 18 yr. 7 mo. + 22 yr. 2 mo. 19 wk. 4 da. + 7 wk. 5 da. + 8 wk. 9 mi. 169 rd. + 84 rd. + 3 mi. 67 rd. 7 yd. 1 ft. + 83 yd. + 19 yd. 2 ft. 18 gal. 1 gt. + 16 gal. 2 qt. + 15 gal. 3 qt. 5 pk. 3 qt. +6 qt. +7 pk. 1 qt. 24 bu. 3 pk. + 24 bu. 3 pk. + 24 bu. 3 pk. 12 qt. 1 pt. + 12 qt. 1 pt. + 12 qt. 1 pt. + 12 qt. 1 pt. 12 qt. 1 pt. x 4. 56., Qmt..25 rd, x. 7. 24 bu. 3 pk. x 3. 57. 15 wk. 3 da. x 5. 5 pk. 8 qt. x 9. 58. Tyr. 3 mo. X 10. 18 gal. 1 qt. x 8. 59. 40 min. 35 sec. X 2, 33 yd. 1 ft. x 6. GOs,0 ibe ine La. 61. 25 ft. — 18 ft. 7 in. 62. 50 min. 13 sec. — 27 min. 30 sec. 63. 12 yr. 1 mo. — dyr. 11 mo. 64. 50 wk. 4 da. — 18 wk. 6 da. 65. 15 mi. — 8 mi. 148 rd. 288 7a. 72. 73. 74. 75. ARITHMETIC. 66. 33 yd. 1 ft. — 18 yd. 2 ft. 67. 240 gal. 1 gt. — 94 gal. 2 qt. 68. 83 pk. 3 qt. — 59 pk. 1 qt. 69. 170 bu. 1 pk. — 85 bu. 2 pk. 70. 1385 qt. 1 pt. — 67 qt. 1 pt. 87 qt. + 2. 76. 253 yd. 1 ft. + 10. 50 min. 85 sec. + 5. 77. 387 gal. + 6. 156 yr. 9 mo. + 9. 78. 222 bu. 3 pk. + 9. 73 wk. 2 da, + 3. 79. 150 qt. + 4. 50 mi. 185 rd. + 7. 80. 75 bu. + 8. 81. 87 qt.+ 48 qt. 1 pt. 82. 50 min. 35 sec. + 10 min. 7 sec. 83. 78 bu. +9 bu. 3 pk. 84. 5 lb. 1 oz. +9 oz. 85. 14 ft. 2in. +1 ft. 5 in. MEASUREMENTS. 680. How many square yards in a room 6 yards long, 5 yards wide? How many square yards in a room 18 feet long, 15 feet wide? 681. Slate Exercises. Calculate the number of square yards in the following. First reduce each side to yards. 1. 2 3 4. 5 18 yd. by 21 yd. . 54 ft. by 63 ft. . 72 in. by 108 in. 19 yd. by 47 yd. . 67 yd. by 89 yd. 10. 54 in. by 72 ft. 38 ft. by 36 yd. 27 ft. by 96 ft. 54 ft. by 72 in. 48 ft. by 45 ft. oO OM 2H MEASUREMENTS, 289 682. First, indicate the operations; then cancel. 11. Find the number of square yards in a room 18 ft. 4 in. long, 22 ft. 6 in. wide. 18 ft. 4 in, = 182 ft. = vs yd. = - yd. 22 ft. 6 in. = 224 ft. = =! yd, = & yd. 5 Area = a x ° sq. yd. Canceling, re = ge = 453 sq. yd. 12. How many square yards in a room 13 ft. 1 in. long, 27 ft. wide ? 18 fe.1in. = 157, in, = oa yd, 27 ft. =9 yd. 157 157 x 9_ 157 Area = ( 15” y 9) sq. ya. eae yd, rea C ) sq y 36 ; 394 sq. yd 4 13. How many square inches in 12 panes of glass, each 5 inches long, 7 inches wide? 14. A piece of cloth is 48 yards long, 24 inches wide. How many square yards does it contain? 15. A merchant imports 8 pieces of cloth, 36 yards to the piece. How many square yards of cloth are there, if it is 32 inches wide? 16. A board fence 6 feet high surrounds a lot 25 feet front by 100 feet deep. How many square feet of boards in the front fence? Inthe back fence? In each side fence? In the whole? (Make diagrams.) 17. A room is 18 feet long, 15 feet wide, 12 feet high. How many square feet in the floor ? Draw a rectangle to represent the ceiling. Write the dimensions in their proper places, and write in the centre the number of square feet in its sur- face. Draw diagrams of the four walls; give dimensions and surface of each. 290 ARITHMETIC. 18. How many faces has a cube? If one edge of a cube meas- | ures 4 inches, how many square inches in the entire surface? Suppose you wish to make a cube out of a single piece of pasteboard. Make a drawing to show the shape of the piece needed, without allowing anything for overlapping parts. 19. The United States government charges a duty of 4% per square yard on imported cotton cloth. What duty must the. importer pay on a piece containing 24 yards, # yd. wide? 20. What will be the cost at $1 per square yard for flagging a sidewalk 12 feet wide and 30 feet long? CHAPTER VIII. DECIMALS, — BILLS. — DENOMINATE NUMBERS.— MEASURE- MENTS. — PERCENTAGE, — INTEREST, DECIMALS. 685. Changing Oommon Fractions to Decimals. Slate Exercises. Reduce the following common fractions to decimals; 7.e, per- form the indicated division : 1. 1+ 800 8. 3 15. 345 2. 1+ 40 9. z2ty 16. ;is 3. ae ip eee 17. 315 4, 25 ieee 18. 13, ayo te toe tos 6.8, 13. 23 20 ant 7. so 14. +f 21. ine7 686. Changing Decimals to Common Fractions. What is the denominator of a decimal fraction ? What prime numbers are contained in10? What are the only factors of 10? The prime factors of 100? Of 1,000? Can zo, be reduced to lower terms? Why? Can 78, be reduced to lower terms? Why? Can 7495, be reduced to lower terms? How can we tell by merely looking at a decimal whether or not it can be reduced to a common fraction of lower terms? 291 292 ARITHMETIC. 687. Slate Exercises. Reduce the following to common fractions — lowest terms. Do not find the greatest common divisor. 22. 0076 BPR Uva rey ae 33. .027 24. .0275 34. .00365 25. .44 35. ..90 26. .03125 36. .0009 27. .486 37. .816 28. .3750 38. .15625 29. .37500 39. .0375 30. .144 40. .00625 31. .0006 41. .096 ADDITION OF DECIMALS. 688. Add the following, reducing the common fractions to decimals : 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 183 + 9.084 + 25,1, + 163 + 2.09 + 86,1, + .0975 275-95 + 58.64 + 8.6796 + 301 + 82 + 99 + 68723 S47, + I3z hg + Syed + r3bby + 684.1 + $4174 250 + 1875.93 + 163 + SA, + o%& + 608.94 + .0005 8.6796 + 96.8 + 188 + 250,1, + 341, + 1876 40,8 + 7.2832 + 86.3 i 128.46 + 23, + 41.5 + 84 540 + 1.82 + .576 + 7885 + 68:5 + 8954 + + 7.51 5.808 + .25 + 567.8 + 5 4896 + 49.795 + 8.3, 7.08 + 23.04 + 8:2; + 848 + 3ole + rh, + 7.00019 8.999, + 84 + 507 + 28 49, + =% + 6.8819 + 3.1416 DECIMALS. 293 SUBTRACTION OF DECIMALS. 689. Give answers in decimals: 275.8 — 8125 38738 — 99.0127 108/00) a . 62.365 — 483 1982 — 13.6431 BY. (hip 24 ve 9, 88 58. 2,845 — 34545 59. 168.3, — 54.8759 60. 18.42—.576} 61. 1,84714 — 344,9, MULTIPLICATION OF DECIMALS. Give answers in decimals: . 24.75 x 34 982 x .00046 . 1482, x 12.5 3801 x .012 wx X 1.48 67. 19.5 x .000484 68. 1.876 x 33 69. 3.48 x 4.8665 TO easel 71. 192.38 x .288 DIVISTON OF DECIMALS. 691. Give 3 places of decimals in quotient, exclusive of ciphers. Divide without writing products (Arts. 385, 616): 72. 7.3845 + .29 . 840,753 + 4.18 4.054 + 18.25 123.5 + 884 AT] + 5.825 8126 + .0134 12.845 = .0047 8756 + 4.3822 8 + 122 . 15.3678 + .9125 82. 83 84. 85. 86. 87. 88. 89. 90. 91. 48.45 + .089 . 89562.478 + 4279 346.25 + 64.8 9.1342 + 208.3 1784 + 29.57 843.71 + 1.127 83.087 + 5.37 137.84 - 7.91 38.9008 + .523 81074 + .009157 294 ARITHMETIC, 692. Solve by short division. When ciphers are canceled in the divisor, what change must be made in the decimal point of the dividend ? 92. 18.756 = 300 102. 48.64 200 93. 48.36 + 4,000 103. .00531 + 90,000 94. .4824 + 12,000 104. 96.51 + 60 95. 11.011 + 700 105. 87.5 + 500 96. 3.6504 + 90 106. 183.275 + 10,000° 97. 45.63 + 1,500 107. 1.7632 + 1,600 98. 130.13 + 1,100 108. 1.5639 + 130 99. .8712+ 60 109. 6144+ 120 100. 3.075 + 5,000 110. .976 + 800 101. .07056 = 140 111. .8008 + 7,000 MISCELLANEOUS. 694. Slate Exercises. 1. Find the cost of 24,400 bricks @ $6.25 per M. Ans. $6.25 x 24.400 = $6.25 x 24.4. (How do we divide by 1,000?) 2. 760 pounds of hay @ 95 cents per cwt. (100 lb.). 48,600 laths @ $2.80 per M. | 39,250 stamped envelopes @ $21.30 per thousand. 1,875 pounds of straw @ 68 cents per cwt. 108,745 Philadelphia bricks @ $22.00 per M. 14,860 oranges @ 75¥¢ per 100. 2,576 eges @ 181 per doz. 4,500 cigars @ $35 per M. 10. 28 doz. wax candles @ $13.50 per gross (144). i i DECIMALS. 295 695. Solve by cancellation where possible : 38,648 lb. of wheat @ 90 per bu. (60 lb.). 11. 12. 13. 14. 15. 16. 93.8). 696. Perform indicated operations. 18,964 Ib. 48,576 Ib. 69,104 Ib. 74,816 lb. of coal @ $5 per ton (2,000 lb.). of oats @ 36f per bu. (82 lb.). of rye @ 911¢ per bu. (56 lb.). of corn @ 481 per bu. (56 |b.). 360 meters of cloth @ $1.10 per yd. (1 meter = 39.37 inches). 17. Cost in United States money of 386 hats @ 24 francs each (1 franc = 19.3f). 18. 480 meters of cloth @ 1.10 marks per meter (1 mark = Change divisor to whole number, making corresponding change in the dividend. Cancel. 19. 21. 22. 23. 24. 25. 7 34.2 X ofp ae 35 DET 239.4 .249 x 3.92 .098 soe 12 288 6876 x .27 081 ele OO 19.3 3.1416 x 2.3 £7804 28. 29. 30. i) Hho 6 x 188 OD X. he T2op 234 450 x 23.8 1.19 p40 x Gul 49 x 100 576 X 6.3 14.4 x 25 TD OU L ese G 306 x 8.75 .9 X 68 296 ARITHMETIC. 697. Reduce to common fractions — lowest terms: 31. 32. . 2 34. . 43. 44. 45. 46. 35. 36. 37. 38. 47. 48. 49. 50. 699. Blackboard Exercises. Write answers at sight - 1. ro ee ee ee ao fF WO wo KF OS OMDMWAT Pw D 244 -+ 153 132 x 6 424 — 131 81h 42 502 + 204 801 — 401 5i x 5d 242+ 2 3864- 3 17i+ 4 . 214+ 5 . 488-— 6 . 182+ 7 . 244+ 8 . 362+ 9 0064 ou 062 8334 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 39. .009 40. .044 41. .7612 42. 0374 bl. ae. 52. ah 5. pS uae 54. 35 544 — 392 488 — 4 621 + 232 123 x 6 662 + 334. 334 — 162 13x 1% 80 + $$ 72 + & 56 + 80 75 + 1t 90 + 12 98 + 13 81. + 1% RECTANGLES. 297 MEASUREMENTS. 700. How many square inches in each of the following rect- angles? First change each dimension to inches. 1. 42 in. by 36 in. 6. 9 ft. by 11 ft. 2.) (Linpy 18 in. 7. 27 in. by 30 in. 3. 3 ft. 1 in. by 4 ft. 2.in. 8. 65 in. by 92 in. 4. 5 ft. 3 in. by 6 ft. 4 in. 9.071 tt. din. by: Arya. 5. 12 ft. by 18 ft. 10. 3 yd. by 6 ft. 6 in. 7OL. How many square feet in each of the following rect- angles? First change each dimension to feet, or to feet and a fraction. 11. 18 ft. by 24 ft. 16. 3 ft. by Li yd. 12. 36 in. by 4 ft. 17. 42 in. by 4 ft. 13. 6 yd. by 8 yd. 18. 25 ft. by 17 ft. 6 in. 14. 1 yd. by 48 in. 19. 42 in. by 48 in. 15. 32 ft. by 4 ft. 20. 13 yd. by 15 yd. 702. How many square yards in each of the following rect- angles? Change each dimension to yards, or to yards and a fraction. 21. 18 yd. by 25 yd. 26. 36 yd. by 24 in. 22. 15 yd. by 1 yd. 1 ft. 27. 17 ft. 6 in. by 32 in. 23. 27 ft. by 36 ft. 28. 22 ft. 9in. by 18 in. 24. 54 ft. by 2 ft. 6 in. 29. 108 in. by 90 in. 25. 24 yd. by 27 in. 30. 180 ft. by 54 in. 298 ARITHMETIC. SPECIAL DRILLS. 703. Give sums: 185+ 89 450 + 690 56 + 256 680 + 350 394+ 77 870 + 260 39 + 461 940 + 480 704. Give differences : 224 — 185 1,089 — 274 3831 — 286 1,197 — 786 442 — 369 1,258 — 680 554 — 487 1,476 — 828 705. Give products : 82 x 25 25 x 73 64 x 25 25 x 55 98 x 4 89 x 5 78 x 6 67x 7 706. Give quotients : 792— 9 415+ 5 ei Bpeomal 406+ 7 192 + 88 380 + 76 693 + 63 486 = 54 707. Give answers: 123 x 5 112 x 4 103 x 3 93 X 2 3X 88 4x 7§ DX 62 6 x 5§ 576+ 76 85 + 646 768+ 48 56 + 575 1,200 — 610 1,460 — 780 1,820 — 390 1,210 — 240 46 X 334 39 X 383} 26 X 333 19 X 833 975 = 25 850 + 25 675 + 25 | 825 -~ 25 45,x 7 333 x 8 2755 X 9 1, x 10 274 + 815 783 + 306 459 + 740 624 + 585 458 — 69 315 — 87 672 — 95 818 — 29 63 x 12 54 x 11 15 Keo 36 no 300 + 33} 4331 + 334 666% + 332 5881 + 331 11 x 2,5 12 x 33 11 x 43 10 x 5% ee a a REVIEW. 299 708. Oral Problems. 1. I sold 875 bushels of wheat to one miller and 87 to another. How many bushels did I sell? 2. Bought goods to the amount of $4.29. How much change from a $5 bill? 3. What will be the cost of 89 tons of coal at $5 per ton? 4. If 49 hats cost $147, what is the cost of one hat? 5. 567 marbles are divided among 9 boys. How many does each receive ? 6. How many yards in 5 pieces of cloth, each containing 12% yd.? 7. Divide 292 by 7. 8. What will be the cost of a barrel of flour at $5.25 and 8 lb. of sugar at 6 #? 9. When silk is 75¢ per yd., how many yards can be bought for $9.75? 10. If 23 yd. ribbon cost 42 cents, what will 33 yd. cost? 11. How much must be paid for 55 lb. of raisins, at 8¢ per lb.? 12. Find the cost of 320 lb. of hay at 60 per hundred pounds. 13. If eggs are sold at the rate of 18 for 25 cents, what will be the cost of 6 dozen eggs? 14. Three men require 22 days to do a certain piece of work. How long would it take 11 men to do the same work? 15. A father earned $14.60, his son earned $7.80. What were the earnings of both? 16. How many yards of fence will be required to enclose a rectangular field 98 yards long and 50 yards wide? 17. A farmer divides his farm of 425 acres into fields of 123 acres each. How many fields has he? 300 ARITHMETIC. 18. There are 36 inches in a yard. How many yards are there in 324 inches? 19. The product is 925, the multiplier is 25. What is the multiplicand ? 20. What will be the cost of 46 tons of hay, at $123 per ton? 21. What is the weight of 25 firkins of butter, each contain- ing 56 pounds? | 22. At $1.75 per yard, how many yards of cloth can be bought for $49? 23. What price was paid for 20 sheep, at $8.75 per head? 24. A man saved $320 per year for 5 years. How much more would he require to make $ 2,000 ? 25. Mr. Jones sold a lot for $675, thereby losing $85. What did he pay for it? 709. Slate Problems, 1. The width of a room is 2 of its length. How many square feet in the floor, if the width is 15 ft. ? 2. If 2 lb. 6 oz. of tea cost 95 cents, how many pounds and ounces can be bought for $ 2.35 ? 3. What will be the duty on 175 kilograms of wool at 33 ct. per lb.? (1 kilogram = 2.2046 Ib.) 4. John and James went out together. John had 38 cents. When one of the boys had spent 18 cents and the other had spent 16 cents, they had 24 cents left between them. Find the amount of money James had. Find 4 of the sum of 2 and 3. What is 2 of the difference between ¢ and 3? What fraction added to ? gives }? Change 1,3, hour to seconds. 15 ft. Sees cee REVIEW. 301 9. £ of what number equals 180? 10. The half of a number added to its fourth 213 part equals 213. What is the number ? 4 4 11. A farm is sold for $5,700, at a loss of 5 of the cost. What was the cost? , 12. When it is noon at Philadelphia, it is 15 seconds and 10 minutes past 5 p.m. at Paris. What time is it at Philadelphia when it is noon at Paris? 13. A, B, and C buy a house. A furnished + of the cost, B 4, and © $1,200. What did A A poe ES A and B pay, respectively ? 4 + $1200. 14. A room is 221 ft. long and 18 ft. wide. What will it cost, at.5¢ per yard, for a strip of moulding around the walls? 15. How many square yards of carpet would be needed for the floor of the above room ? 16. How much is the fraction $ increased or diminished when 2 is added to each of its terms (numerator and denominator)? 17. After James has spent # of his money and + of the re- mainder he has but $1.50 left. How much had he at first ? 18. A man buys oranges at $1.20 per 100. How many would he have to sell, at 25% per dozen to gain $3.18? 19. From a piece of cloth measuring 281 yards, there have been sold 22 yd., 68 yd., 18% yd. If the remainder is worth $13.10, what is the value of the whole piece? 20. A man left for charitable purposes $3,600, which was 2 of his money. The remainder was divided equally among 8 rela- tives. How much did each relative receive? $ 3600. $2 $ Charitable purposes Eight relatives 302 ARITHMETIC, REVIEW. 710. Supply missing numbers : 1. $18,432.65 2. $26,459.88 3. $93,259.80 9,876.04 6,087.90 10,059.77 632.95 12,364.58 5,387.04 27.88 3,030.30 20 5.63 999.99 23.50 99 6,875.84 681.19 04 365.93 32,065.88 87 ? 793.20 2.90 6.50 2,684.39 83.15 25.19 ? 700.07 308.12 15,909.75 4,862.99 4,321.00 123.40 ? 87 6.15 $50,000.00 $ 76,543.21 $ 202,020.20 711. Complete the following table of public school attendance: CoLORED. WHITE, x __|| Aggre- Male. | Female. | Total. || Male. | Female. | Total. ore AUDULN IS tween he 40 35 75 || 1,706| 1,753 | 3,459 || 3,534 Binghamton . . 21 20 2320 Ieee i Brooklyn... 839 797 54,647 | 54,439 Cohoes . . . .| — — a LSG2y Vigo HATAITA Rete int 40 61 23174 2211 New York. . . 806 806 98,029 | 98,304 Rochester . . . 31 38 8,258} 8,597 PLOT Cuter ies, hy, 5 5 1,094 992 Saratoga Springs. | .35 34 1,088) 1,116 fy OUKOTS aasvuee de 15 15 1378) L718 REVIEW. SHORT METHODS. 713. Sight Exercises. . 68 x 25 neo x 49 e244 x 15 - 82x 123 o FF WW WO . 88x 12} - 64 fa C25 Ly Meo OL 12. 8. 66 X 334 13. 9. 48 x 75 14. 10. 24x62) 16. 714. Slate Exercises. 9,347 x 25 863 x 75 8,123 x 124 6,483 x 334 8,128 x 125 9,847 x: 250 9,347 x 22 9,347 x 75 6,483 x 662 6,488 x 374 715. Oral Problems. 1. What will be the cost of 49 Ib. of coffee at 25 ¢ per lb. ? 2. I paid $14.75 for eggs at 25% per doz. How many dozen did I buy? 96 xX 25 25 X 81 48 x 374 92 x 50 82 X 334 16. 17. 18. 19. 20. 303 88 x 25 25 X 97 16 x 874 66 x 662 16 x 663 it, 12; 13. 14. 15. 16. 17. 18. 19. 20. 4,896 x 871 1,284 x 622 75 xX 2,468 334 X 3,870 662 x 3,456 162 X 1,266 8,408 x 62} 8,875 x 87h 1,995 x 123 7,314 x 250 3. What will be paid for 104 bu. of wheat at 8719 per bushel ? 4. How many bushels of corn at 624% per bu. can be bought for $150? 304 ARITHMETIC. 5. How much will be paid for 99 yd. of ee goods at 831¢ per yd.? 6. How many yards of carpet at 662% per yard can be bought for $ 84? 7. Find the cost of 15 doz. collars at 121 ¢ each? 8. Paid $24 for cuffs at 162% per pair. How many dozen pairs were bought? 9. What will be the cost of 128 lb. of tea at 75¢ per lb. ? 10. A bale of cotton at 61f per lb. cost $25. What was the weight of the cotton ? 11. A farmer sold hay at 75 per cwt., receiving for it $39. How many cwt. did he sell? 12. How many bbl. of mess pork at $12.50 per bbl. can be bought for $175? 13. What will be the cost of 96 yd. of carpet at $1.25 per yd.? 14. When wheat sells at $1.124 per bu., how many bushels can be bought for $198? 15. At $3.50 each, what will be paid for 84 coats? 16. Find the cost of 28 hats at $2.75 each? 17. A real estate agent sold 97 lots at $250 each. How much did he receive for them? 18. What will be the cost of 248 horses at $125 each? 19. At 4 cent each, how many pen-holders can I buy for $ 5.76? 20. Paid $3,675 for cows at $75 each. How many were bought? APPROXIMATIONS. ; 805 716. Slate Exercises. Multiply 1875 by 21. Do not place the multiplier under- 3750 neath the multiplicand. 39,375 Ans. 1. 3,456 x 31 Gl Goyee LOL 2. 7,465 x 81 Zab hOs hao OL 3. 2,345 x 41 $11,689) xP 4, 5,482 x 91 9.1) 4,892 x 71 5. 9,284 x 51 10. 10,754 x 121 717. 2468 x 18. Write the product by 8 one place to the 19744 right (above or below). 44 424 Ans. 11. 8,734 x 13 15. 57614 x 21 12. 4,075 x 18 16. 345x15 X41 13. 9,485 x 14 Tie 40 x Sica L 14. 5,832 x 19 18 20x Ge OL APPROXIMATIONS. 719. Give approximate answers, at sight (Art. 521): 1. 232 lb. of tea @ 50L¢. 24 horses @ $124.95. 64 yd. of carpet @ 8733, ¢. 485 bu. of wheat @ 993. 96 lb. of coffee @ 247. 840 yd. of dress goods (@ 33 360 yd. of oil cloth @ 663£. 48 owt. of straw @ 623%. 92 hats @ $1.49}. 128 lb. of lard @ 123%, g, 5 16 OM AHA Pe ww _ sf 306 Inair © ND ARITHMETIC. 720. Give approximate answers in whole numbers: 11. $27 + 2413 21 12. $299.96 + $1.492 22. 13. $24.05 + 375% 23 14. $15.03 + 128¢ 24 15. $60+$ 2.4918 25 16. $32 + 333¢ 26 17. $69.95 + 871¢ 27 18. $60+ 62,1.¢ 28 19. $64 + 6611 29 20. $27.95 + $1.75 30 721. Sight Exercises. Give products : 360 x .25 8. 840x .075 560 x .125 955) 960. 9.005 240 x .875 10. 1,200 x .001 400 x .625 11. 1,500 x .002 480 x .75 12: 96 x .84 820 x .875 13. 840 x .022 720 x .025 14. 1,500 x .06 722. Give quotients : 240 + .5 8. 87+ .05 860 + .75 - 48+ .005 AB + 195 10. 72+ .025 23 4.25 11. 92+ .002 360 + .875 12. 93+ .034 100 + .625 13. 54+ .021 154 + 875 14. 182+ .06 - 17.3 x 3.98765 256.008 x .249875 15. 16. 17. 18. 19. 20. 21. 15. 16. ty 18. 19. 20. 21. » 20.1234 X 15.98 aoa Wipe alae by . 86.4 x .996 . 33.833 x 5.004 . 799.387 x .125 eo eos . 7.33 X 11.0083 . 64.002 x .38750 400 x .04 165 x .062 176 x .064 3,300 x .004 880 x .124 105 x .8 210 x .10 76 + .04 88 = .004 65 + .123 84 + .8 11 + .064 42 -+ .62 93 + .5 723. Give results: DENOMINATE NUMBERS. elite 6. Ay 75 12. 95x14 @ He 13. 18 x 2§ 74 X 24 2S aera 7. © 49 14. 16 x 34 ie 15. 14x 54 3, 63x19 nano haben a 11 V7 Oe 96 x 27 . 22+ 28 fe BD See A 1S sores 23 10. 72x14 19. 60+ 2} Be nex oo Ba 8 11. 38 x 1} 20. 40+ 62 DENOMINATE NUMBERS. 724. Oral Problems. 1. What will be the weight of 16 hams that average 10 lb. 5 oz. each ? 2. From a chest of tea containing 54 lb. there were sold 27 lb. 70z. How many lb. remain? 3. Seven bushels of potatoes are divided among 8 persons. How many pecks and quarts does each receive ? 4. How many square inches in the surface of a sheet of paper measuring 11 inches by 13 inches? 5. How many feet and inches in $ yd.? 6. What decimal of a pound is 14 oz. ? 7. A man buys a bushel of hickory nuts. After he sells 2 pk. 4 qt., what fraction of the bushel has he left? 8. A dealer puts 80 gal. of milk in cans holding 1 qt. 1 pt. each. How many cans does he fill? 9. At $20 per month, how much rent will a man pay in 1 year and 5 months? 10. 75 hundredths of a pound is how many ounces ? 308 ARITHMETIC, 11. How many feet in 5 rods? 12. 1 gal. 3 qt. 1 pt. of milk is divided among 5 people. How many quarts and pints does each receive? 13. What fraction of 2 lb. 3 oz. is 1 lb. 4 oz.? 14. Three-eighths of a ton is how many pounds? 15. Change 9 hours 36 minutes to the fraction of a day. 725. Slate Problems. 1. 382 hams weigh 458 pounds. What is the average weight? 14 lb. 5 oz. 32458 lb. 138 10 lb. remainder. 16_ 160 oz., new dividend. 0 Ans. 14 lb. 5 oz., average. 2. 595 gal. of oil are put into 14 barrels. How many gal. and qt. does each contain? 3. If there are 42 gal. and 2 qt. in a barrel of oil, how much oil will there be in 15 barrels? 4. In the written number 54,372, the value expressed by the 5 is how many times the value expressed by the 2? 5. A piece of cloth containing 57 yd. is divided equally among six persons. What is the length of each one’s share? 6. How many minutes in 1 day, 1 hour, and 1 minute? 7. July 1 is the last school day. How many days’ vacation will there be, if school begins Sept. 6? 8. How many hours and minutes are there from half-past 3 Saturday afternoon to a quarter before 9 Monday morning? 9. How many steps, 2 ft. 6 in. long, must a man take in walking 1,200 yards? FRACTIONS. 309 10. A man owns a plot of ground 420 ft. long, 240 ft. wide. How many rods of fence will be required to enclose it ? 11. A train goes from Jersey City to Washington, 228 miles, in 4 hours 12 minutes. How many miles an hour does it travel? How long does it take the train to go one mile? 12. On Monday a boarding-house uses 3 gal. 2 gt. of milk; on Tuesday, 4 gal.; on Wednesday, 3 gal. 3 qt. 1 pt.; on Thurs- day, 4 gal. 2 qt.; on Friday, 6 gal.; on Saturday, 5 gal. 2 qt. 1 pt.; on Sunday, 3 gal. 1 pt. How much is used during the week, and what is the average per day? 13. June 21 the sun rises at New York at 4.23 a.m. and sets at 7.40 p.m. How long is the night? 14. From 3% bu. take 37 pk. 15. What is the length in rods of a fence surrounding a field 206 ft. 3 in. wide and twice as long? REVIEW FRACTIONS. 726. Slate Exercises. Add: 293 + 17+ 62,5 Weigelebaa. geal ay, 5 432 + 30+ 632+ 8144 114, + 258+ 428 + 82+ 91 83.8, + 914 + 7018 + 6% +37 244801474 6914 93814 14+ 214 + 821 + 484 4 542 927. + 68.5, + 33 + 724.89 75519 + 951 + 303 + 524 13,5 60x45 + 493% + 183% + 65 + 90} OM AFT Pw Do 310 ARITHMETIC, 727. Subtract: 11. 481 — 258 16. 12. 942 —184 17. 13. 6954, — 3313 18. 14. 577, — 2617 19. 15. 100.3, — 5043 20. 728. Multiply : 21. 481 x 2 26. 22. 82x 411 27. 23. 168 x 127 28. 24. 4,3 x 122 29. 25. 24 X 24 X 2h 30. 729. Divide: 31. 75-2 36. 32. 7+ 5 37. 33. 188.7. +5 38. 34. 1718-44 39. 35. 183 -- 24 40. 730. Perform indicated operations 41. A2. 43. 44. (15+ 7) + (68 + 4) (} x 20) — (4g X 23) eS 16 BL 2of 4 lot lt 234 + (84+ 18) 45. 46. 47. 48. 126.8, — 832 99,8, — 61% 84,7, — 15%, 23g5 — O75 91219 — 68428 $x 12x 354 231 x 102 88 x 93 vo X 14 x 34 162 X 8 X 834 73 + 3,3, 373 +15 128.4, + 25 5 -+- 94 7 + 123 524 x (1d — He 7m GES uae tite 9) SM a a LB ics 8 7 of (3$ — 25 + 95) REDUCTION. Se DENOMINATE NUMBERS. 732. Slate Exercises. Change : 1. 48 pounds and 9 ounces to ounces. 2. 34 rods and 3 yards to yards. 3. 2 miles to yards. 4. 3 days and 17 hours to hours. 5. 24 minutes and 15 seconds to seconds. 6. 8 tons and 1675 pounds to pounds. 7. 43 gallons and 8 quarts to quarts. 8. 75 gallons to pints. 9. 19 bushels and 3 pecks to pecks. 10. 19 bushels and 3 pecks to quarts. 11. =, ton to pounds and ounces. 12. .03125 ton to pounds and ounces. 13. Z yard to feet and inches. 735. Slate Exercises. Change: 975 ounces to pounds and ounces. 396 inches to yards. 517 hours to days and hours. 1,694 seconds to minutes and seconds. 9,314 pounds to tons and pounds. 987 pints to gallons, quarts, and pints. 1,485 quarts to pecks and quarts. 185 pecks to bushels and pecks. 840 hours to weeks. 312 739. ARITHMETIC. DENOMINATE NUMBERS. . Slate Exercises. 13 lb. 6 oz. 5lb. 9 oz. 25 |b. 10 oz. 19 yd. 1 ft. 2 ft. & yd. 1 ft. 8 hr. 40 min. 25 min. 5 hr. 9 min. 5 min. 80 sec. 11 min. 25 sec. 9 min. 18 sec. Leveson. 2 ft. 6 in. Aisle tt. 4 ins Subtract : Ait ot 4 lb. 7 oz. 15 yd. 1 ft. 9 yd. 2 ft. 17 hr. 9 hr. 50 min. 40 min. 30 sec. 6 min. 45 sec. 1 yd. 1 ft. 1 in. 2 ft. 9 in. 10. 10. 18 gal. 3 qt. 9 gal. 1 qt. 2 qt. 11 bu. 3 pk. 6 bu. 2 pk. 2 pk. 1 pk. 6 qt. 1 pk. 7 qt. 5 qt. ?) Sowks5 da. 6 wk. 6 da. l wk. 3 da. NG Bd A a Relay at ay: 4T. 983 1b. 1756 |b. . 20 gal. 1 qt. 6 gal. 3 qt. 89 bu. 2 pk. 67 bu. 3 pk. 3 pk. 2 qt. 2 pk. 7 qt. 11 wk. 1 da. 9 wk. 5 da. 5 T. 896 Ib. 1984 lb. REVIEW. Fad Bs: CANCELLATION. 740. Slate Problems. . Indicate operations, and cancel where possible : 1. If 56 men can pave a street in 24 days, how long will it take 32 men to pave it? 2. When a vessel sails 168 miles a day, she completes her voyage in 14 days. In what time would she complete it if she sailed 196 miles a day ? 3. If a field would support 64 sheep for 21 days, how long would it support 48 sheep ? 4. If 42 men could build a wall in 24 days, how many men could build it in 18 days? 5. If 21 horses are worth as much as 35 cows, how many horses are worth as much as 55 cows? 6. A girl that wrote 36 letters to a line, took 15 lines in writing a piece of dictation. How many lines would a girl that wrote 30 letters to a line, require for the same dictation ? 7. If a boy that steps 27 inches at a time takes 1,000 steps in going home from school, how many steps will be taken by a boy that steps 30 inches? 8. If 1,920 bricks will build a wall 15 yards long, how many bricks will be required for a similar wall 24 yards long? 9. A train going 44 miles an hour, went a certain distance in 9 hours. How long would a train take that went 36 miles an hour ? 10. Find the cost of one-eighth of a barrel of flour (196 lb.), at the rate of 11 cents for 34 pounds. 11. Six men can do a certain piece of work in eighteen days. How long would it take eighteen boys to do the same work, if “one man can do as much work as two boys? 314 ARITHMETIC. 12. Ifa certain quantity of flour will last 48 persons 57 days, how long will it last 88 persons? 13. Divide 2 of 56 by 12 times 32. ee DENOMINATE NUMBERS. 742. Slate Exercises. Multiply : 1. 12 1b. 7 02.x8 6. 4yd.1ft.x5 2. 3 hr. 10 min. x 7 7. 7 min. 18 sec. x 10 | 3. 4. 985 Ib. x 11 8. 9 gal. 3 qt. x 2 | 4. 7 bu. 8 pk. x9 9. 2 ft. 9in. x8 | 5. 8wk.4da.x4 10. lyd.1 ft. 6 in. x6 | 743, Divide: 11. 91b. 20z.+2 16. 18 yd. 2 ft. +7 12. 31 gal. 2 qt.+9 17. 19 ft. 2 in. +10 13. 19 hr. 21 min. +3 18. 34 T. 936 lb. +4 14. 26 bu. 1 pk.+5 19. 17 wk. 1 da.=6 15. 41 min. 44 sec. + 8 20. 52 yd. O ft. 9 in. + 11 744, Divide: 21. 18 lb. 4 oz. by 4 lb. 9 oz. 22. 16 yd. by 2 yd. 2 ft. 23. 2 da. 3 hr. 36 min. by 6 hr. 27 min. 24. 47 min. 42 sec. by 5 min. 18 sec. 25. 84 yr. 7 mo. by 12 yr. 1 mo. 26. 19 da. 3 hr. by 2 da. 3 hr. 27. 3 mi. 40 rd. by 125 rd. MEASUREMENTS. 315 28. 103 T. 808 lb. by 8 T. 1,234 Ib. 29. 52 gal. 2 qt. by 3 gal. 2 qt. 30. 68 bu. 1 pk. by 5 bu. 1 pk. 31. 30 ft. 8 in. by 1 ft. 11 in. 32. 52 yd. 9 in. by 4 yd. 2 ft. 3 in. 33. 51 wk. 3 da. by 2 wk. 6 da. MEASUREMENTS. 745. Slate Problems. Make a diagram in each case: 1. A lot 25 ft. by 100 ft. has on it a house 25 ft. by 55 ft. How many square feet are there left for a yard? 2. How many square feet are there in the floor of a room 24 ft. long, 18 ft. wide? 3. How many square yards are there in the ceiling of the same room ? 4. Find the number of square yards of plastering needed for the end wall of a room 18 ft. wide, 9 ft. high, after deducting for two windows each 6 ft. high, 44 ft. wide. 5. How many square yards of plastering will be needed for the opposite wall of the same room, 18 ft. wide, 9 ft. high, after deducting for a door 74 ft. high, 6 ft. wide? 6. Calculate the number of square yards of plastering needed for two side walls of a room 24 feet long, 9 feet high, after deducting for a fireplace 6 feet square on one side. 7. A house 30 ft. by 60 ft., with an addition 15 ft. square, is built upon a lot 100 ft. square. How many square feet of ground are covered by the building? How many square feet remain for a garden? 316 ARITHMETIC. 8. Measure the top of a brick and calculate the number of square inches in its surface. How many square inches in the surface of the bottom of the brick? Measure one side, and cal- culate its surface. How many square inches are there in the surface of the opposite side? How many square inches in each end? 9. Measure a crayon box, and calculate the number of square inches in each face. 10. Calculate the number of square feet in the floor of the class-room. In the ceiling. In each side wall. In each end wall. PERCENTAGE. 746. Per cent means hundredths. Six per cent means six hundredths, ;§5, or .06. It is written 6%. 747. Oral Exercises. 1. What is ;8, of 200? 11. 4% of 125 2. Find .06 of 300 12. 7% of 500 3. 6 per cent of 400 13. 5% of 240 4. 6% of 50 14. 1% ot 600 5. 6% of 150 15. 49% of 600 6. 6% of 250 16. 1% of 600 7. 6% of 125 17. 24% of 600 8. 6% of 75 18. 34% of 400 9. 6% of 60 19. 4% of 400 10. 6% of 160 20. 9% of 90 748. What fraction equals: 21. 25% 25. 20% 29. 62% 22. 121% 26. 50% 30. 374% 23. 331% 27. 61% 31. 624% 24. 162% 28. 34% 32. 874% PERCENTAGE. 317 749. Find: 33. 50% of 96 42. 300% of 140 34. 25% of 72 43. 150% of 140 35. 121% of 120 44. 250% of 140 36. 61% of 48 45. 125% of 140 37. 334% of 36 46. 1% of 140 38. 162% of 126 47. 1% of 350 39. 841% of 72 48. 2% of 350 40. 100% of 140 49. 39% of 350 41. 200% of 140 50. 4% of 350 750. Slate Problems. 1. A house is valued at $24,500. How much taxes must the owner pay at the rate of $22.40 per $1,000 valuation ? 2. A consignee sells a lot of cotton for $1,872.50. He receives 2% of this amount as commission. How much is his commission ? 38. I loan $600 at 6% interest per year. How much interest should I receive from Jan. 1, 1892 to Jan. 1, 1894? 4. How much will it cost me to insure goods to the amount of $18,760 at one per cent? 5. A dealer imports books worth $548.40, on which he pays duty to the government at the rate of 25%. What is the amount of the duty ? 6. Highty per cent of a class of 55 pupils are promoted. How many are not promoted? 7. A man buys a house for $16,000 and sells it at a profit of 3 per cent. How much does he gain? 318 ARITHMETIC, 8. A clerk spends for rent 18 per cent of his income of $1,850 per year. What rent does he pay? 9. A girl spelled correctly 95 per cent of 60 words. How many did she miss? 10. Tea costing 40 cents per pound is sold at a profit of 50 per cent. What is the selling price? 751. Oral Problems. 1. What per cent does a boy receive if he solves 16 examples of the 20 given out? 2. What is the interest on $200 at 4% for 2 years? 3. If 2% yd. of calico cost 22 cents, how many yards can be bought for 60¢ ? 4. What part of a ton is 125 pounds? 5. How old, Dec. 1, 1892, was a boy born Sept. 1, 1879? 6. What is the cost of 3,500 bricks at $20 per M? 7. How many sheep, at $5 each, should be given in exchange for 12 horses, worth $200 each ? 8. Reduce J, to a decimal. 9. One hundred fifty marbles are divided among a certain number of boys. Each receives 12 and there are 6 remaining. How many boys are there? . 10. At 3 oranges for 5 cents, what will be the cost of 4 dozen oranges ? 11. 75 men can do a certain piece of work in 9 days. How long will it take 45 men to do the same work? 12. If it takes 24 yards of carpet, a yard wide, to cover a floor, how many yards ? yd. wide will be needed for the same floor ? FEDERAL MONEY. 319 BILLS. 753. PHILADELPHIA, Sept. 24, 1894. Mr. Harrison JARVIS To Wu. Hart & Son, Dr. =e To 50 lb. of Pipe @, 51g “3 Faucets ‘ 75“ “1 Sink 4 | 75 “ 33 days’ Labor ‘“ $4.75 — 1. Copy and complete the above bill. 2. R. W. Jones has done 34 days’ work, @ $3.50 per day, for Charles Johnston. He charges for 850 ft. lumber at $2 per hundred; 5 lb. of nails at 9% per lb.; 8 locks @ 50¥; 2 bolts at 10¢. Make out his bill. 3. A gardener furnishes 3 rose-bushes at 75%; 4 grape-vines at 50%; 11 fuchsias at 830%; 25 pansies at 10%. He charges $3.25 per day for 23 days’ labor. Make out his bill. 4. An upholsterer charges $3 per day for repairing some fur- niture. He supplies 6 lb. of hair at 50¢ per lb.; 17 yd. of plush at $1.75 per yd.; 3 papers of tacks at 10%; cord, gimp, etc., 47%. He works 4 days. Make out his bill. 5. Make out and receipt a bill for four articles bought to-day by John R. Brown from Smith and Robinson, grocers. 6. Make out a bill containing ten items bought by Mrs. S. W. Robb, at different times during October, 1894, from Frederick Loeser & Co., dealers in dry goods. 7. Make out a bill for labor done and materials furnished by Anthony Jones, gardener. 320 ARITHMETIC. INTEREST. 754. Oral Exercises, What will be the interest on $100 for 1 year at 4% ? On $200 for a year at 5%? On $300 for a year at 6% ? On $400 for a year at 7% ? On $250 for a year at 4%? At 4% per year, what will be the interest : On $200 for 1 year? On $300 for 2 years? On $100 for 3 years? On $200 for 14 years? On $200 for 1 year 6 months? . What will be the interest on $200 for 3 years at 5%? - On $300 for 2 years at 6% ? . On $400 for 6 years at 3% ? . On $100 for 5 years at 7% ? . On $250 for 2 years at 4% ? . On $100 for 1 year 6 months at 6%? 17. On $200 for 3 months at 4% ? At 4% per year, what will be the interest: 18. On $200 for 6 months? 19. On $3800 for 4 months? 20. On $400 for 3 months? 21. On $300 for 2 months? 22. On $150 for 1 month? 23. Find the interest on $ 24 for 1 year at 5%. 24. On $36 for 1 year at 4%. 25. On $67 for 1 year at 3%. a a Gi.2 OC Het 2635-19 et INTEREST. PAS 755. Slate Exercises. Find the yearly interest on: 1. $286.50 at 4% 11. $1,257 at 7% 2. $485 at 6% 12. $168 at 33% 3. $375.40 at 5% 13. $244 at 54% 4. $379 at 3% 14. $890 at 73% 5. $486 at 44% 15. $63.75 at 4% 6. $186.75 at 4% 16. $937.50 at 6% 7. $199.50 at 2% 17. $980.40 at 5% 8. $636 at 34% 18. $159.60 at 24% 9. $84.70 at 6% 19. $1,357.37 at 7% 10. $93.25 at 8% 20. $2,146.18 at 44% Find the interest on: 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. $ 290 for 2 years at 4% $ 1,400 for 3 years at 44% $ 2,840 for 4 years at 5% $ 1,250 at 6% for 3 years $5,360 at 54% for 2 years $380 at 3% for 44 years $ 780 for 1 year 4 months at 6% $ 2,560 for 2 years 6 months at 5% $ 1,025 for 3 years 3 months at 4% $ 1,296 for 7 months at 7% $ 648 for 5 months at 5% $275 for 4 months at 3% $1,000 for 11 months at 6% 322 ARITHMETIC. AREAS OF RIGHT-ANGLED TRIANGLES. 756. The square shown in the diagram is divided into two parts by a diagonal. One side of the square measures 10 feet. 1. Mark in each triangle its area. Square. Rectangle. 2. Divide a rectangle 20 ft. by 12 ft. into two parts by a diagonal. Mark in each triangle its area. 3. Draw a right-angled triangle 3 in. by 4in. Calculate its area in square inches. 4. How many square yards in the surface of a right-angled triangle whose base measures 30 feet, and whose perpendicular measures 224 feet. Perpendicular Find the area in square feet of the following right-angled triangles. (Change each dimension to feet.) Base 20 yd., perpendicular 380 ft. Base 16 in., perpendicular 3 ft. Base 30 in., perpendicular 1 yd. Base 3 ft. 6 in., perpendicular 5 ft. Re eg 00 ae = 03: - Ot Base 2 yd. 1 ft., perpendicular 1 yd. 9 in. 10. Base 50 yd., perpendicular 36 yd. 758. Multiply. Do not reduce to improper fractions. 128 x 45 51 4 times 123 = 51; 3 of 123 = 41. uaa 55L 1. 182 x 64 6. 2. 258 x 84 qs 3. 164 x 54 8. 4. 361 x 94 9. 5. 223 x 6. 10. 759. Divide. Short Division. 11. 15) 752 _ 19 12. 18) 169.8, 20 13. 14) 2958 21 14. 12) 9765. 22 15. 16) 1953 23 16. 21) 2314 24 17. 26) 2634 25 18. 24) 5044 26 SHORT METHODS. SHORT METHODS. . Base 1123 ft., perpendicular 30 yd. . Base 90 in., perpendicular 2 ft. . Base 121 yd., perpendicular 134 yd. . Base 1 rod, perpendicular 74 ft. . Base 334 ft., perpendicular 18 ft. 6 in. 163 x Tt 483 x 121 37% X 10% 268 x 92 323 X 87 . 17) 1993 . 18) 2214 . 12) 8652 . 18) 8924 . 14) 5673 . 28) 702% . 25) 8052 . 22) 4653 323 324 ARITHMETIC. REVIEW. 763. Answers in fractions: RO ti atans Lee pha Les (SUM Dlidy ste eed implity 34 1b oR 2. Add 4, ye 20> a's) ao0- jee Re at txt 3. Reduce 2 ' 8 4 to a simple fraction. 144-4-3 4. Divide (4+4+2) by ¢xHx #8) 5. Simplify +3 of 3 of 4. 1—i 14 6. TRS ee en? 1 7. Find the value of q ers 8. (244 14) + (24+ 33) =? 9. Find the value of 24 times the quotient of (8 — 21) + (23 ie 3): 10. 83¢4+14—7245—-4=? 764. Answers in decimals : 11. Divide the sum of .736 and 1.2854 by their difference. 12. Divide .1 by .2 and .35 by 35, and find the product of the quotients. 13. Reduce <74, to a decimal, and divide it by .3125. 14. Divide .12096 by .082. 15. Multiply .00273 by 3,000.456, and divide the product by .08. 16. Divide 12.8125 by .000625. LY 18. 19. 20. REVIEW. B20 Divide 51.5 by 412, and 412 by 51.5. Multiply 31.5 by 27.9, and divide the product by 9.765. 4.25 844 Reduce 0021 x 8.004 Find the value of 024 765. Reduce to lowest terms: 1. of B. Ay 9. Hy 2. Hf 6. sits 10. 3848 8. $4 1. $48 11. 98h 4. oH 8. Hig 12. $485 CHAPTER IX. DENOMINATE NUMBERS. — SURFACES AND VOLUMES. — PERCENTAGE. — INTEREST. REDUCTION OF DENOMINATE NUMBERS. 766. Reduce 5 gal. 3 qt. 1 pt. to pints, 5 gal. 3 qt. 1 pt. In 5 gal. there are 20 qt. Adding the 3 qt, we _4 have 23 qt. Multiplying by 2 to reduce to pints, 23 qt. and adding in 1 pt., we get the answer, 47 pints. 47 pt. Ans. 767. Reduction Descending, Slate Exercises, Reduce to pints: 1. 16 gal. 1 qt. 1 pt. 6. 314 gal. 2. 27 gal. 2 qt. 7. 9 gal. 23 qt. 3. 16 gal. 8. 10 gal. 2 qt. 1 pt. 4. 16 gal. 1 pt. 9. 27 gal. 1 pt. 5. 34 gal. 3 qt. 1 pt. 10. 4 gal. 3 qt. 14 pt. 768. Change 67 pt. to gallons, quarts, and pints. 2| 67 pt. Ars qt. 1 pt. 8 gal. 1 qt. 1 pt. Ans. 769. Reduction Ascending. Change to gal., etc.: 11. 156 qt. | 16. 277 pt. a ote 17. 139 pt. 13. 408 pt. 18. 171 qt. 14. 1,502 pt. 19. 63 qt. 15. 63 pt. 20. 711 pt. 326 DENOMINATE NUMBERS. 770. Reduction Descending. 21. 22. 23. 24. 25. 26. 27: 28. 29. 30. 31. . 4wk. 6 da. 11 hr. to hours. . # of a week to hours. Change 17 yd. 1 ft. 9 in. to inches, 4 mi. 100 rd. 4 yd. to yards. 74 bu. 2 pk. 7 qt. to quarts. 156 lb. 11 oz. to ounces. 63 yd. O ft. 3 in. to inches. 19 bu. 0 pk. 3 qt. to quarts. 11 rd. 34 yd. to feet. 63 gal. 3 qt. to pints. 3 bu. 6 qt. to quarts. 17 T. 369 lb. to pounds. 15 hr. 16 min. to seconds. . wy of a mile to yards. . .00125 T. to ounces. 771. Reduction Ascending. Change : 36. 1,876 in. to yd., etc. 43. 37. 475 oz. to lb., etc. 44. 38. 729 qt. to bu., etc. 45. 39. 8,675 min. to da., etc. 46. 40. 4,972 lb. to T., ete. 47. 41. 972 rd. to mi., etc. 48. 42. 117 pt. to gal., etc. 49. 9,483 sec. to hr., etc. 877 qt. to bu., ete. 1,495 oz. to lb., etc. 373 in. to yd., etc. 216 qt. to gal., ete. 876 rd. to mi., etc. 319 pt. to gal., etc. 50. 3,520 yd. to mi. 328 ARITHMETIC. 772. Oral Exercises. 1G a Cont oar wn & © How many hours in 2 of a day? . How many hours in 4 of a day? How many minutes in 4 of an hour? How many hours and minutes in 4 of a day? How many quarts and pints in 3 of a gallon? . How many hours and minutes in .2 day? How many quarts and pints in .375 gallon? Change .3 day to hours and minutes. Change .625 bu. to pk. and qt. What part of a gallon is 1 pt.? . What part of a gallon is 3 pt. ? . What part of a gallon is 1 qt. 1 pt.? . What decimal of a gallon is 1 qt. 1 pt.? . What decimal of a gallon is 2 qt. 1 pt.? . What part of 2 gallons is 2 qt. 1 pt.? . Change .375 bu. to pt. and qt. . What decimal of a bu. is 4 qt.? . What fraction of a day is 3 hr. 20 min.? . Reduce 960 min. to hours. 20. How many minutes in a day ? 773. Slate Exercises. 1. What decimal of a ton is 3 lb.? 2. What fraction of a day is 12 min. 30 sec. ? 3. 4. Reduce .03125 day to minutes. Reduce 3), of a day to minutes. DENOMINATE NUMBERS. 329 5. What decimal of a day is 9 minutes? 6. What will be the cost of 15 T. and 750 lb. coal at $5 per ton? 7. If coal is $5 per ton, how much can be bought for $18.76 ? 8. If7 T. 296 lb. coal cost $35.74, how much will I have to pay for 18,748 lb. ? 9. A man pays $48.92 for 9 T. 1,568 lb. coal. How many tons, etc., would he receive for $ 73.11? 10. Change 2 ft. 7 in. to the fraction of a yd. 11. Reduce 3 pk. 4 qt. 1 pt. to the decimal of a bu. 12. How many pk., qt., etc., in .9375 bu. ? 13. If .1875 of a gal. of cologne cost $1.125, what will 1 pt. cost ? 14. Find the cost of 42 gal. 3 qt. 1 pt. oil, at 16% per gal. 15. Reduce 14 of a gal. to qt. and pt. 16. What part of 3 T. is 1 T. 960 |b. ? . 17. A man raised 189 bu. 2 pk. and 2 qt. of rye. He sold 119 bu. 2 pk. 4 qt. What fraction of his crop did he sell ? 18. 10 bu. 1 pk. of seed is packed in 8 bags. How much is there in each bag? 19. What decimal of a day is 21 hr. 14 min. 24 sec.? 20. How many feet in a mile? COMPOUND ADDITION. 774. Add the following : 1. 17 \b. 3802. 2. 18 bu. 3 pk. 7 qt. 4 lb. 9 oz. 9 bu. 2 pk. 4 qt. 23 lb. 12 oz. 14 bu. 1 pk. 6 qt. 15 oz. 2 pk. 330 ARITHMETIC. Se LO Yara tt. 0): T.lL2 eros, 17 yd. 4 in. 8 T. _980,1b. INSTA ee ba 476 lb. 11 in. 1 T. 1,830 Ib. 4.11 da. 5\hr, 19 min: 8. 2 wk. 5 da. 12 hr. 23 da. 40 min. 6 da. 15 hr. 17 hr. 50 min. 5 wk. 2 hr. 5 da. 20 hr. 6 min. 2 da. 19 hr. 5. 93 gal. 3 qt. 1 pt. 9. 18 mi. 100 rd. TA gal. 34 rd. 18 gal. 1 qt. 29 mi. 2 qt. 1 pt. 6 mi. 160 rd. 6. 5 hr. 30 min. 20 sec. 10. 47 yr. 11 mo. 45 min. 33 sec. D yr. 9 mo. 6 hr. 11 min. 5 sec. 7 mo. 10 hr. 38 min. 30 sec. 22 yr. 5 mo. 11. 487 T., 316 T. 1,816 lb., 247 lb., 43 T. 811 1b.,19 T. 25 Ib. 12.) 88 .1b}'15:0z,; 9. lb. 5.02; 18 Ib.) 22 Ibsi1L oz.; & Ib. 8 iog;, 12 oz. 13. 8 hr. 15 min. 5 sec., 37 min. 52 sec., 5 hr. 48 min., 23 hr. 59 min. 5 sec. 14. 72 gal. 3 qt. 1 pt., 17 gal. 1 qt., 2 qt. 1 pt., 90 gal. 1 pt. 15. 7 yd. 2 ft. 11 in., 19 yd. 6in., 105 yd.,4 yd. 2 ft. 2in., 1 ft. 16. 93 mi. 300 rd., 87 mi. 154 rd., 194 rd., 3 mi. 175 rd., 9 mi. 17. 82 yr.1 mo., 19 yr. 10 mo., 25 yr. 9 mo. 6 da., 8 mo. 15 da. 18. 4 wk. 6 da. 17 hr., 20 wk. 5 da., 4 da. 11 hr., 9 wk. 5 da. ST ah 19. 5 hr. 13 min. 23 sec., 16 hr. 27 min. 30 sec., 48 min. 5 sec., 24 sec. 20. 8 bu. 3 pk. 7 qt., 5 qt., 2 pk. 1 qt., 4 bu. 6 gt., 3 bu. 1 pk. 1 qt. Se ee DENOMINATE NUMBERS. Bal 775. Find answers: UG WARE PIL oul itey A + se VER Layo Pes: OZ. 22. 14 bu. 2 pk. 4 qt. + ? 18 bu. 1 pk. 1 qt. ase 1 yde2 ft.7 in. AY 26. 10 hr. 15 min. 80 sec. ae 24 hr. OT ReAO TL LoLOribs an 495° Teh od2 Ip, 28. 9wk. 6 da. 11 hr. oe 15 yd. 2 ft. 2 in. 21 wk. 3 hr. 24. 19 da. 14 h. 40 min. 29. 84 mi. 24 rd. 1" a 30 da. 100 mi. 15 rd. 25. 14 gal. 2 qt. 1 pt. 30. 13 yr. 9 mo. ots af 18 gal. 1 qt. 20 yr. 31. 83 lb. 4 oz. +? = 100 lb. 32. 16 bu. 2 qt. + ? = 25 bu. 1 pt. 33. 1 ft.4in.+?=9 yd. lin. 34. 47 da. 15 min. + ? = 60 da. 35. 93 gal. 3 qt. 1 pt. +? = 150 gal. COMPOUND SUBTRACTION. 776. Subtract: 36. 83 yr. 3 mo. 15 yr. 9 mo. 37. 62mi. 84 rd. 19 mu 159ird: 38. 76'T. 295 Ib. Seek OF 2 lb, 39. 100 lb. 332 40. 41. 42. 46. 47. 48. 49. 50. ARITHMETIC. 52 wk. : 43. 16 yd. 9 in. 13 wk. 8 da. 7 hr. _T yd. 1 ft. 11 in. 19 gal. 1 pt. 44. 100 bu. 8 gal. 3 qt. 42 bu. 3 pk. 7 qt. 18 hr. 5 min. 45. 45 da. 1 hr. 1 min. 40 min. 25 sec. 6 da. 6 hr. 6 min. From 27 bu. 1 pk. 5 qt. take 18 bu. 8 pk. 7 qt. From 100 gal. 1 qt. take 83 gal. 2 qt. 1 pt. From 22 hr. 15 min. 20 sec. take 15 hr. 45 min. 40 sec. From 17 lb. 2 oz. take 18 lb. 8 oz. From 100 bu. take 74 bu. 2 pk. 1 qt. COMPOUND MULTIPLICATION. 777. Add 3\b.9 oz. Add 4 gal. 3 qt. 1 pt. 3 lb. 9 oz. 4 gal. 3 qt. 1 pt. 4 gal. 3 qt. 1 pt. Multiply 3 lb. 9 oz. by 2. Multiply 4 gal. 3 qt. 1 pt. by 3. 778. Multiply: . 13 bu. 3 pk. 6 gt. by 2. 59. 2 pk. 7 qt. by 10. . 20 gal. 2 qt. 1 pt. by 3. 60. 3 qt. 1 pt. by 11. - T lb. 10 oz. by 4. 61. 4 yr. 6 mo. by 12. . Shr. 15 min. 15 sec. by 5. 62. 5 wk. 6 da. 12 hr. by 16. . 23 bu. 8 qt. by 6. 63. 4 T. 250 Ib. by 18. . 82 gal. 1 pt. by 7. 64. 3 yd. 1 ft. 6 in. by 22. . 25 |b. 4 oz. by 8. 65. 2 mi. 15 rd. by 382. . 83 min. 38 sec. by 9. 66. 4 hr. 15 min, 20 sec. by 9. DENOMINATE NUMBERS. $3 67. 31 gal. 2 qt. by 42. 71. 1 bu. 2 pk. 3 qt. by 13. 68. 4 qt. by 37. 72. 4 yd. 2 ft. 9 in. by 15. 69. 43 sec. by 215. 73. 21 hr. by 24. 70. 4 wk. 6 da. 20hr. by 19. 74. 3 yr. 11 mo. by 14. 75. 1 gal. 1 qt. 1 pt. by 30. COMPOUND DIVISION. 779. Divide: 76. 15 lb. 9 oz. by 3. 88. 13 wk. by 5. 77. 2 1b. 3 oz. by 5. 89. 74 mi. 80 rd. by 4. 78. 2 gal. 1 qt. by 3. 90. 69 yr. by 12. 79. 5 bu. by 4. 91. 27 bu. 3 pk. 4 qt. by 2. 80. 7 hr. by 6. 92. 76 gal. 3 qt. 1 pt. by 3. 81. 17 lb. 7 oz. by 3. 93." hr. Liminv 7 sec. by 9: 82. 37 bu. 3 pk. 6 qt. by 2. 94. 33 wk. 3 da. by 12. 83. 67 yd. 2 ft. by 4. 95. 36 yd. 6 in. by 7. 84. 33 da.15 hr. 57 min. by 3. 96. 45 bu. 6 qt. by 6. 85. 563 gal. by 6. 97. 20 da. 13 hr. 4 min. by 16. 86. 22 hr. 20 min. 20sec. by 4. 98. 15 gal. 3 qt. by 18. 87. 112 T. 125 lb. by 5. 99. 54 yd. 1 ft. 4in. by 20. 100. 41 wk. 4 da. 1 hr. + 4. 101. 457 hr. 37 min. 30 sec. + 9. 102. 147 gal. 3 qt. 1 pt. +18. 103. 157 bu. 3 pk. 6 qt. + 7. 104. 175 yd. 2 ft. 6 in. + 10. 105. 188 mi. 12 rd. 2 yd. + 6. 106. 311 da. 21 hr. 36 min. + 12, 334 ARITHMETIC. 107. Divide 180 da. 3 hr. 4 min. by 16. 11 da. 6 hr. 11 min. 30 see. 780. Dividing 180 days by 16, 16)180 da. 3 hr. 4 min, we get 11 days quotient and 4 20 days remainder. Reducing 4 days 5) By (remainder) to hours and adding 3 hours, the 24 next dividend is 99 hours. This — gives 6 hours quotient, 3 hours 99 hr. remainder. Reducing to minutes 3 hr. (remainder) and adding 4 minutes, the next 60 dividend is 184 minutes. This 184 fats gives 11 minutes quotient, and 8 ——7 minutes remainder. Reducing, we 24 have 480 seconds for next dividend. 8 min. (remainder) Dividing, as before, the last quo- 60 tient is 30 seconds. 480 Bec. 0 108. Divide 236 gal. 1 qt. by 18. 109. 384 yd. 9 in. by 21. 781. 15 yd. 2 ft. 9 in. .21)334 yd. 0 ft. 9 in. 124 19 yd. (remainder) 3 57 ft. 15 ft. (remainder) Eb: 189 in. 0 110. 825 lb. by 48. 116. 288 hr. 9 min. by 54. Lee Tby 25. 117. 863 gal. 2 qt. 1 pt. by 47. 112. 483 mi. by 32. 118. 83 wk. 1 da. by 72. | 113. 84 yr. by 24. 119. 1,188 T. 910 lb. by 81. | 114. 462 bu. by 36. 120. 1,629 yd. 1 ft. by 96. 115. 1,078 yd. by 63. 121, 1,867 gal. 13 pt. by 125. DENOMINATE NUMBERS. 335 782. Oral Problems. 1. How many tons and pounds of coal in 40 bags, each con- taining 80 pounds? 2. If it takes 3 hr. 20 min. to hoe a row of corn, how long will it take to hoe 3 rows? 3. A man puts up 34 pounds of tea into 4 oz. packages. How many packages does he make? 4. 3 pecks 3 quarts of apples are divided among 9 children. What quantity does each child receive? 5. What part of a day is 30 minutes? 6. If there are 24 gallons of wine in 12 bottles, how many pints are there in each bottle? 7. What is the weight of two packages each containing foitb, 11 oz, ? 8. What part of an hour is 40 seconds? 9. What is the rent of a house for 1 year 9 months at $16 per month? 10. If 3 gal. 2 qt. 1 pt. of milk are taken from a can contain- ing 10 gal., how much is left in the can? 11. 5 hams weigh 614 1b. What is the average weight? 12. There are on an average 41 pupils in a class. How many are there in 14 classes? 13. At 371 cents per yard, how many yards can be bought for $6.75? 14. Find the cost of 16 bbl. of flour at $ 6.124 each. 15. $1.65 is equally divided among 15 boys. What is the share of each? 16. A floor containing 404 square yards is 7 yards long. How many yards wide is it? 17. How many ounces in 57 pounds? 336 | ARITHMETIC, 783. Slate Problems. 1. If a watch gains 1 min. 17 sec. per day, how much will ‘it gain during March and April? 2. How many bu., pk., and qt. in 1,449 lb. corn, weighing 56 lb. to the bu. ? 3. A chain, 97 yd. 8 in. long, contains 1,000 links. Find the length of one of the links. 4. A farmer sold out of 5 bu. of peas the following quan- tities: 3 pk. 6 qt.; 4 pk; 4 pk. 3 qt.; 1 bu. 1 pk. 1 qt. How much has he still to sell? 5. A man walks on Monday 15 mi. 161 rd.; Tuesday, 10 m1. 84 rd.; Wednesday, 19 mi. 15 rd.; Thursday and Friday, 12 mi. 121 rd. each day; Saturday, 14 mi. 240 rd. What distance per day does he average? 6. If the sun rises at 5 hr. 10 min. A.m., and sets at 6 hr. 42 min. P.M., how long is the day? How many hours and minutes of night ? 7. An iron rod is 12 ft. 6 in. long. From it are cut 73 bolts, each 13 in. long. How much is left? 8. A man rows a mile in 10 min. 30 sec. How long would he take to row 27 miles at the same rate? | 9. A man rows 51 miles in 23 hr. 5 min. and 30 sec. How long does he take to row a mile? 10. If I lost $50 by selling a lot for two-thirds of its cost, what would I have lost if I had sold it for three-fourths of its cost ? | 11. At the rate of $2.75 per day of 10 hours, how much should be given a man that works from a quarter before 8 until 5 minutes past 11? 12. If a railroad train travels 18 miles in 40 minutes, how far will it travel, at the same rate, in 74 hours? REVIEW. SPECIAL DRILLS. 784. Give sums: 163 + 137 256 + 184 149 + 312 458+ 197 42+-35-+ 77 63 + 19+ 54 87 + 22+ 48 91+ 63+ 17 785. Give differences: 400 — 163 501 — 875 275 — 137 650 — 488 185 + 546 668+ 193 ~ 167 + 734 476 + 155 4,170 + 470 1,260 + 850 2,140 + 680 3,450 + 390 540 — 384 361 — 149 455 — 358 662 — 176 7,310 — 6,850 8,610 — 7,680 5,000 — 4,670 4,960 — 4,380 618 — 495 455 — 128 648 — 509 856 — 147 786. Give products: 11x 15 12x 14 Se 14x 14 48 x 162 32 X 374 24 x 624 36 X 662 787. Give quotients: 165 +15 168 + 14 169 +138 294 + 14 6162 + 162 8374 + 3874 7334 + 334 6874 + 624 788. Give answers: 132 x5 142 x4 14,3, x 7 137, x 8 on = lon a = aloo O op a b& Coas oolbo crib | 21 x 15 22 x 14 dl x 18 41x 14 185 + 45 136 + 68 220 + 44 196 + 49 24 + 22 18 + 41 49 + 1% 35 + 34 28 < 75 40 x 874 39 x 832 49 x 25 9334 + 662 975 +75 6123 + 874 925 +25 21 x 3% 22 x.4,% 18 x 32 17 x 44 388 ARITHMETIC, 789. Oral Problems, 1. How many ounces in 11,3, lb.? 2. 258 yd. equal how many ft.? 3. A dealer bought 652 tons of coal and sold 476 tons. How much had he left? 4. Sold my wheat for $347 and my oats for $154. How much did I receive for both? 5. 402 yd. of ribbon are cut into 7 pieces. Find the length of each piece. 6. How many sq. yd. in a-floor 52 yd. long and 54 yd. wide? 7. What will be the cost of 14 Ib. of lard at 14¥ per lb.? 8. At 11 each, how many lead pencils can I buy for 27¢? 9. What part of a 196-lb. barrel of flour is contained in a 49-lb. bag? 10. At 45¥ per yd., how much lace can be bought for $1.35? 11. A woman has saved $8383. How much more must she save to have $1,000? 12. What will be the cost of 16 Ib. of sugar at 43¢ per Ib.? 13. Spent $2.56 for dry goods and $1.84 for groceries. How much did I spend for both? 14. Find the cost of 8 lb. 10 oz. of butter at 32¢ per lb. 15. At $.875 per yd., how much ribbon can be bought for p.75? 16. Ifit takes 1% yd. of cloth to make a jacket, how many can be made from a piece of cloth containing 30 yd. ? 17. A boy paid 50¥ for the use of a boat for 34 hours. What was the price per hour? 18. If 13 pounds of raisins cost $1.69, what is the cost of 1 pound? REVIEW. TABLE. 339 FOREIGN COMMERCE OF THE UNITED STATES. Exports AND Imports, 1875-1891. 790. The following table shows the values of the exports and the imports of merchandise during each year from 1875 to 1891, inclusive. IMPORTS AND Exports oF MERCHANDISE. Year ending June 30. Exports. Imports. Excess of exports. Excess of imports. _— | |s S — —_ | e 1875 1876 1877 1878 1879 1880 1881 1882 1883 1884 1885 1886 1887 1888 1889 1890 1891 Total. $513,442,711 540,384,671 602,475,230 694,865,766 710,439,441 835,638,658 902,377,346 750,542 257 823 839,402 740,513,609 742. 189,755 679,524,830 716,183,211 © 695,954,507 742,401,375 857,828,684 884,480,810 $533,005,436 460,741,190 451,323,126 437,051,532 445,777,775 667,954,746 642,664,628 724,639,574 723,180,914 667,697,693 577,527,329 635,436,136 692,319,768 723,957,114 745,131,652 789,310,409 844,916,196 Find the excess of exports or of imports for each year. Find the total exports and the total imports for the 17 years ending June 30, 1891, and the difference between the excess of exports and excess of imports for the same period. 340 ARITHMETIC. SHORT METHODS. 791. 4,846 7,854 x 3% 274 14,588 8 times 54,978 7 times 2,9072 + of 3 times 6,108 4 of 7 times Multiply by 3, then add ¢ of this 218,1665 Ans. product. 1. 247 x 44 6. 6,305 x 95% 2. 1,896 x 53 7. 8,762 x 233 3. 1,234 x 654 8. 12,3845 x 322 4. 3,742 x 745 9. 7,890 x 1053 5. 4,053 x 74 10. 67,890 x 2344 792. Multiply 8,654 by 99. 865,400 Subtract the number from 100 times BEGITABI UNS: tis mia 11. 7,885 x 99 21. 2,684 x 25 12. 9,427 x 99 22. 9,321 x 334 13. 6,073 x 99 23. 8,693 x 124 14. 5,483 x 99 24. 4,862 x 662 15. 2,761 x 999 25. 3,025 x 374 16. 8,305 x 999 26. 3,464 x 622 Lie ogo x 1 oe 27. 4,872 x 874 18. 999 x 3,859 28. 860 x 24 19. 9,832 x 990 29. 6,318 x 162 20. 7,543 x 990 30. 9,204 x 75 LONG TON. 341 793. Divide. Do not write products (Arts. 385, 616): 31. 41,874,365 + 9,999 33. 300,200,100 + 31,416 32. 123,456,789 + 1,987 34. 254,637,809 = 26,543 35. 837,029,456 + 16,074. 794. Write answers: (Art. 385.) e 23146 38. 11223 40: 98643 49. 633386 7809 1984 28643 16895 68400 72063 87631 72084 37. ——— 39. 41... 43. ——— o709 5999 17025 10203 795. Avoirdupois Weight. Long Ton. In selling iron, coal at the mines, ores, etc., and in calculating the duties at the U. 8. custom houses upon imported goods, the following table is used: 28 pounds (lb.) 1 quarter (qr.) 4 quarters 1 hundredweight (cwt.) 20 hundredweight 1 ton (T.) lewt.=112 1b. 1 T. = 2,240 lb. 796. The ton of 2,240 pounds is called a long ton. Unless otherwise specified in a problem, the cwt. of 100 lb. and the ton of 2,000 lb. are to be taken. 1. Reduce 25 'T. 13 cwt. 2 qr. 25 lb. to pounds. (Long ton.) 2. Change 100,000 pounds to tons (long), cwt., qr., lb. 3. Find the duty at 1)5% per lb. on an invoice of tin weigh- ing 33 T. 7 ewt. 20 lb. (Long ton.) 4. What is the total weight in tons (long), etc., of 19 barrels of soda-ash weighing 13 cwt. 2 qr. 10 lb. each? 5. Find the weight of the rails required for 100 miles of double track (four rails), the weight of a rail being 18 pounds per running foot. What will be the cost of the rails at $21 per long ton ? 6. A coal dealer buys 175 (long) tons of coal. How much does he receive for it at $5 per ton of 2000 pounds? 342 ARITHMETIC. MEASUREMENTS. 800. What is the length in inches of a row of four envelopes, each five inches long, placed end to end? What is the length in feet and inches? 1. What is the width in inches of four such rows just touch- ing each other? What is the width in feet? How many envelopes are there? How many sq. in. in each envelope? How many sq. in. are covered by all of them? 2. How many envelopes 5 inches by 3 inches would cover the top of a table 4 feet 2 inches long and 2 feet 6 inches wide? 3. Draw a rectangle to represent a floor 24 feet long 18 feet wide. Draw rugs 6 feet long, 3 feet wide, and see how many will be needed to cover the floor. 4. How many boards 12 feet long, 6 inches wide will be required for a floor 8 yards long, 6 yards wide? If the boards run lengthwise, now many boards in length are used? How many boards wide? 5. Given the area in square inches of a surface to be covered with envelopes, and the area in square inches of an envelope, how is the number of envelopes ascertained ? If the area of the surface to be covered is given in square feet, how must we proceed ? MEASUREMENTS. 343 6. Given the dimensions of a surface to be covered and the dimensions of the articles to be used for covering, how can we indicate the operations to be performed, without actually doing the work? How about the denominations used ; yards, feet, etc. ? 7. How many bricks 8 inches by 4 inches will be needed for a walk 24 yards long, 6 feet wide, making no allowance for waste ? First indicate the work. Cancel. 8. How many paving tiles } foot square will cover a hearth 6 feet long, 3 feet wide? Make a diagram. 9. How many boards 12 feet long, 9 inches wide will be required for a close fence 120 yards long, 6 feet high? 10. Find the number of paving stones 9 inches by 3 inches, in a street 100 rods long, 10 yards wide. 11. A man buys a piece of ground 300 feet long, 150 feet wide. He builds a house, 50 feet by 30 feet, and a shed 12 feet by 13 feet. How many square yards will he have left for a garden ? 12. There are 160 square rods in an acre. How many square yards are there in an acre? 13. Give the dimensions, in yards, of a field that will contain just an acre. Of one that will contain two acres. 14. Draw a rectangle 2 inches x 3 inches. Draw one twice the size. What are the dimensions of the latter? A plot 100 ft. x 100 ft. is how many times as large as a plot 20 ft. x 25 ft.? 15. How many square feet are there in a fence 10 feet high around a lot 250 feet long, 200 feet wide? 344 ARITHMETIC. 16. The owner of a piece of ground 250 feet long, 200 feet wide, takes 10 feet from each side to make a gravel walk, and uses the remainder for a garden. Give the dimensions of the garden and its area in square feet? How many square feet are taken up by the walk? How many square feet in the whole piece of ground ? 17. How many square feet of flagging would be required for a sidewalk 10 feet wide outside a lot 250 feet long, 200 feet wide? If 250 running feet of sidewalk, 10 feet wide, were laid on two sides of the lot, and 200 running feet on each of the other sides, would the job be finished ? 18. At $80 per acre what is the value of a field 80 rods long, 70 rods wide? What will it cost to fence the field at 20% per running yard? 19. A room is 24 feet long, 18 feet wide, 12 feet high. Draw, touching each other, four rectangles representing the four walls. Write the dimensions of each wall. What are the dimensions of the large rectangle made up of the four smaller ones? Give the area in square feet. In square yards, e TIME BETWEEN DATES. 345 20. Show by a diagram the shape of a piece of paper that when folded will entirely cover a box 12 inches long, 6 inches wide, 4 inches high. Write the dimensions. 801. This is called the “development” of the box. What is the area of the paper in square inches? 21. Make a diagram of a room 24 feet long, 18 feet wide, 12 feet high, showing the surface that is generally plastered. How many square yards of plaster will be needed for the above room, making no allowance for doors, windows, etc. ? 22. What is the length of a rectangular field 60 rods long that contains 60 acres? 23. To contain 48 square yards, how many yards long must be a piece of carpet 27 inches wide? 24. I have bought 24 yards of dress goods, 27 inches wide. How many square yards does the piece contain? How many yards of lining 32 inches wide will contain the same number of square yards? 24 yards long. ? yards long. # yd. | 18 sq. yd. | = | 18 sq. yd. § yd. 25. How many square yards are there in 27 rugs, each 63 inches long, 45 inches wide? TIME BETWEEN DATES. 802. Oral Problems. 1. How many hours from 8 o'clock Saturday afternoon to 9 o'clock Sunday morning? 2. How many days from May 1 to June 1? 3. A boy takes a spoonful of medicine every hour. If he takes the first dose at 2 o’clock, at what hour will he take the sixth? The second? The fourth? 4. A man begins work on the morning of the 6th and ends on the evening of the 11th. How much does he earn at $3 per day? 346 ARITHMETIC. 6. An importer receives some cases of goods numbered con- secutively. How many cases are there if the lowest number is 29 and the highest number is 53 ? 6. How many posts 6 ft. apart will be needed for a fence 120 ft. long? For a fence 6 ft. long? 12 ft. long? 7. Find the time from Jan. 1 to Jan. 31, counting the first and the last day. Omitting both days. 8. How many days from July 4 to August 15, inclusive ? 9. How many chapters from the 25th to the 49th, exclusive? 10. A girl begins at the 146th problem and solves all those on two pages. If the last is the 172d problem, how many does she solve ? 803. In finding the time between dates, either the first or the last day is excluded; that is, from the lst to the 21st is con- sidered 20 days. How many days from March 4 to Sept. 1? March 4 to March 31, 27 days. Excluding March 4, there remain in se ea os m the month 31 —4, or 27 days. To this add ay the number of days in April, May, June, June Suis July, and August. Since March 4 is ex- July Sea cluded, we take 1 day in September, mak- Aug. SLU ing the total 181 days. Sept. Tene: Ans. 181 days. 804. How many days from 7 1134 Jan: | to Febalor 16. Feb. 29 to April 1? 12. Oct. 31 to Dec. 30? 17. May 21 to July 4? 13. Sept. 30 to Dec. 16? 18. April 7 to May 27? 14. Noy. 1 to Dec. 19? 19. June 10 to Aug. 1? 15. March 16 to April 25? 20. July 4 to Sept. 1? TIME BETWEEN DATES. 347 805. Slate Problems, (Take note of leap year.) How many days from: Feb. 6, 1892, to Oct. 1, 1892? Oct. 14, 1892, to March 8, 1893? Jan. 1, 1892, to April 19, 1892? Dec. 23, 1891, to March 8, 1892? Sept. 3, 1892, to Feb. 1, 1893? March 16, 1892, to Dec. 25, 1892? June 8, 1892, to Nov. 29, 1892? Aug. 17, 1892, to Jan. 3, 1893? April 4, 1892, to July 4, 1892? May 16, 1892, to Oct. 14, 1892? _ fe 11. How much wages at $4 per day should a man receive rom Monday, Jan. 2, 1893, to Feb. 28, inclusive, no pay to be eeceived for Sundays or Washington’s birthday? 12. A man borrowed $100 April 4, and returned it November 25. How many days’ interest did he owe? (Do not include both days.) 13. May 1, 1898, fell on Monday. Upon what day of the week did July 4 fall? | 14. How many days does vacation last if it begins on the morning of Saturday, July 2, and school commences on the first Tuesday of September ? 15. A man borrows some money June 16, and agrees to return it in 60 days. On what date should he pay it? 16. A traveler starts upon a trip Aug. 24, 1894, and reaches home again Feb. 10, 1895. How long is he away? 348 ARITHMETIC. 806. When the difference between dates is more than a year, it is cus- tomary to take 30 days to each month. (See Appendix, Part IIT.) Find the difference in time between March 8, 1879, and Jan. 1, 1898. | 1893 1 1 Writing 1893, Ist month, Ist day, we subtract jong 9 98 8 1879, 3d month, 3d day. "1S a ee Answer, 13 years 9 months 28 days. 17. George Washington was born Feb. 22, 1732. How old was he at the signing of the Declaration of Independence, July 4, 1776? 18. Abraham Lincoln was first inaugurated president, March 4, 1861. How long had he served at his death, April, 15, 1865? 19. The battle of Lexington took place April 19,1775. The treaty of peace was signed Sept. 3, 1783. How many years, months, and days between the two events ? 20. How many years elapsed between the discovery of America by Columbus, Oct. 12, 1492, and the landing of the Pilgrims, Dec. 21, 1620? 21. General Harrison fought the Battle of Tippecanoe Nov. 7, 1811. He was inaugurated president 29 years 3 months 27 days later. Give the date of his inauguration. 22. How long was it after the treaty with England, signed Dec. 24, 1814, that the Mexican treaty was concluded, Feb. 2, 1848? 23. General Taylor died July 9, 1850. How long did he live after the capture of Monterey, Sept. 24, 1846? 24. President Garfield was born Noy. 19, 1831. How old was he at his inauguration, March 4, 1881? 25. The last battle of the Mexican war took place Sept. 14, 1847. The Battle of Bull Run was fought 13 years 10 months 7 days later. What was the date of this battle? 26. Find the time between July 4, 1776, and Jan. 1, 1894. INTEREST. O49 PERCENTAGE. 807. Oral Exercises. 1. Find 4% of $125. 6. 334% of 1 day. 2. 25% of 16. 7. 622% of $12. 3. 6% of 200 cows. 8. 9% of $23. 4. 1% of 150 lb. 9. 75% of 3 gal. 5. 20% of 65 yd. 10. 14% of $400. 808. Slate Exercises. 1. Find 6% of $576. 9. 25% of $156. 2. 41% of $340. 10. 1% of $156. 3. 25% of 1,876 bu. 11. 4% of $156. 4. 121% of 864 cows. 12. 50% of 480 hr. 5. 50% of 482 yd. 13. £% of 480 hr. 6. 331% of 576 soldiers. 14. 1% of $1,420. 7. 162% of 696 gal. 15. 31% of $66. 8. 61% of $4.96. 16. 74% of 360 days. INTEREST. 809. In computing interest, the year is considered as com- posed of 12 months of 80 days each. 810. Oral Exercises. Find the interest on: 1. $90 for 2 mo. at 4%. 2. $60 for 60 da. at 6%. 3. $100 for 2yr.6 mo. at 5%. 4. $120 for 30 da. at 5%. 5. $300 for 90 da. at 3%. 6. $100 for 1 yr. 3 mo. at 4%. 7 . $50 for 3 yr. at 6%. 8. $100 for 2 yr. 4 mo. at 6%. 9 10 . $60 for 40 da. at 6%. - $120 for 120 da. at 5%. 350 ARITHMETIC. 811. Find the interest on $63 for 4 yr. 5 mo. at 5%. p69 is called the principal. 5 = rate. 4 yr. 5 mo. = time. B12 i Tne vse principal X rate < time (in years) 100 4 years 5 months = 4,5 yr. = 23 yr. 21 BOB x 2 x 28 $95.60 _ gig 9) 4 Ans. AO We 4 4 Notre. — The divisor 100 is canceled by placing a decimal point before 21. Find the interest on $160.50 for 3 mo. 15 da. at 6%. OU29 6 7 56175 160.50 x —~ ne : = 1.89050 x 95 = 2.808 + 0p ya 2 Ans. $2.81. 2 Find the interest on $69.75 for 1 mo. 17 da. at 4%. 00775 4 AT 0669075 x —— = .86425. Ans. 36 cents. 100 * 360 20 813. Slate Exercises. Find the interest on: 1. $192 for 3 yr. 7 mo. at 5%. 2. $60 for 2 mo. 12 da. at 4%. 3. $240 for 1 yr. 1 mo. at 6%. 4. $14.40 for 5 yr. 5 mo. at 5%. 5. $36 for 77 days at 44%. REVIEW. BOL $99 for 21 months at 6%. $192 for 2 yr. 4 mo. at 5%. $600 from Jan. 1 to Jan. 16 at 4%. $1,200 from July 1, 1891, to Jan. 1, 1893, at 6%. 10. $57.60 from Oct. 4, 1890, to Feb. 4, 1894, at 5%. eo Oo ~~ OH REVIEW. 814. Oral Problems. 1. What part of 4is $? (16 twentieths, 15 twentieths.) 2. 16 is how many hundredths of 64? 3. What per cent of 25 is 5? 4. What part of 2 lb. 1 oz.is11b.? (83 oz., 16 oz.) 5. Divide 4 gal. by 3 pt. 6. How many pencils at 4 mills each can be bought for a dollar ? 7. Write 5 as a decimal. 8. Divide 34 by 200. 9. At 20¢ per qt., what will be the cost of 2 gal. 8 qt. 1 pt. of maple syrup ? | 10. Find the cost of 4 T. 400 lb. of coal at $5 per ton. 11. A man puts 4 |b. 8 oz. of tea into 12 oz. packages. How many packages does he make ? 12. 4 pecks 3 quarts of apples are given to some children. If each child’s share is 5 quarts, how many children are there? 13. If it takes 3 hours 20 minutes to hoe a row of corn, how many rows can a man do in 2 days of 10 hours each? 352 ARITHMETIC. 14. How many square inches in the surface of a sheet of paper 1 ft. 8 in. long, 11 in. wide? 15. How many pieces of paper 2 inches square will exactly cover a slate 12 inches long, 8 inches wide? 815. Slate Problems. 1. What part of 6 hr. 17 min. 5 sec. is 38 hr. 15 min. 25 sec. ? 2. Ifaman walks at the rate of 3 mi. and 96 rd. per hour, how far will he walk in 3 hr. and 20 min. ? 3. What is one-ninth of 28 bu. 3 pk. and 7 qt.? 4. Three men buy a house for $1,200. A furnishes $600; B, $400; C,$200. They sell the house for $1,500. How much money should each receive? 5. If 5 T. and 1,000 Ib. of coal cost $30.25, how much will be paid for 7 T. and 320 lb. ? 6. At 25¢ per hour, how much should a man receive that works 8 hours and 86 minutes? 7. At $45 per month, what is the rent of a house for 2 yr. 7 mo. and 8 da. ? 8. If 2 1b. 4 0z. of tea cost $1.35, what will be the cost of 1) Ib: 12 oz.? 9. How many sq. in. in a paving tile 6 in. square? How many sq. in. in a rectangle 4 ft. by 3 ft.? How many paving tiles 6 in. by 6 in. would cover a surface 4 ft. by 3 ft. ? 10. A merchant imports 360 yd. of dress goods, 27 inches wide, costing 30% per yd. What will the duty be at 8% per square yard, and 50 per cent of the cost in addition ? 11. A man pays $60 interest per year. How much interest does he pay in 3 yr. 7 mo. 10 da.? 12. Find four-ninths of 28 bu. 3 pk. 7 qt. MEASUREMENTS. 353 APPROXIMATIONS. 816. Give an estimate of the answer (Art. 521): 1. If3 T. and 1,988 lb. of coal cost $19.97, what will be the cost of 8 T. and 1 1b.? (Nearly 4 tons cost nearly $20.) 2. At $500 per year, what will be the rent of a house for 1 year 11 mo. and 29 da.? (Nearly 2 years.) 3. Find the cost of 5 bbl. sugar, averaging 299 lb. each, at 415 ¢ ner lb, 4. What is the interest on $199.86 at 6%, for 5 mo. 28 da. ? 5. If 11 men and 2 boys can finish a piece of work in 234 days, how long would it take 23 men and 5 boys? 6. What decimal of 639 acres is 321 acres? 7. What will be the cost of 20,060 bricks at $19.90 per M.? 8. A farmer sells 5,584 lb. of rye at 87% per bu. of 56 |b. How much does he receive? 9. If 9 1b. and 15 oz. of tea cost $7.95, what will be the cost of 21 lb. and 1 oz.? 10. Paid freight on 1,987 lb. at 70% per hundredweight. How much did I pay ? MEASUREMENTS. 817. Make diagrams when necessary. 1. A man has a lot 100 feet by 200 feet. How many square feet will he have left for a garden after he builds a house 25 feet by 60 feet? 2. One wall of a room is 24 feet long and 12 feet high. There is a door in it 8 feet high, 44 feet wide. How many square yards of plastering will be needed to cover the wall? 354 ARITHMETIC, 3. A brick is 8 inches long, 4 inches wide, 2 inches thick. — How many square inches are there in the surface of the widest face? In the surface of one side? In the surface of one end? 4. How many bricks laid on the widest face will be needed for a walk 288 inches long, 96 inches wide? 5. How many bricks laid on the side will be needed for a walk 24 feet long, 8 feet wide? 6. Make a diagram of a piece of paper that when folded will just cover the six faces of a brick 8x42 inches. How many square inches of paper would be needed ? 7. How many square feet are there in a roll of wall paper 24 feet long, 18 inches wide? 8. How many rolls 24 feet long, 14 feet wide, would be required to paper the ceiling of a room 45 feet long, 36 feet wide, making no allowance for matching or waste? 9. The owner of a piece of ground 200 feet wide, 300 feet long, divides it into lots 25 feet by 100 feet. How many lots are there ? 10. Make table of: 818. Square Measure. square inches (sq. in.) 1 square foot (sq. ft.) square feet 1 square yard (sq. yd.) square yards 1 square rod (sq. rd.) 160 square rods 1 acre (A.) acres 1 square mile (sq. mi.) 11. The owner of a piece of ground 600 feet long, 150 feet wide, builds a fence 6 feet high around the plot. How many square feet of fence are there? 12. What would be the cost of building 1,800 feet of fence 6 feet high at $1 per square yard? MEASUREMENTS. 350 13. A farm is one mile square. How many 40-acre fields does it contain? 14. How many yards of fence will be needed to enclose the plot of ground shown in the following diagram ? 12 rods 15. The above field was originally a rectangle, but the owner sold one piece 5 rods by 3 rods, and a second piece 3 rods by 7 rods. How many square rods did it contain at first? What is its present area? 16. How many acres in a field in the shape of a triangle whose base and perpendicular measure 40 rods each? 17. Calculate the number of square yards in the field shown in the accompanying diagram. 18. The owner of a field 160 yards long, 121 yards wide sold |. 24 yd. (6 ya. from one corner a triangular piece 40 yards long, 801 yards wide. Find the number of square yards in the part remaining. (Make diagram.) 15 yd. I s l { J ! 356 ARITHMETIC. 19. How many acres are there in a triangular plot of ground when the base measures 80 yards and the perpendicular measures 604 yards? 20. What are the dimensions of the box that can be made of a piece of paper of the size shown in the following diagram ? How many square inches of paper are needed for such a box (making no allowance for pasting) ? 4 in. 8 in. DENOMINATE NUMBERS. (Long Measure.) 819. Slate Exercises. 1. Change 43 yd. to rods and a fraction. 2. Change 43 yd. to rods and yards. 3. Change 43 yd. to rods, yards, and feet. 4. Change 43 yd. to rods, yards, feet, and inches. 5. Change 72 yd. to rods, etc. 6. Change 66 yd. to rods. DENOMINATE NUMBERS. 357 820. Change to rods, etc.: 7. 49 yd. 14. 1,836 in. 8. 147 ft. 15 ean 9. 1,764 in. 16. 52 yd. 10. 812 rd. 17. 492 yd. TLS o0-yd: 18. 49 yd. 1 ft. 6 in. 12. 51 yd. 19.. 1484 ft. 13. 153 ft. 20. 1,782 in. 821. Change 1,581 in. to rods, ete. 12) 1581 in. 3)131 ft. 9 in. 4)43 yd. 2 ft. 9 in. 2a ine 7rd. 9 half-yards 2 ft. 9 in. 7 rd. 43 yd. 2 ft. 9 in. Grd. eva *2, thai din, + 1ft. Gin. =#4 yd. 7rd. Syd. 1ft. 3in. Ans. 822. To find how many rods in 43 yards, we divide by 53. 53 yards = 11 half-yards. 43 yards = 86 half-yards. As there are 11 half-yards in a rod, 86 half-yards will be equal to 7 rods and 9 half-yards, or 7 rods 44 yards. Changing } yd. to 1 ft. 6 inches, we obtain the answer as above. 823. Change to rods, etc.: 21. 1,488 in. 24. 2,796 in. 27. 3,453 in. 22. 984 in. 2a Lokal try: 28. 1,278 in. 28. 1,345 in. 26. 1,470 in. Boul oO hon: 30. Change 5 rods to inches. 31. Change 990 inches to rods. 32. How many inches in 7 rods 1 yd.? 33. Change 1,422 inches to rods. 358 ARITHMETIC. 824. Add: 34. 4rd.3 yd. 1 ft. 36. 5rd. 4 yd. 2 ft. 9rd. 4 yd. 2 ft. 5 yd. 1 ft. 3 rd. eet Ger Gird lavas BERANE Recah S7 hl rd26 yonder 3 rd. lsat: 4 rd. Aly § 2 yd. 2 ft. 5 yd. 1 ft. 4 rd. ger 6 rd. 38. From 8rd. 1 ft. take 2 rd. 2 ft. 39. Find the difference between 3 rd. 1 yd. 1 ft. and 16 rd. 40. Multiply 5rd. 4 yd. 2 ft. by 4. 41. Multiply 11 rd. 2 ft. by 10. 42. Divide 30 rd. 5 yd.2 ft. by 8. 43. Divide 34 rd. 2 yd. by 9. SOLID CONTENTS. 825. How many one-inch cubes can be placed on the bottom of a box 3 in. long, 4 in. wide? 1. If the box is one inch high, how many will it hold? If the box is 2in. high? 3 in. high? 2. A cube one inch long, one inch wide, one inch high con- tains a cubic inch. How many cubic inches in a box 38 in. long, 4 in. wide, 1 in. high? Ina box 8 in. long, 4 in. wide, 2 in. high? In a box 4 in. long, 4 in. wide, 4 in. high? 3. If you had 24 one-inch cubes, how could you pile them to make a solid with six rectangular faces ? If the pile was 2 inches high, how many cubes would there be in each tier? How many square inches would the lower tier cover? SOLID CONTENTS. 359 How could the 24 cubes be arranged to make a pile 3 inches high? 4. Can you give a rule for finding the contents of a box 6 in. long, 8 in. high, 4 in. wide? 5. How many cubic inches of water would a tin box hold, the dimensions of the box being 5 in. by 34 in. by 4in.? 6. How many one-inch cubes could be placed in a box one foot long, one foot wide, one foot high ? 7. How many cubic inches in a cubic foot? 8. How many one-foot cubes could be placed in a cubical box one yard long, one yard wide, one yard high? How many cubic feet in a cubic yard? How many cubic inches in a cubic yard? 9. How many cubic inches in a solid, 3 yd. long, 2 ft. wide, 6 in. high? How many cu. ft.? How many cu. yd.? Indicate the operations necessary in each case to obtain the correct answer. What should be done with the denominations in each case before beginning the work of obtaining the solid contents? 10. A cord of wood contains 128 cu. ft. If the wood is cut into 4-ft. lengths, what should be the other two dimensions of a regular pile to hold just a cord? 11. How many cubic feet of air in a room 24 ft. long, 18 ft. wide, 12 ft. high? 12. A gallon contains 231 cu.in. Give the dimensions of a tin box that will hold exactly a gallon. 13. Find the solid contents of a piece of timber 25 ft. long, 8 ft. wide, 5 ft. thick. Is it larger or smaller than a piece 4 ft. wide, 4 ft. thick, and 28 ft. 6 in. long? 14. How many cubic yards of earth will have to be removed in digging a cellar 18 ft. wide, 55 ft. long, 6 ft. deep? What will be the cost at 60% a load (1 cu. yd.)? 360 ARITHMETIC. 15. Give the width of a wagon body 18 in. high, 6 ft. long, that will hold, when full, a cubic yard. 16. About how many gallons are there in a cu. ft.? 17. A bushel contains 2,150.4 cu.in. About how many cu ft. in a bushel? 18. Find (by cancellation) the capacity in gallons of a tank | ft. 9 in. long, 1 ft. 3 in. wide, | ft. 10 in. deep. 19. Find (by cancellation) the capacity in bushels of a bin 1 yd. long, 2 ft. 4 in. wide, 5 ft. 4 in. high. 20. How many cords of wood (128 cu. ft.) are there in a pile 24 ft. long, 4 ft. wide, 12 ft. high? 21. A brick is 8 in. long, 4 in. wide, 2 in. thick. How many bricks are there in a pile 90 bricks long, 60 bricks wide, 30 bricks high? What are the dimensions of the pile? How many cu. in. in 1 brick? In the pile? 22. Find the number of bricks in a wall 24 ft. wide, 48 ft. high, 1 ft. thick, making no allowance for mortar, ete. 23. How many bricks are there to a cu. ft. ? 24. Allowing 20 bricks to a cubic foot when laid in mortar, how many bricks will be needed for a wall 24 ft. wide, 50 ft. high, 20 in. thick ? 25. What will be the cost of building a stone wall 40 rods long, 4 ft. high, 1 yd. thick, at $6.40 per perch of 243 cu. ft.? APPROXIMATIONS. 826. Give sight answers in-whole numbers: 1. If there are about 74 gal. to a cu. ft., estimate the number of gallons in a tank 5 ft. long, 3 ft. wide, 4 ft. high. 2. If there are about 14 cu. ft. in a bushel, estimate the con- tents in bushels of a bin 5 ft. x 3 ft. x 4 ft. DENOMINATE NUMBERS. 361 3. Give the dimensions of a tank of 150 gal. capacity. 4. Give the dimensions of a bin that will hold 100 bushels. 5. At 20 bricks laid in mortar to the cu. ft., give the length and the height of a wall 1 ft. thick that can be built with a thousand bricks. 6. At $1 a load (1 cu. yd.), give the dimensions of an excavation that can be made for $100. 7. A cu. ft. water (about 74 gal.) weighs 624 lb. About what does a gallon weigh? Pond FH FSO OD UWA eT PF WY DW HO Bought for $36; sold for $40. Gain per cent? Bought for $40; sold for $36. Loss per cent? Cost 36¢; selling price 40%. Gain per cent? Cost $24; gain 10%. Selling price? Selling price 70%; loss 75%. Cost? Buying price 70%; gain 75%. Selling price? Cost $20; selling price $29. Gain %? Cost $20; selling price $290. Gain %? Cost $20; selling price $20.90. Gain %? Cost $20; selling price $20.09. Gain %? . Selling price $300; loss $100. Loss %? - Cost $300; gain $100. Gain %?°* . Selling price $175; cost $150. Gain %? . Selling price $375; gain 25%. Profit? . Cost $36.50; selling price $28.50. Loss % ? . Selling price $33.95; loss 3%. Cost? . Cost $75.50; loss 54%. Selling price? . Selling price $20.16; gain 5%. Cost? . Selling price $64; profit $16. Gain %? . Cost $37.50; selling price $42. Gain %? . Selling price $26.88; loss 62%. Loss? . Cost $4. gain 1%. Selling price? . Selling price $41.16; gain5%. Cost? . Selling price $29.83; loss 5%. Loss? 391 392 ARITHMETIC. 25. Cost $19.50; loss 6%. Selling price? 26. Cost $84; selling price $184. Gain %? 27. Selling price $700; gain 250%. Cost? 28. Cost $324.80; gain 175%. Selling price? 29. Selling price $6.50; loss 133%. Loss? 30. Cost $346.50; selling price $339.57. Loss %? 31. Selling price $17.64; loss 2%. Cost? 32. Cost $4,613; gain 134%. Profit? 33. Selling price $26.69; gain 61%. Profit? 34. Cost $8,766; loss 7%. Loss? 35. Selling price $50.00; profit $4.20. Gain %? 36. Cost $37.50; gain $5.40. Gain %? 37. Selling price $205.20; loss $45.90. Loss %? 38. Cost $25.60; gain 64%. Selling price? 39. Selling price $17.35; loss $%. Loss? 40. Profit $36; gain4%. Cost? 41. Loss $28.17; loss 3;3,%. Selling price? 42. Cost $3,864.25; loss 8%. Loss? } 43. Selling price $89.37; profit $6.62. Gain %? 44. Cost $82; loss $2. Loss %? 45. Selling price $22.85; gain 62%. Cost? 46. Profit $47.25; gain 74%. Selling price? 47. Loss $38.46; loss 4%. Cost? 48. Cost $75.52; gain 831%. Profit? 49. Selling price $49.95; loss $4.05. Loss %? 50. Cost $3,879; loss 8%. Loss? PERCENTAGE. 393 INTEREST. 873. In calculating interest, take 30 days to a month, 12 months to a year. 874, Principal x fe x Time (wn years) = Interest. 100 Notre. — Change given time to years. 2 yr. 6 mo. =2)hyr. =$ yr. 1 yr. 7 mo. =19mo. =1$ yr. 4mo.10da.=41} mo. = 3 yr. = yr. 1734 lyr. 5 mo. 15 da. = 17} mo. = — yr 5 mo. 17 da. = 167 da. = 487 yr. 875. Slate Exercises. Find the ipiareat on: 1. $750, for 24 years, at 6%. $ 84.75, 34 months, at 4%. . $308.25, from Oct. 1 to Oct. 21, at 5%. $464.75, 8 mo. 12 da., at 6%. . $360, 33 da., at 5%. $94.43, 2 mo. 5 da., at 7%. $400, 1 yr. 1 mo. 1 da., at 43%. $720, 21 da., 7%. $1,000, 8 da., 5%. $630, from April 1, 1890, to Jan. 16, 1892, at 6%. 1892 —1—16 1890 —4— 1 11. $394.50, 2 yr. 1 mo. 7 da., at 6%. 12. $1,560, 3 yr. 4 mo. 9 da., at 44%. OO AFT P w DN _ 2 394 ARITHMETIC. 13. $960, 11 mo. 24 da., 2%. 14. $86.40, 1 yr. 9 mo. 20 da., 5%. 15. $108.36, 4 yr. 7 mo. 10 da., 33%. 876. Amount = Principal + Interest. Find the amount: 16. $813, from April 19, 1889, to March 4, 1894, at 6%. 17. $960, from Jan 1, 1898, to Dec. 21, 1894, at 4%. 18. $27.84, for 3 yr. 6 mo. 9 da., at 6%. 19. $48.90, for 17 da., at 6%. 20. $144, for 2 yr. 5 da., at 32%. 21. $834.76, for 15 mo. 27 da., at 44%. 22. $5,760, for 1 yr. 5 mo. 29 da., at 5%. 23. $2,346.50, for 7 yr. 13 da., at 3%. 24. $1,892, for 3 yr. 5 mo., at 7%. 25. $150.40, for 1 yr. 2 mo. 3 da., at 6%. _ 877. Interest-bearing Demand Notes. 26. San Francisco, Jan. 7, 1893. On demand, I promise to pay William Britt, or order, Seven Hundred Sixty-five 549% Dollars, value received, with interest at 6 per cent. $ 765545. ARTHUR TOWNSEND. How much money will be required to pay the above note, with interest, July 15, 1894? 27. A demand note, dated Sept. 25, 1892, with interest at 8% from date, is paid Jan. 2, 1895. How much was due, the face of the note being $750? : 28. Find the amount due March 4, 1894, on a note for $365.84, dated May 20, 1892, with interest from date at 7%. INTEREST. 395 29. Find the amount necessary, Oct. 16, 1896, to pay a note of $1,240, with interest at 6% from Aug. 15, 1892. 30. An interest-bearing note for $87.60 is dated April 38, 1886. How much is due on it for principal and interest Jan. 2, 1894? Rate 44%. 878. Oral Problems. 1. Find the interest on $300, for 1 year 7 months, at 4%. $12 per year is how much for 7 months? 2. On $60, for 33 days, at 6%. $3.60 for 360 days is how much for 33 days? 3. On $120, from Jan. 1, 18938, to July 1, 1894, at 5%. 4. How long will it take $100 to produce $15 interest at 6%? 5. At what rate per cent will $50 produce $6 in 2 years? 6. What is the interest on $300, at 6%, from Feb. 1 to Feb. 21? 7. What part of a year is 72 days? 8. Find the interest at 4%, for 90 days, on $150. 9. On $240, for 36 days, at 5%. 10. What is the amount of $200, for 3 years 1 month, at 6%? 11. How long will it take $1 to make $1 interest at 5% ? 12. How long will it take any sum to double itself at 6% ? 13. How long will it take $14.90 to double itself at 4% ? 14. At per cent per month, find the interest on $90 for 16 months. 15. 5% per year is 1% for how many days? 16. 41% per year is 1% for how many days? 17. Find the interest on $75, at 5%, for 72 days. 396 ARITHMETIC. SPECIAL DRILLS. 884. Give sums: 495 + 99 99914295 $2634+$6.37 $5.45+ $9.99 99+576 5764999 $4564+$2.84 $9.99+ $6.78 685+ 99 999+685 $649+$3.12 §$12.68+ $0.99 99 +599 599+999 $3.58+$5.67 $0.99+$13.33 885. Give differences : 565-— 99 1424-999 $7.00-$2.63 $15.44— $9.99 488— 99 1,575—999 $640—$8.56 $9.44 — $6.45 794— 99 1,684—999 $9.61—$4.49 $7.88 -— $4.89 898— 99 1,598—999 $815 —$5.58 $9.53 — $2.99 886. Give products: 24 x 21 21 x 31 41 x 41 19 x 374 03 X 21 02 X 31 32 x 41 18 x 624 42 x 21 43 x 31 21 x 41 22 X 662 51 Xx 21 dl x 31 42 x 41 33 xX 75 887. Give quotients: Poe 25 228 +19 9A + 331 656 + 16 18 + .25 234 +18 36 + .334 544-17 24 + .75 238 +17 16 + .662 558 + 18 36 + .75 336 + 16 66 + 662 418+19 888. Give answers: 147 x 7 1334 x 8 13 x 134 20 x 82 152 x 6 212 x 4 14 x 12% 21 x 73 55 + 5 Ti + 3 15 + 3h 63 + 34 142-124 63-8 20 ++ 88 64 + 54 REVIEW. 397 889. Oral Problems. 1. What will be the cost of 48 yards of cloth at 871 per yard? 2. How many square yards in a piece of carpet 48 yards long, 27 inches wide? 3. How many yards of carpet 27 inches wide will be needed to cover a floor containing 48 square yards? 4. Paid $3.45 for groceries, $1.50 for dry goods, and 99% for sundries. What is the total? 5. From a chest containing 254 pounds of tea, 84 pounds were sold. How many pounds remain? 6. At 8719 per peck, what will I receive for 4 bushels of potatoes ? 7. 831 yards of cloth are divided into 9 pieces. How many yards are there in each piece? 8. I buy hardware to the amount of $6.37. I give the store- keeper two $5-bills. How much change should I receive? 9. What will be the cost of 24 yards of calico at 42¢ per yd.? 10. What will I have to pay for 19 base-balls at $1.25 each ? 11. At $1.874 per yard, what will be the cost of 120 yards of silk ? 12. For $120, how many yards of silk can I buy at $1.873 per yard? 13. What will be the cost of a ton of hay at 971 per ewt.? 14. A square field requires 320 rods of fence. How many square rods are there in the field? 15. How many acres are 6,400 sq. rd.? 16. At 48¢ per yd., how many yards of calico can I buy for 95 ¢? 17. If slate pencils oost 2 mills each, how many will I receive for $4? © ‘ 398 ARITHMETIC. 18. At $5.00 per ton, how many pounds of coal can be bought for 1¢? 19. Find the cost of 8 T. 480 lb. coal at $5 per ton. 20. At $5 per ton, how many tons and pounds of coal can I buy for $10.80? 21. How many square yards are there in a field 41 yards Wng, 42 yards wide? 22. If I pay 15¢ for 34 yards of muslin, what is the price per yard? 23. How many acres of land are there in two farms contain- ing, respectively, 347 and 495 acres? 24. At871¢ each, how many base-balls can be bought for $56? 25. If one man can do a piece of work in 24 days, and an- other man can do it in 48 days, how long will it take both, working together ? APPROXIMATIONS. 890. Give approximate answers at sight (Art. 521): 1. Find the interest of $150, at 4%, from Jan. 1, 1893, to Dec. 30, 1895. (Nearly 3 years.) 2. What is the weight, at 574 lb. per cu. ft., of a cake of ice 4 ft. by 2 ft. by 14 ft.? (Nearly 60 lb. per cu. ft.) 3. Find the amount of goods sold, the commission at 27% being $11.75. (About 3%.) 4. What % of 497 is 249? 5. What % of 31% is 1134? 6. Cost of 19,987 ft. boards at $30.05 per M.? 7. How much will be paid for 4 bbl. sugar, each containing 299 lb., @ 5,¢ per lb.? 8. 18.0327 + 4.5026. 9. 8348 + 328. 10. 74 A. 155 sq. rd. land at $79 per A.? 891. Slate Exercises. REVIEW. | 899 SHORT METHODS. 7,854 x 3 9,365 x t 1,9631 Deduct 1. 1,170% Deduct 4, 5,8901 Ans. 8,1943 Ans. Multiply 6,578 by 92. 65,780 = 10 times number. 2,192 2 = 4 number (Deduct). 63,5874 Ans. 892. Find products: 1 2 3 a 5. 6 7 8 9 176 x 13 11. 4,844 x 94 273 x 12 © 12. 8,960 x 8% . 4,554 x $ 13. 3,245 x 7Z . 1,001 x 49 14. 9,060 x 114 3,248 34 15. 658 x 994 . 6,776 x & 16. 658 x 993 . 2,307 x $ 17. 725 x 1194 . 7,284 x Z 18. 347 x 798 . 5,631 x 7% 19. 418 x 891 9,657 x 44 20. 543 x 492 . Multiply: - 418 x 99 26. 724 x 86 x 45 . 674 x 874 27. 484 x 1° x 93 36°. 999° 25 28. 576 x 914 x 124 48 x 125 x 71 29. 95 x 36 x 19% 64 x 77 x 33h 30. 74x 31x13 x 9% 400 ARITHMETIC. 895. Divide. Do not write products (Arts. 385, 616): 1. 611,463,874 + 87,659 6. 703,205,104 + 71,685 2. 279,864,597 + 45,678 7. 928,812,701 + 18,789 8. 387,250,005 + 34,567 8. 575,646,828 + 59,764 4. 800,700,900 = 68,439 9. 1,234,567,890 + 169,375 5. 453,211,687 + 576,258 10. 3,126,045,000 + 483,729 896. Write answers (Art. 385): 11, 876,459 14, £12,000 17, 208,040 94.317 69,999 17,613 1p, 768,154 15, 458,124 1g, 208,018 82.915 123,456 126,748 654,817 375 005 862,304 73,295 16. 59 687 87,925 MEASUREMENTS. 897. Slate Problems. 1. How many boards 16 ft. long, 8 in. wide, will be required for a tight fence 8 ft. high, around a piece of ground 240 ft. long, 180 ft. wide? How many posts, 6 ft. apart, will be needed? 2. A room is 18 ft. long, 15 ft. wide. The walls and the ceil- ing contain 930 sq. ft. What is the height of the room? 3. What will it cost to cover a table 6 ft. long, 24 ft. wide, with baize 3 yd. wide, at 75% per yd.? =n es or - MEASUREMENTS. 401 4. 160 square rods make lacre. How many square yards are there in an acre? About how many yards long is a square field containing 1 acre? 5. A 40-acre field is 160 rods long. How many rods of fence are needed to enclose the field? 6. A room 80 ft. long, 24 ft. wide, 15 ft. high, contains 40 persons. How many square feet of floor space are there for each occupant? How many cubic feet of air space are there for each? 7. How many yards of carpet, 27 inches wide, would be needed for the floor of such a room? 8. How many bundles of laths, each bundle covering 3 square yards, would be needed for the walls and ceiling of the above room, no allowance being made for doors and windows? 9. A farmer owned a rectangular piece of ground 380 rods long, 27 rods wide. He sold three lots 18 x 8 rods, 12 x 3 rods, 15 x 9 rods. Find the number of square rods in the original piece. Mark in the diagram the area of each lot sold, and the area of the part remaining. 10. How many rods of fence will be needed to enclose the part remaining? 402 ARITHMETIC. 398. 1 gallon = 231 cubic inches. 1 bushel = 2,150.4 cubic inches. lcord =128 cubic feet. 11. How many gallons will fill a tank 22 feet long, 14 feet wide, 9 feet deep? Indicate operations. Cancel where possible. 12. How many cords of wood are there in a pile 48 feet long, 16 feet wide, 12 feet high ? BANK DISCOUNT. 900. Wm. Brown and Sons receive the following note in set- tlement of their account with Thomas Tierney : St. Paun, May 30, 1893. Thirty days after date, I promise to pay to the order of Wm. Brown and Sons, Three Hundred Fifty-Four 743, Dollars, value received, at the Park National Bank. $ 3045455. THomas TIERNEY. 901. This money is payable 33 days after May 30, 3 days of grace being allowed. If Wm. Brown and Sons desire to use the money at once, they may have the note discounted at a bank. In this case, the bank deducts from the face of the note ($354.75), the interest thereon for 33 days, and pays over the difference (proceeds). Face of note $354.75 Discount (Int. for 33 da.) 1.95 (at 6%) Proceeds $352.80 902. Slate Exercises. Find the discount at 6% on the following, allowing 3 days of grace in each case. (See Appendix, section 1305.) A 30-days note for $75. 15-days note for $183.60. 60-days note for $275.40. 20-days note for $96. 4-months note for $336. co Pw Ne BANK DISCOUNT. 403 Find the proceeds, at 7%, on 6. A 6-months note for $180. 7. A 3-months note for $36.90. 8. A 24-days note for $795.60. 9. A 90-days note for $180. 10. A 72-days note for $1,000. 903. In computing the discount on a note, the banks ascertain the exact number of days. A 3-months note, dated February 1, is payable three days after May 1, which is May 4. The discount is taken for 27 +31+30+4=92days. —.- 5 men mow 46 acres days 12 men mow 90 acres 45 x 12 If 5 men do the work in a certain time, 1 man will require 5 times as many days. We place 5 in the numerator (asa multiplier). To cut 1 acre, he will take j. of the time required to cut 45 acres. Place 45 in the denominator (as a divisor). 12 men will take +, of the time 1 man requires. Place 12 in the de- nominator. To cut 90 acres will require 90 times as long. Place 90 in the numerator. 438 ARITHMETIC. 8. If 12 horses eat 60 bushels of oats in 6 days, how many bushels will 24 horses eat in 3 days? Make bushels the last term. 12 horses in 6 days eat bu. l horse in 1 day eats }60 24 horses in 3 days eat 975. This example can be solved more easily. 6 days’ food for 12 horses will supply how many horses for 1 day? 3 days’ food for 24 horses will supply how many horses for 1 day? 9. If 24 men use 240 lb. of beef in 2 weeks, how many pounds will 18 men use in 8 weeks? 24 men in 2 weeks use 240 lb. 10. If 6 printers can print 1,656 sheets in 9 days, how many sheets will 15 printers print in 10 days? 11. How much will it cost to feed 520 sheep for 36 days, if it costs $128 to feed 160 sheep 48 days? 12. In what time will 8 masons build a wall 84 ft. long, working 10 hours a day, if 12 masons build a wall 96 ft. long in 8 days, working 8 hours a day? 13. How much money must I lend for 1 year and 3 months, when the rate of interest is 5 per cent, in return for $60 lent me for 9 months, which I borrowed at 4 per cent? 14. If 27 men build 54 rods of wall in 6 days, how many rods will 32 men build in 9 days? 15. If 50 men can do a piece of work in 90 days, working 8 hours a day, in how many days will 72 men do it, working 10 hours a day ? 16. If $350 earns $42 interest in 3 years, how much will _ $225 earn in 5 years? 17. Ifa wall 34 feet high could be built by 68 men in 15 days, how many men could build a wall 32 feet high in 8 days? PARTNERSHIP. 439 18. Ifa ship’s crew of 500 men have provisions to serve for 48 days, at the rate of 27 ounces a day for each man, how many men will the same provisions serve for 60 days, allowing each man 30 ounces a day ? . 19. How many hours a day must 9 men work so that they may do as much in 16 days as 12 men can do in 16 days of 8 hours each ? 20. If 30¢ is paid for 6 lb. 14 oz. of bread, when wheat is $1.14 per bu., what should be paid for 23 lb. 12 0z., when wheat is $1.32 per bu. ? Notr. — Reduce weights to ounces, or to pounds and fractions. 21. If 3 men can do as much work as 7 boys, how long will it take 28 boys to do as much work as 16 men can do in 24 days? 22. A crew of 16 men have provisions for 36 days, allowing 20 ounces to each man per day. After sailing 10 days they pick up 10 shipwrecked sailors. How long will the provisions then last at the daily rate of 16 ounces per man? PARTNERSHIP. 977. Slate Problems. 1. Band C gain by trade $182. What is the gain of each, B having put in $300, and C $400? The gain of $700 is $182. Whatshould $300 gain? What should $400 gain? 2. A, B,and C invest $720, $340, and $960, respectively. The profits are $101. What is each one’s share? How many dollars of capital produce $101 profits? 3. Two men hire a pasture for $45. One puts in 15 cows, the other puts in 12 cows. What should each pay ? 4. A and B hire a boat for 50 days, paying $30. A uses it 27 days, B uses it 23 days. How much should each pay? 5. Our standard gold coin consists of 900 parts gold, 90 parts silver, 10 parts copper. What is the quantity of each metal in 50 pounds of coin? 440 ARITHMETIC. 6. Gunpowder is composed of 15 parts of saltpeter, 2 of sulphur, and 3 of charcoal, mixed together. How many pounds of each are there in 72 pounds of powder? 7. Three farmers hired a threshing-machine for $54. A used it to thresh his crop of 900 bu., B to thresh his crop of 828 bu.; C, 672 bu. How much should each pay ? 8. A,B, and C rented a warehouse. A stored in it 2,400 bales cotton; B, 1,500; 0, 1,100. A fire destroyed 1,800 bales. How much of the loss should each sustain? 9. X and Y rent a field for $32. X puts in 8 horses for 6 months, and Y 10 horses for 8 months. How many dollars should each pay ? 8 horses for 6 months = how many for one month? 10 horses for 8 months = how many for one month? 10. M and N entered into partnership. M puts $200 into the business for 5 months, and N, $300 for 4 months. They gained $132. Find the share of each. REVIEW DECIMALS. 978. Slate Exercises. 1. Express as decimals 75/5, zygy, and 345. 2. .3895 + 86.7 + 209.0043 + .81 + 3.075 + 27. 3. Divide 34,020.072 by 5.309. 570 + .005 =? 4. Multiply 80.037 by 10. Seventy-three one hundred-thousandths by one hundred. .2054 x 1,000 = ? 5. Subtract 48.8067 from 53.07. .0539 x 26.08 =? DECIMALS. 441 6. The smaller of two numbers is 8.5307, and their sum is 25.07. Find the larger number. 7. Express .39, 6.175, .00036, and 74.0005 as common frac- tions (or mixed numbers). 8. Divide .826 by 100; 548.71 by 10,000; and fifty-nine ten- thousandths by one thousand. 9. Find the difference between 9.84 and 38.005, and the con- tinued product of 83.09, .734, and 5.007. 10. Reduce 6 shillings 9 pence to the decimal of a pound sterling. 11. Express as decimals seven hundredths, forty-three ten- thousandths, and ninety-one millionths. 12. Change 7, 845, 74s, and s4,, into decimals. Find their sum. 13. Express .42796 as a common fraction, and the sum of 34, z3y, and =373, as a decimal. 14. 3.009 x .07 x .0907. 15. Divide .0075 by .15, and .00044408 by .0112. 16. Divisor, 403.6; quotient, 2.709. Dividend? .085 x .0056 4 00007 18. Change 69 rods to the decimal of a mile. 19. Change .4285 month (80 days) to days, hours, etc. 20. How many meters, each 39.37 inches, in 3 miles 220 rods? 17. What is the value of 21. Change .1875 bu. to quarts. 22. What decimal of a pound is 13 oz.? 23. Reduce 4 ft. 14 in. to the decimal of a rod. 24. How many links of 7.92 in. each in a 4-rod chain? 25. A chain is 66 ft. What decimal of an acre is 1 sq. chain? 449 ARITHMETIC. APPROXIMATIONS. 981. Give approximate answers at sight (Art. 890): 1. 487% 1s what per cent of 960? 2. If17 bu. 87 lb. of corn cost $8.75, what will 52 bu. cost? 3. About how many cords of wood in a pile 25 ft. long, 4 ft. wide, 5 ft. high? 4. How many bushels (14 cu. ft.) can be placed in a bin 6 ft. long, 5 ft. wide, 4 ft. high? 5. How many acres in a field 50 rods long, 80 rods wide? 6. About how many yards are there in the side of a square field containing 1 acre (4,840 sq. yd.) ? 7. At 74 gal. to cu. ft., about how many gallons will a tank hold 6 ft. long, 4 ft. wide, 3 ft. high? 8. 64.3 + 0987 =? 9. About how many dollars are equal to £199 17s. 6d.? 10. A mark = 23.8%. How many marks in $100? BONDS AND STOCKS. 982. Slate Problems. 1. The people of a certain town wish to build a street rail- road. A-company is formed. Five hundred shares of stock, of the par value of $100 each, are sold. At the end of 6 months it is found that the profits are $2,000. How much should the owner of 10 shares receive? 2. Profits thus distributed are termed dividends. What % semi-annual dividend is declared on the stock of the above rail- road? ‘To what per cent interest per year is it equal? 3. Mr. H. has $4,500 in the savings bank, on which he receives a low rate of interest. Hearing of the success of the BONDS AND STOCKS. 4483 new road, he gives that amount for 30 shares of the stock. What price does he pay per share? What per cent of the par value ? 4. Ifthe semi-annual dividend is again 4%, how much more income does Mr. H. receive from the railroad stock than he would obtain from the savings bank in six months, interest 4 per cent per annum? 5. What per cent, for six months, does the stock pay on his investment of $4,500? What % per year? 6. If he sells the stock (380 shares) at 1641 (per cent), how much more does he receive for it than it cost him? 7. Which investment will pay better, one in a gas company paying 6 per cent dividends annually, their stock selling at 150, the other in a bank paying 7 per cent dividends annually, stock selling at 175? 8. What annual dividend should be declared on railroad stock bought at 125, so that the buyer will receive 4% per annum on his investment? What semi-annual dividend ? 9. What will be the cost of 17 shares of canal stock, par value $50, at 937, and 143 shares gas stock, par value $10, at 1023 ? 10. If the above stock is purchased through a broker, what commission does the latter receive at 4% on the par value? 11. A railroad company needing more money to extend its road, issues bonds, promising to pay the holder the face value in twenty years, with interest at 4%. If these bonds are sold at 95, what rate of interest on the money invested does the owner of a bond receive? 12. Government 4 per cent bonds sell for 1164. What ver cent interest is received on the amount invested ? How is it that these bonds bring higher prices than railroad bonds? 444 ARITHMETIC. 13. Can you state a difference between stocks and bonds as to the rate of income received from each ? Bonds are redeemed at maturity. How about stocks? If a railroad prove unsuccessful, which claims are first met, those of the stockholders or those of the bondholders? 14. Why is it necessary sometimes to employ a broker to pur- chase stocks or bonds? What is his fee called ? 15. Mention some other persons, not owners, through whom buyers regularly make purchases. 16. What is the base in the following? (a) Insurance; (0) taxes; (c) brokerage; (d) commission; (e) interest; (f) discount; (g) stocks; (A) bonds. 17. At $24.50 per thousand, what will have to be paid in taxes by the owner of property assessed at $18,750? 18. Mr. Cartwright owns a house and lot worth $36,000. The tax rate is 21%, and his tax bill is $540. What is the assessed value of the property? What per cent of the actual value is the assessed value? 19. If the property in the last problem were assessed at its real value, what should be the rate to make he Cartwright’s tax bill the same? 20. For insuring his property, Mr. Cartwright pays a yearly premium of $135. If the rate is #%, for how much is his property insured? 21. Reduce 1,674 feet to rods, etc. 22. A man paid $8,575 for bank stock at 245. How many shares, par value $100, did he buy? Ifa quarterly dividend of 24% is declared, how much should he receive ? 23. Reduce 7,481 inches to rods, etc. 24. A woman deposited $100 in a savings bank Jan. 1, 1892. On the first of July, interest at the rate of 4% per annum was COMPOUND INTEREST. 445 calculated, and entered on the depositor’s bank book. Jan. 1, 1893, interest on the new principal was placed to the credit of the depositor. The same was done July 1, 18938. How much was there to the woman’s credit at the date last mentioned? 25. Reduce 3,793 feet to rods, etc. COMPOUND INTEREST. 983. Find the amount of $375, for 1 year, at 6%. Consider- ing this as a new principal, find the amount for a year, same rate. Find the amount of this last principal for 3 months. 26. What is the amount of $375, for 2 years 3 months, at 6%, compound interest? 27. What is the amount of $375, for 2 years 3 months, at 6%, the interest compounded semi-annually ? Principal, $375. 3% 11.25 6 months’ interest. 386.25 Amount 6 months. 3% 11.5875 6 months’ interest. Amount 1 year. 3% 6 months’ interest. Amount 1} years. etc., etc., etc. Find the “compound interest”’ on $375, for 2 years 3 months, at 6 per cent, compounded semi-annually. 28. What is the amount of $100, at compound interest, for 3 years, interest at 6%, compounded annually? 29. Find the compound interest of $1,800, at 4%, for 2 years, interest compounded quarterly. : $ 1,800.00 1%, 18,00 1,818.00 1%, 18.18 etc., etc., etc. 446 ARITHMETIC. 30. Find the difference between the simple interest of $100, for 2 yr. 3 mo., at 5%, and the compound interest for the same time, interest compounded semi-annually. $ 100.00 . 22%=25 2.50 Divide by 4, and put first quotient figure 102.50 one place to the right. 24% 2.5625 105.0625 21 97 2.6266 $ 107.6891 (four places of decimals are sufficient.) 984. Compound interest is allowed by savings banks. It is not collectible on notes or other debts. REVIEW. 985. Oral Problems. 1. A capitalist wishes to realize 5% on money invested in stock. What must be the annual dividend on stock costing 300, in order to produce this rate? 2. What will be the taxes on property assessed at $25,000, the rate being $16 per $1,000? 3. Find the compound interest on $1,000, for two years, at five per cent, interest compounded annually. 4. What will be the net cost of an article marked $8, on which a discount of 50, 25, and 10% is allowed? 5. Find the “list” price of an article sold for $10 after a discount of 50 and 50 per cent had been deducted. 6. Paid 90¢ for an article. The discount is 25 and 25 per cent. What is the list price? 7. One boy can do a certain piece of work in 2 hours, a second boy requires 3 hours, a third needs 6 hours. How long will it take the three working together? 8. Sold a cow for $60, losing 25%. What was the loss? 9. Sold a cow for $60, gaining 25%. What was the gain? EXCHANGE. 447 10. Sold two horses at $240 apiece. On one I gained 20%, on the other I lost 20%. Did I gain or lose on both, and how much ? Suaaestion. — $ 240 in the first case represents 120% of cost of horse. The gain is 20%, which is 4 of selling price, or $40. The loss in the other case is 20%, which is what part of the selling price ? Do not find the cost. 11. John has $60, James has $80. James has what per cent more money than John? John has what per cent less money than James? 12. 2is what per cent of 4? 41s what per cent of 4? 13. Two men working together can finish a piece of work in 8 days; one can doit in 12 days. How long would the other take to do the work? 14. How many yards of cloth at $3.75 per yard can be bought for $90? EXCHANGE. 992. If I wish to pay a bill in a distant city, ought I to enclose the money in a letter? Why? Can money be sent by express? Can the telegraph be used in paying money at a distance? What is a money-order ? Can I buy from the postal authorities a money-order payable in Europe? What will be the cost of a money-order for $ 85, payable in San Francisco ? What is the largest money-order that can be purchased? What is a check? Can you tell why a draft rather than a check is used in paying a bill at a distance? Pupils should be encouraged to look up answers to the fore- going. 448 ARITHMETIC. Bills of exchange are either foreign or domestic. A domestic bill of exchange is called a draft, the term bil of exchange being generally applied only to foreign bills. DOMESTIC EXCHANGE. 993. Slate Problems. William F. Smith, of Memphis, Tenn., owes John M. Thomson, of New York, $3,475.86. He purchases from a Memphis banker, Joseph E. Washington, a sight draft for the above amount on the Chemical Bank of New York. The following is the form of the draft: $3,475,85,. Mempuis, Tenn., Aug. 9, 1893.' At sight, pay to the order of John M. Thomson Three Thousand Four Hundred Seventy-five and 88 Dollars, value _ received, and charge to the account of To CuEMiIcAL BANK, JOSEPH EK. WASHINGTON. New York. 1. What must William F. Smith pay for the above draft, the rate being $1.50 premium per $1,000? (A draft for $1,000 costs $1,001.50.) 2. Find the cost of a Boston draft on New York for $1,875, at 12¢ discount per $1,000. (A draft for $1,000 costs $999.88.) 3. What will a St. Louis merchant have to pay for a draft on New York for $2,460.53, at 50% premium per $1,000? 4. If the rate of exchange is 50% discount per $1,000, what is the face of the sight draft on Boston, that can be bought in New York for $1,000? 5. When the premium is $1.25 per $1,000, Mr. Brown pays $1,634.04 for a draft on Louisville. What is the face of the draft? REVIEW. 449 6. At4% premium, find the cost of a sight draft for $ 1,843.60. $ 1,843.60 Aches 2.30 Add. 7. At 75¢ discount per $1,000, how much will cost a sight draft on Milwaukee for $946.75? $946.75 50 perM. 473 as 25 per M. 237 8. Paid $632.18 for a sight draft on Milwaukee. What was the face of the draft, the discount being 3,%? | 9. I sent a commission merchant $1,000 to buy grain. How much will he spend for grain, if his commission at 14% is included in the amount sent? (Let « = amount spent for grain. ? = commission.) _ 10. A farmer ships produce to a commission merchant, which the latter sells for $339.66, charging 2 per cent commission. For the remainder of the money he buys groceries and dry-goods, charging 2 per cent commission on the amount spent. What is the cost of the goods purchased ? REVIEW. 994. Slate Problems. 1. A joiner worked on Monday 9 hr. 45 min., on Tuesday and Wednesday 10 hr. 45 min. each day, on Thursday and Fri- day 10 hr. 15 min. each day, and on Saturday 6 hr. 45 min. What was the average length of his day’s work? 2. A watch that loses 85 seconds in an hour was set right at noon on Monday. What time did it show at 6 p.m. the fol- lowing Thursday ? 3. There are 5 boys whose heights are 4 ft. 9 in., 5 ft. 1 in., 4 ft. 5in., 3 ft. 11 in., and 4 ft. 4 in., respectively. What is their average height ? 450 ARITHMETIC. 4. A man hada plot of ground 20 yards long and 12 yards wide, which he planted in cabbage. How many plants did he require, if the rows, which ran lengthwise, were 2 feet apart and 2 feet from the fence surrounding the plot, and the plants in the rows 16 inches from each other and from the fence ? Get the correct number of rows, and the correct number of plants in a row. How many plants would have been needed if the rows ran crosswise ? 5. How long would it take a person to count a million silver dollars, at the rate of 100 a minute, and working 8 hours a day? 6. The front wheel of a wagon is 13 ft. 4 in. in circumference. How many revolutions will it make in a journey of 14 miles? How many more revolutions will it make than the hind wheel, the circumference of the latter being 17 ft. 6 in.? 7. The wheels of an engine being 16 ft. 8 in. in circumfer- ence, and the number of revolutions 150 per minute, how far does it goin an hour? Give answer in miles and rods. 995. Circular Measure. 60 seconds (/7) —-1 minute (’) 60 minutes 1 degree (°) 360 degrees 1 circle. 8. If the equatorial circumference of the earth is 25,000 miles, how many miles apart are two places on the equator, the distance between them being 20°? 9. What is the length of a degree on a circle whose diameter is 18 feet? The circumference = diameter x 3.1416. 10. The 60th parallel of latitude is a circle about one-half as long as the equator. How far due east of Christiania is St. Petersburg, both situated on this parallel, the former being 10° east of Greenwich, and the latter 30° east ? 11. How many miles north of the equator is a place in lati- tude 46° 22' 30"? Take 694 miles to a degree, EXCHANGE. 451 12. Two places in latitude 45° are 22° 30! apart, measured on that parallel. Find the distance in miles, assuming the 45th parallel to be a circle .7071 times the length of the equator, and considering the length of the latter to be 25,000 miles. 996. Time Drafts. $ 9875875. New Orxrans, June 15, 1893. At three days’ sight, pay to the order of John D. Hallock, Nine Hundred Eighty-seven 55>, Dollars, value received, and charge to account of To Natronat Park Bank, FRANK PHILLIPS. New York. When Mr. Hallock receives the above, he presents it to the National Park Bank for acceptance. The proper bank official writes across the face of the draft in red ink “ Accepted,” with the date, say ‘“‘ June 18, 1893,” and signs his name. Three days thereafter, plus three days of grace, or June 24, the draft will be payable. 997. Sight drafts are usually not allowed days of grace. Time drafts are generally allowed three days of grace. (See Appendix.) 998. The premium on the above draft at $1.50 per $1,000 is calcu- lated on the face of the draft, and amounts to $1.48. 999. Since it is not payable until six days after acceptance, the inter- est (or bank discount) for that time is deducted. Interest on $987,565, for 6 days at 6% = $.99. Cost of draft = $987.65 + $1.48 — $.99 = $988.14. N.B. Take 6% as the interest rate, unless a different rate be expressed. 1000. Slate Exercises. 1. What will I have to pay for a 90-days draft on San Fran- cisco for $840, at $1.75 premium per $1,000? 2. Face $400; 30 days’ sight; discount 4%. Cost? 3. Face $560; 60 days’ sight; premium 50 per $1,000. Find cost. 4.52 ARITHMETIC. 4. What will be the cost of a sight draft for $ 625.38 at 71 ¢ discount per $1,000? _ 5. Find the cost of a 60-days draft for x dollars, premium 25 ¢ per $1,000. 6. Find the cost of an z-day draft for $ 1,200, discount 4%. 7. Find the cost of a 30-days draft for $1,600, premium x dollars per $1,000. 8. Paid $1,188.90 for a 60-days draft, at 1% premium. What was the face of the draft ? 9. A time draft for $1,800 at $1 premium per $1,000, cost $1791.90. At how many days’ sight was it drawn? 10. At what rate did I purchase a 90-days draft for $900, its cost being $884.70? LONGITUDE AND TIME. Notr.— This topic should be taught in connection with the study of Mathematical Geography. The globe should be used to show the pupils that all places on the same meridian have the same time, that a difference in longitude of 15 degrees produces a difference in time of 1 hour, and that the more easterly of two places has the later time. | 1001. Oral Problems. 1. The difference in time being 1 hour for each 15 degrees, find the difference in longitude between two cities differing in time 34 hours. 2. Two places differ in longitude 61 degrees. What is their difference in time? 3. London is 75° east of Philadelphia. When it is 1 o’clock at Philadelphia, what is the time at London? 4. When it is 2 p.m. at London, what is the time at Phila- delphia ? —— LONGITUDE AND TIME. 453 5. When it is noon at a city 25 degrees west of Vienna, what is the time at the latter place? 6. How many degrees of longitude correspond to a time dif- ference of 3 hours 40 minutes? 7. What is the difference in longitude between Philadelphia, 75° west longitude, and St. Petersburg, 30° east longitude ? 8. When it is 3 p.m. at St. Petersburg, what is the time at Philadelphia ? 9. Washington is in 77° west longitude, and uses “standard time,”’ that is, the time of 75° west longitude. What is the difference between the correct time at Washington and its clock time ? 10. A town in 84° west longitude uses standard time (of 90°). What is the correct time when the clocks are striking 12, noon? 1002. Slate Problems. 1. Find the difference in longitude between two places dif- fering in time 3 hr. 44 min. 2. Two places differ in longitude 37°18’. What is their difference in time? 3. Chicago is 87° 35! west of Greenwich. What is the dif- ference in time between the two places? Is it earlier. or later than noon at Chicago when it.is noon at Greenwich? Why? What is the standard time at Chicago when it is 1 P.M. at Greenwich? 4. When a captain’s observation of the sun shows that it is exactly noon, the ship’s chronometer, keeping Greenwich time, reads 30 minutes past 2p.m. How many degrees west of Green- wich is the vessel ? 5. Find the difference in time between two places in longi- tude 74° 31' and 93° 14! west of Greenwich, respectively. 454 ARITHMETIC. 6. When it is noon at a place 11° east of Greenwich, it is 1.30 p.m. at another place. Find the longitude of the latter place. 7. A train ran from New York to San Francisco, 3,313.5 miles, in 3 da. 12 hr. 17 min. How many miles per hour did it average ? 8. If for $6 I can have 1,200 pounds carried 36 miles, how many pounds can I have carried 24 miles for the same money ? 9. At 80% per ounce, what is the value of 86 ingots of silver, each weighing 2 lb. 10 oz. 15 pwt.? 10. Find 30 per cent of 27 yards 8 inches. 11. The solid contents of a block 12 feet 6 inches wide and 3 feet 9 inches thick are 27 cubic yards 1 cubic foot 810 cubic inches. Required its length. 12. A farmer sold 237 bushels 3 pecks of wheat, which was 48 per cent of his crop. How many bushels, pecks, etc., did he have left? (48% is given; you have to find what %? What part of 48% added to itself will give the required per cent? Do not find the whole crop.) 13. How many spoons, each weighing 2 ounces 12 penny- weights, can be made from 4 pounds 4 ounces of silver? 14. A man travels due west, on the 45th parallel of latitude, 84 miles per hour for 24 hours. How many degrees has he traveled, the length of a degree being 48.96 miles? REVIEW. 1003. Oral Problems. 1. A puts $600 into business; B, $400; the profits are $500. What is the share of each ? 2. Two boys hire a camera for 26 weeks, paying $5.20. How much should be paid by the boy that uses it 12 weeks? 3. New Orleans is 90° west of Greenwich. When it is 2 P.M. at the latter place, what is the time at New Orleans? EXCHANGE. 455 4. Find the discount, at 6%, on a note for $300, that has 48 days to run. 5. What will be the cost of 84 yards of muslin at 49% a yard? 6. Two men hire a pasture for $84. One puts in twice as many head of cattle as the other. What should each pay? BILLS OF EXCHANGE. Exchange for £180 17s. 6d. New York, Dec. 14, 1895. Sixty days after sight of this First of Exchange (Second unpaid), pay to the order of John W. Moran & Bro., One Hun- dred Highty pounds sterling, seventeen shillings, six pence, Value received, and charge the same to account of To JAMES Lennon & Co., London. No. 39. PETER COMERFORD & Son. 1005. Slate Exercises. 1. Find the cost of the above bill at $4.87 per &. £200 = $974.00 20 = £180=§ 10s. = 2.435 £4 5s; = 2s. 6d. = oy $ 2. What would be the cost of a cable transfer of £251 11s. 9d., at $4.881 per £? £250 = $1,221.25 +1 of £1,000 l= 10s. = 1s. = 6d. = 3s. = The newspapers give quotations of foreign exchange for sight and 60-day bills, also for cable transfers. 456 ARITHMETIC. -_ 1006. The New York quotations for French exchange give the number of francs for $1. i, Paris cable transfers 5.164 @ 5.153. Paris bankers’ 60 days 5.183 @ 5.18}. Paris bankers’ sight 5.163 @ 5.164. 1007. The quotations for German exchange give the value in U.S. money of 4 Reichmarks (or marks). Reichmarks (4) 60 days 954 @ 95}. Reichmarks (4) sight 953 @ 954. 3. Find the cost of a sight bill on Paris for 1,000 frances, at 5.164 francs for $1. 4. Find the cost of a 60-days bill of exchange on Berlin for 1,874.85 marks, at 951¢ for 4 marks. 5. What will be the face in marks of a sight bill of exchange on Berlin that can be bought for $1,000, at 9519 for 4 marks? 6. A New York merchant pays $1,637.50 for a 60-days bill on Paris. What is the face of the bill, the rate of exchange being 5.183 francs for $1? 7. At $4.88 per £, what will be the face of the cee bill on London that can be bought for $1,500? 18750 £307 s., etc. 1399.99 _ 18750 61)18750 ABB 61 450 61 £23 remainder 20 460s., new dividend. 8. Bought goods in London amounting to £437 5s. 10d. less 4%. How much will I have to pay in Boston for a sight bill of exchange at $4.884, to settle the account? 9. What will be the cost in Chicago for a 60-day bill on Paris, that will pay for the following articles? Rate, 1 franc = 192. 18 pieces silk, 44 meters each, at 25 francs per meter, less 74%. 3 pieces of cloth, 50 meters each, at 20 francs per meter, less 5%. Packing charges, 60.50 francs. EXCHANGE. | 457 10. I wish to send a sight bill of exchange on Berlin in pay- ment of the following invoice : 4 cases musical instruments amounting to 3,598.60 marks, less 10, 5, and 24%. Freight to Hamburg, 165 kilos, at 4.80 marks per kilo. At 95£¢ for 4 marks, what will be the cost of the bill of exchange ? CHAPTER XIII. PARTIAL PAYMENTS. — RATIO AND PROPORTION. — SQUARE ROOT. —SURFACES AND VOLUMES. PARTIAL PAYMENTS. 1008. U.S. Rule. DututH, Mrinn., Jan. 5, 1889. On demand, I promise to pay to the order of Owen McGee Three Hundred Dollars, value received, with interest at 7 per cent. $3005. J. RANDOLPH PAGE. Endorsements: May 20, 1889, $100; Oct. 30, 1889, $100; March 6, 1890, $50. How much was due Jan. 5, 1891? Find amount of $300 Jan. 5, 1889, to first payment May 20, 1889, 4 mo. 15 da. (by compound subtraction). $307.88 Deduct first payment, 100.00 Balance May 20, 1889, $ 207.88 Interest on $ 207.88 to Oct. 30, 5 mo. 10 da., 6.47 | Amount, $ 214.35 Less second payment, 100.00 Balance Oct. 30, 1889, $ 114.35 Interest on $114.35 Oct. 30 to March 6,4 mo.6da., ° 2.80 Amount, $117.15 Less third payment, 50.00 Balance March 6, 1890, $67.15 Interest on $67.15 March 6 toJan. 5,9 mo. 29 da, —_—3.90 Due Jan. 5, 1891, $71.05 458 PARTIAL PAYMENTS, 459 1009. Slate Exercises. Norr.— Find time by compound subtraction. 1. How much is due June 8, 1896, on a demand note for $1,200, with interest at 6%, dated June 3, 1898, bearing en- dorsements of payment of $500, Sept. 18, 1894; $600, Jan. 38, 1895? 2. A demand note for $600, bearing interest at 5%, was given Feb. 18, 1892. A payment of $250 was made May 28, 1898; one of $150 was made Oct. 8, 1898. How much is due Jan. 23, 1895? 3. Note for $2,000; interest, 7%; dated April 15, 1891. Endorsements : $50, Sept. 20, 1891; $100, May 26, 1892; $1,000, June 20, 1893. How much is due Dec. 27, 1894? 1010. Face of note, $ 2,000.00 Interest from April 15 to Sept. 20, 1891, 5 mo. 5 da., ; 60.28 Amount due Sept. 20, 1891, $ 2,060.28 If the $50 payment were deducted, and interest computed on the balance, $2,010.27, the maker would be charged interest on $10.27 more than the face of the note, and this the law does not allow. Interest is taken on $2,000 until next payment, May 26, 1892, 8 mo. 6 da., 95.67 Amount due May 26, 1892, $ 2,155.95 As the two payments are not large enough to meet the interest now due, the interest is again calculated on the original $2,000 from May 26, 1892, to June 20, 1893, 1 yr. 24 da., 149.33 Amount, $ 2,305.28 Less $50 + $100 + $1,000 (three payments), 1,150.00 Balance due June 20, 1893, $1,155.28 Interest on $1,155.26 to Dec. 27, 1894, 1 yr. 6 mo. 7 da., 122.87 Due Dec. 27, 1894, $ 1,278.15 1011. By the United States rule for partial payments, the amount of the principal is found to the time when the payment, or the sum of two or more payments, equals or exceeds the interest. From this amount deduct the payment or sum of payments. Use the balance then due as a new principal, and proceed as before, 460 ARITHMETIC. 4, ApBany, N.Y., March 5, 1893. One year after date, I promise to pay John Harrigan, or order, Nine Hundred Dollars, value received, with interest at six per cent. $ 9005%%5 ANDREW T. SULLIVAN. Endorsed as follows: June 5, 1893, $10; Sept. 5, 1893, $50 ; Jan. 5, 1894, $120. What was due March 8, 1894? 1012. In the United States courts, and in those of some of the states, interest for a portion of a year is taken by days, upon the basis of 365 days to the year. To make the work easier for the pupils, however, the year of 360 days should be used in the examples given, and the time between dates should be found by compound subtraction. PRESENT WORTH AND TRUE DISCOUNT. 1016. Problems are frequently met with in books, in which the “ pres- ent worth” is asked of a sum of money payable at a future date. 1017. 1. What is the present worth of $150 payable in 1 year 6 months, interest 6% ? By this is meant what sum at 6 % interest will amount to $ 150 in 1 year 6 months? Or, Given the amount ($150), rate 6%, time 1 yr. 6 mo., to find principal. x + (a X 85 X 14) = 160. 1018. By “true discount” is meant the difference between the sum payable at a future time and its ‘present worth.” 2. What is the “true discount’ of $150, payable in 1 year 6 months, interest 6% ? The amount $150, rate 6%, time 1 yr. 6 mo., are given. Find the interest. Let # = principal Amount = @ + (% X 785 X 14) = 150 Interest = amount — x MENSURATION, 461 SURFACES AND VOLUMES. 1024. Slate Problems. 1. If a piece of cloth is 20 yards long and 3 yd. broad, how broad is another piece of cloth 12 yards long ay, contains as many square yards as the former? 2. An iron beam 16 ft. long, 24 ft. broad, and 8 in. thick, weighs 1,280 lb. What is the length of a similar beam whose breadth is 34 ft., thickness 74 in., and weight 2,028 lb. ? 3. What will it cost to carpet a room 224 ft. long by 153 ft. wide with carpet 24 ft. wide, costing $1.50 per yd.? 4. What is the length of a box 62 ft. wide and 74 ft. high, that will exactly contain 12 boxes 4% ft. long, 34 ft. wide, and 23 ft. deep? _ 5. What is the value, at $120 per acre, of a square field whose side 1s 35.25 chains ? 10 sq. chains = 1 acre. 6. What is the area in square feet of a triangle whose base is 18 ft. 4 in., and whose altitude is 11 ft. 10 in.? 7. What is the area of a circle whose diameter is 7.5 feet, the area of the circle being .7854 times the area of the square that will just enclose it? 8. Find the capacity, in bushels, of a bin 22 ft. ahs 14 ft. wide, 12 ft. high? 9. How many gallons will a tank hold, its dimensions being 4 ft. lin. by 3 ft. 8 in. by 2 ft. 3 in.? 10. How many square yards are there in the walls and the ceiling of a room 21 ft. long, 18 ft. wide, 12 ft. high? Make a diagram. 11. A tank 54 ft. by 6 ft. by 7 ft. can be emptied by two pipes, one of which discharges 9 gallons per minute and the 462 ARITHMETIC. other 7 gallons per minute. How long will it take each to empty the tank? How long will it take both together? 12. A parlor is 18 feet long, 15 feet wide. Make a diagram showing how carpet 27 inches wide can be laid without cutting the carpet lengthwise. Which would be the better way to lay carpet 30 inches wide in the above room ? 13. Calculate the number of running yards of carpet 30 in. wide needed for the floor of the above room, including 44 yards wasted in matching the pattern. Find the cost of carpeting the room at 95 cents per running yard for carpet, 5 cents per square yard for lining, and 10 cents per running yard for sewing and laying. 14. A rug 18 feet long, 15 feet wide, is placed in the centre of the floor of a room 21 feet long, 18 feet wide. What is the width of the strip left uncovered? Find the area of the uncov- ered space ? 15. A room is 18 feet wide, 24 feet long, 9 feet high. There are two doors 4 feet wide, 74 feet high; two windows 4 feet wide, 6 feet high; and a fire-place 5 feet square. How many square feet of plastering will there be on the walls and ceiling, deduct- ing for a baseboard 12 inches wide? How many running feet of baseboard will be needed ? Draw “development” of the above room, showing the four walls and the ceiling, and locating the doors, the windows, and the baseboard. Do not use baseboard where it is not required. 16..At the rate of $1,400 for a pile of lumber 25 ft. long, 20 ft. wide, 10 ft. high, what is the value of a pile 50 ft. long, 40 ft. wide, 20 ft. high? 17. Ifit costs $14 to paint the walls and the ceiling of a room 25 ft. long, 20 ft. wide, and 10 ft. high, what will it cost to paint the walls and the ceiling of a room 50 ft. long, 40 ft. wide. and 20 ft. high? ) INVOLUTION. 4638 SQUARE ROOT. 1029. Squaring a number is multiplying the number by itself. The square of 8 = 8 X 8= 64. 1030. The square of a number is indicated by writing a small 2a little to the right of the upper part of the number. 5? = 25, 12? = 144. What is the square of 4? Of 6? Of 7? Of 9? Of 10? Of 11? 2=? B=? Square 13. 15. 21. 16. 19. 1#@=? 17?=? 24°=? 33?=? 1O31. The square of 25 = (20+ 5) x (20+ 5). 20 +6 20 +5 Multiplying by 20 207+ 20x5 Multiplying by 5 20x 5 +5 202 + 2(20 x 5) + 5? = 400 + 200 + 25 = 625. 1032. The square of the sum of two numbers is equal to the square of the first + twice the product of the first by the second + the square of the second. 13?=(10+38)?= 10°+2(10x3)+3?=? 18*= (10 +8)? = 100 +1604 64=? O71? = (20+ 7)? = 400 +2804 49 =? 1033. Oral Exercises. Square : 1. 14 4. 22 Typ OL 10. 32 13. 24 pei ta, 5. 31 8. 61 11. 42 14. 33 Be EMA 6. 41 9. 23 12. 52 15. 43 1034. The square root of 41s 2; of 9is 8; of 16 is 4; of 25 is 5. 1035. Give the square root of 86. Of 64. Of 81. Of 121. Of49. Of 100. Of 144. 464 ARITHMETIC. 1036. The sign of square root is +/. V8 0. VIEL ei Veo ny oes 1037. Find the square root of 169. 10?= 100. 20?=400. The square root is between 10 and 20; it is, therefore, 10 + a second number. 169 = 10? + 2(10 x second) + second?. 169 = 100 + 20 x second + second ?. 20 x second + second? = 69. From this it appears that the second number is 3, since 20 x 3 + 3? = 69. 1038. It may be shown in this way: : _10 (first number) 169 10? = 100 Trial divisor — twice 10 20) 69(3 second number) 60 9 3= 9 Ans. 10 +3 = 13. 1039. Find the square root of 2,116. 40 (first number) 2,116 40° 1,600 40 x 2 = 80, trial divisor ) 516(6 second number) 480 36 = 6? Ans. 46. 1040. Instead of multiplying the trial divisor by the second number, and then ascertaining whether the remainder is the square of the second number, the second number is added to the trial divisor and this sum is mul- tiplied by the second number. 40 (first number) 2,116 1,600 (2 x 40) +6 = 86) 516(6 second number) 516 Ans. 46. REVIEW. 465 1041. In practice, the work is shortened by omitting the ciphers. First, point off in periods of two figures, commenc- AWA ing at the right. Find the greatest square in the first on org period, and place the root in the quotient. Subtract 21'16 the square from the first period. Bring down the next 16 period. Multiply the first quotient figure by 2,and 86) 516 use it as a trial divisor. Place the second figure in 516 the quotient. Affix it also to the trial divisor. Mul- (ian tiply the two figures in the trial divisor by the second quotient figure. 1042. Slate Exercises, Extract the square root: ee boo 6. 1,296 11. 2,809 16. 5,625 2. 256 7. 1,225 127)72:916 pe AG etet: 3. 324 8. 1,764 13. 3,721 18. 7,056 4. 576 9. 1,936 14. 3,969 19. 8,281 5. 676 10. 2,601 15. 5,184 20. 9,025 REVIEW. 1043. Slate Exercises. Divide (Arts. 385, 616): 1. 4,270,978,096 + 564,347 4,3875,621,423 + 856,789 2,171,008,895 - 721,985 . 86,409,429,120 + 876,008 4,518,821,072 + 752,134 . 57,681,954,968 + 768,437 8,817 832,184 ~~ 607,432 . 40,333,410,989 + 568,709 8,462,706,614 = 567,843 10. 58,531,676,960 + 678,432 oO PR WO ND oanrt Write answers (Art. 385): 1, 450,000 4, 700,000 901,020 86,432 59,084 98,642 , 500,000 5. 683,427 385,098 72,356 67,805 76,057 588,217 701,380 673,217 64,587 58,437 85,607 466 ARITHMETIC. 1044. An Invoice (English). Invoice of 3 bales Linen Goods forwarded by rail to Glasgow, for shipment thence per 8.8. Anchoria to New York, to order, and for account and risk of Messrs. Robinson & Oo. yd. [R]Co. | #2 | 30 pes. Bord. Crash 1500 | 1g jj £11 | 14 [43 ag) peaked “iahhs 1500 | 2 « #3 | 60 “ Checked G. C. 3000 | 15° #4 | 60 “ ms « 2889 | 253 « A £ Less 24% disct. aut A £ 1. Find the duty in U.S. money at 50% ad valorem. £ =$4.8665. 2. What is the cost in English money of crockery amounting to £166 18s. 4d. less a discount of 5 and 5% ? RATIO. 1045. Ratio is the relation which one number has to another of the same kind. 1046. The first term of the ratio is called the antecedent - the second, the consequent. The ratio of 3 to 6, $9 to $18, 15 cows to 30 cows may be expressed 2, 7%, 43. They are each equal to 4. 1047. Oral Exercises. lixpress the ratio in lowest terms: 1. 275 On se 4. 3 quarts to 4 gallons AL Seis Notr. — The denominations must 2. $19 to $95 be the same. 3. $36.50 to $18.25 3 quarts to 16 quarts = is RATIO. 467 5. 6 pecks to 5 bushels 8. 1 gallon to 500 cu. in. 6. 20 mills to 1 dollar 7. 7 tenths to 3 fifths 10. 1 shilling (24.33) to 1 dollar 1048. Sight Exercises, 9. 1 mark (23.8) to 1 france (19.3 ¢) fs Be: Bi Leica Leia G4 nies Oo ae hae 18 386 PS R36 Yaak a = OG wee, rane ew) i138 165 PA yo Peyaaels LAE og Plb 2 imarks’ : 3bu. $24 ie? 21 marks 3 BS qt. _ 30% 10. 5+22—?+88 lgal. ?¢ 11. 6 horses + ? horses = $600 + $900 12. 1 ft.+? yd. =15¢+ 90% 13. 1 qt. 1 pt. +1 pt. =?¢+4¢ 1049. Oral Problems, 1. One line is a rod long, another is 53 ft. long. What is the ratio of the first to the second ? 2. What is the ratio of 7 hours to 1 day? 3. A pound of coffee costs 30%, of sugar 6%. What is the ratio of their respective prices? 4. A walks in 4 hours as far as B in 5. What is the ratio of A’s speed to B’s? 468 ARITHMETIC. 5. HE earns in 6 days as much as D earns in 8 days. Find the ratio of H’s daily earnings to D’s. 6. One wheel makes 800 revolutions in 2 minutes, the second requires only 1} minutes to make the same number. Find the ratio of the number of revolutions made by the first wheel in 1 minute to the number made by the second wheel in the same time. 7. A circle whose diameter is 1 ft. has a circumference of 34 ft. What is the ratio of the diameter to the circumference? 8. One train goes 40 miles an hour, a second goes 45 miles an hour. What is the ratio of the speed of the first to that of the second ? 9. A window is 6 ft. 4 in. high by 4 ft. 2in. wide. What is the ratio of the height to the width? 10. A father is 36 years old, his son is 9. What was the ratio 6 years ago of the father’s age to that of the son? 1050. Slate Problems. (Be sure your answer is correct.) 1. One line is 3 rods 4 yards long, the length of another is 5 rods 1 ft. Find the ratio of the first to the second. 2. One candle lasts 4 hours 20 minutes, another lasts 3 hours 15 minutes. Find the ratio of the first to the second. 3. A pound of coffee costs 2549; 1 lb. of sugar costs 5,3, What is the ratio of price of sugar to that of coffee? 4. M walks in 1 hour 47 min. as far as N walks in 2 hours 3 minutes. What is the ratio of M’s speed to N’s? 5. P earns in 19$ days as much as Q in 183 days. What is the ratio of Q’s daily earnings to P’s? Of P’s to Q’s? 6. One wheel makes 600 revolutions in 84 seconds; a second makes 300 revolutions in 84 seconds. What is the ratio of the speed of the first wheel to that of the second? INTEREST AND DISCOUNT. 469 7. The circumference of a circle is 12.5664 ft., and its radius is 2 ft. What is the ratio of the diameter to the circumference ? 8. One train goes 40 miles in 50 minutes, another goes 24 miles in a half hour. What is the ratio of the speed of the sec- ond to that of the first? 9. One window is 6 ft. 8 in. x 4 ft. 2 in.; a-second is 4 ft. 8 in. x 2 ft. lin. What is the ratio of the area of the second to that of the first? 10. A mother is now 35 years old, and her son is 3 years and 6 months old. Fourteen months ago, what was the ratio of the mother’s age to that of her son? 11. A farm costing $4,750 was sold for $5,750. What is the ratio between the profit and the cost? 12. A man can do a piece of work in 42 days. What part of it can he do in a day and a half? What decimal? What per cent? 13. What is the ratio between a ton of 2,000 pounds and one of 2,240 pounds? INTEREST AND DISCOUNT. 1052. Slate Exercises. (Solve the first ten by aliquot parts.) Find the amount: 1. $1,875.25, 3 yr. 5 mo. 15 da., 44%. 2. $487.50, 1 yr. 10 mo. 25 da., 6%. 3. $1,206.84, 2 yr. 1 mo. 16 da., 5%. 4. $595.00, 7 yr. 7 mo. 7 da., 7%. 7mo.=7y of 7 yr. 7 da. = what part of 7 mo. 470 ARITHMETIC. $763.25, 8 mo. 11 da., 4%. $685.70, 19 mo. 5 da., 34%. $1,563.00, 8 mo. 20 da., 5%. $998.45, 87 da., 4%. $2,575.50, 149 da., 3%. 10. $693.27, 214 da., 6%. OM WH A 1053. Find the value of z: 11. Principal, $240; rate, x; interest, $32.04; time, 2 yr. 11 mo. 18 da. 12. Principal, x; rate, 6%; amount, $717.40; time, 3 yr. 3 mo. 4 da. 13. Principal, $360; rate, 3% ; interest, $48.87; time, 2. 14. Principal, $288; rate, 24% ; amount, $307.22; time, 2. 15. Principal, z; rate, 6%; interest, $13.10; time, 4 mo. ll da. 16. Principal, $270; rate, x; amount, $273.27; time, 3 mo. 19 da. 1054. Distinguish between “term” and “time.” Term of a 90-day note is 93 days. (See Arts. 939 and 941.) ce 18. 19. 20. 21. 22. 23. 24. 25. Term, «; face, $600; discount, $6.30; rate, 6%. | Term, 33 days; face, x; proceeds, $397.80; rate, 6%. Time, 90 days; face, $300; proceeds, x; rate, 6%. Term, 21 days; face, $600; discount, $2.45; rate, 2. Time, 4 mo.; face, $200; discount, 7; rate, 6%. Term, 132 days; face, x; proceeds, $2,689.50; rate, 6%. Term, x; face, $150; proceeds, $147.75; rate, 6%. Term, x; face, $1,650; discount, $4.95; rate, 6%. Time, 69 days; face, x; proceeds, $469,380; rate, 6%. LONGITUDE AND TIME. 471 REVIEW. 1055. Find products: 1 648x4 11. 1,864 x 250 2. 976x138 1208 3 9600% OL 3. 1,648 x 874 13. 1,576 x 624 4. 2,592 x 918 14. 176 x 232 5. 2,416 x 875 15. 1,128 x 375 6. 874x9¢ 16. 895 x 444 7 848x125 17. 864 x 486 8. 375 x 999 185/975 X313 9. 192 xX 25 19. 372~x 64 10. 457 x 16 20. 483 x 42 LONGITUDE AND TIME. 1056. Norr.— Making diagrams, as shown below, may assist the pupil to solve the problems. 1. Given the longitude of A as 95° east, and that of B as 74° east, and the time at A as 1: 30 p.m., to find the time at B. Since the latitude of B has no bearing upon its time, both places may be located upon the same line running east and west. Time difference =? hours. Time =? Time 1: 30 p.m. B A | = Hast 0° 74° 95° Longitude difference = 21°. Locate the prime meridian (that of 0°), then the meridians of 74° and 95° east. Mark above the last two the names of the places, Band A. Write above A its given time, 1: 30 p.m. To find the time at B, we must find the difference of time between B and A. The difference in longitude is 95° — 74° = 21°. The difference in time is (21 + 15) hours. Notr. — Remember that the more easterly of the two places has the later time. 4792 ARITHMETIC. 2. A is situated in 71° west longitude, B in 107° west longi- tude. What time is it at B, when it is noon at A? Time diff. =? Time? 12M. te] A West —— DEEPER EME Re STL ae LOT? (ey 0° Long. diff. =? 3. Find the longitude of B, whose time is 8:10:80 a.m., when it is 7:15 a.m. at A, whose longitude is 156° 48! west. Time difference =? 7:15 aM. 8:10:30 a.m. A B = West ——_—_—_ |_—_—_|_—_ East ; 156° 487 Longitude =? 0° Longitude difference =? Since B has the later time, its location is east of A. The difference in time, being nearly an hour, shows the difference in longitude to be nearly 15°. Find the exact difference. Is it to be added to 156° 48’ or subtracted from it, to give the longitude of B? 4. When it is 2:40 a.m. at A, in 57° 24' west longitude, it is 10 a.m. at B. Find the longitude of B. Time difference = 71 hours. 2:40 a.m. 10 a.m. A B West a a a East 57° 24! cI @ hd Longitude =? Longitude difference = 15° x 71 = 110°. If we go 110° eastward from A, we shall reach the prime meridian after going how many degrees and minutes? How many more degrees and minutes must we travel to reach B? Is B in east or in west longitude? 5. When it is noon at B, what is the time at A, the former being in longitude 44° east, and the latter in longitude 57° west? Time difference =? Time =? 12 m. A B West ———_| cc — East 57° Oe 44° Longitude difference = 101°. Why? SQUARE ROOT. 473 1057. Find the longitude or the time: Longitude of A. Time at A. Longitude of B. Time at B. 6. 63° east 9 A.M. 54° east ? 7. 57° 25! east ? 83° 20' east 1:45 p.m. 8. 156°48! west 3:15 p.m. ? 4:10 pa. 9. ? 11:42 a.m. 56° 25! west 1:27 P.M. 10. 2° 15! west 6:53 A.M. 67° 48! east ? 11. 27°10! east ? 27° 10! west 12 M. 12. ? 4:10 P.M. 18° 4! east 11:30 a.m. 13. 74°56! west 3:50 a.m. ? 11 a.m. 14. 4! 30" east 8:47 aM. 90° 15’ west ? 15. ? 10:30pm. 32°30! east 6:48 P.M. SQUARE ROOT. 1058. Find the square root: 1. os 6. rer 9. ooo? 2. % 6. 2%, 10. 7224 3. aT 7. S245 TL. 2304 4. 2 8. sear 12. 2033 Notre. — Before extracting the square root of the following, reduce tho mixed numbers to improper fractions. 13. 121 16. 4124 19. 1561 14. 112 17. 6285 20. 264, 474 ARITHMETIC. SPECIAL DRILLS. 1060. Find sums: 11+ 55+ 99 46 + 22+ 88 33 + 76 + 37 66 + 42+ 54 92+18+ 48 36+ 71+ 57 89 + 28 + 64 53 + 47+ 35 LOGL. Find answers: 150 — 23 —48 162—(26+ 61) 174—41—35 165 — (28-4 47) 172—19—66 154—(86+44) 163 —38—43 171 —(82+ 33) 183 — 87 —42 184—(89+ 35) 155—49—24 180—(18 + 28) 161—12—71 173—(57+17) 181—47-— 33 153 —(45+ 31) 1062. Find products: 36 x 31 36 X 29 45x 41 45 x 39 54 x 51 54 x 49 63 x 61 63 x 59 1063. Find quotients: 576 +18 693 + 21 608 -- 19 848 + 16 448 +14 533+ 13 924 + 22 943 + 23 1064. Find answers: 3815 x 144% 3878 x 14% 971+ 3 85i-+ 11 32 x 804 45 y 1914 1054 + 12 109i 4 25 + 84+ 58 66 + 15 + 96 27 + 19+ 87 69 + 73 + 38 1 2xopL 72 x 69 SLL 81 x 79 600 + 24 675 + 75 825 + 75 525 + 75 Tix Th 81x 8} 155i-+ 7 2002 +18 45 + 56 -L 68 75 + 34 + 86 41 + 65 + 59 52+ 39+ 78 90 x 89 90 x 91 99 x 99 101 x 99 225 -- 124 150+ 61 825 + 374 750 + 624 182 x 54 124 x 64 194 + 22 252 + 34 REVIEW. 475 1065. Oral Problems. 1. Paid 92% for coffee, 48¢ for butter, and 18¢ for lard. How much was my bill? 2. I had $150. Spent $28 for a suit of clothes and $48 for tools. How much was left? 3. What is the area of a field 36 yd. by 31 yd.? 4. 600 hours = how many days? 5. What is the cost of a cow if I pay $630 for 15? 6. How many ounces in 294 lb. ? 7. 1094 lb. sugar are divided among 4 people. What is the share of each? 8. At 1,9,¢ per lb., how many lb. iron can I get for $5.70? 9. What will be the cost of 51 tons iron at $17 per ton? 10. What will be the average age of 9 boys, each 12 years old, and 6 boys, each 10 years old ? 11. At 42 miles per hour, how long will it take a train to go 882 miles? 12. At 25¢ per hour, what will a man earn in 18 days of 10 hours ? 13. What will be the net price of an article whose catalogue price is $20.00, the discount being 90 and 10%? 14. A man had $181 in bank. What will be his balance after taking out $47 and $33? 15. How many feet in 14 rods? 16. 77 yards = how many rods? 17. How many sq. yd. are there in a floor 102 yd. long and 64 yd. wide? 18. Cost of 372 eggs at 15¢ per doz. 476 ARITHMETIC. 19. A man owns 3 farms containing 65 acres, 86 acres, and 98 acres, respectively. How many acres does he own? 20. What is the area of a piece of glass measuring 8} by 61 inches ? 21. What is the value in U.S. money of 50 marks at 23,8, cents? 22. How many francs will a calf cost, if 18 are worth 630 francs ? 23. A man spends $1,740 per year. What is the average amount spent per month? PROPORTION. 1068. A ratio is generally expressed by the sign (:). This is another form of the division sign (+). 1069. Two equal ratios form a proportion. 3+9=13 + 39; or, 3:9 = 13: 39; or, 3:9::138:39. 1070. Supply missing term: 1¢ 21 Laie! x Me Fhe) 4: 2590 2. 834+ 16=4+2 5.538 = a= 12-420 rR ay Ged B22 Sh 6. ?:19::28: 76 7. 11b.1 0z.:2 lb. 4 02.::178: xf. 8. 8 qt. 1 pt.+1 gal. =2f + 80¥. 9. 4 bottles : x bottles = 6 pints : 15 pints. 10. x men: 9 men = 16 acres : 36 acres. PROPORTION. 477 1071. The first and the last term of a proportion constitute the extremes; the second and the third, the means. 5 . 18 Sis 9 ‘ ah 5 and 27 are the extremes. 15 and 9 are the means. The proportion is read: 5 is to 15 as 9 is to 27. 1072. The proportion 3:4::2%:y may be. written ; x Clearing of fractions, we have 3y = 42; 1.., the product of the extremes is equal to the product of the means. Solve the following: 1073. Make the product of the extremes equal to the product of the means, after canceling any factor common to an extreme and a mean. 3 per itr Rog DAY | %=15, Ans. 2. Sieh B80 z= 18, Ans. 4 3. 3: Abi: a: IB ; 4. 3:4¢::5:2 9. 2:a::4:2 Sepa elo a ee 10. £:8::%:24 Gece lO 1S x RM ei ae ie? 7 3:d4::2:44 Lo eg eee leo Bet ne? ease ea 13u eu Oa: 45a 1074. Oral Problems. 1. If 9 eggs cost 25%, what will 3 dozen cost? 2. If 7 lb. flour cost 23%, what will be paid for 49 lb.? 3. For $51 can get 12 straw hats. How many can I get for $20? ue 4. A wheel makes 75 revolutions in 5 minutes. How many does it make in an hour? 478 ARITHMETIC. 5. $100 principal gives $6 interest. How much will be the interest of $450 principal ? : 6. A merchant pays 75% freight for 125 Ib. of merchandise. How much will be the freight on 1,000 lb. at the same rate ? 7. A locomotive goes 3 miles in 4 minutes. How far does it go in an hour? 8. 4 horses can eat a certain quantity of hay in 10 months. How long will it last 20 horses ? 9. 12 men can do a piece of work in 15 days. How long will 386 men require? 10. 15 kilos cost 270 francs. What will be the cost of 5 kilos? 1075. Slate Problems. NotE. — Solve by proportion or in any other way. 1. If 9 cows cost $267, what will be the cost of 86 at the same rate? In solving such examples by proportion, we say 9 cows = $ 267, 36 cows = $a. The ratio of the cost, 267 : 2, must be the same as the ratio of the number of cows, 9:36. Making the proportion, we have 9230220) 6a Canceling, x = $1,068. 2. 7 bbl. sugar cost $104.32. Find the cost of 42 bbl. 3. A wheel makes 248 revolutions in 5 minutes. How many does it make in 1 hour 20 minutes? 4. A locomotive goes 1.8 kilometers in 4 minutes. How far does it go in an hour? COMMERCIAL DISCOUNT. 479 5. From 9 kilos (kilogrammes) of yarn are made 42 meters of cloth. How many meters of cloth can be made from 165 kilos of yarn? How many kilos of yarn are needed for 196 meters of cloth? 6. If 17 men receive $357 for a week’s work, how much should 24 men receive ? 7. If 17 men take 27 days to finish some work, how long would it take 54 men? 17 men take 27 days 54 men take x days 17 or 54 men?: 17 or 54 men? :: 27 days: « days. 8. When a sum of money is divided among 48 persons, each receives $27.50. How much would each receive if the same sum were divided among 16 persons? 9. For $85 I can purchase 238 yards of dress goods. How many yards can I purchase for $5? 10. A can do a piece of work in 6 days, B can do it in 7 days. If B’s wages are $2.10 per day, how much should A receive per day?. COMMERCIAL DISCOUNT. 1076. Oral. When the list price is $1, what is the net price after the deduction of each of the following discounts? 1. 30 and 20% 6. 10 and 5% 2. 40 and 10% 7. 20 and 20% 3. 25 and 40% 8. 331 and 10% 4. 50 and 10% 9. 20 and 15% 5. 40 and 20% 10. 30 and 15% 480 ARITHMETIC. 1077. What single discount is equal to each of the following double discounts? 11. 30 and 30% 16. 30 and 10% 12. 20 and 25% 17. 40 and 5% 18. 25 and 20% 18. 50 and 20% 14. 15 and 30% 19. 40 and 15% 15. 40 and 30% 20. 50 and 15% 1078. Slate Exercises. Which is the better discount for the buyer? 21. 30 and 20%, or 40 and 10%. 22. 50 and 10%, or 40 and 20%. 23. 20 and 20%, or 30 and 10%. 24. 20 and 15%, or 830 and 5%. 25. 30 and 15%, or 25 and 20%. 26. 30 and 30%, or 50 and 10%. 27. 40 and 30%, or 20 and 50%. 28. 40 and 5%, or 30 and 15%. 29. 20 and 50%, or 60 and 10%. 30. 40 and 15%, or 30 and 25%. 1079. Find the value of x: List price, $250; selling price, $x; rate, 40 and 10%. List price, $ 800; selling price, $684; rate, x and 5%. List price, $2; selling price, $90; rate, 334 and 10%. List price, $600; selling price, $378; rate, 30 and x%. List price, $16; selling price, $x; rate, 30 and 20%. SQUARE ROOT. 481 List price, xf; selling price, 27¢; rate, 50 and 10%. List price, $5; selling price, $3.20; rate, x and 20%. List price, $x; selling price, $73.50; rate, 30 and 80%. List price, $200; selling price, $v; rate, 25 and 20%. 10. List price, $1.50; selling price, 60%; rate, 50 and «%. OM AA SQUARE ROOT. 1080. Find the square root of 425,104. 65 2 | 42!51'04 19)5 651 130|2 2604 Ans. 652. LOS1. In finding any figure of the root after the first, we multiply the other figure or figures by 2 for a trial divisor. 1082. Find the square root of 20,857,489. 4567 20'85'74'89 g5 «A 85 mi Re 912\7 63889 Ans. 4,567. 1083. Find the square root of 1. 64,516 6. .702244 2. 734.41 7. 264.7129 3. 1.8769 8. .23775376 4. 718.24 9. .093636 5. 14.1876 10. .004761 489, ARITHMETIC. 1084. Slate Problems. 1. What is the profit on 9 boxes of oranges, each containing 20 dozen, bought at $1.10 per hundred and sold at the rate of 18 for 25¢? 2. How long will it take a train to go 176 miles at the rate of 3,520 feet per minute? 3. If .0875 of an acre of land is worth $9, what is 3, acre worth? | 4. At £1 1s. 7d. per barrel, how many barrels of flour can be bought for £161 17s. 6d. ? 5. A,B, and C buy a house for $7,500. A furnishes $2,000; B, $2,500; C, the remainder. The yearly rent, less expenses, is $576. To what amount is each entitled ? 6. If 580 tiles, each 6 inches square, will cover a certain area, how many tiles, each 4 inches long and 3 inches wide, will be needed to cover the same area? 7. A man receives $1,500 commission on his yearly sales. What is the amount of his sales, if he is allowed + per cent com- mission ? 8. At what rate per cent will $360 produce $3.06 interest in 2 months 12 days? 9. Find the square root of 25.00400016. 10. What will be the capacity, in gallons, of a tank 9 feet long, 6 feet 8 inches wide, and 6 feet 5 inches deep? 11. What decimal multiplied by 312.5 will give the sum of 8, zs, %, 09375, and 2.46? 12. A dealer bought a lot of coal at $4.95 per ton. What was the total cost if he gained $142.50 by selling it at $5.25 per ton? FRACTIONS. 488 QL + 4.5, 18x38 14. The front wheel of a wagon measures 18 feet in circum- ference. What is the distance traveled in miles, rods, yards, etc., when the wheel has made 527 revolutions? 15. Write in words .349, 300.049, 34%, 300549. 13. Find the value of —4 of 63. 16. If a bar of silver weighing 4 lb. 6 oz. 12 pwt. is worth £13 8s. 4d., what is the value (an English money) of a similar bar weighing 7 lb. 9 oz. 12 pwt.? 17. A and B form a partnership. A furnishes $5,000; B, $10,000. During the year A draws $1,500 of the profits and B draws $1,000. At the end of the year the entire business is disposed of for $20,000. What amount should each receive ? 18. What per cent is gained on en article bought for 20 per cent less than its value and sold for 20 per cent more than its value? 19. A person loans $750 to M and $1,200 to N at the same rate. From the latter he receives half-yearly $9 more interest than from the former. What is the annual rate of interest ? 20. A 4-months note for $375, drawn March 19, was dis- counted at a bank June 4. Find the proceeds. Rate, 6%. 21. M can do a piece of work in 4 days, N can do it in 5 days, 0 in 6 days. How long will it take the three together to do the work? REVIEW FRACTIONS. 1085. Slate Exercises. 1. Divide the sum of 63 and 1% by the difference between 24 and 34, 2. What is the difference between the sum of # and 3 and the product of $ and 54? A84 ARITHMETIC. 3. What is the product of the sum and the difference of 44 and 64? 4. Subtract 2 of + from 44; and find the value of 5% of 16s. 6d. 5. Add 78, 8 of zy, & of 72, and 43. 6. Reduce $ of asq. rod to the fraction of an acre, and find the value of ;& of a ton in pounds and ounces. 34 — 24 34+ 2h 7. Reduce 58&%6& to its lowest terms, and to its simplest form. 8. Addi, 3, #,and4; multiply the sum by %; and sub- tract the product from 1. 9. Find the value of 954 meters at 43 francs per meter. 10. Divide 21 by 34, and add the quotient to 55. 11. Multiply 2,2, by 163, and divide the result by 14 of 22. 12. Reduce 7s. 6d. to the fraction of a £, and 7hr. 12 min. to the fraction of a day. 2+ 4 of 54 Foy 14. Add together £2 and + of 53 shillings. 15. What fractional part of 7 A. 127 sq. rd. is 5 A. 81 sq. rd.? 16. What must be added to $ of to make it equal to 7% of 32? 17. 2 of a number is 148. What is the number? 18. If? of a field is worth $325, what is the field worth? 19. If 2 of a house is worth $4,900, what is the value of 4? 20. If 3; of a ship is worth £1,278 2s. 6d., what is 5; worth ? fs = ay = £1,273 2s. 6d. Deduct = 13. Reduce to its simplest form SQUARE ROOT. , DOMESTIC EXCHANGE. 1087. Slate Exercises. Find the value of x. 485 (Do not allow days of grace in the case of “sight” drafts.) FACE oF DRAFT. x . $1,800 x $200 $ 600 . $1,000 . $1,200 $800 $400 RATE oF EXCHANGE. $100 20% per M. premium par x 1% premium $1 per M. discount 38% premium 75¢ per M. discount par x $2 per M. premium 1088. Slate Problems. TERM. sight 6 days 60 days sight 30 days Dp z 93 days 24 days 9 days Rate oF Cost oF IntEREST. DRAFT. x 6% $499.50 6% $1,778.85 $701.75 6% x 6% $598.95 6%» $999.25 z $1,178.30 6% $796.80 az $400.80 APPLICATIONS OF SQUARE ROOT. 1. How many inches in the side of a square table top con- taining 529 square inches? 97 sq. in. (Reduce area to square inches.) 2. The surface of a square piece of board contains 3 sq. ft. What is the length of one side in feet and inches? 3. How many rods long is a square field containing 90 acres? How many yards of fence would be needed to enclose it? 486 _ ARITHMETIC. 4. Land surveyors use a measure called a chain. What is its length in feet, 10 square chains being equal to an acre? It is subdivided into 100 “links.” Find the length of a link in inches and decimal. » 5. The surface of the six equal faces of a cube is 1,350 sq. inches. What is the length of the cube? 6. Carefully construct a right-angled triangle, base 4 inches, perpendicular 3 inches. Measure the hypotenuse. Take the square of the length of each side, and endeavor to show the relation between the square of the hypotenuse and the squares of the other two sides. 7. Construct a right-angled triangle, base 3 in., perpendic- ular 14 in. Measure the hypotenuse, and see if the relation between this hypotenuse and the other two sides of this triangle is the same as that found in the other triangle. 8. A right-angled triangle has a base 12 inches long; its perpendicular is 34 inches. What is the length of the hypot- enuse ? 9. The hypotenuse of a right-angled triangle is 25 inches; its perpendicular is 7 inches. What is the base? 10. The base of a right-angled triangle is 12 feet; the hypot- enuse is 13 ft. Find the perpendicular. 1089. Draw a right-angled triangle (Fig. 1). Upon each side construct a square (Fig. 2). From the upper portion of the lpg Fira. 1. Fia. 2. Fia. 3. Fia. 4. Fia. 5. COMPOUND INTEREST. A487 largest square C, cut a right-angle triangle of the same dimen- sions as those of the original triangle m. Cut another triangle of the same dimensions from the left-hand portion (Fig. 3). Place one of these triangles below the remainder of the square C, and the other at the right, as in Fig. 4, and the resulting polygon will be exactly equal in surface to the two squares A and B (Fig. 5). COMPOUND INTEREST. 1091. Slate Exercises. 1. Find compound interest of $2,048, for 3 years, at 5%, interest compounded semi-annually. $ 2,048 Do not use unnecessary figures. 75 each 6 mo. first 4 year 51.20 $ 2,099.20 second year 52.48 Carry to only 4 places of decimals. $ 2,151.68 etc. etc. 2. Compound interest of $1,864, at 4%, for 2 years, interest compounded quarterly. 3. Compound interest of $1,500, at 6%, for 3 years, interest compounded semi-annually. 4. Amount of $800 at compound interest, for 3 years, at 3%, interest compounded semi-annually. $ 800.00 19 8.00 first 4 year { 1 4 4.00 $812.00 0, second 4 year { Fe ie $ 824.18 4 19 8.2418 third 4 year { i 4.1209 $ 836.5427 etc. etc. 488 ARITHMETIC, STOCKS AND BONDS. 1092. Slate Problems. Brokerage is calculated on the par value. The dividends are based on the par value. 1. Find the cost of 240 shares mining stock, par value $10, at 873, brokerage 4%. 2. Paid $11,460 for 120 shares R. R. stock, par value $100, brokerage 4%. What was the value of the stock per share? 3. Bought 150 shares canal stock at 874, brokerage 1%, pay- ing for it $5,265. What is the par value per share? 4. How much brokerage is paid by the buyer of 275 sharer bank stock, par value $100, brokerage 1% ? 5. A broker sells for a customer 200 shares stock, par value $25, at 1024. If he retains 1% brokerage, how much does he pay over to the former owner of the stock ? 6. A man buys 60 shares bank stock, par value $100, at 40, no brokerage. If the annual dividend is 18%, what is his income therefrom ? What per cent does he receive on his investment ? 7. A manufacturing corporation makes $20,000 a year over all expenses. The stock consists of 4,000 shares, par value $50. What rate of dividend can be declared ? What per cent on his investment does a man receive who has bought his stock at 175, no brokerage? 8. A railroad company’s stock consists of 10,000 shares, par value $100. Its profits for the year are $47,500, out of which must be paid the interest for the year on $200,000 worth of bonds, at 5%. What rate of dividend can be paid the stock- holders ? 9. A capitalist bought 360 shares stock, par value $25, at 1683. He paid therefor, including brokerage. $15,176.25. What was the rate of brokerage? SQUARE ROOT. 489 10. A broker sold 250 shares, par value $100, at 107%. He deducted brokerage, and paid over the proceeds, amounting to $26,875. Find the amount of the brokerage and the rate per cent. 11. A woman invests $35,050 in stock at 175, brokerage 1%. If the annual dividends are 74%, what is her income from the investment ? 12. Mr. Tower pays $104 for a $ 100 five per cent bond. At the end of six years, the bond is redeemed at par. What rate of simple interest does he receive on his investment of $ 104? SQUARE ROOT. 1094. Slate Exercises. Find square roots to three decimal places: | he DOL) 3. 38 5. 350 (by aye 9. 1,874 2. 14 4. 74 6. 758 8. 1,384 10. 4,000 1095. Oral Exercises. What is the square of 1? Of.38? Of.11? Of LZ? 1096. How many decimal places in the first two answers? In the last two? V/01 =? V.09 =? 0121 =? V.0144 =? 1097. Slate Exercises. Find square roots to three decimal places: 1.0 Ger 4 9. 3.6 Lae Lee lak 2. 40 6. 9 10. 1.60 14. 64 18. .144 saa Taal 11. 2.50 LGsoek 193169 4341 8. 2.5 12. 3.60 16. 10.0 20.000 490 ARITHMETIC. MEASUREMENTS. 1099. Slate Exercises, Find the missing side of each of the following ten right-angled triangles : 1. Base, 15; perpendicular, 8; hypotenuse, 2. 2. Base, 35; perpendicular, x; hypotenuse, 37. 3. Base, 2; perpendicular, 15; hypotenuse, 39. 4. Base, 20; perpendicular, 21; hypotenuse, 2. 5. Base, x; perpendicular, 45; hypotenuse, 53. 6. Base, 56; perpendicular, 2; hypotenuse, 65. 7. Base, 55; perpendicular, 48; hypotenuse, 2. 8. Base, x; perpendicular, 14; hypotenuse, 50. 9. Base, 63; perpendicular, x; hypotenuse, 65. 10. Base, 112; perpendicular, 15; hypotenuse, 2. 1100. Slate Problems. 11. One parallel side of a trapezoid measures 160 yd., the other measures 200 yd., the area is 32,400 sq. yd. Find the perpendicular. 12. One parallel side of a trapezoid is 20 rods, the perpendic- ular is 15 rods, the area is 225 sq. rods. Find the length of the other parallel side. 13. One parallel side of a trapezoid measures x rods, the other measures x + 6 rods, the perpendicular is 10 rods, the area is 150 sq. rods. Find the length of the parallel sides. 14. Find the area in acres of a right-angled triangle, the length of the sides being 24 rods, 7 rods, 25 rods. REVIEW. 3 49] 15. A court yard 84 ft. by 36 ft. is to be paved with flag- stones measuring 6 ft. by 3 ft. How many stones will be needed ? What will be the cost of the work at $1.25 per sq. yd.? pane 1 chain = 66 feet. | 1eicHaine. Norr. — A right angle contains 90 degrees. 16. How many rods of fence will be needed to enclose the field shown in the diagram ? 12 chains, 70 rd. 17. Find the length of the fourth sideof [ys the following piece of ground. E How many yards of fence are needed to yh ranr enclose it? How many acres does it contain? 18. What is the length of the diagonal of a rectangular field 90 yd. wide, 120 yd. long? 19. The dotted line in the accompanying diagram indicates a path through the field. How many yards are saved by taking the path instead of following the road? Sandi 86 chains. 27 chains. Road. 20. Find the length (in rods and a decimal) of the diagonal of a square 40-acre field. REVIEW. 1101. Oral Problems. To the following ten problems the wrong answers are very frequently given. 1. Sold a horse for $250, losing $50. What is the loss per cent ? 2. If 3 boys solve 3 problems in 3 minutes, how long will it take 6 boys to solve 6 problems? . 3. Two boys go fishing; one brings 7 cakes for lunch, the other brings 5 cakes. A third boy joins them at noon, and pays 12¢ for his share of the dinner. How should the first two divide the money received ? 492 ARITHMETIC. 4. If 100 per cent is gained by selling an article for $1, how much would be gained by selling it for $2? 5. A boy had a slate 5 inches by 7 inches. He buys one twice as large. Give the dimensions of the new slate. 6. A man wishes to put up on the front of his lot a fence 30 feet long. If the posts are 6 feet apart, how much will they cost at 25 ¥ each? 7. One-half the money received by a newsboy is profit. What per cent does he make? 8. 50 per cent of a number multiplied by 30 per cent of the same number equals 60. What is the number? 9. Three-fourths per cent of a number is 90. What is the number ? 10. An importer receives some cases of goods numbered con- secutively. How many cases are there, if the number of the first is 28, and of the last 75? 1102. Slate Problems. 1. If a bar of silver weighing 2 lb. 3 oz. 6 pwt. is worth £6 18s. Td., what is the value in English money of a oe bar weighing 15 lb. 7 oz. 4 pwt.? 2. A quantity of provisions would last a ship’s crew 20 days, allowing each man 2 lb. 4 oz. daily. What should each man be allowed so as to make the provisions last 4 days longer? 3. If 40 men are able to do a piece of work in 10 hours, how many extra men must be employed to finish it in 8 hours? 4. Ifit requires 40 yd. carpet 2 ft. 9 in. wide to cover a floor, how many yards of carpet 2 ft. 6 in. wide would be needed ? 5. How long will it take a train to go 112 miles, at the rate of 46 miles in 1 hour 20 min. 30 sec. ? 6. Change 1,759 yards to rods, yards, ete. REVIEW. 493 7. If a beam 5 ft. 6 in. long, 10 in. wide, and 8 in. thick, weighs 924 lb., find the length of another beam of the same material which weighs 3,024 lb., and whose end is a square foot. 8. A field 110 yd. long and 44 yd. wide contains an acre. What is the area of a field 220 yd. long and 88 yd. wide? Of one 440 yd. long and 176 yd. wide? 9. A ship with a crew of 32 men has provisions that will serve for 45 days, at a daily allowance of 3 lb. for each man. If it then picks up another vessel's crew consisting of 16 men, what must be the daily allowance, to make the provisions last for 40 days? } 10. If a steel bar 12 ft. long, 4 in. broad, and 24 in. thick, weighs 480 lb., what is the weight of another steel bar 18 ft. long, 3 in. broad, and 2 in. thick ? 11. If 8 horses eat 13.5 bushels of oats in 9 days, how many days will 15.75 bushels last 14 horses? 12. A person deposits in two banks $750 and $1,200, respec- tively, at thesamerate. The latter sum draws $18 more interest per year than the former. What is the rate per cent? 13. Two men have saved $2,000 each. One has loaned $1,400, at 4%, and the remainder at 5%. What rate must the other man receive for his money in order to get the same interest ? 14. I owe $8,625, payable in 3 years 4 months. I have at present $7,500. What rate of interest must I receive to pay my debt at maturity ? 15. A person loans 4 of his capital at 5%, and the other half at 4%. He receives annually $40 more interest from the former than from the latter. What is his capital? 16. A certain sum loaned at 4% produces $30 less interest than a sum $400 greater, loaned at 5%. How muoh is loaned at each rate? 494 ARITHMETIC. 17. A capitalist has placed 4 of his money at 4%, and his remainder at 5%. His income is $2,940 per year. What is his capital ? 18. Change 13,576 inches to rods, ete. 19. Three men buy a lot for $600. After selling it A receives $ 220 as his share of the proceeds, B receives $280, and C $3800. How much did each invest originally ? 20. D receives 4 of a sum of money, E 4, and F the remain- der. E’s share is $90 more than D’s. What is the share of F? 21. A man receives $593.70 as the proceeds of his note. 638 days thereafter he pays the bank $600. What rate of interest has the bank charged on the $593.70 loaned ? What is the rate of bank discount on the $600 note? 22. A tank is fed by two pipes, one of which can fill it in 2 hours, and the other in 3 hours. A third pipe can empty it in 1 hour. If, when the tank is full, the supply pipes and the exhaust pipe are all set to work, in what time will it be emptied ? 23. What per cent is gained on oranges bought at 20 cents per dozen and sold at the rate of 10 for 25 cents? EXACT INTEREST. Exact interest is used by the Government in its calculations. 365 days are taken to the year. 1104. Slate Exercises, 1. Find the exact interest of $280 from April 14 to Sept. 6 at 4%. Time 145 days. Ans. $280 x —*- x 14. 100 365 2. Find the exact interest on $76.65 from March 4 to Dee. 15 at 6 per cent. MEASUREMENTS. 495 On $384 at 733 per cent for 75 days. On $ 438 at 5% from Jan. 1 to March 15. On $ 109.50 at 42% for 87 days. On $847.60 at 5% from April 29 to Sept. 20. 7. On $584 at 32% from May 16 to Dee. 1. Oo on FP & 1105. Unless “exact” or “accurate” interest is specified, use 360 days to the year. MEASUREMENTS. 1. What is the area of a triangle whose sides measure 15, 16, and 17 inches, respectively ? 15 From the half sum of the three sides subtract each 16 side separately. The square root of the continued prod- iA uct of the half sum and the three remainders will be the 2)48 area. ea V2EXIXBEXT= 24—16=8 04—17=7 12,096 = 109.98 sq. in. Ans. 2. Find the area in square feet of a triangle whose sides measure 386 ft., 84 ft., 91 ft. 3. Find the area of a triangle whose sides measure 21, 28, and 35 rods, respectively. B 4. In the following field, AB measures 59 rods; BC 52 rods; CD, 25 rods; AD, 60 rods; and the diagonal, AC, 65 rods. 4 C Find the area of the field in square rods. 5. Find the area of an isosceles tri- D angle whose base is 30 yards, its equal sides measuring 25 yaras. : 6. What is the altitude of an isosceles triangle, base 96 ft., equal sides 64 ft.? Find its area. 4.96 ARITHMETIC. 7. Find the area of an equilateral triangle, each side being 6 ft. 8. Find the area of a right-angled triangle, base 42 ft., hypotenuse 70 ft. 9. Find the area of an isosceles triangle, altitude 48 ft., equal sides 50 ft. 10. Place two equilateral triangles, sides 2 inches, base to base, making a rhombus. Find its area, also the length of each di- agonal. 11. Find the radius of a circle whose circumference is 1382 ft. (3.1416 x diam. = circum.) 12. Find the area of a circle whose radius is 4 inches. (Area = circumference X 4 diameter.) 13. Find the area of a circle whose radius is x inches. 14. Find the radius of a circle whose area is 314.16 sq. in. 15. Find the area of a circle whose circumference is 15.708 ft. PARTIAL PAYMENTS. 1108. Merchants’ Rule, Brooktyy, N.Y., June 19, 1894. On demand, I promise to pay William R. Budd, or order, Two Thousand Four Hundred Fifty-four 343, Dollars, value received, with interest at 6 per cent. $2,454,705. ARTHUR TOWNSEND. The following payments are endorsed on the note: July 5, 1894, $200. July 29, 1894, $450. Sept. 18, 1894, $700. Oct. 25, 1894, $300. Find the amount due Jan. 2, 1895. PARTIAL PAYMENTS. 497 If no payments had been made, there would be due $ 2,454.75 And interest from June 19 to Jan. 2, 197 days, 80.60~ Total due, $ 2,535.35 The credits are: Payment July 5, 1894, 200.00 Interest on $200, July 5 to Jan. 2, 181 days, 6.03 Payment. July 29, 1894, 450.00 Interest on $450, July 29 to Jan. 2, 157 days, 178 Payment Sept. 18, 1894, 700.00 Interest on $700, Sept. 18 to Jan. 2, 106 days, 12.37 Payment Oct. 25, 1894, 300.00 Interest on $300, Oct. 25 to Jan. 2, 69 days, 3.45 Balance due, $851.72 1109. By the merchants’ rule, interest is calculated on the face of an interest-bearing note from its date until settlement, and interest is allowed on all credits from their payment until settlement. 1110. Slate Exercises. 1. A note for $500, with interest at 6%, is dated July 25, 1893. Payments are made: $100, Sept. 18; $200, Feb. 5, 1894. How much is due April 1, 1894? 2. Find amount due Sept. 15, 1894, on a demand note for $1,875, with interest at 6%, dated Jan. 18, 1894. Payments of $1,000 and $500 were made March 30 and June 17, respectively. 3. June 12, 1892, Robert Colgate bought goods amounting to $600. Dec. 31, 1892, he paid $300; April 5, 1893, $200; June 1, 1893, he settled the account. How much did he pay on that date, if he is charged 6% on the purchase from its date, and is allowed 6% interest on his payments? 4. T. J. Minturn loaned Chas. A. Dorsey $500, Sept. 1, at 6%. Payments of $200 each were made Oct. 1 and Nov. 1. How much is due Dec. 1? 498 ARITHMETIC, Dr. Witson T. Jones. Cr. 1893. 1893. Feb.| 5/To merchandise, | 840]00]Mar.| 9] By cash, 500 | 00 Dec. | 31 | To interest to date, Sept. | 13 | By cash, 200 | 00 SEY Bye cashe Dec. | 31] By interest to date, 5. Find the amount paid in settlement of the foregoing account, Dec. 31, 1893. Interest 6%. 6. A merchant’s books show the following debits: Feb. 13, merchandise, $725.00; April 14, merchandise, $603.00. The credits are: April 5, cash, $600; Aug. 29, cash, $300. How much is due Oct. 5, interest 6%? L111. The merchants’ rule is frequently used where the transactions all take place within a year. The exact number of days is taken, and the interest is calculated on the basis of 360 days to the year. CHAPTER XIV. EQUATION OF PAYMENTS. — MENSURATION OF SURFACES AND VOLUMES. — BOARD MEASURE. — ANNUAL INTER- EST. —GOVERNMENT LANDS. — METRIC SYSTEM, EQUATION OF PAYMENTS. 1114. Oral Problems. 1. A friend loans me $800 for 6 months without interest. How long ought I to loan him $400 to cancel the obligation? 2. In what time would the interest on $450 be the same as the interest for 8 months on $600? 3. W borrows from X $200 for 5 months and $400 for 2 months. How much money should W loan X for one month in return for the accommodation ? 4. A man offers a lot for $600, payable $300 in 2 months, and $300 in 4 months. How much credit should be given toa buyer who wishes to pay the $600 at one time? 5. Mr. Jones has bought $600 worth of goods on 6 months’ credit, and $300 worth on 8 months’ credit. For what time should he give a note (without days of grace) for the whole amount, $900? 1115. Slate Problems, 1. In what time would the interest on $1,000 be the same as the total interest on the following amounts: $100 for 1 month, $200 for 2 months, $300 for 3 months, $400 for 4 months? 499 500 ARITHMETIC. Interest on $100 for 1 month = Interest on $100 for 1 month 4 “200 “ 2 months = We + SOW a ae “ “ 300 “c 3 a3 — « 4c 900 “ce il oc « « 400 « 4 a aa “cc « 1600 “ 1 “cc ec 46 1000 “cc 2 66 = (73 “e 3000 cc i “cc 2. A person owes $400 payable in 4 months, and $500 pay- able in 18 months. What would be the average time for the payment of the whole indebtedness of $900? The debtor is entitled to the use of $400 for 4 months, which is equal to the use of $1,600 for 1 month. He is also entitled to the use of $500 for 13 months, which is the same as $6,500 for 1 month. He is entitled, in all, to the use of $8,100 for 1 month, which is equal to the use of $900 for how many months? L116. By equation of payments is meant a method of ascer- taining at what time several debts due at different times may be settled by a single payment. The time thus found is called the average time, or the equated time. 3. Find the average, or equated, time for the payment of the following : $600 due in 2 years $500 due in 14 years $300 due inl year $400 due in 9 months 4, $250 due in8 months $450 due in 6 months $500 due in 8 months $600 payable in cash 250 x 8 = 450 x 6 = 500 x 3 = 600 x 0=0 1800 x ? = EQUATION OF PAYMENTS. 501 5. $ 200 due in 15 days $300 due in 30 days $A00 due in 45 days 6. $840 to be paid in four equal installments in 1, 2, 3, and 4 months, respectively. 7. $960 to be paid + in 2 months, 4 in 4 months, 4 in 5 months, and the remainder in 6 months. 8. A debt to be paid ;4; in 2 months, + in 8 months, 4 in 4 months, and the balance in 12 months. 9. $6,000; 4 to be paid in cash, + of the remainder in 3 months, another fourth in 6 months, and the balance in 9 months. 10. On what date should the following account be paid in full? Bought, July 1, goods to the amount of $300 payable in cash, to the amount of $800 payable in 30 days, and to the amount of $1,000 payable in 60 days. 1117. Miscellaneous. 11. A farmer sold 300 bu. wheat at 921 per bushel, 100 bu. at 90, 400 bu. at 95, 200 bu. at $1. What was the average price ? 12. Three men hire a pasture for $84. One puts in 15 cows for 12 weeks, the second puts in 20 cows for 6 weeks, the third puts in 18 cows for 10 weeks. What amount should each pay? 13. A and B form a partnership. A furnishes $2,000, B $3,000. After a year A furnishes an additional $1,000. At the end of 2 years the business is disposed of for $7,100. How much should each receive? SuecEstion: A receives his $3,000 and how much of the profits? Should he receive as much as B, who had $3,000 in the business the whole time? 502 ARITHMETIC, 14. How many bushels of bran worth 40 cents per bushel should be mixed with bran worth 30 cents per bushel to make 100 bushels worth 36 cents a bushel ? x = number of bushels at 40 ¢ 100 — x = number of bushels at 309 40 « = value (in cents) of one kind 30(100 — x) = value of other kind Total value = how many cents? 15. How many bushels of corn worth 60% per bushel should be mixed with 80 bushels of corn worth 50 per bushel to make a mixture worth 52¢ per bushel? 16. A can doa piece of work in 20 days, B can do it in 80 days. They work together and receive $5 per day as the wages of both. What should be A’s share of the total amount received? How long does it take both together to do the work? What would A receive per day if he did the work alone? 17. A partnership is formed between A with a capital of $1,500 and B with a capital of $2,500. Six months thereafter, they take in C with a capital of $4,000. How should a profit of $3,500 be divided at the end of the year? 18. Three merchants shipped a cargo of iron by sea. A sent 180 tons, B sent 105 tons, C sent 315 tons. During a storm the sailors were obliged to throw overboard 180 tons to save the vessel. What portion of the loss should each merchant sustain ? 19. If pure milk is reduced in value from 24 per gallon to 20¢ per gallon by the addition of water, how many quarts of water have been placed in a can that contains 40 quarts of the adulterated article ? 20. Find the entire surface of a cube whose edge measures 7 inches. 21. What is the edge of a cube whose entire surface contains 726 square inches? SURFACES. 503 MENSURATION OF PLANE SURFACES. 1124. Slate Exercises. 1. Find the circumference of a circle whose radius is 2. (Diameter x 3.1416.) 2. Find the area of a circle whose radius is 2. (4 circumference x 4 diameter.) 3. Find the area of a circle whose diameter is 2. 4. Find the area of a circle whose circumference is x. 5. What is the area of a circle whose diameter is 86 feet ? 6. What is the radius of a circle whose area is 158.9384 sq. yd.? 7. What-is the circumference of a circle whose area is 198.95 sq. rods? 8. Find the area of a square whose diagonal is z. 9. Find the area of a square whose diagonal is 150 rods. 10. Find the area of an isosceles triangle, its base being 56 meters, equal sides 100 meters. 11. Find the area of an equilateral triangle whose side is 12 feet. 12. Find the area of a triangle whose sides are 50 yd., 60 yd., 70 yd. 13. What is the area of a circle whose cir- 40 rd. cumference is 10 feet? (The square of the circumference x what = area?) 14. Find the area of the rhomboid, Fig. 1. Fig. 1. 504 ARITHMETIC. 15. Of the rectangle, Fig. 2. 16. Of the rhombus, Fig. 3. 17. Of the trapezoid, Fig. 4. 25 rd. Fie. 2. | Fie. 3. Fia. 4. 18. Of the trapezium, Fig. 5. 19. Of the rhombus, Fig. 6. 20. Find the altitude, AB, of the following triangle, Fig. 7: (First find the area.) 30 yd. A xy < 4 og 2 x, |: § % : 4 5a 24 rd. 7 ft. Fia. 5, Fie. 6. Viex7; 21. Find the diagonal (in rods) of the square whose area is 5 acres. 22. Find the area of a hexagon, composed of six equilateral triangles, each side being 6 inches. Fig. 8. Fig. 9. Fia. 10. 23. What is the area of the circle circumscribed about the above hexagon, Fig. 9? 24. What is the area of the square inscribed in a circle whose diameter is 10 feet, Fig. 10? REVIEW. SPECIAL DRILLS. 1129. Give sums: 112+91+485 129+ 62+ 98 182+ 138+67 114+ 21+49 43+131+61 26+172+81 75+ 193+ 23 1382+494-+77 1130. Give answers: 150 —23-+-48 154— 36+ 44 155—49-+ 24 1538 —45+31 172+19—66 184+ 39—385 181+47—33 151+ 46— 24 1131. Give products: 44 x 20 44 x 22 44x18 63 x 28 63 x 82 54 x 38 54 x 42 88 x 48 1132. Give quotients: 676+ 13 602 + 14 690+ 15 672 +16 527 +17 738 + 18 950 + 19 924 + 21 1133. Give results: 84 x 13 48 x 27 36 x 94 48 x 192 2111 13 2142 + 14 1304 + 21 1554 + 22 95+ 144+ 79 63-+117+97 91+126+4382 63+ 143+ 24 183 —(72—87) LTS (5 nes) 165—(47 —28) 182—(48—38) 83 X 52 26 x 58 26 x 62 Lax OS 704 + 22 966 — 23 768 + 24 975 + 25 36 x 498 32 X 59f 49 x 49 _ 58 X 58 505 68+-56-+174 63+ 34+ 186 91+59+165 78+39-+183 161-4+79—12 174+41—36 E7i-f ooh ode 175-433 —46 Uti bP TL Gio TL C82 45 x 88 887 + 27 961 + 31 992 + 32 759 + 33 1624 + 25 1587 + 31 182 + 32 173 + 5 506 ARITHMETIC. . 1134. Oral Problems. 1. A has 96 sheep; B has 28 sheep more than A. How many sheep have both? 2. There are 56 pupils in one class, 48 in a second class, and 52 in a third class.) How many pupils are there in the three classes ? 3. March 29 is what day of the year 1894? 4. How far is a man from his starting-point, if he travels due east 150 miles, due west 23 miles, due east again 48 miles? 5. A body falls 16 ft. in the first second, three times as far in the second second, five times as far in the third second. How far does it fall in three seconds ? 6. The base of a right-angled triangle is 12 ft., the perpen- dicular is 16 ft. What is the hypotenuse? 7. At $35 per month, what will be the rent of a house for 16 months? 8. A field containing 169 square rods is 18 rods long. How many rods of fence will be needed to enclose it? 9. 25 packages of sugar weigh together 874 lb. How many pounds are there in each? 10. At 45 miles per hour, how many hours, minutes, and seconds will it take a train to go 230 miles? 11. How many years have elapsed since the invention of gun- powder, 1356? 12. What profit is made on an article bought for $175, less 12%, and sold for $200? 13. How many square rods in a field 71 rods long, 81 rods wide? 14. Assuming a kilo to be 24.1b., how many kilos will be equal to 143 lb.? MEASUREMENTS. 507 15. A degree of longitude in latitude 45° is about 70% of the length of a degree on the equator. Calling the latter length 69 miles, how long is a degree of longitude in latitude 45°? 16. At $44 per acre, how much land can be bought for $ 968? 17. A number of marbles divided among 29 boys gives each 16 marbles, and leaves a remainder of 26. How many marbles are there? 18. What is the monthly salary of a clerk who receives $1,500 per year? 19. How many revolutions in a mile, 5,280 ft., are made by a locomotive wheel 16 ft. in circumference? 20. How many feet of fence are there around a lot 49 ft. wide, 87 ft. long? 21. How many bricks 8 in. by 4 in. by 2 in. would make a cubic foot ? 22. 13 is one factor of 1,001. Find the other two prime factors. 23. What are the three equal factors of 348? 24. What is the square root of 1,225? 25. At 44 miles per hour, how long will it take a man to walk 374 miles? 26. What will be the cost of 9 dozen hats at $1.331 each? MEASUREMENTS. 1137. Find the area of each of the following triangles and its altitude. When the area of a triangle is known and the length of the base, how can its altitude be calculated ? 508 ARITHMETIC. Base, 51 ft.; other sides, 20 ft. and 37 ft. Base, 21 yd.; other sides, 13 yd. and 20 yd. Base, 148 rods; other sides, 39 rods and 118 rods. Base, 28 chains; other sides, 17 chains and 25 chains. Base, 75 inches; other sides, 20 inches and 65 inches. Ce So 5. = Ne 1138. Find the areas of the following quadrilaterals: 6. Given AB, 17; BC, 10; CD, 20; DA, 18. AC= 21. 7. Given 4B, 25; BC 89;.CD, 34) DA, 50s ACG: 8. Given AB) 37; "BC, 1b CD ao DA Lia aaa 9. Given 48,111 ;) BC 45.°CD) 2b) (AT GA Clee 10: Given’ A 5,113; BON CD60) DAP TI aC, B 1139. Slate Problems. 1. A and B rented a field for a year for $175. A put in 6 horses for the whole time, B put in 5 horses for 11 months and 3 horses for 5 months. How much of the rent had each to pay ? 2. A bankrupt surrenders property worth $1,287 for the benefit of three creditors to whom he owes $750, $1,125, and 1,245, respectively. How much should each creditor receive? 3. Four persons rented a pasture for 26 weeks. K put in 50 sheep and L 60 sheep for the whole time, M put in 70 sheep for 20 weeks, and N 90 sheep for 22 weeks. How much of the rent, $180, had each to pay? REVIEW. 509 4. A employs a capital of $2,500 in business, and at the end of 3 years takes into partnership B, who furnishes $4,000. Four years later they are joined by C, with a capital of $5,000. At the end of 12 years from the commencement of the business, the profits, amounting to $15,000, are divided. What amount should each receive ? A’s money is in the business how many years? B's, how many years? o's, how many? 5. Four butchers rent a field, and pay for 6 months’ rent $152.50. The first puts in 20 oxen for 10 weeks and 50 sheep for 8 weeks; the second, 25 oxen for 8 weeks and 30 sheep for 7 weeks; the third, 18 oxen for 10 weeks and 10 sheep for 12 weeks; the fourth, 30 oxen for 12 weeks. What share will each have to pay, counting 3 sheep equal to 1 ox? 6. A wall 700 yards long was to be built in 29 days. At the end of 11 days, 18 men had built 220 yards of it. How many extra men had then to be put to work, so that the wall might be completed in the given time? 7. If 5 needlewomen can do a piece of work in 11 days of 9 hours each, how long will it take 3 needlewomen to do two such pieces, supposing them to work 103 hours each day? 8. If 14 men can mow 168 acres in 12 days of 8 hours 15 minutes each, how many acres can 20 men mow in 11 days of 7 hours 48 minutes each ? 9. If 12 men can do a piece of work in 20 days, what num- ber of men will be required to do four times as much work in a fifth part of the time? 10. A ship sailed with a crew of 60 men, and provisions for 34 days, and 10 days afterwards, 12 persons were received on board from a sinking vessel. How long would the provisions last the 72 persons then on board? __ How long would the provisions last the 60 persons at the time the sink- ing vessel was met? 510 ARITHMETIC. 11. If 76 boards, each 14 feet long and 10 inches wide, are worth $19.76, how much would 50 such boards be worth ? 12. If 7 men receive $126 for 5 weeks’ work, how much should they receive for 9 weeks’ work? 13. If for 7s. 6d. I can buy 9 Ib. of raisins, how many pounds ean I buy for £56 16s. ? 14. A field of grain was to be cut down by 40 men in 10 days. Hight of the men, however, failed to come. take the others to do the work ? TABLE. Assessments. Real and Personal, For State Purposes. Tax Levies. For County Purposes. For City Purposes. How long did it 1140. Brooklyn Assessments and Taxes for 10 years. of Valuation. rn fm nf | a | ES | ¥ 298,936,506 |$ 874,088 1883 1884 1885 1886 1887 1888 317,853,850 330,683,762 362,009,202 383,851,674 407,454,028 428, 483,681 452,758,601 466,914,249 483,738,129 733,669 889,559 929,273 907,663 940,517 1,344,023 949,253 589,178 888,297 1,323,861 1,307,090 1,412,623 1,398,310 1,682,120 1,997,414 2,009,518 2,159,879 2,240,613 1,242,476 |$ 5,632,795 6,287,462 7,383,911 7,180,990 8,266,643 8,503,581 9,298,236 8,709,541 9 241,130 10,324,617 es ee eee ee eee ee Find for each year the total tax levy, and the tax rate in dol- lars, cents, and mills per $1,000 of assessed value. Find the average assessment per year; the average tax levy for state, county, and city purposes; and the average tax rate. SURFACES OF SOLIDS. SURFACES OF PRISMS AND CYLINDERS. 1141. Slate Exercises, Nors.— The pupils should be encouraged to make cardboard models of the forms studied. 1. Find the convex surface of a square prism, one side of its base being 4 inches and its height 6 inches. Draw the development. Notr. — The convex surface is the surface exclusive of the bases. 2. Find the convex surface of a triangular prism, each side of whose base measures 4 inches and whose altitude is 6 inches. Draw the devel- opment. 3. Find the convex surface of an hexagonal prism, each side of its base being 4 inches and its altitude 6 inches. Draw the development. 4. Can you show that the convex surface of a prism is. found by multiplying the perimeter of the base by its altitude (height) ? 5. Find the convex surface of a cylinder, the diameter of its base being 4 inches and its height 6 inches. 6. How do you find the entire surface of a prism or cylinder? 7. What is the entire surface of a cube whose side is 7 inches? Of a cube whose side is x inches? 8. The entire surface of a cube is 216 sq. in. What is the length of one side ? 9. The convex surface of a cube is 144 sq. in. entire surface. ‘sy lb ARITHMETIC, 10. Find the entire surface of a square prism, one side of whose base measures 4 inches, and whose altitude is 6 inches. 11. The convex surface of a square prism is 600 sq. ft., the altitude is 15 ft. What is the length of one side of the base? 12. The entire surface of a square prism is 1,650 sq. in. One side of the base measures 15 inches. What is its convex surface ? What is its altitude? 13. Find the entire surface of a square prism whose convex surface 1s 040 sq. in., and whose altitude is 15 inches. 14. What is the entire surface of a cylinder whose base has a diameter of 1 foot, and whose altitude is 1 foot? SURFACES OF PYRAMIDS AND CONES. 15. The convex surface of a square pyramid consists of how many equal triangles? Find the convex surface when one side of its base meas- ures 4 inches and its slant height (AX) 6 inches. Draw the development. 16. The convex surface of a pyramid is equal to the perimeter of the base multiphed by what? 17. Find the entire surface of the above pyramid. 18. Calculate the entire surface of a square pyramid. whose slant height is 18 inches, the area of its base being 144 sq. in. 19. Find the entire surface of a triangular pyramid whose three convex faces and the base are equilateral triangles, each side measuring 2 inches. 20. Draw the developed convex surface of a cone, the diameter of whose base is 4 inches, and whose slant height is 6 inches. Calculate the convex surface. eX i 21. How many square inches of paper would be required to cover the side and the base of a cone 6 inches in diameter at the base, and having a slant height of 10 inches? VOLUMES. 513 22. Calculate the slant height of a cone whose altitude is 12 inches, the diameter of its base being 10 inches. What is its convex surface? 23. What is the entire surface of a cone, the diameter of whose base is 6 inches, and its slant height 10 inches? Draw the development. 6 in. 24. A semi-circular piece 4 Ne of paper 6 inches in diam- eter is folded into a hollow cone (without overlapping). What will be the diameter AB of the mouth of the cone (the base)? What will be the slant height BC? C VOLUMES OF PRISMS AND PYRAMIDS. OF CYLINDERS AND OONES. 1145. Slate Exercises, SuaGEstion. — Have the pupils construct of cardboard a hollow square prism of convenient size, and a pyramid having base and altitude respectively equal to those of the prism. Let them use sand or water to ascertain how many times the contents of the pyramid must be taken to exactly fill the prism. Volume of prism or cylinder = area of base x altitude. Volume of pyramid or cone = area of base x + altitude. 1. Find the volume of a square pyramid, the area of the base being 9 square feet and the altitude 6 feet. 2. What is the volume of a square pyramid whose altitude is 12 inches, one side of the base being 10 inches? 3. The base of a prism is a triangle whose sides measure 3, 4, and 5 inches respectively. Find the solidity, its altitude being 10 inches. : 4. The base of a prism 19 feet high is a rectangle whose sides are 9 feet and 13 feet. How many cubic yards does it contain ? 514 ARITHMETIC. 5. Find the volume of a prism whose bases are equilateral triangles, each side being 4 ft., and the height of the prism being 12 ft. 6. How many cubic feet are there in a stone roller 6 ft. long, 8 ft. in circum- ference ? 7. Find the volume of a cone whose altitude is 18 meters, diameter of base 6 meters. 8. How many gallons of oil (231 cu. in.) will fill a cylindrical tank 54 ft. high, radius of base 8 ft.? 9. Measure accurately the interior dimensions of a quart or a pint cup, and calculate its volume. Note. — How many cubic inches in a quart, liquid measure? 10. Measure the interior dimensions of a peck or a bushel, and calculate its volume. 11. Pour a quart or a pint of water into a paper box having a rectangular base, and calculate the number of cubic inches of water in the box. What would be the depth of a quart of water in a box whose base measures 51 by 3 inches? LUMBER MEASURE. 1147. Lumber is measured in board feet. A board foot is 1 foot long, 1 foot wide, 1 inch thick. A board 16 feet long, 1 foot wide, 1 inch thick, contains 16 board feet. A board 16 feet long, 9 inches wide, 1 inch thick, contains (16 x 3) board feet, or 12 board feet. A board of the same length and width, 2 inches thick, contains (12 x 2) board feet, or 24 board feet. In practice, the term board foot is seldom used, the word foot alone being generally employed. LUMBER MEASURE. 515 1148. Find the number of feet (board feet) in each of the following boards and planks: 16 feet long, 12 inches wide, 1 inch thick. ee = Se ee PF Ww NY HF OS 15. CO Xe TP ww o 14 12 14 16 12 14 16 14 Pae, 12 ce ce ce 6 iss ce ce 1 ep ~The) Seep (by ese [Ney Se) Sep ss) sy SS 16. What is the cost, at $30 per thousand feet, of 15 planks, each 16 feet long, 9 inches wide, 3 inches thick ? 17. Find the number of (board) feet of lumber required to floor a dock 36 feet long, 17 feet 6 inches wide, the planks being 24 inches thick. 18. Find the duty, at $1 per thousand feet, on the following lumber imported from Canada: 13 feet long, 8 inches wide, 1 inch thick; 150 boards, 60 planks, 40 scantlings, 15 feet long, 5 inches wide, 4 inches thick. 14 feet long, 9 inches wide, 2 inches thick; 516 , ARITHMETIC. 19. At $18 per thousand, what will be the cost of the boards necessary to enclose a field 160 yards long, 120 yards wide, with an open fence 4 boards high, each board 6 inches wide, and 1 inch thick ? MENSURATION. 1150. Slate Problems. Area of circle = 4 circumference x 4 diameter. Area of sector = 4 arc x 4 diameter. 1. Find the area of asemicircle whose radius = __-------. ‘ is 20 feet. ne i i 2. How many square inches are contained in Le a sector of 60°, the radius of the circle being 15 \ ¥ A inches? ao 6Q ° 3. A square is inscribed in a circle 10 inches in diameter. Find its area. x = side of square, x?= area. Find 2? from the right-angled triangle, without finding the value pf te. 4. What is the difference between the area of a circle of 10 inches diameter and that of the inscribed square ? 5. The sides of the above inscribed square are chords of arcs of 90°. Find the length of an arc of 90°, and of its chord. 6. A segment of a circle is that portion of the J surface included between an arc and itschord. Find © ti the area of a sector of 90° and the area of the seg- ment, the radius of the circle being 10 inches. 10 7. Calculate the area of a circle whose radius is 1 inch. Of a circle whose radius is 2 inches. What is the ratio of the two areas ? 8. What is the ratio between the area of a circle whose radius is 1 inch and that of a circle whose radius is 3 inches? The area of a circle = square of radius x ? MENSURATION. 517 9. How many square yards are there in a circular walk, the radius, AJB, of the inner edge of walk being 10 feet, and that of the outer edge, AC, being 15 feet ? 54) (Find the difference between the area of a circle of 15 ft. radius, and that of a circle of 10 ft. radius.) C 10. A circular flower-bed 20 feet in diameter is surrounded by a walk 5 feet wide. How many square feet of surface does the walk contain ? (If you have to subtract 100 times 3.1416 from 225 times 3.1416, how can you shorten the work ?) 11. How many square inches are there in the surface of a frame 3 inches wide, around a looking-glass 6 inches in diameter? (Area = ? x 3.1416.) 12: What is the ratio between the surface of the above frame and that of the looking- glass ? (Indicate operations and cancel.) 13. What is the area of a walk 5 feet wide around the out- side of a square plot containing 400 sq. ft.? (What is the area of the large square, including the walk ?) . 14. The outer edge of a walk 5 feet wide, surrounding a plot of ground, measures 120 feet, the inner edge measures 80 feet. How g many square feet does the walk contain ? 120 + 80 2 D (The “average” length of the walk is = 100 ft.; that is, its length measured on a line along the center of the walk.) 15. Find the ratio between the area of a triangle whose sides measure 16, 30, and 384 feet, respectively, and the area of another whose sides are 32, 60, and 68 feet. 51S ARITHMETIC. SURFACE OF SPHERE. 1151. Take a wooden hemisphere and drive a tack into the center of its curved surface. Commencing at the tack, carefully wind a waxed cord about the curved surface, in the way a boy winds a top. When this surface is exactly covered, cut the cord. 4 . Ps ea aK Wind the same cord around a tack driven into the plane sur- face of the base of the hemisphere, pressing it closely to the sur- face. When the latter is entirely covered, just one-half of the cord will be used. If a sphere is cut through in any direction, the section made will be a circle. The section formed when the sphere is cut through the center is called a great circle. The above experiment shows that the surface of the hemi- sphere is equal to that of two great circles of the same sphere. 1152. The surface of a sphere is equal to that of four great circles. Since the surface of a great circle of the sphere is 4 diameter < 1 circumference, the surface of the sphere is 4 diameter x 4 circumference < 4 = diameter of sphere X the circumference. Calling the radius of a circle #, and using the Greek letter aw instead of 3.1416, we have Diameter of circle = 2 R. Circumference of circle = 27f. Area of circle = 7R. (Lof2R x tof 27h.) Surface of sphere = 47’. CUBE ROOT. 519 1153. Slate Exercises. 16. ‘Find the surface of a sphere whose radius is 1 inch. Of a sphere whose diameter is 2 inches. Of a sphere whose circumference is 6.2832 inches. 17. At 10 cents a square foot, what will be the cost of gilding a sphere 12 inches in diameter ? 18. Find the ratio between the surface of a sphere 1 foot in diameter, and the convex surface of a cylinder 1 foot high, the diameter of the base 1 foot. 19. What is the ratio between the surface of the above sphere and the entire surface of the cylinder? 20. Find the surface of a sphere whose circumference is 20 inches. CUBE ROOT. 1155. To cube a number is to employ it three times as a factor. The cube of 4, written 4°, is 4 x 4 x 4, or 64. Find the cube of 1, 9, 6, 3, 5, 8, 2, 7. To find the cube root of a number is to find one of the three equal factors of the number. wee The cube root of 343, written 348, is 7. The cube of 25, 20 + 5, is equal to the following: We have seen (Art. 1031) that (20 + 5)? = 207+ 2 xk 20 x54 5 Multiplying by 20+ 5 we have Product by 20 = 20? + 2x 20?x 5+ 20 x 5? Product by 5= 20? 5+ 2x 20x 5? 4 58 (20 + 5)8 = 20°? + 3 x 20?x5+ 3x 20x 5? + 58 which may be written in this way, 20 + [(3 x 20) + (3 x 20 x 5) + 57] x 5. a 520 ARITHMETIC, 1156. Extract the cube root of 15,625. We see by inspection that the cube root is between 20 and 30; thatis, 20+. Sub- tract from 15,625 the cube of 20, 8,000. The remainder, 7,625, is equal to the second number multiplied by the sum of three times the square of the first (1,200), ete. Using (20)' = 20 +5 3 X 20? = 1,200 &8X20x5 = 300 5S= 25 1,525 15,625 8,000 7,625 7,625 remainder 1,200 as a trial divisor, the second number is seen to be 6 or less. Taking 5 as the second number, we add to the 1,200 three times the product of the first and second (300), and the square of the second (25), making a total of 1,525. Multiplying this sum by the second number, we get 7,625, which is equal to the difference between 15,625 and 8,000. The second number is, therefore, 5, and the cube root of 15,625 is 25. 110,592 40+8 110,592 408 = 64,000 3 x 40? = 4,800 46,592 3x40 x8 = 960 82 — 64 5,824 46,592 Ans. 48. 658,503 3 x 80? 83 = 3x80 xX7 = Tie 19 200 1,680 49 20,929 Ans. 87. 8 7 658/503 512 146,503 146,503 In the last example we point off three places, beginning at the right, and find the greatest cube in the first period, placing its cube root as the first figure of the answer. 1157. Find the cube root of the following: 1. 2197 2. 9,261 3. 32,768 4. 68,921 5. 148,877 6. ib 8. 9. § 10. 238,328 421,875 551,368 512 29 1331 27 11. BYP, 12. 8,375 13; 1244 14. 188% 15. 5454 MENSURATION. VAT VOLUME OF SPHERE. 1158. Cut up a sphere (a round potato, for instance) into a number of ° small pieces, passing the knife in each case through the center of the sphere. Hach piece is a solid, having for its base a portion of the surface of the sphere, and for its altitude the radius of the sphere. When the pieces become very numerous, the base of each may be con- sidered a plane, and the solid a pyramid. The volume of each pyramid is equal to the base x} altitude; and the total volume of all, which is the volume of the sphere, is equal to the total surface of all the bases, which is the surface of the sphere, multiplied by 4 altitude, that is, + radius. Surface of sphere = 47 f’, therefore, volume of sphere =47f’? XL R=47rh’. 1159. Slate Exercises. 1. Find the volume of a sphere whose radius is 3 inches. 2. Ifthe diameter of a sphere is 3 inches, what is its volume? 3. What is the ratio between the volumes of two spheres whose diameters are 1 foot and 2 feet, respectively ? 4. Find the ratio between the volume of a sphere 1 foot in diameter, and that of a cube whose side is 1 foot. 522 ARITHMETIC. 5. The radius of a sphere is 18 inches. What is the circum- ference of a great circle? The surface? The volume? 6. What is the weight of an iron cannon-ball 12 inches in diameter, considering the weight of a cubic foot of water as 1,000 ounces, and considering iron 7.5 times as heavy as water ? 7. Find the ratio between the volume of a sphere 4 inches in diameter, and that of a cylinder 4 inches in altitude, radius of base 4 inches. Nors. — Indicate the volume of each, and cancel. s. A man has a cubical block of hard wood, its side measur- ing one foot, which he wishes made into a sphere one foot in diameter. What decimal part of the block is cut away ? The volume of the sphere is about what fraction of the volume of the cube? CUBE ROOT. 1162. Find the cube root of 9,938,375. When the root contains 3 Mes i bo Sil more than two figures, con- 9'938!375 tinue, as shown in the accom- panying example, taking for Si divisor three times the square 8X 20?= 1200 1988 of the first two figures con. OX 20X1 = 60 sidered as tens, plus three times 1+—= Lila the product of the first two 3 x 210? = 182300 6773875 figures considered as tens by By BIO ae B50 the third figure, plus the square Be OB of the third figure. i = 135 470 677 375 1163. Find the value of the following: 1. 1,442,897 3. 3,723,875 5. V12.977875 2. 1,906,624 4. »/39,651,821 6. 66.923416 ANNUAL INTEREST. 523 ANNUAL INTEREST. 1171. Slate Problems. Derroit, Micu., June 1, 1890. Four years after date, without days of grace, I promise to pay to the order of Daniel W. Lawler, Six Hundred Dollars, value received, with annual interest at six per cent. $ 600%. GEORGE OXNARD. 1. Find the amount due June 1, 1894, no payments of prin- cipal or interest having been made. 1172. When the maker of a note fails to keep his contract to pay interest annually, the laws of some states, including Michigan, permit the collection of swmple interest on the deferred payments of interest. Principal, $600.00 Interest, 4 years, at 6%, 144.00 3 years’ interest, at 6%, on the Ist year’s interest, $36, 6.48 PV te: ‘ Me oe) Boe te ap OCLs - ive a PME tet POR aan, ;: : Amount due June 1, 1894, $ 2. Find the amount due, at 5%, for 5 years, on a note for $1,200, annual interest being unpaid. 3. What is the amount of a note for $720, at 4 years, at 4$%, annual interest unpaid after the first year? 4. The maker of a note for $900, with annual interest at 7%, makes the first and the second interest payments when due. How much will he owe at settlement, 6 years after the date of the note ? 5. Find the difference between the amount due at 6% for 3 years on a note for $300, annual interest unpaid, and the amount of the same sum placed at compound interest for the same time at the same rate. 6. Find the amount due March 1, 1899, on a note for $500, dated March 1, 1893, with interest at 6%, annual interest unpaid after the third year. 524 ARITHMETIC. U.S. GOVERNMENT LANDS. 1173. In surveying government lands, a line is run east and west, called the base line, and one perpendicular to it, called the principal meridian. Parallel lines are run north and south, and east and west, 6 miles apart, forming squares, called townships. The row of townships adjoining the prin- cipal meridian is called Range 1 East or West, according to its location. The row of townships north of the base line is called Township 1, North; the row above, Township 2, North, etc. The township in the diagram marked by a star (*) is designated 3 T. 8, R. 2 HE. (third township south of base line, in the second range east of the Tadentaele pore] | “es | | {Teen [ote LSAM er y ADNVY TOWN HIP|4 | a NORTH ie 7.0 ag Wis w fa) Be 9 aoa heats ioe ee] Price ae] we p> A Wee | ara a ion at eae QO; | O9}/9192!10 m—-m—-m m——-m—-M——m o};n] — YNio;Fia BASE Pad LINE principal meridian). 1174. A township, which contains 36 square miles, is divided into sec- tions one mile square, numbered as in the diagram, No. 1 being found at the northeast corner. TOWNSHIP 2020 |2e|araofes ENCES EEE SIX MILES tS) Each section contains 640 acres. SECTION N ONE MILE = ONE MILE ONE MILE S 1175. Sections are divided into half-sections (320 A.) and quarter-sec- tions (160 A.), and the latter are subdivided into half quarter-sections (80 A.) and quarter quarter-sections (40 A.), METRIC SYSTEM. 525 1176. Slate Problems, 1. Find the cost of the S.W. 4 of the N. 4 of sec. 13, T. 7 N., R. 4 E., at $1.872 per acre. 2. What will be the cost of fencing, at 75% per rod, the W. 4 of the N.W. 4 of sec. 36? 3. Mr. Thompson owns sec. 1, and his brother owns sec. 30 of the same township. What is the length of the shortest lne between the boundaries of the two farms? 4. A road runs east and west between townships 4 and 5, south. Another road runs north and south between R. 7 and 8 east. How far is it by road from the north-east corner of T.55., R. 10 W., to the north-west corner of T. 7 N., R. 8 E.? 5. How many feet of boards, 6 inches wide, would be needed to build an open fence, 4 boards high, around the N. 4 of the S.W. 4 of sec. 16? 6. The owner of secs. 19 and 20 has sold the W. 4 of N.W. 4 of sec. 19; also the N. 4, the N. 4 of S8.H. 4, and the S.E. 4 of the 8.E. 1 of sec. 20. Draw a map of the land he still owns, and calculate its area. METRIC SYSTEM. 1177. The metric system, which is used in nearly all the countries of continental Europe, is based upon the meter. The length of the meter is one ten-millionth part of the length of the meridian from the equator to the poles — about 39.37 inches. 1178. The subdivisions of the meter are denoted by the Latin prefixes milli (z54,), centi (;1,), deci (7). For the multiples, the Greek prefixes deka (10), hecto (100), kilo (1,000), and myria (10,000) are used. 1179. It will be nuwticed, in the table below, that small letters are used for the abbreviations of the Latin prefixes of the 526 ARITHMETIC. subdivisions, and capital letters for the Greek prefixes of the multiples. The following is the table of 1180. Measures of Length. 10 millimeters (mm.) 1 centimeter (cm.) 10 centimeters 1 decimeter (dm.) 10 decimeters 1 meter (m.) 10 meters 1 dekameter (Dm.) 10 dekameters 1 hectometer (Hm.) 10 hectometers 1 kilometer (Km.) 10 kilometers 1 myriameter (Mm.) 1181. The units of this table in common use are the centi- meter, the meter, and the kilometer. 1182. A person who wishes to buy 124 meters of cloth, would not ask for 1 hectometer 2 dekameters 4 meters, any more than a New York mer- chant would tell a person who owes him $38.75 that his bill is 3 eagles 8 dollars 7 dimes 5 cents. 1183. Long distances are expressed in kilometers. The thickness of wire is given in millimeters. 1184. Problems. 1. What will be the cost in francs of 880 m. 75 of dress goods at 2 f. 60 per meter? (880.75 meters @ 2.60 francs.) 2. How many square meters in a piece of carpet 26 m. 50 long, 85 cm. wide? 3. How many square meters in a circle whose diameter is 15 meters? 4. Anareisa surface 10 meters long, 10 meters wide. How many ares in a field 135 meters long, 69 meters wide? 5. Find the area in ares of a right-angled triangle whose base is 245 meters, hypotenuse 875 meters. METRIC SYSTEM. Lapiyy 6. A stere is a cubic meter. What will be the cost, at 8 f. 50 per stere, of a pile of wood 10 meters long, 1 meter wide, 3m. 25 high? 7. A cube one decimeter each way contains a liter (1.), which is the principal unit of dry and liquid measure. How many liters’ capacity has a tank 10 m. 50 long, 8 m. wide, 6 m. 50 high? 8. How many bottles, each containing 0 1. 75, can be filled from a hogshead containing 222 1. ? 9. How much will be received for 36 bags of beans, each con- taining 68 liters, at 1 mark 25 per dekaliter? 10. A liter of water weighs a kilogram (1,000 grams). How many kilos of oil would a tank contain, its dimensions being 5 meters X 4 meters x 3 meters, the weight of the oil being 92% of the weight of water? 11. Assuming the length of the meter as 39.37 inches, what is the length of the kilometer ? 1185. Greater accuracy is assured in operations requiring multiplication and division by indicating the operations beforehand, and performing the division last. Length of meter in yards =. 1 mile = 1,760 yd. 1 km, = 1,000 m. 39.37 x 1,000 _ _ 3,937 361,760 36x176 12. Mt. Blanc is 4800 m. high. How many feet high is it? mile. Ans. = 1186. In the following ten problems call the meter 40 inches. Give answer in two decimal places. 13. How many cubic inches in a liter? (See problem 7.) How many quarts? 14. How many bushels in a hectoliter? How many gallons? 528 ARITHMETIC. 15. How many pounds in a kilo, when a cubic foot of water weighs 1,000 oz.? (See problem 10.) 16. What would be the circumference of the earth in miles if the meter measured 40 inches? (The meter is zggq5500 Of What part of circumference ?) 17. How many square yards in a square meter? 18. How many acres in a hectare? (See problem 4.) , 19. How many rods in a hectometer ? 20. How many cubic feet in a stere? (See problem 6.) 21. How many troy grains (7,000 to ay. lb.) in a gram? (See problem 15.) 22. How many kilometers in a mile? 1187. Measures of Surface. 100 sq. mm. = 1 sq. cm. 100 sq. cm. =18q. dm. 100 sq. dm. = 1 sq. m. = 1.196 sq. yd. 1188. The square meter is the principal unit of surfaces, such as walls, ceilings, floors, etc. 100 centiares (ca.) = 1 are (a.) = 119.6 sq. yd. 100 ares = 1 hectare (Ha.) = 2.47 acres. 1189. The are is the principal unit of surface of small plots of land. The area of a farm is expressed in hectares, of a country in square kilo- meters. 1190. Measures of Volume. 1,000 cuimm?==)1 cusem: 1,000 cu. cm. =1 cu. dm. 1,000 cu. dm. =1 cu. m. = 35.316 cu. ft. L191. The principal unit is the cubic meter. METRIC SYSTEM. 529 1192. The stere (cubic meter) is used for measuring wood. 10 decisteres (dst.) = 1 stere (st.) = 35.316 cu. ft. 10 steres = 1 dekastere (Dst.) The stere is the only unit used. 1193. Dry and Liquid Measures. 10 milliliters = 1 centiliter 10 centiliters =1 deciliter Dry. Liquid. 10 deciliters =1 hter(l.) = .908 qt. = 1.057 qt. 10 liters =I1dekaliter 1.1385 pk.= 2.642 gal. 10 dekaliters =1hectoliter 2.837 bu. = 26.417 gal. 10 hectoliters = 1 kiloliter 10 kiloliters = 1 myrialiter 1194. The liter and the hectoliter are the principal units. 1195. Table of Weight. 10 milligrams (mg.) 1 centigram 10 centigrams 1 decigram 10 decigrams 1 gram (gr.) 10 grams 1 dekagram 10 dekagrams 1 hectogram 10 hectograms 1 kilogram (kilo) 2.2046 lb. 10 kilograms (Kg.) 1 myriagram 10 myriagrams 1 quintal 10 quintals 1 tonneau (ton) 1196. The kilo is the ordinary unit. Heavy articles are sold by the tonneau. CHAPTER XV. ALGEBRAIO EQUATIONS.—TWO UNKNOWN QUANTITIES, — THREE UNKNOWN QUANTITIES.—PURE QUADRATIOS, — AFFECTED QUADRATICS. | ADDITION OF ALGEBRAIC QUANTITIES. 1199. Sight Exercises. Add: 1. 2 fours 2. 6 hundredths 3. $4 4. 8¢ 5. Tx 3 fours 8 hundredths $5 5¢ Ax 4 fours 10 hundredths pT 8¢ 22 5 fours 12 hundredths $8 9¢ 5a ~? fours ~? hundredths $? o¢ 2a 6. — 2a M+ 8e@. 8) —Say 9. Jabe) TO. BA aye — 4a + 42 —4 ry 15 abe — S2yz — 6a + 52 — wy 6 abe — xyz — Ta +102 — 22y abe — ld2yz —19a +? @ hey ? abe ht Ye 1200. In the quantities 2a, 3x, 5xzy, l5abc, the numbers 2, 3, 5, 15, are called coefficients. When no coefficient is expressed, 1 is understood. Thus, abe =1labe. Where no sign is expressed, + is understood. 1201. What a person has may be represented with a plus sign (+) placed before the amount; debts may be shown by a minus sign (—) placed before the amount. 530 ALGEBRAIC EQUATIONS. 531 A has $500; B owes $300. If they unite their fortunes, what will they be worth together? 4+. $500 — $300 + $200 Both together are worth $ 200. The sum of + 500 and — 300 is + 200. 1202. If A had $3800 and B owed $500, the firm would be $200 in debt. (+ $3800) + (— $500) = — $200. 1203. Add: 1 —2a 2. Tx 8 —5ay 4. — 9Yabe &. — 24xyz —4a —4z —42y l5abe 5 xyz —6a — 2x xy 6 abe LY2 7a 5 2 2 xy — abe 15 xyz —5a 62 —? xy if ? 1204. Can you give the rule for addition where the quantities have different signs? Which sign does the sum take? 1205. Add: 6. 82+14, —7x#+9, — 23, 42—5, —2za, and 8x2+11. 82+14 —Tx+ 9 — 23 4x— 5 —22 _ 82+il 7 4a+32, —2a, —Txa—3a, —52, —9a+ze. 8. —8b+c, 4a+6), 55—9c, —8a, —2a—3b+4e. 9. 42—8, —x2+4, —12—38, Tx+16, —5x—10. 10. 42+ 23, —8x+ 241, —2x4+4+11, —x+5, 92-38. 532 ARITHMETIC. SUBTRACTION OF ALGEBRAIC QUANTITIES. 1206. Oral Problems. 1. The thermometer in the morning was 33 degrees, at noon it was 52 degrees. What was the difference in temperature? 2. In December the thermometer was 10 degrees below zero. In July it was 90 degrees above. What was the difference in temperature ? 3. Two cities are in the same latitude. One is in 34° east longitude, and the other in 17° west longitude. What is their difference in longitude? 4. What is the difference in longitude between two cities on the equator, one being in 56° west longitude, and the other in 47° west longitude ? 5. A boy makes 40¥ one day and 50f the next. How does he stand at the end of the two days? 6. How would he stand if he made 40 one day and lost 50¢ the next day? 7. A man traveled from the town M, 60 miles due north, and then traveled 50 miles due north. How far is he from his starting-point ? 8. One day a man goes 50 miles due north; the next day he travels 70 miles due south. How far is he then from his starting- point? 9. On Monday A is worth $250; on Tuesday he is worth $150. What has he lost in a day? 10. A man has $150 Jan. 1. Feb. 1 he owes $250. What has he lost in a month? 1207. The degrees above zero on a thermometer may be indi- cated by a plus sign (+); those below, by a minus sign (—). ALGEBRAIC EQUATIONS. 5338 What is the difference between + 52° and + 33°? Between + 90° and — 10°? Show by a diagram. 1208. A has $600, B owes $400. What are they worth together ? (+$600) + (— $400) =? ‘How much better off is A than B? (+ $600) — (— $400) =? 1209. In subtracting algebraic quantities, change the signs of the subtrahend, and proceed as in addition. 1. From 8a take 2a. 5. From — 8a take — 2a. 8a 6. From — 2a take 8 a. —2a Ans. 6a 7. From — 2a take — 8a. 2. From 2a take 8a. 8. From 2a take — 8a. 2 beam 9. From 82+ 14 take x+ 10. Ans. — 6a 82+14 3. From — 8a take 2a. Seal — 8a 10. From 52—8 take —382—9, — 2a ee Oc 11. From 2 — 28 take 5a — 87. 4. From 8a take — 2a. 12. From 7x+ 16 take 92 — 4. 8a 13. From 62 take 22 —5. + 2a Ans. 10a 14. From 82 take 9z+8. 15. From 8zx+2a—5 take x—a—4Y. 16. From 7y—22+6 take —8y+6b—z. 17. From c—d-+e take ct+d—f. 534 ARITHMETIC. REMOVING PARENTHESES. 1210. From 8 take the difference between 49 and 25. 84 — (49 — 25) = what? Would the result be the same if we should write the above 84 — 49 — 25? What sign must be changed? 1211. Write the following without parentheses: 1. 57+ (83 — 16) = 74 4. (17—8)—(16— 14)=7 2. 92— (63 + 25) = 4 5. 75+4x (15 —10)=95 3. (48—10)+(24—5)=52 6. 75—4x (15—10)=55 1212. Is there any change made in the signs of the first? In the signs of the second? Ofthe third? Ofthe fourth? Ofthe fifth? Of the sixth? 1213. Solve the following equations. Prove the correctness of your answers. 1. 6(22—5)=52+12 NotE. 6(2%—5) means 6 times (2a —5), or 12a—30. 2. T(¢+2)=382+50 4. 3(16 — 2x) = 4(18 — z) 3. 53 +2)+16=61 5. 15(a —8) = 2(189 — 162) 6. 38—(11—9xz)=10x Removing the parenthesis, we have | 38 —11+4+9a¢=10" Transposing, 9a—10%=—38+411 or, —x2=—27 Bringing — a to the right side of the equation, and — 27 to the left side, we have (+) 27=(+)# In practice, however, when the result is such as the above, — x = — 27, the signs of both members are changed, and the result becomes x= 27 ALGEBRAIC EQUATIONS, 535 7. 2(a —1)—2(2"—19)'= 3(# — 8) 8. 6(22—5)—5x=12 9. 5x —6(22—5) =~ 12 Ou Onna EOC Tanner ae | Clear of fractions by multiplying both members of the equation by 10, and observe which sign must be changed to preserve the equality. 1214. =2 When x = 6, the above may be written B2—-6 42-4 2 5 Clearing of fractions, 15 2— 30—(8x—8)=20 =2 Removing the parenthesis, 15z—30—82#+ 8= 20 Transposing, 15x—8xr=20+4 30—8 or, (27=42 z=6 Notre. — The horizontal line between the numerator and the denominator of the foregoing fractions has the effect of a parenthesis, the entire quantity above the line being divided by the number below. 18 24—4 5 o—* = (18 - 6) +2 1 of (24 — 4) at =(40—4) +5 536 1215. Solve: ii; 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. ARITHMETIC. Tx—8 =z? _ z+2 yee 40—5ae 52492 24=3-+4 2124—(5+422)+4 28 §¢4+9=22+4+ (82-42) 1 H 5 x x ae ARNG ce? f2—120=— +10 r—20=(7+15)4 Mae ad bk 9(82 4+1)—-4=4(92+5)+3 5a—6 2 3 = -——— x-+ 5 ALGEBRAIC EQUATIONS. 537 1216. Slate Problems. 1. A certain number is multiplied by 52; 7 is subtracted from the product; the remainder is divided by 16, giving a quo- tient of 3. What is the number? 2. Three-eighths of what number is 60 less than the number itself? 3. Four persons are of the same age. If the first were 4 of his age older, the second 4 of his age older, the third + of his age older, and the fourth 4 of his age older, the sum of their ages would be 99 years. What is the age of each? 4. A man spends } of his earnings on board and lodging, + on clothing and repairs, and 4 on sundries. At the end of the year he has $280 left. What are his yearly earnings? oe, @ ==+—+= + 280. (2 ras oa ) 5. A boy gave 4 of his marbles to one companion, and 1} of them to another. He then bought 4 as many as he originally had, and had 4 marbles more than he had at first. How many did he have at first ? 6. A father’s age and a son’s age added together amount to 138 years. Twelve years ago the father was twice as old as the son. How old is each now? Let « =son’s age 12 years ago. 2a = father’s age then. 7. John has 80 cents, and William has 60 cents. How many ane will William have to give John so that the latter shall have 4. times as much money as the former? After William gives John a cents, the former has (60 — x) cents, and the latter has (80 + a) cents. 8. In how many years will a man, now 25, be double the age of his 11-year-old brother? Let « = number of years. 25+ 2 and 1l+a= ages after x years. 538 ARITHMETIC. 9. A man has a cask of 60 gallons’ capacity. He draws off one-fourth of its contents, and then fills it. If it takes 24 gal- lons to fill it, how many gallons did the cask originally contain? 10. A number is divided by 3, and 401s subtracted from the quotient, leaving a remainder of 104. What is the number? 11. The difference between two numbers is 430. When the greater is divided by the less, the quotient is 4, and the remainder is 76. What are the numbers? Let x = less. greater 4, 76. less less 12. A person pays $103 with 29 $2 and $5 bills. How many are there of each denomination? 13. A father is 80 years older than his daughter. In 4 years, his age will be four times her age. What are their present ages? «and «+ 30=present ages. «+4 and a + 34=ages 4 years later. 14. The product of two numbers is 180. If the smaller num- ber be increased by 8, the product of the two numbers will be 225. What are the numbers? smaller = 2; WS = greater. 15. A man’s wages are $1 per day more than his son's, For 88 days’ work, the father receives $12 more than. the son earns in 40 days. Find the wages of each. 16. The sum of two numbers is 47; their difference is 17. What are the numbers? 17. A mother is 41 years old. Her son’s age is 5. In how many years will the son’s age be 4 of his mother’s? TWO UNKNOWN QUANTITIES. 1217. Preliminary Problems. 1. I paid a dollar for two 25 balls and five bats. How much did I pay apiece for the latter? ALGEBRAIC EQUATIONS. 539 2. When three times one number is added to five times another, the sum is 84. If the second number is 12, what is the first number ? 3. A girl paid 75¢ for + pound of tea and 21 pounds of coffee. The coffee cost 20% per pound. What was the price of the tea per pound? 4. A man sold pigs at $5 each and lambs at $8 each, receiv- ing $42. He sold 4 lambs. How many pigs did he sell? 5. Four times a father’s age added to twice his daughter’s age amounts to 180 years. The girl is 10 years old. What is the father’s age? 6. Eight peaches and seven pears cost 44%. The peaches cost 2¢ each. What is the cost of a pear? 7. Two pieces of cloth and eleven pieces of silk contain 152 yards. ‘There are 10 yards in each piece of cloth. How many yards in each piece of silk? 8. Two-thirds of a yard of linen and three-fourths of a yard of lace cost 40%. The price of the lace is 82¢ a yard. Find the price of the linen. 9. Three and one-half times one number added to four and one-third times a second number equals 60. The second number is 9. What is the first number? 1218. Slate Exercises. Find the value of the unknown quantity : 1. 8x+7y =44. When x= 2, find the value of y. 2. 3y+ 52=34. Find the value of 2; y =8. Soar Lebo ten) 10 le eae 4. l4¢+7y=98 w=31; y=? 5. 24+ $2=40. 2=82. 540 ARITHMETIC, 9x2—25y=8. uses |? 8hy +442 = 60. z= 9. 162—192 = 49. z= 0. Ty— 82=18. y = 64. 10. 8224+ 50y=2,600. y= 20. ei ee he 1219. A boy gave 17% for 3 lemons and 4 oranges, another boy paid 25¢ for 3 lemons and 8 oranges. How much did the lemons cost apiece ? x = cost of lemons su+4y=17 (1) y = cost of oranges dx+8y= 25 (2) Subtracting (1) from (2) 4y=8 The oranges cost 2¢ each y=2 How much apiece was paid for the lemons? 11. If 8 coats and 14 vests cost $78, and 2 coats and 14 vests, at the same rate, cost $66, how much does 1 coat cost? What is the price of a vest? 12. Given 42+7y=53 (1) Qa+8y = 25 (2) to find the value of y. First multiply (2) by 2, making it 4z+6y=50. Why? 13. What is the value of z in equation (1), when the value found for y is substituted therein? Substitute the same value for y in equation (2) and find the value of z. 1220. Find the values of w and y in the following equations: 14. #+ y=165. 2z2+3y = 38. 15. 22+2y=30. 2+ 3y = 27. 16. 2x4+3y=18. 4x+38y= 24. 17. 22+3y=40. 38x2-+2y = 35. 1221. Given ( ALGEBRAIC EQUATIONS. 541 18. 72+5y= 82. 22+ 2y = 28. 19. 52+9y=14 94+5y=14. 20. 34+5y=17. 8xt2y=17. x+8y=46 eye ae es a To find values of x and y. Multiply (1) by 7, T2+2ly = 3822 (2) 7a— 4y= 22 Subtract. 25 y = 300 y= 12 Substituting this value of y in (1), we have 21. 22. 23. 24. 25. 26. 27. 28. x +36 = 46 x =46—36=10 Answers.2 = 10) y= Laie x+y=18 Add or subtract. z—y= 4 42+38y=17 (1) Multiply (2) by 2 and subtract. 22— "y= 1 (2) 82+4y=48 Add. x—4y= 0 382+ 5y=18 (1) Multiply (1) by 7 and (2) by 3. Ta+3y=18 (2) Subtract. 4x+5y= 32 Add. 62—Sy=-—2 382+4y=3 (1) Multiply (2) by 2. Add. 122—2y=8 (2) 52=6y+5 Transpose. 32=d5y—4 . 38xz+5y+ 8=0 29. y—2x=—8r—-1 24%z— y—12=0 2y—4x2=y+2+9 542 ARITHMETIC. 30. ia aH 17 Clear of fractions. ——-+ —4— 20 4 x 8 31. da+thy=42 35. 444+ 32y = 67 datty=11% T4a—5hy=12 32. 232—Ty= 8x£+51 36. 3( «+ 7)=9(y—9) lly=152+2 4(3 2 —8)=17y—155 33. x+y = 100,000 37. 2(a—11)— 2(y—9)=6 5a , 4y zr+9 32 —— + —4+ — 4,640 =— 100 ' 100 y—3 15 3sa2+7 z—4,y-—1 34, ——_=5 38. Z_—=5 oY & 3 a 4 (Becht ea ol | 5y+3 3 4 39. 2a+d5y+3_¢ 3%—4y—2 Bo a deel re ae x—2y+2 1222. Slate Problems. 1. The sum of two numbers is 87. Twice the first added to three times the second is 96. What are the numbers? (Let x = first number; y = second number.) 2. The difference between two numbers is 28. Five times the first less twice the second is 197. What are the numbers? (a —y = 28; 5a—2y =197.) 3. The product of the first of two numbers by 5, added to the product of the second by 3, gives 37. The product of the first by 6, diminished by 5 times the second, equals 10. Find the numbers. 4. Divide 65 into two parts whose difference shall be 19. (Let x and y=parts. Solve also by one unknown quantity.) ALGEBRAIC EQUATIONS. 543 5. A person pays $103 with 32 bills, some of them $2 bills, the others $5 bills. How many of each does he use? 6. For 25 head of pigs and sheep, a farmer received $145. How many of each did he sell, if he sold the former at a each, the latter at $5 each? 7. 10 oranges and 4 peaches cost 88%; 6 oranges and 7 peaches cost 82¢. Find the cost of an orange. Of a peach. 8. 5 pounds of tea and 3 pounds of coffee cost $3.75; 8 pounds of tea and 1 pound of coffee cost $5.05. What is each worth per pound? 9. A farmer buys a certain number of horses at $125 each, four times as many cows at $45 each, eight times as many sheep at $10 each, and half as many pigs at $5 each, spending $1,550 for all. How many of each does he buy? 10. A man paid 75¢ for 2 pounds of raisins and 8 pounds of cheese. 5 pounds of raisins and 2 pounds of cheese at the same price would have cost 94%. What did each cost per pound? 11. The sum of two numbers is 19. The sum of the second number and ten times the first, minus the sum of the first and ten times the second, equals 45. What are the numbers? 12. Reduce 5 to an equivalent fraction, the sum of whose numerator and denominator shall be 126. x=numerator; y = denominator. 5 4 eae ~ 126. Oa T Wi 13. What fraction equivalent to 38 has 147 for the difference between its numerator and denominator? (« —y=—147. Why?) 14. 10 pounds of coffee at 830% per pound are mixed with z pounds of coffee at 25¢ per pound. What is x equal to, when the mixture is worth 26% per pound? 25 x +(10 x 30) = 26 (10 +2). 544 ARITHMETIC. 15. A grocer mixes green tea costing 60% per pound with black tea costing 40% per pound. He uses 100 pounds in all, and the mixed tea costs him 48% per pound. How many pounds of each does he use? Let «=number of pounds of black tea; y= number of green. Then 3+ y =number of pounds of mixed tea. z+y=100; 402+ 60y = 48 (x+y). THREE UNKNOWN QUANTITIES. 1223. 1. Given the following: 8z2+2y— z2=12 (a) o2—4y+3z2=16 (6) 22+38y+22=385 (ec) to find the values of x, y, and z. (a) multiplied by 5, 15z+10y— 5z2=60 (d) He AS 15*a—12y+ 92=48 Subtract, 22y—14z2=12 (d) an equation containing only two unknown quantities. (6) multiplied by 2, 10zx— 8y+ 62= 382 (c) i tO. 10z+15y+10z2¢= 175 Subtract, —28y— 42=— 143 (e) an equation containing only two unknown quantities. Compare the two equations (d) and (e), which contain the same two unknown quantities. (d) multiplied by 2, 44y—282= 24 (e) ‘ Kat — 16ly— 28z=—1,001 Subtract, 205 y = 1,025 y ea ALGEBRAIC EQUATIONS. 545 Substituting this value of y in (d), we have 110—14z=12, —14z=— 98, z=7. Substituting values of y and z in (a), we have 38z+10—T=12, 324=9, x=8. Ans. += 8, y=5,| | teat 2. Find the values of the unknown quantities in the following equations : z—8y+2z2= 3 (a) Qa+ yt+3z2= 22 (6) 52+2y+72=851 (ec) Multiply (a) by 2, and subtract from (6). Multiply (a) by 5, and subtract from (¢). This gives two equations, each of which contains two unknown quantities. Compare these two resulting equations, and eliminate y. 3. Se—2Zy+ z= 10 (a) 382+ 8y—5z=120 (6) ta—3y—2z2= 8 (ce) Eliminate z by comparing (a) and (6), multiplying the forme by 5. Compare (a) and (c), multiplying the former by 2. 4 182-— 4y4+15z= 317 Tz+ 2y— 82= 89 2la2—l1l7y+ 9z2z=—104 & — 84+ y—12z2=— 259 Tzaz— 4y+25z= 418 182+ 2y—412=— 500 ad 546 ARITHMETIC. e. X42t¥_14 AA 3 ada Medea DUE GY 2 6 : RO Ula Be eB ty 2 5 ly cA ah 5 4 31 3 By ue 10 anes 8. 2+ 3 4y—32 194.22 by 9 ore 9. SE Dk fog diame 2 4 4 6 2e-bays Qari Sy 4 Oly aie yy 2 8 4 16 1224. Slate Problems. 1. A man placed 3 of his capital at 5% and the other third at 6%. At the end of a year, capital and interest amounted to $31,600. What was his capital? 2%, 5 x 6 ‘ ox —— and 2x ——- = interest. 3 * T00 an 3 * 00 intere 2. A has 18 chestnuts more than B. If each finds 4 more, A will have four times as many as B. How many chestnuts has each ? 3. Two mechanics earn together $8 per day. One works 23 days and the other 17 days, for which they receive together $166. What does each earn per day? 4. The sum of the first and the second of three numbers is 55, of the first and the third 62, of the second and the third 88. What are the numbers? s ALGEBRAIC EQUATIONS. 547 5. The sum of two numbers is 53. Four times the first is 20 more than twice the second. Find the numbers. 6. A certain sum of money is divided among four persons. The first takes + of it, the second takes 4 of the remainder, the third takes 2 of what then remains, the fourth receives the balance, $24. What is the share of each of the other three? 7. A merchant sold a lot of goods for $510, thereby losing 3 of their cost. What did the goods cost ? 8. A man collected a bill for a physician and deducted 5 of the amount for his services, If he gave the physician $147, what was the amount collected ? 9. Divide 1301 acres of land among three persons, giving the first 274 acres more than the second, and the second 133 acres more than the third. 10. A merchant has sold 4 of a piece of cloth, and has remain- ing 16 yards more than + of the piece. How many yards did the piece contain originally ? 11. A servant is engaged for a year for $280 and a suit of clothes; he leaves at the end of six months, and receives $130 and the suit. What is the value of the clothes? MULTIPLICATION OF ALGEBRAIC QUANTITIES. 1225. Multiply 7+ 3 by x+4. The product is equal to x times (2 + 3) + 4 times (x + 3), “(r+ 3)=24+32 4(4 + 3)= 47+12 (c+ 3)(2+4)=27+72%+412 Ans. Nore. —-2” is read x square. The 2 is called an exponent. 548 ARITHMETIC, Multiply (7+ 7) by («+ 8). z+iT x+8 Product by 2, V+ Product by 8, 8x + 56 2 +152+56 Ans. 1226. Multiply: 1. ( +5) by (4+ 2) 4. (22+8) by ( +9) 2. ( +8) by (&+9) 5. (3%+1) by ( +7) 8. (22+5) by (4+ 2) 6. (2%+1) by (22+1) 1227. (x —5) x (w+4)=? x —5d zx +4 x(a—5) 2—5e 4(a— 5) 4x — 20 x’? —xz—20 Ans. 1228. Find products: Nore. — (w — 3) (w + 9) means « — 3 multiplied by x + 9. 7. (e—38)(@+9) 10. (u+5)(e«—5) 13. (2xr—6)(8x2+8) 8. («—6)(«x+7) 11. (2e—6)(a4+1) 14. (82+6)(22—3) 9. (x—5)(a+5) 12. @—6)(2441) 15. (2x+43)(22—8) 1229, (2 -—5)(x—4)=? The product is equal to 2(«—5)—4(x—5); that is, that 4(2 — 5) is to be subtracted from a(x — 5). a(x —5)=2?—52; 4(a— 5) = 44 — 20. Placing the subtrahend under the minuend, and changing the signs of the former (Art. 1209), we have 2 —52x — 42+ 20 (x - 5)(a@— 4) = 2*?—927+20 Ans. ALGEBRAIC EQUATIONS. 549 1230. (x —7)(x—9) =? Using either as a multiplier, place one under the other. Commencing with 2, say «x # = 2%, —9xa=—9a. Taking —7 as a multiplier, say 2X(—7)=—72, (—9)X(—7)=63. Combin- ing, we get the product. x— 9 x— 7 v— 92 ret halite aes ead ota Ans. 2? — 162+ 638 1231. Note that the multiplication of a + (positive) quantity by a + (positive) quantity gives a + (positive) product; that (+) x (—) or (—) X (+) gives a — (negative) product; and that (—) X (—) gives a+ (positive) product. as follows: This is usually stated 1232. Lrke signs produce +, and unlike signs produce —. 1233. Give results: 16. («—7)(x«—7) 20.( x+7)( x—6) 17. («—5)(«@—9) 21.( x—4)( x—T7) 18. (7+5)(@+5) 22. (2x—4)(8x—6) 19. (x—3)(4+8) 23. (2%+6)(8x2—7) PURE QUADRATICS. 2+6_ 32°— 66 24. (2u+7)(82+8) 25. (2x—3)(8 x— 2) 26. (24—3)(20+38) 27. (2x+9)(42—6) 1234. Given ait to find the value of x. Clearing of fractions, 927+ 54=152* — 3380 Transposing and combining, — 62? =— 884 Dividing by 6, and changing signs, x’? = 64 Extracting square root, z=+8. 1235. Since (— 8) x (— 8) = 64, the square root of 64 may be either +8 or —8. It is written + 8, and is read “ posotwe or negative 8." (It is sometimes less correctly called plus or minus 8.) 550 ARITHMETIC. 1236. Slate Exercises. Find value of 2, y, z, etc. : 1. 2? —13 = 36 11. (x — 3)(x+ 3) = 40 2. 3y°+ 25 = 100 12. (7+ 5)(x+ 5)= 10x + 26 3. 62-18 = 824 87 13. (24+ 4 =824 80 4. 5(2’+17)—32°+63=198 44. 2+ 64—52 5. 5(a?+17)—8(@'-21)=198 45. 3994 18-94 08 436 6 7 seo ein ee 16. (28)? = (4 SP S18 - (e+1)—2#'*=49 17. (x+-7)(a—9) =(x—8)(x—5) ae Pos la lh SOE since is. “44749 3 a ee ee 9 z+7 2—5 Ae et he 8 z—38 2-9 “GE bi) 29 to, 202 _. 80% PY fete Aaah ah z—l «+1 y-5 ytT 1237. Slate Problems. 1. Find the dimensions of a field, the length of which is éwice its breadth, its area being 1,800 square rods. 2. The surface of the six equal faces of a cube contains 96 square inches. Find the length of one edge. 3. One number is fourth-fifths of another, and their product is 80. What are the numbers? 4. One-third of a number multiplied by two-fifths of the same number gives a product of 270. Find the number. 5. Thirty per cent of a number multiplied by forty per cent of the same number gives a product of 500. What is the num- ber ? ALGEBRAIC EQUATIONS. DoW 6. Thirty per cent of forty per cent of a number is 300. What is the number ? 7. The base of a right-angled triangle is $ as long as the perpendicular, and the area of the triangle is 96 square rods. Find the length of the base. What is the length of the hypote- nuse ? 8. The base of a right-angled triangle measures x yd., the perpendicular measures sf yd. What is the length of the hypotenuse? If the hypotenuse measures 15 yd., find the length of the base. 9. The base of a right-angled triangle measures x ft., the hypotenuse measures (7+ 9) ft., the perpendicular measures 15 ft. What is the length of the base? 10. The difference between the squares of two consecutive numbers is 49. What are the numbers? AFFECTED QUADRATICS. 1238. Preliminary Exercises. (a@+1)\(@4+1]l=%7+2241 The square of the sum of two quantities is equal to the square of the first + twice the product of the first and the second + the square of the second. (¢—1)(#-l=2—224+41 The square of the difference of two quantities is equal to the square of the first — twice the product of the first and the second ++ the square of the second. (a+b? =a¢+2ab+6 (m—n? =m’—2mn+n' (10+ 5°? =10+2x10x5+8 (10 — 3)? = 10?— 2x 10x 3+3? 552 ARITHMETIC. 1239. Oral Exercises. Square : 1. 2+8 4. x<+10 2. x—T7 5. a—b 3. 2—9 6. z+y 1240. Sight Exercises. Extract the square root of 1. +6249 2. 2 —1427+49 3. 2?7—182+81 4. 27+202z+100 5. @+2ab4+0? 1241. The square of (x + 3) consists of how many terms? Of how many terms does (# + 4)? consist? 1242. Supply term necessary to make a complete square: 1. 2?+62+? 2. x2—122+? 3. v2 —82r+? 4. 2?—162+? 5. 2?+1824+? 1243. Slate Exercises. 7 30—1 10. x-y 8. 40—1 11. 80+5 9. min 12. 60—5 6. v@+2ay+y7 7 @-—2Qryt+y¥ 8. a—2ab+6 9. 2? — 247+ 144 10. 2’+222+4+121 (2 + 5)?? Cir Aa ee 7. 2 —4x+? 8. 2?—1027+? 9. #+1427+? 10. 2?— 222+? Given vt+62=27 What number must be added to the first member of the equa- tion to make it a ‘“‘complete’” square? If a number is added to one member of an equation, what must be done to the other member to preserve the equality ? ALGEBRAIC EQUATIONS. 5a 1244. Extract the square root of .both members of the follow- ing equations, adding to both, where necessary, such a number as will make the first member a complete square. 1 #+624+9=40+4+9 2. 2 —127%+ 36 = 28+ 36 Remember that (+ 7) x (+ 7) = 49, and that (— 7) x (— 7)=49. - V49 =+ 7 or — 7, written +7. 3. 2? —824+16=204 16 % 2@—l4a¢—15 4. 2—162+ 64=— 39+ 64 pS Merkle ep ag hora! 5. 2?+182+?=19+4+? 9. 2+142=51 6. @+22+?= 244? 10. 2—222=48 1245. Given xv*—102= 24. Completing the square, we have 2? — 10x+ 25= 24+ 25 = 49, Extracting the square root of both sides, we have z—5=+7, s=7+5=12 or —7+5=—2. Ans. 5 or — 2. 1246. Find values of z: ele Oe | 9. 2—24x=0 2. 2 122=108 10. at 82 = 384 3. 2t1+22—48 Peo de 4. 4182-115 12. 27+ 302—175 5. 2@—l4x2=—— 13 13. 7+ 284 = 29 6. 2?—10z2=0 14. 2?°+222=—104 7. 2+202=— 125 15. 2? — 162 = — 64 8. 27+ 262—56 16. a?+4+362= 76 554 ARITHMETIC. 1247. To make the first member a complete square, you added the square of what part of the coefficient of x? 1248. Find values of z: Li ata el 5. 22+ 9x =— 20 x -+ @ + ($) = 12+) 6. 2—llxz=— 28 2. 2° —82=10 7. @+182=—42 a?—32+($)=104+(3 ¢ 2 157=76 3. 7 +52——4 9. 2?—-17z=18 4. 2*°—iz=8 10. 27+19z=— 18 1249. When 2’ has a coefficient, divide both members by the coefficient. 3827+ 92 = 84. Dividing by 3, v’+3 a= 28. Completing the square, #4804 (r= 24+p=Sete Et Extracting square root, 2}+2—=—+41 ~a=ii—3—$=4; or —-i-—$=—-14=-—T7 Ans. 4 or — 7. 1. 62 — 62=36 . ot 9Ja=— 54 6 2. 943+ 92=180 % 827 — 722 —=— 160 3. Ta?-+284—147 8. 727+ 492 —56 4. 42% 402 —— 64 9. 32742le=54 5. 827—162%—504 10. 5a?—25a=— 20 ALGEBRAIC EQUATIONS. 555 1250. Slate Problems. 1. The sum of two numbers is 12; their product is 32. What are the numbers? wand 12—x2=numbers. (12 —.)«# = product. 2. The base of a rectangle is 50 feet longer than its altitude. Its area is 2,400 square feet. How long is the 7 base? Area 7? + 502 2400 sq. ft. 3. The perpendicular of a right- angled triangle measures 15 yards more than the base. The hypotenuse is 75 yards. Find the length of the perpendicular. 1+ 2 [ v2 + (15 + @)? = 75?.] 4. The hypotenuse of a right-angled triangle is 1} 8” times as long as the base. The area of the triangle is 150 square yards. How long is the hypotenuse? [Perpendicular = V(3 a)? — a; area = 1 base x perpendicular. ] ay 5. The entire surface of a square prism is 170 z square feet. Its altitude is 6 feet, and one side of its base is x feet. Find the value 50 +22 of x. SP sd tt 6. A garden 50 feet long, 40 feet wide, a S has a walk just outside it x feet wide. Find the area of the walk. If the area of the walk is 784 Sa feet, what is its width? 7. A field, ABCD, contains 12 a acres. Its length is 1% times its e breadth. How many rods long is ; the diagonal BC? 556 ARITHMETIC. 8. A flag-staff, AB, 50 feet high, was broken 13 off at the point C. The broken part, resting on C, reached the ground JD, 30 feet from the base of ts the staff. Find the length of the part broken off. i 9. A ladder, CH or DE, placed at a point &, is in a street 58 feet wide D_ between the opposite > C houses,. just touches &, ; . the top ofahouse, DB, 3 60 feet high on one side es 33 of the street, or the top of a house, A RB CA, 56 feet high on the other side. Find the length of the ladder. DE = 60+ (582) = Ch bb) 10. A&C isatriangle. The side AB B measures 13 feet; the side BC, 4 feet; ue ¥ AC, 15 feet. Find the altitude BD. A iD eo Bie AR tT eer : 11. ALCDisatrapezium. 4 B= 84 ft. ; BC= 20 ft.; CD=40 ft.; DA = 26 ft. ¥ E 5 The perpendicular BF’ measures 16 ft. Find the length of the diagonal AC and of the perpendicular 4D. D CHAPTER XVI. ELEMENTARY GEOMETRY. — PROBLEMS IN CONSTRUC- TION. PRACTICAL APPLICATIONS. —CALCULATION OF HEIGHTS AND DISTANCES. — MENSURATION, ELEMENTARY GEOMETRY. 1251. Angles, When two straight lines meet at a point, they are said to form an angle. The point at which the linesmeet _~— ia ae is called the vertex of the angle. When two angles are formed by the meeting of two straight lines, they are called adjacent angles. A and B are adjacent angles. C’ and D y / are adjacent angles. are / The angles #' and G, formed by the intersection of two straight lines, are called vertical, or opposite angles. F'and Hare vertical angles HO PH and #, F and G, G and A, H and #) are adjacent angles. When two adjacent angles are equal to each other, each ? oie is said to be a might angle. i OlP Thecanolesa lm yA. I, : NV, O, P are right angles. s An angle that is smaller than a right angle is called an acute angle; one larger /\ oe than a right angle is called an obtuse angle. a @ is an acute angle; £# is an obtuse angle. Angles that are not right angles are called, without regard to their size, oblique angles. 557 558 ARITHMETIC. 1252. Designation of Angles. The angle formed by the lines SZ and 7'U may be called the angle 7. It is frequently better to call it the angle STU or UTS, the letter at the vertex being placed between the two others. Te U The use of the three letters is necessary where there is more than one angle having U its vertex at the same point, as in the accom- V panying figure, where UX, VX, and WX = meet at the point X. ™R 1253. Measurement of Angles. An angle is measured by the are of a circle, the center of the circle being at the vertex of the angle. The angle 123 is meas- ured by the arc YZ; the angle 456 by the arc ab. 1254. Circular Measure, 60 seconds (/7) 1 minute, 60 minutes (7) 1 degree, 360 degrees (°) 1 circle. 1255. The number of degrees in an arc or an angle is deter- mined by a protractor. A SEMI-CIRCULAR PROTRACTOR ELEMENTARY GEOMETRY. 559 To measure an angle, XY YZ, for instance, produce the lines YX and YZ Place the point A of the pro- | x tractor on the-vertex ( Y) of the angle, and the edge AC’ on the line YZ produced. Using the lower line of figures, read off from the protractor the number of degrees at the point where the line YY produced cuts the semi-circle. In measuring the angle DEF’ the line AB " is placed on #//’ the point A on the vertex /. if The number of degrees in this case is read trom i the upper row of figures. LL EXERCISES IN CONSTRUCTION. 1256. Norz.— In the following 100 exercises, the ruler, the compasses, and the protractor may be used. The drawing should be carefully done with a sharp, hard pencil. 1. Draw an obtuse angle formed by two lines, each one inch long. Draw an acute angle formed by two lines, each six inches long. Which is the larger? 2. Fold a piece of paper twice, so that the lines made by the creases will form four right angles. Fold a piece of paper so as to make four 4 angles that are not right angles. 3. The lines GH and JJ intersect at , making four right angles. Which arc is longer, 7 8, or ced? Which contains the greater number of degrees? 4. Draw two lines meeting at an angle of 45°. Two lines meeting at an angle of 90°. Two meeting at an angle of 135°. 5. Draw two lines making two angles, one of which measures 60°. How many degrees does 60° the other angle contain? 560 ARITHMETIC. 1257. Norr.—A line parallel to the right or the left side of the paper is called a vertical line; one parallel to Q the top or the bottom of the paper is called K Pp a horizontal line ; others are called oblique M N lines. KL is a vertical line, MN is a hori- R zontal line, OPand Qf are oblique lines. J, O 6. Toa horizontal line draw a line making two equal adja- cent angles. How many degrees does each angle contain? To a vertical line draw a line making two equal adjacent angles. How many degrees does each angle contain ? To an oblique line draw a line making two equal adjacent angles. How many degrees does each angle contain ? 7. How many degrees are there in a right angle? 8. To an oblique line draw a line making two unequal adja- cent angles. How many degrees are there in the sum of the two angles ? 9. How many degrees in the angle 7, if & ag s contains 75° ? (og V V measures 110°. How many degrees does U measure ? If one of two adjacent angles measures 634°, how many degrees are there in the other angle? How many degrees are there in an angle adjacent to one of 47° 45'? 10. Construct angle 5, 60°; angle 4, 50°. Measure angle 3. How many degrees and minutes will there be 3/4 in angle 5, when 3 contains 494° and 4 contains 833°? When angle 3 contains 36° 30’ and angle 5 contains 79° 45/, how many degrees and minutes will angle 4 contain? 11. Erect a perpendicular at each extremity of a horizontal line. At each extremity of a vertical line. At each extremity of an oblique line. ELEMENTARY GEOMETRY. 561 1258. Norr.—A line making a right angle with another line is said to be perpendicular to it. 12. Construct a square upon a horizontal line. Upon an oblique line. ie 13. Draw two lines intersecting at an angle of 100°. Mark in each of the other three angles the number of degrees it contains. 14. If one of the four angles formed by two intersecting lines measures 90°, what does each of the other three measure? If one measures 60°, what does each of the others measure ? _ 15. At each extremity of a horizontal line draw a line making an angle of 40° with the first line. 16. At each extremity of a vertical line draw a line making an angle of 100° with the first line. 17. At one extremity of an oblique line draw a line making with the first line an acute angle; at the other extremity draw a line making an obtuse angle with the first line. 18. Draw two lines making an angle (6) of 150°. Construct an adjoining angle (7) contain- altitude (slant height) of each of the four triangles the same? 14. Calculate the slant height of each convex face of a rect- angular pyramid whose altitude is 12 inches, and whose base measures 10 inches by 18 inches. Find the convex surface. 1284. Surface of Frustum of Pyramid and Cone, When the upper part of a pyramid or of a cone is cut off by a plane parallel to the base, the remaining part is called a frustum. Da 15. Draw one face of the frustum of a square pyramid. Of the frustum of a triangular pyr- amid. What figure have you drawn? Calculate its surface when the length of the upper side is 4 inches, that of the lower side is 8 inches, and the slant height of the frustum is 10 inches. 16. Draw the developed convex surface of the frustum of a regular triangular pyramid, each side of the upper base measur- ing 1 inch, of the lower base 2 inches, the slant height being inches. i Suaerstion. — Locate the apex of the whole pyramid of which the given frustum forms a part. 17. Find the convex surface of the frustum of a square pyramid, one side of the upper base measuring 2 feet, of the lower base 3 feet, and having a slant height of 4 feet. Find the entire surface. ELEMENTARY GEOMETRY. 597 18. Show that the convex surface of the frustum of a pyramid is equal to one-half the sum of the perimeters of the upper and the lower bases multiplied by the slant height. 19. Draw the pattern of a small shade for a candle. Make the upper opening 14 inches in diameter, the lower one 24 inches in diameter, and the slant height 2 inches. 20. How many square inches of tin will be required to make a pan, its upper base being 9 inches in diameter, the lower base 6 inches in diameter, and the slant height 4 inches? (Do not forget the bottom of the pan.) 1285. The frustum of a cone may be considered the frustum of a pyramid whose bases contain a very great number of sides. The convex surface of the frustum of a cone may, therefore, be found by multiplying the half sum of the circumference of the two bases by the slant height. 21. Find the convex surface of a frustum of a cone, the cir- cumferences of the bases being 15 inches and 20 inches, respec- tively, and the slant height 10 inches. 22. How many square yards are there in the entire surface of a frustum of a cone, the radius of the upper base (7) being 3 yards, of the lower base (4) 5 yards, and the slant height 6 yards? | Circumference of upper base = 27r; of lower base = 27 R. Convex surface = (27r + 27R) x ee = t(r + &) Xslant height. Surface of upper base = mr?; of lower base = 7 R?2. Entire surface = 7 X what? Multiply only once by 3.1416. 23. The diameter, AB, of the upper base of the E frustum of a cone measures 6 feet, CD measures bay 8 feet, the slant height AC measures 9 feet. Find SO the slant height 4A of the part cut from the cone Las in making the frustum. Let HA=2; HC=24+9; AB=6; CD=8. The triangles HAB and ECD are similar. 598 ARITHMETIC. 24. Find the convex surface of the whole cone, HCD, and the convex surface of the part cut off, HAD. 1286. The Sphere. A sphere is a body all points on whose surface are equally distant from the center. The distance from the center to the surface is called the radius of the sphere. The diameter is a line running between two points on the surface and passing through the center. CG, CK, CD, CF, CA, and Cl are radii; AD and FG are diameters. 1287. If a sphere be cut through at any part, the cut surface will be a circle. When the cutting plane passes through the center of the sphere, the circle is called a great circle; other cir- cles are called small circles. FXGC is a great circle; HYIB and JLEZ are small circles. 25. Find the length of an are of 60° of a great circle of a sphere whose circumference is 25,000 miles. 26. Calling the are AJ in the preceding figure, 30°, the angle BCT will measure 30°. Calculate the radius BJ of the small circle when the radius CZ of the large circle is 4,000 miles. (LAA = are of O00) A == Ghordcon60.)) 27. If Z is 60° from G, a point on the equator, find the length of the circumference of the small circle YJ, assuming the cir- cumference of a great circle to be 25,000 miles. 28. What is the ratio between the length of a degree on the small circle YJ, and the length of a degree of a great circle? 29. Calculate the radius of a small circle formed by passing a plane parallel to GCXF through a point on GA 45 degrees from GQ, ELEMENTARY GEOMETRY. 599 1288. Surface of Sphere. We have seen (Art. 1151) that it may be experimentally shown that the surface of a sphere is equal to the surface of four of its great circles. Calling the radius of the sphere A, its surface 1s 4 ft’, \W 4, aK yi 4 ig Ay 7 RR 7 30. Find the surface of a sphere whose diameter is 6 inches. 31. How many square inches are there in the convex surface of a hemisphere whose radius is 3 inches? What is the area of the great circle that forms the base of the hemisphere ? Find the entire surface of the hemisphere. 32. Is there any difference between the convex surface of a sphere and its entire surface? Why? VOLUMES. 1289. Prisms and Cylinders, 1. How many one-inch cubes will cover the base of a box 4 inches by 8 inches? If the box is 2 inches deep, how many one- inch cubes will it contain? How many cubic inches are there in the volume of a right prism whose base is a rectangle measuring 4 inches by 3 inches, and whose altitude is 6 inches? 2. If the above hollow prism were divided into two equal parts by a thin partition extend- ing from a vertical edge to one diagonally oppo- site, how many cubic inches of sand would each part contain ? 3. How many cubic inches are there in the volume of a prism whose base is a right-angled triangle 3 by 4 by 5 inches, and whose altitude is 6 inches? 600 ARITHMETIC. 4. Find the volume of a triangular prism, the area of the base being 6 square inches, and the altitude 6 inches. Find the volume of a triangular prism, each side “JS of whose base measures 6 inches, its altitude being 8 inches. b 5. What are the solid contents of a pentagonal prism formed by fastening together three triangular prisms whose bases contain, respectively, 12, 16, and 18 square inches, the altitude of each being 15 inches? 6. Ifa very great number of triangular prisms of the same height are united so as to form a cylinder whose base contains 12.5664 square inches, and whose altitude measures 5 inches, what are the solid contents? 1290. Pyramids and Cones. With a center at C, and a convenient radius, describe an arc AB. Mark off four equal portions v, w, «, and y; and draw the equal chords. Cutting out CAvwey, with an additional narrow strip along Cy for gumming, and creasing C iy along the lines Cv, Cw, Cx, and Cy, we can fold the paper into a square pyramid. Measure its altitude, and make a square As _ prism of equal altitude and with an equal base. A Filling the pyramid with sand, and pour- -; os ing the sand into the prism, it will be found that the latter will contain the contents of the former three times; that is, the volume of a square pyramid is one-third that of a square prism having an equal base and an equal altitude. The same ratio will be found true in the case of a triangular, or any other pyramid, as compared with the corresponding prism, and of the cone as compared with a cylinder. 1291. The volume of a pyramid or of a cone is equal, there- fore, to the area of the base multiplied by one-third of the altitude. ELEMENTARY GEOMETRY. 601 1292. Frustums of Pyramids and Cones, 7. Find the volume of a square pyramid whose altitude is 12 inches, one side of the base measuring 6 inches. Find the volume of a square pyramid whose altitude is 6 inches, one side of the base measuring 3 inches. 8. Find the volume of the frustum of a square pyramid whose altitude is 6 inches, one side of the upper base 3 inches, and one side of the lower base 6 inches. 9. A square pyramid whose altitude measures 18 inches, and each side of whose base measures EB 15 inches, is divided into two parts by \ a plane, GHZ, parallel to the base, iT the distance, HJ, of the plane from the vertex, 4, being 6 inches. The ratio between the edge, #B, of the whole pyramid and the edge, 4 ay: HG, of the part cut off will be equal to that between HV and #X; that is, 6:18=1:3. The same will be B the ratio between BC and GH, and the latter will be one-third of 15 inches long, or 5 inches. Find the volume of the large pyramid and that of the small pyramid. \ | aK 10. Each side of the upper base of the frus- tum of a square pyramid measures 5 inches; each side of the lower base measures 15 inches; the perpendicular distance between them meas- ures 12 inches. Find the solid contents. Find the convex surface of the above frustum. Find its entire surface. Note. — What is the slant height? 602 ARITHMETIC. 11. Find the total volume of three square pyramids, the alti- tude of each being 12 inches, and the areas of their bases being 25 sq. in., 225 sq. in., and 75 sq. in., respectively. V75 15 5 12. Find the number of cubic feet in a block of stone whose shape is that of a frustum of a square pyramid 4 feet high, each side of the upper base measuring 3 feet, and each side of the lower base 5 feet. 1293. The volume of the frustum of a pyramid is equal to the sum of the volumes of three pyramids, each having an altitude equal to that of the frustum ; the base of one of them being equal in area to that of the lower base of the frustum, the base of a second being equal in area to that of the upper base of the frustum, and the base of a third being a mean propor- tional between the area of the other two. Base of first =3 x 3 sq. ft.; of second, 5x 58q. ft.; of third, V9 x 25 sq. ft. = 15 sq. ft. Norr.— The mean proportional between two numbers is equal to the square root of their product. , 13. Find the volume of the frustum of a square pyramid, its upper base containing 64 square inches, and its lower base 196 square inches, its altitude being 18 inches. 1294. Note that the mean proportional between 64 and 196 is 8 x 14, or 112. Since each is multiplied by one-third of the altitude, the opera- tion is shortened by adding together the three areas, 64, 196, and 112, and multiplying their sum by one-third of 18. Calling the altitude A, the side of the large square S, of small square s, the volume V, we have 24 ) ELEMENTARY GEOMETRY. 603 1295. The volume of the frustum of a cone is found in the same way as that of the frustum of a pyramid. + altitude (area upper base + area lower base + area mean proportional). Calling the radius of the upper base r, and that of the lower base R, the area of the upper base will be mr’, of the lower base wA?, of mean propor- tional mr. =1A(mr? + wh? + ark) Since w (or 3.1416) is a common factor, we can save time by first adding r?, #?, and rf, and then multiplying by 7. ae: An(r? Le + rk) 14. The diameters of the bases of the frustum of a cone meas- ure 8 and 15 inches, respectively; the altitude is 18 inches. Find the volume. 15. How many cubic inches of water will a pan hold, whose lower base is 12 inches in diameter, whose upper base is 16 inches in diameter, and whose depth is 6 inches? How many gallons? 1296. The pupils should make a frustum of a square pyra- mid of convenient size, and the three corresponding pyramids, as given in the rule. Fill the latter with sand, and pour the contents of all three into the frustum. To make the frustum, draw two concentric circles. Lay off equal arcs, AB, BD, DE, EF. Draw the chords and radii from the extremities of each chord. Draw the chords ab, bd, de, and éf. Cut 4 out, after constructing a square fom either the upper or the lower base, and taking care to provide flaps for pasting. 604 ARITHMETIC. To get a mean proportional between ab and AB for one side of the base of the third pyramid, lay off a line 7/7 equal in length to ab+ AB. On this line construct a semicircle. ee Make 1X equal to A, and at X erect a perpendic- ular KM. KM is a mean proportional between f ye ab and AB. 1297. Oblique Prisms, We have seen that a rectangle, ABCD, and a parallelogram, — EFGH, are equal in area when the bases, AB and #F’ and the altitudes, i & 7a 7 AD and HX, are equal, each to each. Vib This can be shown by cutting both A BE F out of paper, and by shifting the tri- angle HEX to the right side of the parallelogram. 1298. In a somewhat similar way, we can show that an oblique prism is equal te a right prism that has an equal base and an equal altitude. Make from a potato or a tur- nip an oblique prism having rectangular bases, and change it to a right prism of the same height by cutting and shifting a portion. 1299. The volume of any prism (or cylinder) is found by multiplying the area of the base by the altitude. 1300. In the same way it can be shown that the volume of any pyramid or cone is equal to the product of the area of the base by one-third the altitude. 1301. The Sphere, A sphere may be considered as made up of a great number of pyramids whose bases together make the surface of the sphere, ELEMENTARY GEOMETRY. 605 and whose vertices all meet at the center of the sphere, making their altitudes each equal to the radius of the sphere. The volume of a sphere is equal, therefore, to its surface x 4 radius. Surface = 4 great circles = 47h’; Volume = 47? x ‘ = tie 16. Find the volume of a sphere whose radius is 9 inches. 17. What is the volume of a sphere whose diameter is 9 inches ? Find the volume of a cone whose altitude is 9 inches, diameter of base 9 inches. How does the volume of the cone compare with the volume of the sphere? How does the volume of the sphere compare with the volume of a cylinder 9 inches in diameter and 9 inches high ? 1302. Take a clay sphere of a convenient size. Make a paper cylinder that will exactly contain it, the height of the cylinder being equal to the diameter of the sphere. Make a hollow cone of the same diameter and altitude. Place the sphere in the cylinder, carefully fill the cone with water, and pour it into the cylinder, which should then be filled to the top, showing that the volume of the cylinder is equal to that of the sphere and the cone together. APPENDIX. Se eeeneneaR a3 Gen Sa | TABLES. 1303. Measures of Extension. Lona MEASURE. 12 inches (in.). . . 1 foot (ft.) SPCCuL Aa pire 2 1 yard (yd.) Ber VEC Matera nthe 1 rod (rd.) SoU TOdeaet ie) 2h. 1 mile (mi.) A furlong (fur.) = 4 mile. Surveyors’ MEASURE. SPAR DCE A a ge 1 link (li.) BOLIN ke eka: 1 chain (ch.) SUA CEE Te. IEEE aR Ee 1 mile 1 chain = 4 rods = 66 feet. SquaRE MEASURE. |e nS ee 1 square foot 9 sq. ft. . l square yard 304 sq. yd . . 1 square rod POUIS08 Pde mn oh 1 acre 640 acres . . 1 square mile 10 square chains = 1 acre. CuBic MEASURE. Di2e cunin. . .)., L.cubic foot Bide OTM Ges iic eo Bb 1 cubic yard lecord ~=128 cu. ft. 1 bushel = 2150.4 cu. in. 1 gallon = 231 cu. in. Measures of Capacity. Dry MEASURE. 2 pints (pt.) . . . 1 quart (qt.) Siquarte mis, 1 peck (pk.) 4 pecks.\. 4. 6. 1 bushel (bu.) Liquip MEASURE. 2 pints (pt.) . . . 1 quart (qt.) 4quarts .... 1 gallon (gal.) A gill is 4 pint. . The capacity of tanks, vats, etc., is frequently expressed in barrels of 31% gallons. 1 qt. dry measure =674 cu. in. 1 qt. liquid measure = 57$ cu. in. Measures of Weight. Troy WEIGHT. 24 grains (gr.) 1 penny weight (pwt.) 20 pennyweights, 1 ounce (oz.) 12 ounces 1 pound (1b.) Troy weight is used in weighing gold, silver, precious stones, etc. APOTHECARIES WEIGHT. 20 grains (gr) . 1 scruple (Dd) dscruples ..... 1 dram (3) Sra nigu ets 1 ounce (3 ) 12 ounces ... . 1 pound (fb) 608 Apothecaries’ weight is used in prescriptions. Drugs are bought and sold by avoirdupois weight. The grain, the ounce, and the pound apothecaries’ weight are the same as the corresponding denominations of troy weight. Avorirpupots WEIGHT. 16 ounces (0z.) . 1 pound (1b.) 2000 pounds. ... 1 ton (T.) 1 lb. avoirdupois = 7000 grains. 1 lb. troy eet LOU eee QM 1 (a3 loz.avoirdupois= 4374 1 oz. troy = 480 “ In weighing ores and coal at the mines and in calculating duties at the U. 8. custom houses, the follow- ing table is used: AO DOUDOS matt emer ue 1 quarter (qr.) 4 quarters, 1 hundredweight (cwt.) 20 hundredweight .... 1 ton (T.) 1 ewt. = 112 lb.) TT) = 2240 Ib. Measures of Value. U.S. Money. DO trate ik ee 1 cent LOU CENTS fe hte ale 1 dollar 1 dime = 10ce. 1 eagle = $10. EneuisH Money. 12 pence (d.) . . 1 shilling (s) 20 shillings . 1 pound (£) 1 farthing = + penny. £1 = $4.8665. ARITHMETIC, The Canadian dollar is equal in value to that of the United States, and is also divided into 100 cents. The French frane (fr.) = 19.3f, is divided into 100 cen- times (c.). The German mark (reichs- mark) (M., m.) = 28.8f, is di- vided into 100 pfennigs (pf.). Circular Measure. 60 seconds ("’) . . 1 minute (') 60 minutes... . 1 degree (°) 360 degrees . 1 circumference Time Measure. 60 seconds (sec.) 1 minute (min.) 6Osminutes 2. ars 1 hour (hr.) 24 HOUrSs sos es ste 1 day (da.) y Gaye wove ue ae 1 week (wk.) 365 days 1 common year (yr.) BOO GAYS vu ae ee 1 leap year Years divisible by 4 and not by 100 are leap years. 1892, 1896 are leap years. Years divisible by 100 but not by 400 are common years. 1700, 1800, 1900 are common years; 1600 and 2000 are leap years. APPENDIX. 609 1304. Time between Dates. 1. In the common method, by compound subtraction, each month is considered one of 30 days, regardless of its length. EXAMPLE: Find the time between May 18, 1895 and March 2, 1899. 1899 3 2 Taking 30 days to the month, the 1895 ee 8: difference in time is found to be 3 3 Se ks) years, 9 months, 19 days. 2. A more exact method is to take first the number of entire years, then the number of entire months, and lastly the number of days. May 13, 1895 to May 13, 1898 = 3 years. May 13, 1898 to Feb. 13, 1899 = 9 months. Feb. 13, 1899 to Mar. 2, 1899 =17 days. Ans. 3 years, 9 months, 17 days. 8. Another method is to find the difference in years and days. May 13, 1895 to May 13, 1898 =3 years. May 13, 1898 to Mar. 2, 1899 = 293 days. Ans. 3 years, 293 days. 1305. Days of Grace. Days of grace are not allowed in California, Connecticut, District of Columbia, Idaho, Illinois, Maryland, Massachusetts, Montana, New Jersey, New York, North Dakota, Ohio, Oregon, Pennsylvania, Utah, Vermont, and Wisconsin. Answers in which days of grace are not included, are inclosed in a parenthesis. When a note falls due on a Sunday or a legal holiday, it is generally payable on the next preceding business day. Some states, however, do not require payment until the next business day following. In these latter cases, banks include the extra days in calculating discount. In a majority of the states, days of grace are not allowed on sight drafts 610 1306. Rates of Interest. ARITHMETIC. The following table shows the rates of interest allowed by law in the several states and territories, the first column giving the rate allowed where none is specified in the note or other docu- ment. States and Territories. Alabama 4°). oy. 2 AYIZON GE. eter ies ATE ONSRBi loys els Py Cig lifornia.:) ecco ne OLOTACOM, ca cee Connecticut Delaware...... District of Columbia Mlorida "0075 sae ns CLeOTgZiat) hy. sare TARNOD vee Ones ae Indiana. ©. .fer5e% Indian Territory. . LOWS ages cota ns is KANG RS We lieth ore Kentucky ..... Louisiana Maryland ..... Massachusetts .. . Michigan.) J 4/4 Minnesota ..... Mississippi ..... Missouri © ©. @ (© ‘¢. Legal Rate. Contract Rate. Any rate up to the one given in the second column is permitted when an agreement is made in writing. Contract Rate. I 0 OD @D DD © ~T O =~T Co 4 a) Cou So) J Cd) (Od) Od -O>. Cr Od9 Od Cn) Od Op. Ct Montanay sien A Nebraska ..... Nevada. sks: New Hampshire . New Jersey... . New Mexico ... New York i North Carolina. . North Dakota Orevon a ase tas Pennsylvania Rhode Island. . . South Carolina. . South Dakota Virgenes eae Washington : West Virginia . . Wisconsin. .... Wyoming. ope..ss 10 DAODA~ATAPHHAHAAWAAANTNAI AAAI a nsw oOOoganrinona -t APPENDIX. 611 PARTIAL PAYMENTS. 1307. Connecticut Rule. Find the amount of the principal to the time when the payment, or the sum of two or more payments equals or exceeds the interest, uf such time rs one year or more from the tume the interest com- menced. From this amount deduct the payment or sum of payments. Use the balance then due as a new principal, and proceed as before. Tf, however, any payments exceeding the interest then due, are made within a year from the time from which the interest is reckoned, the amount of the principal is calculated for one year, and from this amount are deducted the amounts of such payments from the tueme made until the end of the year. When the settle- ment 1s made in less than one year from the tume from which anterest 1s reckoned, the amount of the principal and those of the payments are calculated to the date of settlement. 5 Ae Norwicu, Conn., Jan. 5, 1896. Two years after date, we, jointly and severally, promise to pay to Emerson W. Keyes, or order, Three Hundred Dollars, value received, with interest at 6 per cent. JostaH H. Pitts, $ 300,29. CHARLES W. FIELD. Endorsements: May 20, 1896, $100; Nov. 2, 1896, $100; March 17, 1897, $50. How much is due Jan. 5, 1898? Amount of $300 to Jan. 5, 1897, l year . . . $ 318.00 Amount of $100 from May 20, 1896 to Jan. 5, 1897, Thy mo. $103.75 Amount of $100 from Nov. 2, 1896 to Jan. 5, 1897, 2 A OeutaClegh RA 3h) Sie ean eamog uke AOL OD oun a anes Balance Jan. 5, 1897 Boe 4, Pea dim 9 Meat eS $113.20 Interest on $113.20 to Jan. 5,1898.. . . . .... 6.79 Amount.) > . $119.99 Amount of $50 from March 17, 1897 Be ee 5, "1898, OVO G JIGME DTE oe, A eh aes 52.40 DSR ANPOGLOOG ES 5 cuba) Sool a ao isl eh $67.59 612 ARITHMETIC. 2. DANIELSONVILLE, Conn., April 15, 1896. On demand, I promise to pay to the order of Robert P. Webb, Two Thousand Dollars, value received, with interest. $ 2000,%%. JoHN J. BARNICLE. On this note are endorsed the following payments: $50, Sept. 20, 1896; $100, May 26, 1897; $1,000, June 20, 1898. How much is due Dec. 27, 1899? Face of note . . » ve 2000.00: As the first aphettte $50, is nor ieee aecneh 3 nee the in- terest then due, the interest on $2,000 is calculated to date of second payment, May 26, 1897, lyr. lmo.llda. . . 133.67 Amount die/May: 26, 18970 SI LP Re ae Sree Less payments, $o0 and $1009.) 25) heyaemey een 150.00 Balance May 26,1897. . . . pul other tes UAB LESSis A Interest on $ 1,983.67 to June 20, 1898, 1 yr. 24 ie ss stele Wiad 126.95 Amountidue Jane'20,11898) 4). salu aa) Same Less third payment). Argentine Republic........... Gold and silver..... PORGie an deeiaiel ene $0.96,5 Austria-Hungary ............. (CGR Seek eiche aeentor Crowng ts acanekl steals .20,38 1 Folin) CASO amo bnie echoes e Gold and silver ..... CAT Cesta ae rsye ees icicle eet ste 19,3 isXbhwhy AA seus ee os dete Shela SII? oie a eiierentn wine's BONViIanO) 26ers estas « 47,4 Brazilyacia « 5.12 .13,3 Chin Kiang. 74,9 Fuchau .... 70,9 Haikwan... .78,0 CHING Ree recess sie uese scien se Silveracsauseds acces Tele. ase Hankow .. ALT Niuchwang. CTS Ningpo .. BES Shanghai... .70,0 Swatow .... .70,8 Takanlesce .. ee Tientsin 74,8 GWolombia vende sssoneeeee wees Silvera PSG ier OR ee ae ok AT,4 Cubase ee eco s aacesee Gold and silver..... POO nSeos ee e eee eee 92,6 Grab kertatcsrete sare. cccicle de sien cores OL Scott cee baa cle Crowle inion coke “ .26,8 UCTIBCLOL setae a4 creme e's, vie eca selene Silvers. wen ee arrc ces PSULCh Gn trees Alena eine: 47,4 SUT Phe cis oh apelin en's wSideriees Golde aera scweeeee s Pound (100 piasters)....} 4,94,8 TTA CHAG. a5 AAR OSORIO EO BORe er Goldiarwecce on ate Mia rkterc ss sites td. loetter oe os 19,3 JOT NS ct PRO ee SAE cm ier Gold and silver ..... LNs COB MOREL On OCE 19,3 German Empire .............. Golde Rnewee sedescs IM air Keverne ste eisharet nevarane Dobe .28,8 Speer) ayy UE aad Sodciemapo Hpccig GrOLGUAE Ne Sake cieve crs 8s Pound sietlingy\ vo. a0. <- 4,86, 64 GCTECCER LE ene ia eee we vical Cold and silversseee| Drachinay -cesicee en ekuee 19,3 URE Meyer v's eu-s acis ale crelsieraietacate Gold and silver..... GouUurdeye eee ee eee 96,5 LGN og de pe SoA OEL a eae OEIee Silverio hase RU POC! aisionscearaieny owes 122.5 Wfslivttdctistreceis ste ssita uae ascigs Gold and silver..... Drache atc eee tele 19,3 MODAN Geass ns attra vest siess Gold and silver..... cons, eens Hants ar Liberia. ......-.......--2..--. reli fe RSP aces Wage tr DGUSE Ay rican ncet eens 1.00,0 INSGXIGO Bere e tne cierto ee oimicote.s Silvas soem coerce Dolareeeeces vce FOES INGCHErIANUS se tease oe eos Gold and silver..... IW fey bia aan aoreees or .40,2 Newfoundland................ ra aA ea Wee Miele ota eae cee gies 1.01,4 INOUWHVamere «ccuwialat scien ie croee Gold CO Wises taco estas ae .26,8 ROrsiaerrieyen oe aes c cules cca Silver sesame teams FTA re certs iste eres stein ers 08,7 POF... 2... e ee ee cece e cece cnes Silverds sy. soeeccee : Boley wescantesre es 47,4 Portugal .........-0sssceesees Goldivanaeco eae Milreig.€. calc ciate ts 1.08,0 ELURSIB Bese wierceies oe ieee ote cacs Sil verve sass tne Ruble.... : ae ton en 310 Fee OE Deli Har: es AER re ape AAR ae Gold and silvers.c.e nbesetanec sts iyeriefee « 19,3 BW OU Git nua ttre Go teenth en ccc Golder ors net CEOW iets ce sean keels ke .26,8 Pwitzenlandeoce voce seco nes Gold and silver..... Han Cacitee bart cio crateloa areas 19,3 PUUEKOyieerseth stlathine die elena GOId. aaue wins eaters’ RoW) gan Sno oracle se 04,4 MOTUS AY oe ales sla eam atde ed no nic Goldie en Sameera POSO eer aac tees oe 1.03,4 WWONOZMO Al utas seus ole tore castarcat Gold and silver..... olivaree cme aescate 19,3 sh , Li > i p Vat _ Tera navec utes hi yi Ay ’ oe ray AM Prd > Nd ~ paaleee ye 1 i y i] i - } iyi : ‘ jag it oy Ad Pasha) te ; oe 1a I } \ » Ag INDEX. The Roman numbers refer to pages in the Supplement. Abstract numbers, i. Acceptance, 451, xxvi. Accounts (see Manual), xii. Acurate interest, 494, 495. Definitions, etc., 494, xviii. - Acute-angle, 865, 557, 559, 561, 565. Acute-angled triangle, 365, 570, 571, 584, 585. Altitude of, 584, 585. Area of, 411, 412, 427, 584, 585. Construction of, 865, 565, 584. Addends, ii. Addition, of algebraic quantities, 565, 530, 531. Of decimals, 241, 283, 292, 440, 441, 482. Rule, ix. Of denominate numbers, 236, 270, 285, 287, 312, 329, 330, 336, 858, 406, 430, 433, 449, 484. Of fractions, 199, 200, 201, 202, 208, 204, 205, 249, 255, 262, 277, 309, 362, 364, 482, 433, 483, 484. Principle and rule, 203, ix. Of integers, 211, 212, 217, 275, 302, 339, 429, 430, 510. Cross-adding, 211, 216, 275, 302, 510. Definition, principle, and rule, ii. Adjacent angles, 365, 557, 560, 561. Ad valorem duty, xvi. Algebraic equations, one unknown quantity, 869-380, 534-538, 550-556, Algebraic equations, Continued. Clearing of fractions, 373. Transposing, 377. Removing parentheses, 534. Quadratics, affected, 551-556. Pure, 549-551. Three unknown quantities, 544- 547. Two unknown quantities, 588- 544. Algebraic quantities, addition of, 530, 531. Multiplication of, 534, 547, 548, 549. Square root of, 552, 553. Subtraction of, 532, 533, 584. Aliquot parts, U. S. money, 218, 219, 220, 221, 222, 228, 224, 225, 272, 273, 803, 804, 335, 397, 427, 447, 455, 475, 507. Interest by, 417, 418, 419, 420, 469, 470, xvii. Miscellaneous, 399, 405, 410, 421, 422, 429, 455, 471, 474, 484. Alligation, 502, 5438, 544. Altitude, of parallelogram, 366, 412, 569, 570, 578, 584. Of solid, 511, 593, 594, 595, 596, 604, Of triangle, 367, 412, 495, 496, 504, 508, 556, 569, 578, 584, 585. Amount, addition, ii. Interest, 394, xvi. Percentage, xiii. Analysis (see Cause and effect), 230, 236, 258, 259, 263, 264, 265,270, 271, 299, 300, 3is, 318, 829, 336, 352, 353, 362, 621 622 TNDE Xe Analysis, Continued. 364, 407, 409, 423, 436, 437, 438, 489, 442, 454, 461, 462, 477, 478, 479, 482, 483, 492, 493, 509, 510. Angles, acute, 865, 557, 559, 561, 565. Adjacent, 365, 557, 560, 561. Bisection of, 577. Construction of, 559, 560, 561, 562, 563, 564, 565, 566, 567, 569, 577, 578, 579, 580, 581, 582, 585. Designation of, 558. Exterior, 565. Formed by parallel lines and a secant, 563, 564, 565, 590. Kinds of, 365, 557. Measurement of, 558, 559, 560, 561, 562, 563, 564, 565, 566, 567, 569, 577, 578, 579, 580. Oblique, 557. Obtuse, 365, 557, 561, 566. Of parallelogram, 569, 590. Of polygon, 574, 579. Of triangle, 564, 565, 579. Opposite, 557, 561. Right, 565, 491, 557, 559, 560, 561. Vertical, 557, 561. Annual interest notes, 523, 613- 617, xviii. Partial payments on, 613, 614. New Hampshire rule, 614, 615. Vermont rule, 615, 616, 617. Antecedent, 466, xxi. Apothecaries’ weight, 361, 406, 607. Apothem, of regular hexagon, 591. Of regular octagon, 591. Applications, of algebra, 383, 384, 385, 387, 389, 400, 409, 415, 422, 424, 434, 452, 460, 480, 490, 498, 502, 503, 555. Of percentage (see Percentage). Of square root, 270, 485, 486, 490, 491, 508, 504. Approximations, 226, 282, 305, 306, 353, 360, 361, 398, 442. Arabie notation, 210, 240, i. Arc, 516, 572, 577, 580. Bisection of, 576, 577. Are, Continued. Length of, 450, 451, 572, 580, 590, 598. Area (see Measurements). Of circle, 461, 496, 503, 504, 516, 517, 526, 592. Of rectangle, 246, 247, 248, 262, 276, 283, 284, 286, 288, 289, 290, 297, 800, 301, 315, 338, 342, 343, 344, 345, 352, 353, 054, 855, 356, 389, 390, 397, 398, 400, 401, 411, 412, 413, 434, 435, 486, 461, 462, 475, 476, 482, 504, 506, 517, 525, 526, 550, 555, 584. Of regular polygon, 591. Of hexagon, 504, 591. Of octagon, 591. Of rhomboid and rhombus, 367, 412, 413, 496, 508, 504, 570. Of ring, 517, 592. Of sector, 516, 592. Of segment, 516. Of trapezium, 368, 413, 495, 504, 508, 571. Of trapezoid, 367, 368, 412, 413, 434, 490, 491, 504, 571. Of triangle, oblique-angled, 367, 411, 412, 461, 495, 496, 503. Right-angled, 322, 823, 355, 356, 367, 411, 412, 453, 490, 496, 526, 556. Assessed value (see Taxes), 385, 407, 444, 446,510, 617, 618, xvi. Assets, xxii. Avails (see Proceeds), 402. Average, of accounts, 500. Term of credit, 499, 500, 501. Rules, etc., xxi. Avoirdupois weight, table, 227, 361, 406, 608. Long ton, 541, 608. 577, Balance, xii. Bank check (see Manual), xxvi. Bank discount, definitions, etc., 402, 403, xix. Of interest-bearing notes, 407, 408. Of notes without interest, 402, 403, 404, 420, 421, 455, INDEX. Bank discount, Continued. Proceeds (avails), 402, 403, 404, 420, 421, 483. Term of discount, 402, 403, 404, 421, 424, 470, xix. To find face of note, etc., 424, 425, 470, 494. Barrel, 607. Base, bonds, 444, 488, xxv. Commercial discount, xiv. Commission, 444, xv. Duties, xvi. Insurance, 444, xv. Line (U.S. lands), 524. Of isosceles triangle, 366, 567. Of prism, 592. Percentage, 3838, xiii. Profit and loss, 386, 387, xiv. Stocks, 444, 448, xxv. Taxes, 444, xvi. Bill, 288, 239, 274, 319. Definitions, etc., xii. How receipted, xii. Of exchange, 448, 455, 456, 457, XXvVl. Bins and tanks, capacity of, 359, 560, 361, 389, 402, 406, 433, 435, 442, 461, 482, 514, 527. Bisection of angle, 577. Of arc, chord, and sector, 576, yee Of line, 575. Board, foot, 514, 515, 516. Measure, 515. Bonds and stocks, 888, 442, 443, 444, 445, 488, 489. Brokerage, 4438, 444, 488. Definitions, etc., 442, xxv. Broker, xxv. Brokerage, 443, 444, 488, xxv. Business forms, 238, 274, 319, 394, 402, 407, 448, 451, 455, 458, 460, 466, 496, 498, 523, 613, 614, 615, 616, 617. Definitions, xxiv. Calculation of heights and dis- tances, 436, 587, 588, 589, 590. Canadian money, 608. Cancellation (see Manual), 263, 264, 265, 275, 289, 295, 3807, 623 Cancellation, Continued. 318, 314, 850, 862, 889, 3890, 402, 414, 435, 487. Definitions, etc., 268, 264, v. Capacity of bins and tanks, 359, 360, 361, 389, 402, 406, 483, 435, 442, 461, 482, 514, 527. Capital, xxii. Capital stock, xxiv. Carpeting, 301, 3845, 889, 401, 434, 461, 462, 492. Cause and effect (see Analysis), 436. Chain, surveyors’ 461, 491, 607. Check (see Manual), xxv. Chord, 572. Bisection of, 576, 577. Length of, 516, 572, 582, 590, 598. Circles, area of, 461, 496, 508, 504, 516, 517, 526, 592. Circumference of, 450, 468, 469, 503, 571. Circumscribed, 504, 582. Concentric, 572, 592. Diameter of, 450, 468, 469, 503, 571, 592. Great, 518, 522, 598. Inscribed, 582. Radius of, 469, 496, 503, 571. Small, 598. Tangent, 572, 584. Circular measure, 450, 558, 608. Circumference, 450, 468, 469, 508, 571. Circumscribed, circle, 504, 582. Polygon, 573, 574, 577, 582, 583, 584. Square, 574, 577. Triangle, 574. Clearing of fractions, 373, 477. Coefficient (see Manual), 530. Commercial discount, 410, 411, 421, 422, 431, 446, 456, 457, 475, 479, 480, 481. Definitions, xiv. Commission, 317, 384, 385, 398, 423, 449, 482. Agent, xiv. Definitions, ete., xiv. measure, 441, € Common, denominator, 206, 267, 284, ix. Factor, 252, v. Fraction, vii. Multiple, vi. Prime factor, v. Complex, decimal, 292, 296, x. Fraction, viii. Reduction of, 266, 268, 271, 324, 364, 4338, 484, x. Composite, factor, v. Number, v. Compound, addition, 286, 270, 285, 287, 312, 829, 330, 336, 358, 406, 480, 483, 449, 484. Decimal, x. Denominate number, xii. Division, 285, 288, 307, 308, 314, 315, 333, 334, 335, 336, 351, 352, 358, 406, 408, 4382, 433, 439, 449, 454, 482, 485, 492. Fraction, 266, 271, 324, 564, 482, 433, 484, viii. _ Interest, 445, 446, 487, xvili. Multiplication, 228, 285, 287, 307, 314, 332, 333, 385, 336, 341, 358, 406, 408, 433, 450, 454, 466. Numbers, reduction of, 202, 227, 228, 245, 261, 270, 286, 311, 326, 327, 328, 335, 336, 338, 841, 356, 406, 408, 483, 441, 444, 475, 492, 404. Subtraction, 208, 285, 287, Abpapre WAnbrhs Gainritey 358, 430. Concentric circles, 572, 592. Concrete number, i. Cone, definitions, etc., 594. Surface of, 512, 518, 595, 598. Developed surface, 512, 513, 595, 596. Volume of, 513, 514, 600, 604, 605. Frustum of, 596. Surface of, 597. Volume of, 603. Connecticut rule, partial payments (see Manual), 611, 612, 613. 203, 285, 329, 3577, 445, 288, 348, INDEX. Consequent, 466, xxi. Consignee, Xv. Consignment, Xv. Consignor, Xv. Construction, of angles, 559, 561, 562, 563, 564, 565, 567, 569, 577, 578, 579, 581, 582, 585. Of parallel lines, 562, 563, 564. Of parallelograms, 366, 569. Of rectangle, 366, 367, 570, 571, 578, 584, 592. Of regular polygon, 574, 576, 577, 578, 579, 590, 591. Of rhomboid, 366, 867, 570, 578. Of rhombus, 3866, 867, 569, 578, 579, 588. Of square, 569, 573, 574, 576, 577, 578, 579, 583. Of trapezium, 366, 570, 571. Of trapezoid, 366, 569, 570, 571. Of triangle, acute-angled, 365, 571, 584. Equilateral, .866, 566, 567, 574, 575, 577, 579, 580, 582, 583, 584. Isosceles, 366, 867, 567, 575, 578, 579, 582, 584. Obtuse-angled, 571, 5838, 584. Scalene, 366, 564, 576, 578, 579, 580, 582, 584, 585, 586. Right-angled, 365, 367, 566, 567, 578, 579, 580, 581, 584, 586. Convex surface, of cone, 512, 518, 595, 598. Of cylinder, 511, 593, 595. Of hemisphere, 599. Of frustum, of cone, 597. Of pyramid, 596. Of prism, 511, 512, 592, 595. Of pyramid, 512, 595, 596. Of solids, development of, 511, 512, 518, 594, 595, 596, 597. Couplet, xxi. Creditor, xi. Cross-addition, 211, 216, 2'75, 302, 510. Subtraction, 339. Cube (third power), 519. 560, 566, 580, 365, INDEX. 625 Cube, Continued. Definition, 519, xxiii. Root, 507, 519, 520, 522. Of decimals, 520, 522. Of fractions, 520, Cubie measure, 361, 402, 607. Customs (see Duties), xvi. Cylinder, definition, etc., 593. Surface of, 511, 512, 598, 595. Volume of, 513, 599, 604. Dates, time between, 345, 346, 347, 348, 398, 404, 458, 459, 404, 498, 609, xiii. Days of grace, 402, 4038, 451, 458, 609. Debtor, xi. Decagon, 573. Decimals, addition of, 241, 283, 292, 440, 441, 482. Cube root of, 520, 522. Definitions, etc., x, xi. Division of, 244, 279, 280, 281, 282, 283, 293, 294, 295, 306, 364, 480, 440, 441, 442, 482. Miscellaneous, 324, 325, 364, 440, 441. Multiplication of, 242, 243, 278, 279, 282, 283, 293, 295, 306, 440, Notation and numeration of, 238, 239, 440, 441, 483. Reduction of, 291, 292, 296, 440, el eo. Square root of, 481, 482, 489. Subtraction of, 242, 283, 293, 364, 440, 441. Definitions, ete., i, xxvi. Abstract number, i. Acceptance, 451, xxvi. Account current, xii. Accurate interest, 494. Acute angle, 365, 557. Acute-angled triangle, 565. Addends, ii. Addition, ii. Adjacent angles, 365, 557. Ad valorem duties, xvi. Altitude, of parallelogram, 366, 569. Of prism, 593. Definitions, ete., Continued. Of pyramid, 594. Of triangle, 569. Amount, addition, ii. Interest, 394, xvi. Percentage, xiii. Angles, 547. Annual interest, xviii. ‘Antecedent, 466, xxi. Arabic notation, i. Are, 572. Assets, xxii. Avails (see Proceeds). Average, term of credit, xxi. Time, 500. Balance, xil. Bank discount, xix. Barrel, 607. Base, bonds, xxv. Commercial discount, xiv. Commission, xv. Duties, xvi. Insurance, xv. Line (U.S. lands), 524. Of isosceles triangle, 366, 567. Of prism, 592. Percentage, 383, xiii. Profit and loss, 386, xiv. Stocks, xxv. Taxes, XVi. Bill, xii. Of exchange, domestic, 448, xii. Foreign, 448, xii. Board foot, 514. Bonds, xxv. Broker, xxv. Brokerage, xxv. Cancellation, v. Capital, xxii. Stock, xxiv. Check, xxvi. Chord, 572. Circles, great, 518, 598. Small, 598. Circumference, 571. Coefficient, 530. Commercial discount, xiv. Commission, Xiv. Agent, xiv. Common denominator, ix. 626 Definitions, etce., Continued. Factor, v. Fraction, vii. Multiple, vi. Prime factor, v. Complex, decimal, x. Fraction, viii. Composite, factor, v. Number, v. Compound, decimal, x. Denominate number, xii. Fraction, viii. Interest, xviii. Concentric circles, 572. Concrete number, i. Cone, 594. Consequent, 466. Consignee, Xv. Consignment, xv. Consignor, Xv. Convex surface, 511, 592. Couplet, xxi. Creditor, xi. Cube (third power), 519, xxiii. toot, 519, xxiii. Cylinder, 593. Debtor, xi. Decagon, 573. Decimal, x. Fraction, x. Point, x. Demand note, xxv. Denominate, fraction, xii. Number, xii. Unit, xii. Denominator, 203, vii. Diagonal, 576. Diameter of circle, 571. Difference, subtraction, ii. Percentage, xiii. Discount, stocks, xxiv. Dividend, division, iv. Stocks, 442, xxv. Division, iv. Divisor, iv. Domestic exchange, xx. Draft, 448, xxvi. Drawee, xxvi. Drawer, xxvi. Duties, xvi. Endorser, xxv. INDEX. Definitions, etc., Continued. Equal triangles, 585. Equated time, 500, xx. Hquation of payments, 500, xx. Equilateral triangles, 366, 566. Equivalent triangles, 585. Even number, v. Evolution, xxiii. Exact, divisor, v. Interest, 494. Exchange, domestic, xx. Foreign, xx. Exponent, 547, xxiii. Exterior angle, 565. Extremes, 477. Face of note, xxv. Factor, v. Factoring, v. Fire insurance, xv. Foreign, bills of exchange, 448. Exchange, xx. Fraction, vii. Fractional unit, vii. Frustum, 596. Great circle, 518, 598. Greatest common, divisor, 252, v. Factor, v. Measure, v. Gross, 259. Heptagon, 573. Hexagon, 573. Horizontal line, 560. Improper fraction, viii. Indorser, xxv. Insurance, xv. Insured, xv. Insurer, xv. Interest, xvi. Interest-bearing note, xxv. Invoice, xii. Involution, xxiii. Isosceles triangle, 566. Joint and several note, xxvi. Joint note, xxv. Least common denominator, 203, 255, ix. Multiple, 205, vii. Legal rate of interest, xvi. Liabilities, xxii. Life insurance, xv. Like numbers, i. INDEX. 627 Definitions, etc., Continued. Lowest terms, 202 Maker of note, xxv. Marine insurance, xv. Market value of stock, etc., xxiv. Maturity of note, xxvi. Means, 477. Measure, xii. Minuend, ii. Mixed, decimal, x. Number, 199, viii. Multiple, 204, vi. Multiplicand, iii. Multiplication, iii. Multiplier, iii. Negotiable note, xxvi. Net, price, xiv. Proceeds, xv. Nonagon, 573. Non-negotiable note, xxvi. Notation, i. Arabic, i. Roman, i. Number, i. Numeration, i. Numerator, 203, vii. Oblique angle, 557. Oblique-angled triangle, 565. Oblique, line, 560. Solid, 593. Oblong, 568. Obtuse angle, 365, 557. Obtuse-angled triangle, 565. Octagon, 573. Odd number, v. Opposite angles, 557. Orders of units, i. Parallel lines, 562. Parallelogram, 366, 568. Parallelopipedon, 593. Partial payments, xix. Partners, xxii. Partnership, xxii. Par value, xxiv. Payee, xxv, xxvi. Pentagon, 573. Pentagonal prism, 593. Per cent, 316, xiii. Percentage, xiii. Period, i. Perpendicular lines, 561. Definitions, ete., Continued. Personal property, xvi. Lax; xvi Place of figure, i. BONG, ORY: Poll tax, xvi. Polygon, 578. Power, xxiii. Premium, insurance, xv. Stocks, etc., xxv. Present worth, 460. Prime, factor, v. Numbers, 251, v. Principal, xvi. Principal meridian, 524. Prism, 592. Proceeds, xix. Product, iii. Profit or loss, xiv. Promissory note, xxv. Proper fraction, viii. Proportion, 476. Property, xvi. Tax, Xvi. Protractor, 559. Pyramid, 594. Quadrangular prism, 593. Quadrant, 572. Quadrilateral, 866, 568. Quotient, iv. Radius, 571. Range (U.S. lands), 524. Rate, xiii, xvi. Ratio, 466, xxi. Real property, xvi. Receipt, xii. Reciprocal, of a fraction, viii. Of a number, viii. Rectangle, 366, 568. Reduction, ascending, xiii. Descending, xii. Of denominate numbers, xii. Of fractions, viii. Regular, polygon, 573. Solid, 593. Remainder, ii. Rhomboid, 366, 568. Rhombus, 366, 568. Right angle, 365, 491, 557. Right-angled triangle, 565. Right solid, 593. 28 INDEX. Definitions, etc., Continued. Root, xxiii. Roman notation, i. Scalene triangle, 366, 566. Secant line, 582. Second power, xxiii. Section (U.S. lands), 524. Sector, 572. Segment, 516, 572. Sextant, 572. Share (stock), xxiv. Sight draft, xxvi. Simple, denominate number, xii. Fraction, viii. Slant height, 594. Small circle, 598. Specific duty, xvi. Sphere, 598. Square, 568. (Second power), 463, xxiii. Root, xxiii. Stock, xxiv. Stockholder, xxiv. Subtraction, ii. Subtrahend, ii. Sum, ii. Discounted, xix. Tangent, circles, 572. Line, 582. Taxes, Xvi. Term of discount, 403, xix. Terms of fraction, vii. Third power, xxiii. Time, draft, xxvi. Note, xxv. Township (U.S. lands), 524. Transposing, 377. Trapezium, 366, 568. Trapezoid, 366, 568. Triangular prism, 593. True discount, 460. Underwriter, xv. Unit, i. Of fraction, vii. Unlike numbers, i. Value of fraction, vii. Vertical, angles, 557. Line, 560. Degrees of longitude, length of, 450, 451, 454, 507, 598. Demand notes, 394, 458, 496, | Demand notes, Continued. 612, 618, 614, 615, 616, 617, xx Ve Denominate fraction, 228, 285, 311, 327, 328, 329, 433, 484, xii. Denominate numbers, addition of, 236, 270, 285, 287, 312, 329, 330, 336, 358, 406, 480, 453, 449, 484, Definitions, ete., xii. Division of, 285, 288, 307, 308, 314, 315, 3833, 334, 535, 536, 351, 352, 358, 406, 408, 453, 439, 449, 454, 482, 483, 492, 510. Metric system, 525, 526, 527, 528, 529. Miscellaneous, 227, 300, 301, 307, 308, 309, 328, 329, 335, 336, 341, 351, 352, 358, 362, 364, 406, 408, 483, 482, 483, 484, 492. Multiplication of, 228, 285, 287, 314, 332, 338, 335, 336, 341, 358, 406, 408, 433, 450, 454, 466. Reduction of, 202, 208, 227, 228 245, 261, 270, 285, 286, 287, 311, 326, 327, 328, 329, 335, 336, 338, 341, 356, 357, 406, 408, 433, 441, 444, 445, 475, 492, 494. Subtraction of, 208, 285, 287, 288, 312, 331, 332, 335, 336, 348, 358, 430. Denominate unit, 284, xii. Denominator, 203, 548, vii. Least common, 203, 249, 255, ix. Developed surface, of cone, 512, 513, 595, 596. Of cube, 290. Of frustum, of cone, 596, 597. Of pyramid, 596, 603. Of prism (see Manual), 345, 356, 402, .434, 511, 594. Of pyramid, 512, 595, 596, 600, 603. Diagonal, of polygon, 576, 579. Of rectangle, 322, 491, 504. Of rhomboid, 581. Of rhombus, 496, 504. INDEX. Diagonal, Continued. Of square, 322, 491, 503, 504. Of trapezium, 495, 504, 508. Diameter, of circle, 450, 468, 469, 503, 571, 592. Of sphere, 518, 519, 597, 599, 605. Difference, between dates, 345, 346, 347, 348, 398, 404, 458, 459, 494, 498, 609, xiii. Subtraction, ii. Percentage, xiii, xiv, Xv, XXv. Discount, bank, 402, 403, 404, 407, 408, 420, 421, 423, 424, 425, 455, 470, 4838, 494, xix. Commercial, 410, 411, 421, 422, 431, 446, 456, 457, 475, 479, 480, 481, xiv. Exchange, 448, 449, 451, 452, 485, Xx. Stocks and bonds, xxv. True, 460, 494. Distances, calculation of, 587, 588, 589, 590. Dividend, division, 235, 236, iv. Stocks, 442, 448, xxv. Divisibility of numbers, 252. Division, of decimals, 244, 279, 280, 281, 282, 288, 298, 294, 295, 306, 364, 480, 440, 441, 442, 482. Rule, 280, xi. Of denominate numbers, 285, 288, 307, 308, 514, 3815, 333, 334, 335, 336, 851, 352, 358, 406, 408, 483, 489, 449, 454, 482, 483, 492, 510. Of fractions, 209, 214, 215, 216, 250, 256, 267, 268, 269, 271, 278, 296, 307, 310, 324, 363, 364, 427, 481, 482, 483, 483, 484, Rules, 267, x. Of integers, 235, 2386, 263, 341, 400, 480, 465. Principles and rule, iv. Of lines into equal parts (see Bisection ), 581, 587. Divisor, 236, iv. Domestic exchange, 448, 449, 451, 452, 485. 629 Domestic exchange, Continued. Definitions, etc., 447, 448, xx, =x Vi Drafts, sight, 448, 449, 452, 485. Time, 451, 452, 485. Acceptance of, 451. Definitions, xx, xxvi. Drawee, xxvi. Drawer, xxvi. Drills, sight, 307, 337, 505. Dry measure, 227, 607. Duties, 290, 300, 317, 352, 384, 466, 515. Definitions, etc., xvi. 229, 232, 308, 306, 881, 396, 426, 474, Endorsement (see Manual), 458, 459, 460, 496, 611, 612, 618, 614, 615, 616, 617, xix. Endorser, xxv. Liability of, xxv. English money, 408, 480, 482, 441, 442, 455, 456, 466, 482, 483, 484, 492, 510. Tables, 408, 608. Equal triangles, 522, 584. Equated time, 500, xx. Equation of payments, 499, 500, 501. Definitions, etc., 500, xx. Equations (see Algebraic equa- tions), 369, 530. Equilateral triangle, altitude of, 496, 585, 591. Area of, 496, 503, 504, 600. Construction of, 570, 574, 575, 577, 579, 580, 581, 582, 583, 584. Definition, 566. Equivalent triangle, 585. Erection of perpendicular, at any point in line, 560, 577. At extremity of line, 577. At middle of line, 575. Through point outside of line, 579. Even numbers, v. Evolution, 468, 519. Definitions, etc., 468, 519, xxiii. Exact divisor, v. 630 Exact interest, 494, 495. Rule, 3tc., xviil. Exchange, domestic, 448, 449, 451, 452, 485. Foreign, 455, 456, 457. Definitions, 447, 448, xx. Exponent, 463, 519, xxiii. Face of bill of exchange, to find, 456. Of draft, to find, 448, 449, 452, 485, xx. Of note, xxv. To find, 424, 425, 470, 494, ab ah Face value of stocks and bonds, 388, 443, 488, xxiv. Factors, 204, 205, 250, 251, 291, 507, v. Prime, 291, 507, v. Federal money, 212, 216, 217, 218, 224, 429, Fractional parts of a dollar, oral, 218, 219, 220, 221, 222, 223, 924, 295, 272, 278, 299, 300, 303, 304, 335, 397, 3898, 427, 447, 455, 475, 507. Fence, area of, 289, 348, 354, 390, 435, 516. Fire insurance, 317, 364, 384, 385, 386, 423, 444. Definitions, etc., xv. Foreign coins, value of, 619. Foreign exchange, 448, 455, 456, 457, Xx. Fractional unit, vii. Fractions, addition of, 199, 200, 201, 202, 208, 204, 205, 249, 255, 262, 277, 309, 362, 364, 432, 453, 483, 484. Clearing of, 373, 477. Complex, 266, 268, 271, 324, 364, 433, 484. Compound, 266, 271, 824, 364, 432, 433, 484. Cube root of, 520, xxiv. Decimal, x. Definitions, principles,and rules, Vile vail Seen ke Denominate, 228, 285, 311, 327, 328, 329, 4383, 484. INBEX., Fractions, Continued. Division of, 209, 214, 215, 216, 250, 256, 267, 268, 269, 271, 278, 296, 307, 310, 324, 363, 364, -427, 431, 482, 433, 483, 484, Miscellaneous, 205, 207, 208, 209, 250, 256, 266, 268, 310, 324, 364, 482, 483, 483, 484. Multiplication of, 209, 213, 214, 218, 250, 256, 265, 266, 268, 277, 282, 307, 310, 323, 340, 368, 364, 399, 405, 482, 433, A471, 484. Reduction of, 199, 200, 201, 202, 203, 205, 252, 253, 256, 257, 9258, 284, 325, 482, 484. Square root of, 478, xxiv. Subtraction of, 205, 206, 207, 250, 256, 262, 268, 277, 296, 310, 324, 363, 364, 482, 433, 483, 484. ; French money, 456, 476, 478, 608. Frustum of cone, 596. Developed surface, 597. Surface, 597. Volume, 603. Of pyramid, 596, 597. Developed surface, 596, 6038. Surface, 596, 597. Volume, 601, 602. Furlong, long measure, 607. Geometrical exercises, 365-367, 557-605. Angles, 365, 557, 558, 559, 560, 561, 562, 568, 564, 565, 566, 567, 569, 577, 578, 579, 580, 581, 588, 585, 590. Calculating heights and dis- tances, 587, 588, 589, 590. Circles, 571, 572, 578, 574, 575, 576, 577, 579, 580, 582, 583, 584, 590, 591, 592, 598. Construction problems, 575-584, Mensuration (see Measure- ments ), 600-605. Parallels, 562, 563, 564, 565, 587, 588, 589, 590. Polygons, 5738, 574, 576, 577, 579, 584, 591, 592. INDEX. Geometrical exercises, Continued. Quadrilaterals, 3866, 367, 568, 569, 570, 571, 574, 577, 578, 579, 580, 581, 582, 583 584, 590. Triangles, 366, 367, 564, 565, 566, 567, 570, 5738, 574, 575, 576, 577, 578, 579, 580, 581, 582, 583, 584, 586, 587, 588, 589, 590. German money, 442, 456, 476, 608. Government lands, 524, 525. Grace, days of, 402, 403, 451, 485, 609. Great circle, 518, 522, 598. Greatest common divisor, 252, 253, 258, 482, v, vi. Factor, v. Measure, v. Heights, calculation of, 436, 587, 588, 589, 590. Heptagon, 573, 574. Hexagon, 504, 578. Apothem of regular, 591. Area of, 504, 591. Construction of, 578, 574, 579, 584. Perimeter of, 591. Horizontal line, 560. Hypotenuse, 486, 490, 491, 496, 506, 525, 526, 555, 556, 580, 581, 596. Improper fraction, 266, 267, viii, ja. Indorsement, xix. Indorser, Xxv. Inscribed polygon, 578, 574, 576, 577, 579, 591, 594. Square, 504, 516, 574, 576, 577. Triangle, equilateral, 574, 577, 579, 582. Right-angled, 579, 580, 582. Scalene, 579, 582, 585. Insurance, 317, 364, 384, 385, 386, 423, 444. Definitions, etc., xv. Interest, 320, 321, 349, 350, 351, 352, 353, 384, 3886, 393, 394, 395, 398, 404, 407, 408, 415, 631 Interest, Continued. 416, 417, 418, 419, 420, 421, 423, 431, 488, 442, 443, 445, 446, 458, 459, 460, 469, 470, 482, 487, 493, 494, 495, 496, 497, 498, 528, 610, 613, 614, 615. Accurate, 494, 495. Annual, 528, 613, 614, 615, xviii. Compound, 445, 446, 487, xviii. Definitions, etc., xvi, xvii, xviii, xi Exact, 494, 495, xviii. General method of computing, 301, 393, Xvii. Legal rates of, 610. Method by aliquot parts, 417, 418, 419, 420, 469, 470, xvii. Notes bearing interest, 394, 395, 407, 408, 458, 459, 460, 523, So aia Partial payments, 458, 459, 460, 611, 612, 618, 614, 615, GIG, Olidy xix. Six per cent method, xviii. To find principal, rate, or time, 384, 3886, 395, 415, 416, 417, 470, 482, 489, 4938, 494, xvii. Invoice, 466. Definitions, xii. Involution, 463, 464, 519. Definitions, etc., 463, 519, xxiii. Isosceles triangle, altitude of, 495, 567, 575, 578, 584, 594. Area of, 367, 411, 412, 495, 496, 503. Base of, 567. Construction of, 366, 367, 567, 575, 576, 578, 579, 582, 584. Definitions, 366, 566. Joint and several note, 611, xxvi. Joint note, xxv. Leap years, 608. Least common denominator, 203, 249, 255. Definitions, etc., 205, 255, ix. Least common multiple, 205, 249, 254, 255, 432, 433. Definitions, etc., 205, vi, vii. 632 Ledger account (see Manual), ya Legal rates of interest, 610. Liabilities, xxii. Liability of indorser, xxv. Life insurance, Xv. Like numbers, i. Lines, bisection of, 575. Division of, into equal parts, 581, 587. Horizontal, 560, 561, 562, 563, 576, 581. Oblique, 560, 561, 562, 563, 567, 575, 576, 577, 578, 581. Parallel, 562, 563, 564. Perpendicular, 365, 561, 577,579. Tangent, 582. Secant, 582. Vertical, 660. Link, 441, 486, 607. Liquid measure, table, 227, 607. Long division (see Division). Drills, 232. Longitude and time, 452, 453, 454, 471, 472, 478. Length of degrees of longitude, 450, 451, 454, 507, 598. Long measure, 285, 607. Long ton, 541, 608. Lowest terms, 202, 203; 205, 252, 253, 256, 257, 258,, 284, 292, 296, 325, 432, 484, viii. Lumber measure, 514, 515. Market value of stocks and bonds, SOKA Vs Maturity of note (see Days of . grace), 403, xxvi. Means, 477. Measure, xii. Measurements, miscellaneous, 342, 343, 344, 345, 358, 354, 355, 306, 358, 359, 360, 389, 390, 400, 401, 402, 411, 412, 413, 414, 488, 484, 435, 486, 461, 462, 490, 491, 492, 493, 495, 496, 508, 504, 511, 512, 513, 514. Of arcs, 450, 451, 572, 577, 580, 590, 598. Of angles, 558, 559, 560, 561, INDEX. Measurements, Continued. 562, 563, 564, 565, 566, 567, 569, 577, 579, 580, Of bins, 359, 360, 361, 435, 461. Of boards, 434, 514, 515, 516. Of carpeting, 301, 345, 389, 401. 434, 461, 462, 526. Of chords, 516, 572, 582, 590, 598. Of circles, 461, 496, 5038, 504, 516, 517, 526, 592. Of circular rings, 517, 592. Of circumference, 450, 468, 469, 503. Of cone, 512, 513, 514, 595, 596, 598, 600, 604, 605. Of cylinder, 511, 512, 514, 519, 522, 600. Of degrees of longitude, 450, 451, 507, 598. Of fencing, 289, 343, 354, 389, 400, 401, 4385. Of frustum, 596, 597, 601, 602, 603. Of hexagon, regular, 504, 591. Of hypotenuse, 486, 491, 496, 506, 525, 526, 555, 556, 580, 581, 596. Of lumber, 514, 515. Of octagon, regular, 591. Of parallelopipedon, 290, 316, 345, 358, 359, 360,. 362, 390, 414, 434, 461, 462, 493, 527. Of prism, 511, 512, 513, 595, 599, 600. ; Of pyramid, 512, 513, 595, 596, 600, 601, 602, 604. Of rectangle, 246, 247, 248, 262, 276, 288, 284, 286, 288, 289, 290, 297, 300, 301, 315, 342, 348, 344, 345, 353, 354, 355, 356, 366, 367, 3889, 390, 401, 411, 412, 413, 427, 431, 485, 436, 461, 462, 475, 476, 482, 485, 491, 504, 506, 526, 550, 555, 584. Of regular polygon, 591. Of rhomboid, 367, 412, 418, 503, 584. Of rhombus, 3867, 412, 413, 496, 504, INDEX. Measurements, Continued. Of rooms, 289, 315, 316, 345, 359, 389, 390, 400, 434, 461, 462. Of sector, 516, 592. Of segment, 516. Of sphere, 518, 519, 521, 522, 599, 605. Of tanks, 360, 389, 402, 406, 433, 435, 461, 482, 514, 527. Of trapezium, 367, 368, 413, 495, 504, 508, 556. Of trapezoid, 355, 367, 368, 412, 413, 484, 490, 491, 504. Of triangle, right-angled, 322, 328, 355, 367, 411, 412, 433, 461, 486, 490, 491, 496, 526, 551, 555, 584, 585, 586. Oblique-angled, 356, 367, 411, 412, 461, 495, 496, 503, 504, 508, 556, 584, 585, 586. Of wood, 359, 402, 485, 527. Measures, circular, 608. Of capacity, 227, 361, 402, 607. Of extension, 285, 607. Of time, 226, 608. Of value, 408, 608. Of weight, 227, 361, 406, 607, 608. Mensuration (see Measurements ). Merchants’ rule, partial payments, 496, 497, 498. Metric system, 525, 526, 527, 528, 529. Minuend, 235, ii. Mixed decimal, x. Mixed numbers, 199, 257, 266, 267, viii, ix. Multiple, 204, 205, 249, vi, vii. Least common, 205, 254, 255, 432, 433, vi, vii. Multiplicand, 235, 300, iii. Multiplication of algebraic quan- tities, 547, 548, 549. Of decimals, 242, 243, 278, 279, 282, 283, 293, 295, 306, 364, 440, 441. Rule, xi. Of denominate numbers, 228, 285, 287, 314, 332, 338, 335 336, 341, 358, 406, 408, 433, 450, 454, 466. 6338 Multiplication, Continued. Of fractions, 205, 209, 218, 214, 218, 250, 256, 265, 266, 268, 277, 282, 307, 310, 328, 340, 363, 364, 399, 405, 432, 433, 471, 484. Principles and rule, viii, ix. Of integers, 234, 303, 305, 307, 340, 428, 429, 471. Principles, definition, and rule, iii, iv. Cross-multiplication, 212. Short methods, 229, 260, 276, 298, 303, 305, 323, 337, 340, 363, 396, 399, 426, 428, 429, 471. Multiplier, 235, 300, iii. Negotiable note, 394, 402, 407, 458, 460, 496, 523, 611, 612, 618, 614. Definition, xxvi. Net. price, xiv. Net proceeds, xv. New Hampshire rule, partial pay- ments, 614, 615. Nonagon, 573. Construction of regular, 574, 590. Non-negotiable note, 615, 616, 617, XXvi. Notation of decimals, 239, 240, 441, x. Of integers, 210, i. Roman, i. Notes (see Manual), xxv, xxvi. Demand, 394, 458, 496, 612, 613, 614, 615, 616, 617, xxv. Interest-bearing, 394, 407, 458, 460, 496, 611, 612, xxv. Annual interest, 523, 613, 614, 615, 616, 617, xviii. Joint, xxv. Joint and several, 611, xxvi. Negotiable, 394, 402, 407, 458, 460, 496, 523, 611, 612, 613, 614, xxvi. Non-negotiable, 615, 616, 617, XXVi. Time, 402, 407, 460, 528, 611, 612, xxv, 634 Notes, Continued. Days of grace, 402, 403, 451, 485, 609. Numeration of decimals, 239, 240, 440, 441, 483, x. Of integers, 210, 211, i, ii. Oblique angle, 557, 558. Cone, 594, 604. Cylinder, 593, 604. Line, 560, 561, 562, 563, 567, 575, 576, 577, 578, 581. Prism, 593, 604. Pyramid, 594, 604. Oblique-angled triangle, 565. Area of, 356, 367, 411, 412, 461, 495, 496, 503, 504, 508, 556. Oblong (see R ectangle), 568. Obtuse angle, 365, 557, 559, 561, 565, 566. Obtuse-angled triangle, 365, 565. Altitude of, 569, 578, 584. Area of, 412, 570, 571, 584, 585. Construction of, 571, 583, 584. Octagon, 573. Apothem of regular, 591. Area of, 591. Construction of, 574, 576, 579. Perimeter of, 591. Odd numbers, v. Opposite angles, 557, 561. Oral, exercises, 199, 200, 201, 205, 206, 214, 218, 219, 22 7 221) ‘ 222, 223° 224, 239, 250, 267, 278, 284, 316, 317, 328, 349, 369, 378, 382, 385, 386, 417, 468, 466, 467, 480, 489. Problems, 202, 203, 230, 236, 257, 258, 261, 269, 272, 2738, 276, 284, 299, 308, 304, 807, 308, 818, 335, 338, 345, 346, 351, 352, 397, 398, 411, 423, 427, 436, 446, 447, 452, 453, 454, 455, 467, 468, 475, 476, 478, 491, 492, 499, 406, 507. Orders of units, 308, i. Par sixk. Parallel lines, 562. INDEX. Parallel lines, Continued. Construction of, 562, 5638, 564. Parallelogram, altitude of, 366, 569. Angles of, 569, 590. Area of (see Rectangle, Rhom- boid, etc. ), 412, 604. Construction of, 366, 569, 570, 574, 577, 578, 579, 580, 582, 5838, 584. Definitions, 366, 568. Parallelopipedon, definition, 593. Surface of, 290, 316, 345, 354, 356, 389, 406, 418, 414, 434. Volume of, 358, 359, 360, 362, 390, 402, 418, 414, 435, 454, 461, 462, 493. Partial payments, Connacnent rule (see Manual), 611, 612, 6138. Merchants’ rule, 496, 497, 498, ah United States rule, 458, 459, 460, 2.8 Da Annual interest notes, 613, 614. New Hampshire rule, 614, 615. Vermont rule, 615, 616, 617. Partnership, 352, 407, 409, 439, 440, 454, 455, 482, 483, 501, 502, 508, 509. Definitions, ete., xxii. Par value, 388, 442, 448, 444, 488, Xxiv. Payments, partial (see Partial pay- ments), Xix, Equation of (see Equation of payments), xx. Pentagon, 573. Construction of regular, 574. Percentage, 316, 317, 349, 351, 352, 361, 881, 382, 383, 384, 385, 386, 387, 388, 3898, 407, 408, 410, 411, 421, 422, 423, 431, 482, 442, 444, 446, 449, 456, 466, 479, 480, 482, 483, 488, 489, 491, 492, 494, 506. Bonds, 442, 489, Xxv. 443, 444, 445, 488, INDEX. Percentage, Continued. Commercial discount, 410, 411, 421, 422, 431, 446, 456, 457, 466, 475, 479, 480, 481, xiv. Commission, 317, 384, 385, 398, 423, 449, 482, xiv. Definitions, etc., 316, 381, xiii, xiv. Duties, 317, 352, 384, 466, xvi. Insurance, 317, 564, 385, 386, 423, 444, xv. Profit and loss, 817, 818, 386, 387, 388, 391, 392, 423, 432, 446, 447, 483, 491, 492, 494, xiv. Stocks, 388, 442, 448, 444, 446, 488, 489, xxiv, xxv. Taxes, 317, 385, 444, 446, xvi. To find the base or the rate, 383, 384, 385, 386, xiii, xiv. Perimeter of rectangle, 299, 301, 309, 427, 435. Of square, 435, 485. Of right-angled triangle, 491. Of regular polygon, 591. Of trapezoid, 491, Perpendicular lines, 865, 561, 562, 577, 579. Policy of insurance, xv. Poll tax, 617, 618, xvi. Polygon (see Areas, etc. ), 5738. Angles of regular, 573, 574. Construction of regular, 573, 574. Powers, 463, 519, xxiii. Premium, exchange, 448, 449, 451, 452, 485, xx. Insurance, 385, 484, xv. Stocks and bonds, xxv. Present worth, 460. Prime factors, 251, 291, 507, v. Numbers, 251, 257, 291, v. Principal (see Interest), 350, 393, 415, xvi. Principles, addition, ii. Cancellation, v. Decimals, x. Division, iv. Fractions, viii. Addition of, 203, ix. Subtraction of, 206, ix. 635 Principles, Continued. Greatest common divisor, vi. Least common multiple, vii. Multiplication, iii. Proportion, xxii. Ratio, xxi. Subtraction, ii. Prisms, kinds, ete., 592, 593. Surface of, 511, 512, 595. Developed surface of, 356, 434, 511, 594. Volume of, 518, 514, 599, 600. Problems in interest, 384, 386, 395, 415, 416, 417, 470, 482, 489, 493, 494, xvii. In percentage (see Percentage, etc. ), 383, 384, 385, 386, 398, 407, 430, 481, 482, 447, 492, XI SL Ve Proceeds (see Bank discount), 402, xix. Product, 235, 300, iii. Profit and loss, 317, 318, 386, 387, 388, 391, 392, 428, 482, 446, 447, 483, 491, 492, 494, xiv. Proper fraction, viii. Proportion (see Analysis), 476, 477, 478, 479. Definitions, etc., xxi, xxii. Protractor, 558. Pyramid, definition, etc., 594. Surface of, 512, 595, 596. Developed surface of, 512, 595, 596, 600, 603. Surface of frustum of, 596, 597. Developed surface, 596, 603. Volume of, 5138, 600, 601, 604. Volume of frustum of, 601, 602, 604. Quadrilateral (see Rectangle, Rhomboid, ete. ), 366, 4138, 508, 568. Construction of (see Square, etc. ), 366, 569, 570, 571, 574, 577, 578, 579, 580, 581, 582, 583, 584, 590. Quotient, 235, 236, iv. Radius, of circle, 469, 496, 503, 571, 636 Radius, Continued. Of sphere, 518, 521, 598, 599, 605. Range, U. S. lands, 524. Rate (see Discount, Interest, Per- centage, etc. ), xiii, xvi, xvii. Ratio, 466, 467, 468, 469, 516, 517, 519, 521, 522, 582, 584, 595. Definitions, etc., 466, xxi. Rectangle, area of, 246, 247, 248, 262, 276, 283, 284, 286, 288, 289, 290, 297, 300, 301, 315, 338, 342, 348, 344, 345, 352, 358, 354, 355, 356, 389, 390, 397, 3898, 400, 401, 411, 412, 413, 427, 481, 485, 436, 461, 462, 475, 476, 482, 504, 506, 517, 525, 526, 550, 555, 584. Definition, 568. Diagonal of, 322, 491, 504. Perimeter of, 299, 801, 309, 427, 435. Reduction of decimals to common fractions, 292, 296, 441, xi. Of denominate numbers, ascend- ing, 228, 231, 285, 286, 287, 311, 326, 327, 328, 341, 356, 357, 408, 441, 444, 445, 484, 492, 494, xiii. Descending, 227, 228, 245, 285, 286, 287, 311, 326, 327, 328, 341, 488, 441, 484, xii. Of fractions to common denom- inator, 205, 205, 249, 255, 267, 1x: To decimals, 291, 296, 351, 364, 440, 441, xi. To higher terms, 200, 201, 205, 258, viii. To lowest terms, 202, 203, 205, 252, 258, 256, 257, 258, 284, 325, 432, 484. To per cents, 381. To simplest form, complex, 266, 268, 271, 824, 364, 433, 483, 484, x. Compound, 266, 271, 324, ' 364, 482, 433, 484, To whole or mixed num- bers, 199, 257, 266, 267, ix. Of mixed numbers to improper INDEX. Reduction, Continued. fractions, 199, 257, 266, 267, 473, ix. Of per cents to fractions, 324, Regular polygons, area of (see Hexagon), 591. Construction of (see Pentagon, etc.), 575, 574. Definitions, 573. Perimeter of, 591. tegular, prism, 593. Pyramid, 594. Remainder, 235, iv. teview, bonds and stocks, 488, 489. Cube root, 522. Decimals, 324, 825, 440, 441. Denominate numbers, 341, 406, 408, 483, 441. Discount, bank, 420, 421, 470. Commercial, 479, 480, 481. Domestic exchange, 485. Fractions, 250,. 256, 277, 309, 310, 328, 324, 340, 362, 363, 364, 899, 405, 429, 432, 433, 471, 483, 484. Fundamental processes, 211, 212, 216, 217, 233, 234, 235, 263, 275, 276, 302, 303, 305, 339, 340, 341, 400, 428, 429, 430, 465, 471, 510. Interest, simple, 417, 418, 419, 420, 469, 470. Compound, 487. Longitude and time, 471, 472, 473. Measurements, 288, 289, 290, 315, 316, 342, 348, 344, 345, 353, 354, 355, 356, 358, 359, 360, 367, 368, 389, 390, 400, 401, 402, 411, 412, 413, 414, 433, 484, 435, 486, 490, 491, 495, 496, 505, 504, 508, 516, 517. Miscellaneous, 207, 208, 209, 225, 226, 2385, 236, 287, 238, 245, 258, 259, 270, 271, 288, 300, 301, 3808, 309, 336, 341, 352, 362, 364, 406, 407, 409, 431, 482, 449, 450, 454, 475, INDEX. 637 Review, Continued. 476, 477, 478, 479, 482, 483, 490, 491, 492, 493, 494, 501, 502, 508, 509, 510. Square root, 478, 481, 489. Surfaces and volumes, 390, 413, 414, 434, 435, 461, 462. thomboid, area of, 367, 412, 413, 503, 570. Construction of, 866, 569, 570. Definition, 366, 567. Diagonals of, 581. Rhombus, area of, 367, 412, 413, 496, 504, 570. Construction of, 366, 496. Definition, 366, 569, 570, 579. Diagonals of, 496, 504, 579. Right angle (see Perpendicular), 365, 491, 557. Right-angled triangle, area of, 322, 323, 355, 356, 367, 411, 412, 433, 551, 555. Construction of, 566, 580. Definition of, 565. Length of sides of, 486, 490, 491, 506, 516, 525, 555, 556, 580, 581, 596. Right cone, 594. Cylinder, 593. Prism, 593. Pyramid, 594. Roman notation, i. Root, cube, 520, 522. Square, 270, 463, 464, 465, 478, 481, 485, 486, 489, 490, 491. Definitions, etc., xxiii, xxiv. Rules, accurate interest, 494, xviii. Addition of algebraic quanti- ties, 531. Of decimals, xi. Of fractions, 203, 255, ix. Of integers, ii. Aliquot-part method of finding interest, 417, xvii. Analysis, 456. Angle, measurement of, 559. Annual interest, 523, xviii. Partial payments of annual- interest notes, 613. N. H. rule, 614. Vermont rule, 615. Rules, Continued. Area of circle, 496, 516, 592. Of sector, 516. Of trapezium, 495. Of triangle, 495. Average term of credit, xxi. Bank discount, 402, 405, xix. Of interest-bearing notes, 407. Proceeds, 402. To find face of note, ete., 424. Base, of right-angled triangle, 486, Per cent and rate given, 383, Xi: Rate and amount given, 3853, xiii. Rate and difference given, 385, xiv. Bill of exchange, cost of, 455. Face of, 456. Board feet, 514. Brokerage, 488. Cancellation, 2638, 264. Cause and effect, 436. Circle, area of, 496, 516, 592. Circumference of, 450, 503, 518. Diameter of, 450, 508, 518. Radius of, 518. Clearing of fractions, 374. Commercial discount, 410, xiv. Common prime factor, Vi. Complex fractions, reduction of, ib a Compound numbers (see De- nominate numbers ), xii. Interest, 445, 487, xviii. Cone, convex surface of, 595. Frustum of surface of, 597. Volume of, 603. Volume of, 515, 600. Connecticut rule, partial pay- ments, 611. Convex surface of cone, 595. Of cylinder, 595. Of frustum of cone, 597. Of pyramid, 597. Of prism, 511. Of pyramid, 595. Cube root of fractions, xxiv. Of integers, 520, 522, xxiv. 638 INDEX Rules, Continued. Cylinder, surface of, 595. Volume of, 514, 604. Dates, time between, 346, 348, 609, xiii. Decimals, addition of, xi. Division of, 280, xi. Multiplication of, xi. Reduction to common denom- inator, Xi. To common fractions, Xi. Subtraction of, xi. To read, 239, 240, x. To write, x. Denominate numbers, reduction ascending, 326, 357, xiii. Reduction descending, 326, xii. Denominator, least common, of fractions, 255, ix. Of decimals, xi. Diameter of circle, 450, 503, 518. Difference between dates, 346, 548, 609, xiii. Discount, bank, 402, 403, xix. Of interest-bearing notes, 407. Proceeds, 402. To find face of note, etc., 424, Commercial, 410, xiv. True, 460. Dividends, stock, 488. Division of decimals, 280, xi. Of denominate numbers, 334. Of fractions, 214, 267, x. Of integers, iv. Draft, sight, cost of, 448, 449, xx. Face of, xx. Time, cost of, 451, xx. Face of, xx. Equated time, xxi. Exact interest, 494, xviii. Face of bill of exchange, 456. Of draft, xx. Of note, 424. Factors, common prime, Vi. Prime, v. Fractions, addition of, 203, 255, ix, Clearing of, 374. Rules, Continued. Complex, reduction of, ix. Cube root of, xxiv. Division of, 214, 267, x. Improper, reduction of, ix. Mixed numbers, reduction to improper, ix. Multiplication of, 265, ix. Reduction of, to higher terms, Viii. To least common denomi- nator, 255, ix. To lowest terms, 252, viii. Square root of, xxiv. Subtraction of, 206, ix. Frustum of cone, surface of, 597. Volume of, 608. Of pyramid, surface of, 597. Volume of, 602. Greatest common divisor, 253, vi. Hypotenuse, length of, 486. Improper fractions, reduction of, ix, Interest, accurate, 494, xviii. Annual, 523, xviii. Compound, 445, 487, xviii. Exact, 494, xviii. Simple, general method, 350, 393, Xvii. Method by aliquot parts, 417, xvii. Six per cent method, xviii. Amount, 894. ; To find the principal, 415, XVii. Rate, 415, xvii. Time, 415, xvii. Least common denominator, 255, 1x. Multiple, 255, vii. Longitude and time, 471. Lowest terms, reduction of frac- tions to, 252, viii. Merchants’ rule, partial pay- ments, 497, xix. Mixed numbers, reduction of, ix. Square root of, 4738. Multiple, least common, 255, vii. INDEX. 639 Rules, Continued. Multiplication, algebraic quan- tities, 549. Decimals, xi. Fractions, 265, ix. Short methods, 399. Integers, iii. Short methods, 276, 308, 305, 340. By 10, 100, ete., iv. Mixed numbers, short meth- ods, 276, 340, 399, 405. New Hampshire rule, partial payments, 614. Notation of decimals, x. Of integers, i. Numeration of decimals, x. Of integers, ii. Oblique solids, volume of, 604. Partial payments, Connecticut rule, 611. Merchants’ rule, 497, xix. United States rule, 458, 459, Ks Annual interest notes, 6138. New Hampshire rule, 614. Vermont rule, 615. Partnership (see Manual), xxii. Percentage, 383, xiii. To find base, 384, xiii, xiv. Rate, 384, xiii. Perpendicular right-angled tri- angle, 486. Power, second, 463, xxiii. Third, 519, xxiii. Prime factors, v. Common, Vi. Principal, to find, 415, xvii. Prism, convex surface, 511. Volume, 604. Proceeds, 402. Profit and loss, 387. Proportion, 477, xxiii. Pyramid, convex surface of regular, 595. Frustum of, convex surface, 597. Volume, 602. Volume of, 604. Radius of circle, 518. Of sphere, 518, 599. Rules, Continued. Rate, discount, 424. Interest, 414, xvii. Percentage, 383, xiii. Ratio, 466. Reduction of decimals to com- mon denominator, xi. To common fractions, xi. Of denominate numbers, as- cending, 326, 357, xiii. Descending, 326, xii. Of fractions to the least com- mon denominator, 255, ix. To decimals, 291, xi. To higher terms, viii. To lowest terms, 252, viii. To simplest form, ix. Of improper fraction to mixed number, ix. Of mixed number to improper fraction, ix. Root, cube, 520, 522, xxiv. Square, 465, 481, xxiii, xxiv. Sphere, surface, 518, 521, 605. Volume, 521. Square root of fractions, xxiv. Of integers, 465, 481, xxiii. Of mixed numbers, 473. Subtraction of algebraic quanti- ties, 533. Of decimals, xi. Of fractions, ix. Of integers, ii, iii. Surface of cone, 595. Of cylinder, 595. Of frustum of cone, 597. Of pyramid, 597. Of prism, 511. Of pyramid, 595. Of sphere, 518, 521, 605. Taxes, 617. Vermont method, 617. Term of discount, xix, xx. Time between dates, 346, 348, 609, xiii. Principal, interest, and rate given, 415, xvii. Discount, 424. Time draft, cost of, 451, xx. Face of, xx. Transposing, 377. 640 Rules, Continued. True discount, 460. Triangle, area, 495. Volume of cone, 514, 600, 604. Of cylinder, 514, 604. Of frustum of cone, 603. Of frustum of pyramid, 602. Of prism, 514, 604. Of pyramid, 514, 600, 604. Of sphere, 521, 605. Scealene triangle, 366, 566. Altitude of, 556, 569, 578, 584. Area of, 495, 503, 504, 508. Construction of, 366, 564, 576, 578, 579, 580, 582, 584, 585, 586. Secant, 582. Second power, 463, xxiii. Section (U.S. lands), 524. Sector, area of, 516, 592. Definition, 572. Segment, area of, 516. Definition, 516, 572. Sextant, 572. Share, xxiv. Short methods, addition and sub- traction, 234, 302, 429. Addition of 99, 999, etc., 396. Bank discount, 420. Commercial discount, 421, 422. Compound interest, 445, 446. Division, 233, 268, 341, 400, 430. Divisors 125, 25, 75, etc., 229, 260, 298, 337, 396, 426. Divisors ending in ciphers, 281, 294. ° Multiplication, 305, 428, 429,471. Fractional multipliers, 399, 426, 429, 471. Multiplier a mixed number, 323, 340, 363, 399, 426, 429. 99, 999, etc., as multipliers, 340, 399, 429, 471. 125, 25, '75, etc., as multi- pliers, 229, 260, 276, 298, 308, 337, 340, 396, 399, 429, 471. Simple interest, 419. Subtraction and addition, 234, 302, 429. INDEX. Short methods, Continued. Subtraction of 99, 999, etc., 396. Sight drafts, 448, 449, 452, 485, XXxvi. Cost of, 448, 449, 452, 485, xx. Face of, 448, 449, 485, xx. Sight exercises, 206, 213, 221, 222, 224, 225, 226, 229, 232, 238, 248, 244, 251, 252, 254, 260, 262, 263, 275, 276, 277, 278, 281, 296, 303, 306, 307, 369, 370, 374, S577, 381, 426, 442, 467, 474, 505. Similar triangles, 586. Slant height, of cone, 512, 5138, 595, 596. Of pyramid, irregular, 596. Regular, 512, 594, 596. Small circle, 598. Solid contents (see Volumes), 358, 359, 360, 362, 390, 402, 413, 414, 434, 4385, 442, 461, 462, 493, 507, 513, 514, 521, 522, 527, 528, 599, 600, 601, 602, 603, 604, 605. Special drills, 260, 298, 337, 396, 426, 474. Sphere, great circle of, 518, 598. Small circle of, 598. Surface of, 518, 519, 605. Volume of, 521, 522, 605. Square, area of, 348, 355, 367, 435, 461, 503, 504,516. —.- Construction of, 569, 578, 574, 576, 577, 578, 579, 588. Definition of, 568. Diagonal of, 322, 491, 503, 504. Side of, 270, 401, 485, 486. Square measure, 354, 607. Square root, of algebraic quanti- ties, 652, 653. Of decimals, 481, 482, 489. Of fractions, 473. Of integers, 270, 465, 481, 489. Of mixed numbers, 478. Rules and definitions, 464, 465, Xxill, xxiv; Stocks and bonds, 888, 442, 443, 444, 445, 446, 488, 489. Definitions, xxv. INDEX. Subtraction, algebraic quantities, 532, 533, 534. Removing parentheses, 534. Decimals, 242, 283, 2938, 364, 440, 441. Denominate numbers, 208, 285, 287, 288, 312, 331, 332, 335, 336, 348, 358, 430. Fractions, 205, 206, 207, 250, 256, 262, 263, 277, 296, 310, 324, 363, 364, 482, 433, 488, 484. Integers, 217, 284, 302, 429. Cross-subtraction, 217, 339. Principles, definitions, and rules, ii. Subtrahend, 235, ii. Successive discounts (see Com- mercial discount), xiv. Sum, 285, ii. Surface (see Areas and Measure- ments). Of cone, 512, 598. Of cylinder, 511, 512, 598, 595. Of fence, 289, 343, 354, 390, 400, 435. Of frustum of cone, 597. Of pyramid, 596, 597. Of parallelopipedon, 290, 316, 345, 854, 406, 418, 414, 434. Of prism, 356, 511, 512, 595. Of pyramid, 512, 595, 596. Of sphere, 518, 519, 599, 605. Of walls of room, 289, 315, 316, 344, 345, 389, 3890, 400, 484, 461, 462. Surveyors’ measure, 607. 518, 595, 596, Tables, apothecaries’ weight, 361, 607. Avoirdupois weight, 227, 361, 406, 608. Long ton, 341, 608. Circular measure, 450, 558, 608. Cubic measure, 361, 607. Dry measure, 227, 607. English money, 408, 608. Liquid measure, 227, 607. Long measure, 285, 607. Square measure, 354, 607. 641 Tables, Continued. Surveyors’ measure, 607. Time measure, 226, 608. Troy weight, 361, 607. U. S. money, 608. Tangent, circle, 572, 584. Line, 582. Taxes, 317, 385, 444, 446, 510, 617, 618. Definitions, xvi. Poll, 617, 618. Vermont method, 617, 618. Term of discount, 402, 403, 404, 421, 424, 470, xix. Third power, 519, xxiii. Time drafts, 451. Acceptance of, 451, xxvi. Cost of, 451, 452, 485. Definitions, xx, xxvi. Face of, 452, 485. Time measure, 226. Time note (see Notes), xxv. Township (U. S. lands), 524. Transposing, 377. Trapezium, area of, 368, 413, 495, 504, 508, 571. Construction of, 570, 571. Definition, 578. Diagonal of, 495, 571. Trapezoid, area of, 355, 367, 368, 4138, 414, 484, 490, 491, 504, 571. Construction of, 569, 570, 571. Definition, 568. Perimeter of, 491. Sides of, 434, 490, 491. Triangle, acute-angled, 365, 565, 570, 571, 584, 585. Altitude of, 584, 585. Area of, 411, 412, 427, 584, 585. Construction of, 865, 565, 584. Isosceles, 366, 566. Altitude of, 495, 567, 578, 584, 594. Area of, 367, 411, 412, 495, 496, 503. Construction of, 366, 566, 567, 575, 576, 578, 579, 582, 584. 642 Triangle, Continued. Oblique-angled, 565. Area of, 367, 411, 412, 461, 495, 496, 503, 556. Obtuse-angled, 365, 565. Altitude of, 569, 578, 584. Area of, 412, 570, 571, 584, 585. Construction of, 571, 583, 584. Scalene, 366, 566. Altitude of, 556, 569, 578, 584. Construction of, 564, 575, 578, 579, 580, 582, 584, 585, 586. Right-angled, 322, 365, 565. Area of, 322, 328, 355, 356, 367, 411, 412, 483, 490, 496, 526, 551, 555. Construction of, 564, 576, 578, 579, 580, 582, 584, 585, 586. Hypotenuse of, 486, 490, 491, 496, 506, 525, 526. Perimeter of, 491. Perpendicular or base of, INDEX. Triangle, Continued. 322, 438, 490, 555, 556, 580, 581, 596. True discount, 460. United States lands, 524. U. 8. rule, partial payments, 458, 459, 460, xix. Usury (see Manual), 610. Value of foreign coins, 619. Vermont rule, partial payments, 615, 616, 617. Taxes, 617, 618. Vertical, angles, 657. Lines, 660. Volume of cone, 513, 514, 600, 604. Of cylinder, 518, 514, 522, 599, 604. Of frustum of cone, 603. Of pyramid, 601, 602. Of parallelopipedon, 358, 359, . 360, 362, 390, 402, 413, 414, 434, 485, 454, 461, 462, 4938. Of prism, 513, 514, 599, 600. Of pyramid, 513, 600, 601, 604. Of sphere, 521, 522, 605. The pupil can make a protractor by pasting one of the above on a stiff piece of paper and carefully cutting it out. NUMBER. Atwood’s Complete Graded Arithmetic. Present a carefully graded course in arithmetic, to begin with the fourth year and continue through the eighth year. Part I. 200 pages. Cloth. 4octs. Part lI. 382 pages. Halfleather. 75 cts. Walsh’s Mathematics for Common Schools. Special features of this work are its division into half-yearly chapters instead of the arrangement by topics; the omission, as far as possible, of rules and definitions; the great number and variety of the problems; the use of the equation in solution of arithmetical problems; and the introduction of the elements of algebra and geometry. THREE Book Series — Elementary, 218 pages. 35 cts. Intermediate, 252 pages. 4o cts. Higher, 387 pages. Half leather. 75 cts. Two Boox Ssries — Primary, 198 pages, 35 cts. Grammar School, 433 pages. Half leather. 75 cts. Sutton and Kimbrough’s Pupils’ Series of Arithmetics. Primary Book. Embraces the four fundamental operations in all their simple relations. 80 pages. Cloth. 25 cts. INTERMEDIATE Book. Embraces practical work through percentage and simple interest. 145 pages. Cloth. 30 cts. Lower Boox. Primary and Intermediate Books bound together. Cloth. 45 cts. Hicuer Boox. A compact volume for efficient work which makes clear all necessary theory. 275 pages. Half leather. 75 cts. Safford’s Mathematical Teaching. Presents the best methods of teaching, from primary arithmetic to the calculus. Paper. 25 cts. Badlam’s Aids to Number. For Teachers. First Series. Consists of 25 cards for sight-work with objects from one to ten. 40 cts. Badlam’s Aids to Number. For Pupils. First Series. Supplements the above with material for slate work. Leatherette. 30 cts. Badlam’s Aids to Number. Jor Teachers. Second Series. Teachers’ sight-work with objects above ten. 40 cts. Badlam’s Aids to Number. For Pupils. Second Series. Supplements above with material for slate work from 10 to 20. Leatherette. 30 cts. Badlam’s Number Chart. 1: x 14 inches. Designed to aid in teaching the four fundamental rules in lowest primary grades. 5 cts. each; per hundred $4.00. Sloane’s Practical Lessons in Fractions. For elementary grades, Boards. 30 cts. Set of six fraction cards for children to cut. 12 cts. White’s Two Years with Numbers. Number Lessons for second and third year pupils. 40 cts, White’s Junior Arithmetic. For fourth and fifth year pupils. Cloth. 50 cts. White’s Senior Arithmetic. 7x gress. For advanced work see our list of books in Mathematics. D. C. HEATH & CO., PUBLISHERS, BOSTON. NEW YORK. CHICAGO. READING. Badlam’s Suggestive Lessons in Language and Reading. A manual for pri. mary teachers. Plain and practical; being a transcript of work actually done in the school-room. $1.50. Badlam’s Stepping-Stones to Reading.— A Primer. Supplements the 283-page book above. Boards. 30 cts. Badlam’s First Reader. New and valuable word-building exercises, designed to follow the above. Boards. 35 cts. oat Bass’s Nature Stories for Young Readers: Plant Life. Intended to supple- ment the first and second reading-books. Boards. 30 cts. Bass’s Nature Stories for Young Readers: Animal Life. Gives lessons on animals and their habits. To follow seoond reader. Boards. 40 cts. Firth’s Stories of Old Greece. Contains 17 Greek myths adapted for reading in intermediate grades. Illustrated. Boards. 35 cts. Fuller’s Illustrated Primer. Presents the word-method in a very attractive form to the youngest readers. Boards. 30 cts. Hall’s How to Teach Reading. Treats the important question: what children should and should not read. Paper. 25 cts. Miller’s My Saturday Bird Class. Designed for use asa supplementary reader in lower grades or asa text-book of elementary ornithology. Boards. 30 cts. Norton’s Heart of Oak Books. This seriesis of material from the standard imagin- ative literature of the English language. It draws freely upon the treasury of favorite stories, poems, and songs with which every child should become familiar, and which have done most to stimulate the fancy and direct the sentiment of the best men and women of the English-speaking race. 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Has lessons in geology, astronomy, world-life, etc. Boards. 7octs. For advanced supplementary reading see our list of booksin English Literature. D. C. HEATH & CO., PUBLISHERS, BOSTON. NEW YORK, CHICAGO, ENGLISH LANGUAGE. Hyde’s Lessons in English, Book I. For the lower grades. Contains exercises for reproduction, picture lessons, letter writing, wses of parts of speech, etc. 40 cts, Hyde’s Lessons in English, Book II. For grammar schools. Has enough techni- cal grammar for correct use of language. 60 cts. Hyde’s Lessons in English, Book II with Supplement. Has, in addition to the above, 118 pages of technical grammar. 70 cts. Supplement bound alone, 35 cts. Hyde’s Practical English Grammar. For advanced classes in grammar schools and for high schools. 60 cts. Hyde’s Lessons in English, Book II with Practical Grammar. The Practical Grammar and Book II bound together. 80 cts. Hyde’s Derivation of Words. 15 cts. Penniman’s Common Words Difficult to Spell. 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Treats salient features with a master’s skill and with the utmost clearness and simplicity. $1.30. Meiklejohn’s English Grammar. Also composition, versification, paraphrasing, etc. For high schools and colleges. go cts. Meiklejohn’s History of the English Language. 78 pages. Part III of Eng- lish Language above, 35 cts. Williams’s Composition and Rhetoric by Practice. For high school and col- lege. Combines the smallest amount of theory with an abundance of practice. Revised edition. $1.00. Strang’s Exercises in English. Examples in Syntax, Accidence, and Style foi criticism and correction. 50 cts. Huffcutt’s English in the Preparatory School. Presents advanced methods of teaching English grammar and compositon in the secondary schools. 25 cts. Woodward’s Study of English. From primary school to college. 25 cts. Genung’s Study of Rhetoric. Shows the most practical discipline. 25 cts. See also our list of books for the study of English Literature. D. C. HEATH & CO., PUBLISHERS, BOSTON. NEW YORK. CHICAGO. ELEMENTARY SCIENCE. Bailey’s Grammar School Physics. A series of inductive lessons in the elements _ of the science. Illustrated. 60 cts. Ballard’s The World of Matter. A guide tc the study of chemistry and mineralogy; adapted to the general reader, for use as a text-book or as a guide to the teacher in giving object-lessons. 264 pages. Illustrated. 1.00. Clark’s Practical Methods in Microscopy. Gives in detail descriptions of methods that will lead the careful worker to successful results. 233 pages. Illustrated. $1.60. Clarke’s Astronomical Lantern. Intended to familiarize students with the constella- tions by comparing them with fac-similes on the lantern face. With seventeen slides, giving twenty-two constellations. $4 50. Clarke’s How to find the Stars. Accompanies the above and helps to an acquaintance with the constellations. 47 pages. Paper. 15 cts. Guides for Science Teaching. Teachers’ aidsin the instruction of Natural History classes in the lower grades. . Hyatt’s About Pebbles. 26 pages. Paper. 10 cts. II. Goodale’s A Few Common Plants. 61 pages. Paper. 20 cts. III. Hyatt’s Commercial and other Sponges. Illustrated. 43 pages. Paper. 20 cts. IV. Agassiz’s First Lessons in Natural History. Illustrated. 64 pages. Paper. 25 cts. Vz. Hyatt’s Corals and Echinoderms. Illustrated. 32 pages. Paper. 30 cts. VI. Hyatt’s Mollusca. Illustrated. 65 pages. Paper. 30 cts. VII. Hyatt’s Worms and Crustacea. Illustrated. 68 pages. Paper. 30 cts. VIII. Hyatt’s Insecta. Illustrated. 324 pages. Cloth. $1.25. XII. Crosby’s Common Minerals and Rocks. Illustrated. 200 pages. Paper, 40 cts. Cloth, 60 cts. ;. XIII. Richard’s First Lessons in Minerals. 50 pages. Paper. ro cts. XIV. Bowditch’s Physiology. 58 pages. Paper. 2octs. XV. Clapp’s 36 Observation Lessons in Minerals. 80 pages. Paper. 30 cts. XVI. Phenix’s Lessons in Chemistry. 20 cts. Pupils’? Note-Book to accompany No. 15. 10 cts. Rice’s Science Teaching in the School. With a course of instruction in science. for the lower grades. 46 pages. Paper. 25 cts. Ricks’s Natural History Object Lessons, Supplies information on plants and their products, on animals and their uses, and gives specimen lessons. Fully illustrated. 332 pages. $1.50. Ricks’s Object Lessons and How to Give them. Volume I. Gives lessons for primary grades. 200 pages. 9o cts. Volume II. Gives lessons for grammar and intermediate grades. 212 pages. 90 cts. Shaler’s First Book in Geology. For high school, or highest class in grammar school. 272 pages. Illustrated. $1.00. Shaler’s Teacher’s Methods in Geology. An aid to the teacher of Geology. 74 pages. Paper. 25 cts. Smith’s Studies in Nature. A combination of natural history lessons and language work, 48 pages. Paper. 15 cts. Sent by mail postpaid on receipt of price. See also our list of books in Science. D. C. HEATH & CO., PUBLISHERS, BOSTON. NEW YORK. CHICAGO. DRAWING AND MANUAL TRAINING. Anthony’s Mechanical Drawing. 08 pages of text, and 32 folding plates. $1.50. Anthony’s Machine Drawing. 50 pages of text, and 1s folding plates. $1.25. Daniels’ Freehand Lettering. 34 pages of text, and 13 folding plates. 8s5 cts. Lunt’s Brushwork for Kindergarten and Primary School. 18 lesson-cards in colors, with teacher’s pamphlet, in envelope. 30 cts. Johnson’s Progressive Lessons in Needlework. Explains needlework from its rudiments and gives with illustrations full directions for work during six grades. 117 pages. Square 8vo. Cloth, $1.00, Boards, 60 cts. Seidel’s Industrial Instruction (Smith). A refutation of all objections raised against industrial instruction. 170 pages. go cts. Thompson’s Educational and Industrial Drawing. Primary Free-Hand Series (Nos. 1-4). Each No., per doz., $1.00. Primary Free-Hand Manual. 114 pages. Paper. 40 cts. Advanced Free-Hand Series (Nos. 5-8). Each No., per doz., $1.50, Modei and Object Series (Nos. 1-3). Each No., per doz., $1.75. Model and Object Manual. 84 pages. Paper. 35 cts. Esthetic Series (Nos. 1-6). Each No., per doz., $1.50. fésthetic Manual. 174 pages. Paper. 60 cts. Mechanical Series (Nos. 1-6). Each No., per doz., $2.00. Mechanical Manual. 172 pages. Paper. 75 cts. Thompson’s Manual Training, No. 1. Treats of Clay Modelling, Stick and Tablet Laying, Paper Folding and Cutting, Color, and Construction of Geometrical Solids. Illustrated. 66 pages. Large 8vo. Paper. 30 cts. Thompson’s Manual Training, No. 2. Treats of Mechanical Drawing, Clay- Modelling in Relief, Color, Wood Carving, Paper Cutting and Pasting. Illustrated. 7o pp. Large 8vo. Paper. 30 cts. Waldo’s Descriptive Geometry. A large number of problems systematically ar ranged, with suggestions. 85 pages. go Cts. Whitaker’ s How to Use Wood Working Tools. Lessons in the uses of the universal tools: the hammer, knife, plane, rule, chalk-line, square, gauge, chisel, saw, and auger. 104 pages. 60 cts. Woodward’s Manual Training School. Its aims, methods, and results; with detailed courses of instruction in shop-work. Fully illustrated. 374 pages. Octavo. $2.00. Sent postpaid by mail on recetpt of price. D. C. HEATH & CO., PUBLISHERS, BOSTON. NEW YORK. CHICAGO. GEOGRAPHY AND MAPS. Heath’s Outline Map of the United States. Invaluable for marking territorial growth and for the graphic representation of all geographical and historical matter. Small (desk) size, 2 cents each; $1.50 per hundred. Intermediate size, 30 cents each. Large. size, 50 cts. Historical Outline Map of Europe. 12x 18 inches, on bond paper, in black outline. 3 cents each; per hundred, $2.25. Jackson’s Astronomical Geography. Simple enough for grammar schools. Used for a bricf course in high school. 40 cts. Map of Ancient History. Outline for recording historical growth and statistics (14x 17 in.), 3 cents each; per 100, $2.25. Nichols’ Topics in Geography. A guide for pupils’ use from the primary through the eighth grade. 65 cts. Picturesque Geography. 12 lithograph plates, 15 x 20 inches, and pamphlet describing their use. Per set, $3.00; mounted, $5.00. Progressive Outline Maps: United States, *World on Mercator’s Projection (12 x 20 in.) ; North America, South America, Europe, *Central and Western Europe, Africa, Asia, Australia, *British Isles, *England, *Greece, *Italy, New England, Middle Atlan- tic States, Southern States, Southern States— western section, Central Eastern States, Central Western States, Pacific States, New York, Ohio, The Great Lakes, Washington (State), *Palestine (each 10 x 12 in.). For the graphic representation by the pupil of geography, geology, history, meteorology, economics, and statistics of all kinds. 2 cents each; perhundred, $1.50. Those marked with Star (*) are also printed in black outline for use in teaching history. Redway’s Manual of Geography. I. Hints to Teachers; II. Modern Facts and Ancient Fancies, 65 cts. Redway’s Reproduction of Geographical Forms. I. Sand and Clay-Modelling; II. Map Drawing and Projection. Paper. 30 cts. Roney’s Student’s Outline Map of England. For use in English History and Literature, to be filled in by pupils. 5 cts. Trotter’s Lessons in the New Geography. Treats geography from the human point of view. Adapted for use as a text-book or asa reader. $1.00 D. C. HEATH & CO., PUBLISHERS, BOSTON. NEW YORK. CHICAGO. A As en ays CAS aN FRR Ha Ae A aes 7 nf ‘ bi ‘ b j ‘ ‘ . ‘ DP, . A es i i i: PITTI TTT TAT \ WA | ae ae 1 GRe Ba a BME BT eal I 1 aaa ea 8 Bell ae I 1 / MAA A | 14 | iN iH] in| ii Ht | | WA INH | i 1) + Wana ri ii | | i} | WW al WALI \| Hae a) WELL A TT |