pepe Rroereess Merny oh Bae #84, oF S56 Sno pee et ge Deseret ste oe Em rene tne op pet eal evince sree oan er maw gaheien pyle cage mypaierey sear eyes name ip tnyea toner pie gg : cane pment Nap enn eae ee oy og ekene worcasy aetna eye: Sreceerseterorae rane Sin ; I ener te ee ids « es fe cee Oto eaten e MATHEMATICS LIBRARY THE UNIVERSITY OF ILLINOIS LIBRARY The Frank Hall collection of arithmetics, presented by Professor H. L. Rietz of the University of Iowa. | SHOFE Sls I08Q | VS aT HEMATECS Ly Return this book on or before the Latest Date stamped below. University of Illinois Library L161—H41 THE SOUTHWORTH-STONE ARITHMETIC A RATIONAL METHOD BOOK Ill. FOR ADVANCED GRADES GORDON A. SOUTHWORTH SUPERINTENDENT OF SCHOOLS, SOMERVILLE, MASS. AND JOHN C. STONE, A.M. ASSOCIATE PROFESSOR OF MATHEMATICS, STATE NORMAL COLLEGE, YPSILANTI, MICHIGAN OU TOAN AAXA TOAD BENJ. H. SANBORN & CO. BOSTON NEW YORK CHICAGO THE SOUTHWORTH-STONE ARITHMETIC. BOOK I. PRIMARY. BOOK II. INTERMEDIATE. BOOK III. ADVANCED. WITH OR WITHOUT ANSWERS. COPYRIGHT, 1904, BY GORDON A. SOUTHWORTH anp JOHN C. STONE. Norwood 48ress J.S. Cushing & Co.— Berwick & Smith Co. Norwood, Mass., U.S.A. MATHEMATICS LIBRARY PREFACE “The Southworth-Stone Arithmetic” is a graded series of three Books each separated into two Parts. The series is designed to cover the work of all the elementary grades in which a text-book is commonly used, beginning with the third-year grade and ending with the last year below the high school. The books have been prepared not by theorists to exploit their peculiar notions, but by teachers of long and successful experience. They follow the order of subjects and the lines of development established by the highest educational authorities. No attempt has been made to follow the so-called “spiral plan,” now decadent; each grade, however, thoroughly reviews and carries forward the work of the preceding grades, new topics being in- troduced in order to stimulate the interest of the student and to develop his power. In the presentation of subjects the inductive method has been employed throughout in a way that calls for study and effort and secures that mathematical training that never comes by mechanical figuring and imitation. This logical development of subjects dif- ferentiates the series from mere books of problems. To secure skill and proficiency in the more important na biecs, 7 abundant exercises for drill and practice have been provided. A profusion of oral and written problems is given in about equal pro- portion. The number to be used must depend upon the need of the student. It will be found that fewer problems carefully solved and logically analyzed will be more valuable than many mechanically performed. Many subjects heretofore treated in arithmetics have been omitted as non-essential or beyond the legitimate work of the ele- ili 464010 iv PREFACE mentary schools. Enough has been given, however, to meet the demands of business and to furnish the requisite mental discipline. The methods employed in all the books of the series have been tested in manuscript in the model or training classes in the State Normal College at Ypsilanti, Michigan. The authors acknowledge their indebtedness to Miss Abigail Roe and Miss Mary Steagall and other teachers in that institution for valuable suggestions growing out of such tests. Especial thanks are due to President L. H. Jones of the College, for his counsel as the work has progressed and for his aid in making the books worthy of adoption and use. Part I of this Third Book of “The Southworth-Stone Arithmetic ” Series presents, with more prominent reference to principles, a brief review of the fundamental processes, common fractions, ratio, and decimals. It gives a complete presentation of mensuration and its applications to the six quadrilaterals, to triangles, to circles, to rectangular prisms, and to the cylinder. Tables of weights and measures in full are given for reference. Problems, both oral and written, abound. Part II takes up the subject of percentage analytically and ina way to show that it is only a restatement of principles and pro- cesses already familiar, under new names. Its application to busi- ness problems, insurance, commission, stocks, bonds, taxes, etc., is fully illustrated. Interest,— simple, exact, compound, — partial payments, bank discount, exchange, proportion, square root and its applications, mensuration of pyramids, cones, spheres, and similar triangles, the metric system, longitude and time, are followed by a large number of practical problems that afford a complete review of all subjects. Definitions of all technical terms used in the book are alphabetically arranged for easy reference and review. (See Index.) INDEX PAGE Accounts, ha pss Addition, of integers, 8, 9 Of fractions, 46-49 Of decimals, 75 Angles and arcs, 87, 88 Bank check, 199 Bank discount, 184-189 Of interest-bearing notes, 188 Bills of exchange, 200-201 Bonds, 195-197 Business forms, 15, 169, 184, 185, 190, 199, 200 Cancellations, 53 Cash account, 15 Check, 199 Circles, 102-105 Commission, 163-166 Comparison of numbers, 35, 36, 38, 61-63 Complex fractions, 57 Compound interest, 182-183 Cones, 228-230 Customs, 180-181 Cylinder, 115, 116 Decimals, 71-81 Decimal system, 2 Definitions, 280 Denominate numbers, 37, 40, 70, 86 Difference between dates, 151 Discount, bank, 184-189 Trade or commercial, 157-160 Successive, 159 Division, of integers, 23-26 Of fractions, 55-57 Of decimals, 78, 79 Drafts, 199-201 Duties, 180-181 Equations, 5) Exact interest, 153 Exchange, 198-202 PaGcE Factoring, 45 Fractions, common, 4,5 Complex, 57 Decimal, 5, 71-81 Changes in form of, 42-44, 74, 79 Added and subtracted, 46-49 Multiplied, 33, 50-53 Divided, 55-57 Practice table, 54, 58 Greatest common divisor, 44 Insurance, 161, 162 Interest, general method, 82, 83 Bankers’ method, 146-149 One dollar method, 149, 150 Choice of methods, 152, 154 Compound, 182-183 Drill table, 156 Exact, 153 Legal rates, 149 Land measure, 95, 96, 236 Leap years, 129 Least common multiple, 47, 48 Longitude and time, 245-247 Measurements of — Ares and Angles, 87, 88; Circles, 102-105 ; Cones, 226, 227 ; Cylin- ders, 115, 116, 226-227 ; Hypote- nuse, 218+ Land,.95, 96,7 236: Lines, 8, 87 ; Lumber, 113 ; Pyra- mids, 223-225; Prisms, 109-111, 114, 2238-225.; Rectangles, 90-96 ; Rhomboids, 100; Rhombus, 106 ; Roofs, 93; Surfaces, 86 ; Spheres, 228-230 ; Trapeziums, 99; Trape- zoids, 100; Triangles, 97-99; Wood, 112 Mensuration (see Measurements). Metric system, 235-244 Mixed numbers, 33 Multiples, 47 vl INDEX PAGE Multiplication, of integers, 17-21 Of fractions, 33, 50-53 Of decimals, {homer tr Notation, integers, 1-8 Decimal, 6, 71-72 Notes, promissory, 169-177 Discounted, 184-189 Partial payments of, U.S, rule, 172-177 Numbers, kinds of, 6, 44 Divisibility of, 45 Numeration, 3, 71-72 Partial payments, 172-177 Percentage, 63, 181-205 Business problems, 141, 144, 145; Bonds, 195-197 ; Commission, 165- 166; Duties, 180-181; Exchange, 198-202; Insurance, 161, 162; Profit and loss, 141, 144, 145; Stocks, 190-194; Taxes, 178-179; Trade discount, 157-160. Powers, 211 Principles of — Cancellation, 53; Decimal system, 73, 78; Division, 24; Fractions, 43, 46, 50; Interest, 147; Multi- plication, 18; Partial payments, 178, 175; Proportion, 206; Re- duction of fractions, 43; Right triangle, 218. Square root, 211-222 Prisms, 109-111, 114, 228-225 Profit and loss, 141, 144, 145 Proportion, 206-210 Pyramid, 223-225 Quadrilaterals, 88-90 Ratio, 35, 36, 88, 61-63 | Rectangles, 90-96 Revenues, government, 180-181 Review Exercises in — Fundamental rules, 138, 14, 32, 34, 39, 41 Fractions, 41, 42, 59, 60, 64-70 Decimals, PAGE 80, 81 Measurements, 91-96, 101, 107, 108, 117-121, 237, 238 Percentage, Miscellaneous, 167-168, 203-205 Interest and bank discount, 122-128, 248-279 189 From examinations, 260-279 Square root, 221, 222 Rhomboids, 100 Right triangles, 218, 219 Roots, 211-222 Rule of three, 207 Short processes, 27, 28 Signs, use of, 29 Similar surfaces, 231 Similar triangles, 234, 235 Similar volumes, 232, 233 Spheres, 228-230 Square root, 211-222 Statement of problems, 22, 31 Stocks, 190-194 Subtraction, of integers, 10-12 Of fractions, 46-49 Of decimals, 75 Successive discounts, 159 Surveyor’s measure, 236 Tables for drill — Fundamental rules, 12 Fractions, 54, 58 Percentage, 140 Interest, 156 Tables of weights and measures, 129, 130 Metric system, 235-244 Taxes, 178-179 Time between dates, 161 Trade discount, 157-160 Trapeziums, 90 Trapezoids, 90 Triangles, 97-99, 218-220 United States money, 7 Weights and measures, 129, 130 Wood measure, 112 THE SOUTHWORTH-STONE ARITHMETIC THIRD BOOK dedaWiaidl a NUMBERS: USE, NAMES, AND NOTATION 1. What need of numbers has a merchant? A carpenter? A farmer? A tailor? A surveyor? A capitalist? 2. oanonnrt Oona F w 10. What does thirteen mean ? Fourteen ? Explain the meaning of all the numbers from 13 to 19. What does the syllable -teen mean ? What does twenty mean? Thirty? Forty? What does the syllable -ty mean ? What do we cail 10 tens? 10 hundreds? 1000 thousands ? Mention some numbers that are named by a single word. Show how other numbers are named. Give some examples. How many different figures are used in expressing numbers by the system we use ? ll. How is it that all numbers can be expressed by only ten figures ? Our system of writing numbers is called the Arabic system, for the system was first introduced into Europe by the Arabs, and they were supposed to be the discoverers. Modern historical research shows that the Hindus were the real discoverers, and hence the system is sometimes called the Hindu system. 1 ae REVIEW: A DECIMAL SYSTEM Oral 1. In 5, 50, 500, 5000, how does the 5 change in value ? 2. What value does the zero have? Why is it used ? 3. The value of a figure depends upon what two things ? 4. What does the figure in the first order at the right represent ? In the second order? In the third ? 5. In 707,070 name the units in each order. 6. Compare the value of each 7 with that of the other 7’s. ' 7. 100 ones=1 ——. 10 ——=1 million. hundreds=10 thousands. 10——=1 hundred thousand. hundreds=50 tens. 10 -—- =] ten thousand. 8. How many units of any order does it take to make one unit of the next order at the left ? 9. In our money system how many cents make onedime? How many dimes make one dollar ? Since in our system of writing numbers, and in our money system, ten units of any order make one of the next higher, we call these decimal systems. Remember that decem is Latin for “ ten.’’ In a decimal system ten units of any order make one of the next higher order. 10. In 347,900,903,531 what is the order of each 9? Compare their values. 11. How do the three 3’s compare in value ? 12. What is the use of the zeros ? 13. Why is this number grouped into periods of threes ? 14. Name each period, beginning with the lowest. REVIEW: READING AND WRITING NUMBERS 3 Read without using the word “and” : — 1. 4,705 6,137,008 42,200,020 10,063,005 2. 27,003 3,000,975 34,003,007 16,100,005 3. 195,006 600,001 93,040,075 26,013,200 4. 70,590 17,080,005 349,000,672 85,003,017 5. 100,054 2,008,500 34,206,000,127 93,090,006 The next three periods after millions are billions, trillions, quad- rillions. 6. Mention something counted in millions. Can you think of any use for billions, trillions, or larger numbers ? Read the following : — 7. The population of New York State in 1900 was 7,268,894. The school enrollment of the state was 1,242,416. 8. The total population of the United States in 1900 was 84,233,069. 9. In the twelve months ending with March, 1903, the exports of the United States were $1,414,786,954, against $1,001,596,683 of imports. 10. In 1900 the United States produced 2,105,102,516 bushels of corn valued at $751,220,034. 11. Write the largest possible number, using these six figures only: 0; 0; 2,.9,.3,.7.°' Tell why you put each figure in the order that you did. Write in figures, putting a comma after each period before filling another : — 12. 3 billion, 108 thousand,11. 15. Ten billion, two million, sixty. 13. 828 million, 7 thousand, 9. 16. One less than a billion. 14. 200 million, 76. . 17. The sum of 18000, 200000, 520. 4 REVIEW : UNITS, FRACTIONAL AND INTEGRAL Oral 1. In the number 6 ft., what is the unit of measure? 2. Name the unit in each of the following : — 6bu.; 12 in.3! 60 ft); 9 eb Any quantity with which another quantity of the same kind is compared or measured is considered a unit. 3. What are numbers called whose units are whole things? ee 4. Numbers whose units are parts of whole things are what? 5. How many fractional units are made by cutting a thing into three equal parts? Into four? How do these units compare in size? Then which is larger, 3 or 2? Why? 6. Give the largest possible fractional unit. Explain your answer. 7. Give avery small fractional unit. How many of these make 1? Write the fraction denoting the following : — NuMBER OF UNITS THE FRACTIONAL UNIT 8. 3 one fourth. 9. 5 one eighth. 10. 2 one fifth. 11; 7 one ninth. 12. Illustrate the following numbers by drawings : — on wile. 2 ak bts 2)i) pliGig wel eleG? peeedr li Beer Gennes 13. Read the numbers in the order of their size, the largest first. 14. How many of each would make 1? 15. Upon what does their size depend ? 16. How many fractional units in 2, 75, H, 7%, 14? 17. How many of each size make 1? Which of the fractions are nearest 1 ? Oral REVIEW: TERMS OF A FRACTION 5 1. The two terms of a fraction are and 2. Which term shows how many units the fraction contains ? 8. Which term shows into how many parts the integral unit has been divided ? 4. From which term do you get the name of the fractional unit ? 5. Give the numerator and denominator and the use of each : — oy aay YL; ee eee 0-5 eee al, 6. Name the fractional unit in each. How many of each make one integral unit ? 7. How many kinds of units in $5,3,? In 34 ft.? 8. An integer with a fraction added is called a number. DECIMAL FRACTIONS 1. Compare 1 with 0.1; 0.1 with 0.01. 2. In our decimal system of writing numbers how may we give to any figure a certain value and then a tenth of that value ? 3. What effect upon the value of a figure has moving it one place to the right ? To the left ? 4. What fractional units must fractions have before they can be expressed decimally ? 5. How do you know the denominator, or size of the unit, in deci- mal fractions ? 6. For what is the decimal point used ? 7. Name the size of the units in each order, beginning at the first order at the right of ones, and reading to the right. 6 REVIEW: READING DECIMALS Oral I. Read the following. WU. Give numerator and denominator. III. Give the value of each figure separately. oe 5. 4.053. 9. 0.1478. 2. 0.54. 6. 0.765. 10. 16.47. ee AVE 7. 0.249, 11. 18.475. 4. 0.003. 8. 0.319. 12. 9.0364. 13. Which are mixed decimals ? 14. Where is and used in reading decimals ? 15. If you should express the above decimals as common frac- tions, how would the number of decimal places compare with the number of zeros in the denominator ? ABSTRACT AND CONCRETE NUMBERS LS tts (One Lih iL Oe oye ames. Which of these numbers are concrete, that is, associated with some- thing? Which are abstract, that 1s, which are used alone without reference to any particular thing ? 2. Classify: $275; 362 1b.; 875; 1000; 600 acres; 90 men. 3. What is the unit in each of these numbers ? 4. Select those having like units, that is, those having units of the same kind and size. 5. Tell what kind of number, and the unit of each: — A fts. B 25k ya.y (O05 18s) 10.865 016 tts ard: 6. Give the integral and the fractional unit in the following: — 43 dozen; 214; 94 quarts; 2.1 seconds. 7. Which of the following have the same integral unit: — t1b.; 40z.; $ton; 2000 1lb.; ton; 4 cwt. 8. Change the unit without changing the value: — 36in.; 6ft; 14 ft.; 120sec., thr; 4 wk. Oral REVIEW: UNITED STATES MONEY iT 1. What is meant by a decimal system ? 2. Why may 1.23 represent dollars, dimes, and cents and not yards, feet, and inches ? 3. How many dimes represented in $12.625 ? 4. How are dimes usually read? What does the 5 represent ? How is it usually read ? 5. Read and explain the use of the zeros: $7.77; $7.07; $7.7; $7.70. Does $7.7 differ in value from $7.70? fiead the following, (1) as dollars, cents, and mills; (2) as dollars and cents; (3) as dollars and thousandths : — 6. $38.19. 10. $0.625. 14. $6425. 7. $5,192, 11. $ 309.083. 15. $290 222, 8. $24,072, 12. $ 400.040. 16. $80,025.95. 9. $36,051. 13. $ 64.375. 17. $8005.025. 18. Write the preceding numbers from dictation. 19. Express as dollars and cents and mills : — $2h; $20yy; $4; $24); $52; B64, 20. Name four silver coins. What is a nickel ? 21. What other metal is coined ? Into coins of what value ? 22. Whatisaneagle? A double eagle? A quarter eagle? 23. Whatisa mint? What is bullion? 24. Are mills ever coined ? Of what use are they ? 25. What is counterfeit money ? | 26. What gives value to paper money ? 27. Are United States coins made of pure silver or of pure gold ? Why is an alloy used ? 28. Find what is meant by 18-carat gold. 1. REVIEW : Give the sum of 7 and 8. how could you find it by counting ? ADDITION At Sight If you had not learned this sum, 2. Find and explain a quick way of adding the following: — lnonwairkoa lao kRaor lm wo aoNwkOa lo rmoRHae 3 17 83 36 64 32 68 95 lomak ie Explain what change you make before adding : — 3 wk. and 14 da. 3. Principle. 2 gal. + 6 pt. 7. Tyr. +96 mo. 8. 9 2 ay 2s Ae gaan OT ts eu Osrins = —— pt.; 4. dyd., 7 ft. and 24 in. 5. In Exercise 4, why not change the 3 yd. and 7 ft. to inches ? Before unlike units can be combined into one sum their units must be made alike. 6. 48 oz. -- 2 lb. = —— Ib. tg t+t=7 6 yd. + 36 in. = in. t+3= 19 Practice until you can give the sum of these numbers instantly : — 10. ke 12. 13. 14. 1857 16. 1B fe LS. 46, 34. 19, 71. 53, 47. 86, 32. 38, 69. AT, 46. 65, 25. 32, 99. 72, 88. 19: 20. 21. 22. 23. 24. 25. 26. 27. 127, 123. 28 900, 140. 29 560, 240. 30 767, 232. 31 808, 191. 32 346, 509. 33 888, 212. 34 694, 106. 35 333, 766. . 8000, 1798, 2000. . 1300, 2000, 175. . 4080, 1507, 6000. . 85, 300, 9000. + ts Ts a5 a: 10.0670: 1870.24: . 25%, 8%, 30%. . 0.41, 0.19, 0.40. . $4.75, $3.25, $7.87 Written REVIEW: ADDITION Give directions for five steps in adding : — I. Arranging the numbers. 378 II. Beginning to add. A92 III. Setting down the sum. 864 PV. i Carrying: 798 V. Checking. 956 3483 Without copying, first add vertically ; then horizon- tally : — if 2. 3. 4. 5. 6. $3.47 $14.69 $193.67 $4769.83 $6483.47 7 $62 48.96 846.84 4892.16 8432.97 8. 946 387.81 932.71 8487.66 6432.98 9. 658 47.94 683.77 6989.84 8469.32 10. 7.89 82.66 765.75 4829.41 9396.48 11. 9.88 68.48 392.50 6832.47 9375.58 12. Find the sum of the five sums of the columns. 13. Find the sum of the six sums of the lines. . Why should these two sums be equal ? Add and check by adding upwards and down- wards : — 15. $475.21 649.85 837.64 246.89 937.48 742.37 —_———— 16. 17. $ 648.93 973.26 387.92 814.78 687.34 968.47 $ 719.63 854.56 784.97 469.38 847.86 952.78 18. $ 963.94 738.42 697.18 346.32 923.76 768.93 ——_—— 19. $ 679.83 759.94 678.90 543.21 783.94 989.76 A 679,458 a 340,276 b 950,673 ¢ 268,479 d 728,735 e 629,876 f 724,894 g 548,975 h 829,386 7 445,876 j 317,872 k 763,874 _1 689,983 m 670,498 n 988,875_0 687,568 p 994,693 gq 849,376_7 649,478 s 384,925 t 569,247 _u 347,964 v 976,394 w Fay 3 ko EGS Be 917,966 y 2997 G2. 20. Make fre- quent use of Col Aa tor practice in rapid add- ing. 10 REVIEW: SUBTRACTION Oral 1. 8 from 17 leaves what? If you had forgotten, how would you have found out that 17 -8 =9? Make a problem in subtraction, using concrete numbers. Which is the swbtrahend, and which the minuend ? The three terms used in subtraction are ; , and The largest term 1s CF a How do you find the third term when you have the difference and ie subtrahend ? 7. Howiis the third term found from the minuend and difference ? 8. Which is the larger number, 3 ft. or 24 in.? Which is the larger quantity ? How can one be subtracted from the other ? 9. A boulder weighs 7000 lbs., a stone block 4 ton. Find the difference in weight and explain the process. Where you can, give the difference, first like the minuend, then like the subtrahend, in the following : — 10. 4 lbs. — 32 oz. = ——. 12. 10 hr. — 240 min. = ——. 11. 60 mo. — 2 yr. = ——. 13. 2T.— 2 lb. = ——. Give in one minute or less the difference between each number and the one below it ; between each number and the one at the right of it. a LL OS AS eT AOS Le ed ee aL i ee ae ee aie) Tie op te: 6 Vb 8 4 Sap b DF 69 tO ALG vai tL La ea 6S Sl Aad os Ae Le ae (45 Dam 0 Sho tel 6 arent ) 4 9 c iRay A TULO HAD {Lae Oe ese Se i hiee 4 58 Ome aL Hi 5 4 a d DAU, 2) EG aaah a Ro heals Kini Maype bate) let ls de se 6 8 4 ein he Pas, Tong O01 Be) 3. A Oral REVIEW: SUBTRACTION 11 1. From 97 count backward rapidly by 6’s; 8’s; 9’s; 12’s, 2. From 200 count backward rapidly by 15’s; by 12’s; by 22’s. 3. Give the difference between 100 and each of the following: — a He OSs 144 er aa voce OelroOr lo OU Of jor 48. 1 Sl 8h b SO ROL NES Lalo 22.605 A253 18 O91 69° 47 G: 80,2009 25 01 6892435) 95° 29») 68 $2 | 84°26) 79° 16 d. DOSE TO ane cla Ob) GOYA 93 etl 66+) 58> Ol e Ole Oo meoavOr Lo s0- 40: 1076 249 1d, 62h 27 5 85 if: BLT O40 28084) 89°) 78. 88. |-80 25 | 97 54. |.98 40 4. Give the difference between each number and the one at its 5. Give the difference between each number and the one below it. 6. ‘From 1000, take 120, 175, 225, 350, .760, 807, - 901. ”. Take each number in the table from 173. From 182. What change from a $ 5.00 bill in payment for :— 8. Oysters, $0.75; 9. Gloves, $1.25; 10. Pens, $0.35; Crackers, 0.88; Scarf, 0.75; Ink, 0.15; Cheese, 0.62? Pin, 2.50 ? Paper, 0.88? Subtract at sight : — ji 0 12. ES 14. 15. 16. 700 3000 60503 25000 35111 36459 325 800 40402 37892 46221 47560 Find what remains after receiving and paying as shown below : — RECEIVED PAID RECEIVED PAID RECEIVED PAID 17. $1.16 $0.93 18. $45.00 §$ 28.00 199. 2.20 ato 0.24 Q.17 95.00 19.00 O19 2.30 0.60 0.25 70.00 23.00 1.25 1.25 12 REVIEW: SUBTRACTION From 683 take 457. . 1. If you try to subtract one order at a time, teharetagt 683 what is your first Serene bL Subtrahend, 457 Pa Ake you had 83 sticks in bundles of 10 each, Remainder, 226 with 3 sticks over, how would you subtract 7 sticks ? 3. How many bundles would remain? How many sticks over would remain ? 4. You would then take 5 tens from what ? 5. What terms may be added to check the work? PROCESS 6. Give directions for each separate step in the process. Written Exercise for Drill Without copying, find quickly the sum of the four differences between : — 1. eand 3.9 and h. 6... cand 7 ye iysa one on aie 2. fandg. 4. handi. 6. jandk. 8. landm. 10. nande. A B C D ée. $ 3764.82 $ 4769.31 $ 5000.37 $ 9000.15 i 927.35 3468.97 689.82 794.38 9g 860.85 385.68 1348.75 1866.75 h. 1527.96 2487.52 946. 2889.43 i. 3784.98 694.39 37.89 648.95 j. 2876.45 1748.64 9586.34 1864.37 k 825.35 4839.87 829.85 624.94 £ 96.47 658.34 1472.98 1739.41 m 849.53 1987.62 468.52 866. n. 276.41 594.83 5500.31 49.75 11. Calling A and B the two sides of an account, find the balance. 12. Do the same with O and D. 13. With Band @. What will balance : — 14. A and C. 15. Dand A. 16. Band D. Oral REVIEW: ADDITION AND SUBTRACTION is) 1. Four parts of 75 are 18, 9,13 and 22. Find the fifth part. 2. 37 gallons are ina tank. While 28 gallons run out, 17 run in. How many gallons remain ? 3. An engine goes forward 25 rd., back 388 rd., forward 60 rd. How far is it from the starting point ? 4. How much farther is it around a 17-foot square than around a 13-ft. square? How did you get your result ? 5. By annexing to 57 the figure 6, how much is added ? 6. Bought a pony and phaeton for $500. Sold the pony for $175, losing $50. What did the phaeton cost ? 7. Having $400 in the bank, a person draws $25, deposits $150, draws $75 and $50. How much remains ? 8. One horse is worth $50 more than a second and $150 more than a third. If the highest priced one is worth $200, what are they all worth ? 9. If you get 5 eggs one day and 6 the next, how many dozen will you get at this rate in a week ? 10. How far around a rectangle 15 feet long and 10 feet wide ? 11. Around a rectangle 18 feet long and 12 feet wide? 12. My book contains 170 pages. I have read 82 pages. How many more have I to read ? 13. IT had $100 ina bank. At different times I drew out $ 12, $18, and $20. How much remained? 14. A farmer had 75 sheep. He sold 45 and bought 17. How many did he then have ? 15. A trader had 29 horses and bought 57. He then sold 69. How many had he left ? 16. Ina farm of 160 acres, 23 was woodland, 47 pasture, and the remainder grain. How much in grain? 14 REVIEW: ADDITION AND SUBTRACTION Written 1. How much remained in bank to Mr. Rich’s credit Saturday night, if he put in and took out the following : — Deposits: $ 26.95, $793.82, $427.96, $ 839.64, $500, $387.28; Withdrawals: $18.56, $ 689.37, $419.28, $ 649.39, $ 600, $125.82? 2. I have on hand at the opening of business, cash to the amount of $846.95. I pay out $84.92, $64.87, and have on hand at night 837.69. What have I received ? 3. I received during the day $249.85, and I paid out $521.75. I had on hand at night $37.62. What had I on hand at the opening of business in the morning ? 4. Thomas Bond begins business January 1, with cash $478.37 and merchandise $1875.28. At the close of the year he has $1487.63 worth of merchandise and $738.29 in cash. How much has he gained or lost during the year ? 5. The sum of two numbers is 346,301. The smaller is 89,795. What is the larger? © 6. What number must be subtracted from one million to leave the difference between 347,698 and 486,931 ? 7. The distance from A to B is 628 feet, from A to C 1426 feet, and from B to D 1648 feet, all in a straight line. How far is it from C to D? Draw a lne and mark off the distance. Find the excess of exports from this country when the exports and imports from 1898 to 1902 were as follows : — YEAR EXPORTS IMPORTS 8. 1898 $ 1,255,546,266 - - $634,964,448 9. 1899 1,275,467,971 798,967,410 10. 1900 1,477,946,113 820,140,714 11. 1901 1,405,375,860 880,419,910 12. 1902 1,360,696,355 696,270,009 13. Find the increase or decrease of imports and of exports from year to year. Oral CASH ACCOUNTS 15 1. A debtor is one who owes another, or is in debt to another. A debit is something owed. 2. A creditor is one to whom another owes a debt. A credit is an amount owed to one’s account. 38. An account with “Cash” is, as it were, an account with one’s pocket book or cash box. Cash is debtor, that is, owes me, for all that is put in, and cash is credited with all that is taken out. DT CASH Cir 1903 1905 Apr. 1| On hand $100 | 00 May 3] By Mdse. bought || $450 | 00 5| To Rent received 50 | 00 4| By Piano bought 350 | 00 7| To Mdse. sold 25 | 00 8} By Clothing bought 25 | 00 10} To Land 725 | 00 11] By Balance 75 | 00 900 | 00. May 22] On hand 75 | 00 4. Cash is charged with having received four amounts, which it owes me, that is, for which it is my debtor. How much on hand at the beginning ? 5. What is the total amount Cash has received, that is, it owes me or 1s debtor for how much ? 6. When I take out $350 with which to purchase a piano, Cash has paid me back how much of what it owes me ? 7. What other amounts has Cash paid me, that is, for what other amounts should Cash be credited ? 8. How much more has Cash received than paid out ? 9. How much more might I have spent so as to balance the footings ? é 10. For what is Cash debtor at the beginning of the next account ? 16 1. Balance the Cash account of Charles Watson. He receives at various times $6.24, $7.36, $8.49, $7.34, $4.21. BALANCING ACCOUNTS $6.75. He pays out $8.75, $9.81, $8.39. Written He has on hand 2. Monday morning a merchant begins business with $247.84 on hand. He receives $24.75, $86.91, $84.28, $97.25, $164.29. He pays out $18.99, $37.49, $64.91, $83.15. hand. Find the balance of each of the following accounts : — aT: $ 987.65 1859.76 6482.91 478.85 698.47 Dr. $ 246.94 839.76 842.94 327.68 946.32 Dr. $ 94.68 37.95 469.38 24.38 6.49 17.32 Cr. $629.55 83.74 968.71 28.46 318.93 Cr. $ 839.75 646.81 794.32 546.78 937.89 Cr: $ 986.84 69.39 74.29 83.62 45.39 169.38 4. Dr. Cr. $4768.82 $468.34 947.61 984.59 847.77 1483.22 3998.64 8372.91 ake Dr. Cr. $698.32 $649.83 316.59 478.88 843.26 694.31 695.98 883.24 831.96 695.64. 10. Bye Cr. $346.85 $249.65 976.87 998.54. 695.79 648.36 949.83 799.35 697.87 869.40 749.78 309.79 Dr. $ 649.81 8439.87 648.38 91.76 Dr. $ 356.78 938.12 45.23 938.85 876.23 $192.48 765.35 362.93 15.56 746.29 126.48 Find the balance on Cr. $135.72 873.54 137.92 7639.85 736.29 11. $129.76 947.34 274.56 1286.54 364.92 3647.10 Oral REVIEW: MULTIPLICATION 17 1. In combining unequal numbers, as 9+8+7+4= 28, what is the process called ? 2. T4+7+747 or 4X7T=28. What is each of these two processes called ? 3. Could the unequal numbers, addends, in Exercise 1, have been combined by a shorter process, as in Exercise 2 ? 4. If you do not know the product of 5 x 8 from memory, how may you find it? Which number is to be multiphed? Which is the multiplier ? 5. What does the multiplier show? Then can it ever be a concrete number, as 5 men, 5 feet, or 5 strokes ? 6. Compare addition and multiplication. 7. What are the three terms in multiplication ? Since the multiplier and multiplicand make the product, they are called factors (makers) of the product. Name quickly the factors, less than 14, that produce :— 8. 65, 72, 77, 78. 11. 96, 99, 104, 108. 9. 81, 84, 88, 91. 12. 110, 117, 121, 130. 10. 48, 52, 54, 63. 13. 132, 143, 156, 169. One of the two equal factors that make a number is called the square root of the number. 4/ means “the square root of,” thus 14. V64=-——; V8l=——; V121=——_; Re ae 15. V9x25=3%x5 0r15; V16 x 25 =—_;; V25 x 86 =—_-; V25 x 49 = —_. 16. V49x36=——_;_ V36x81=——_;_ V9 x 144=—_;; 1/36 x 121 =——. 17. V4x16 x 25=2x4x5 or 40; V4 x 25 x 86 =—_; V9 x 16 x 49 = —_. 18 REVIEW: MULTIPLICATION Oral 1, LSB S B39, fac esiggete i eG eae 2. Give the factors, saying which is the multiplier and which the multiplicand: $42; 63 ft.; 72 yd.; 12 sq. ft. 3. Make two examples; the multiplicand concrete in one and abstract in the other. 4. What kind of a number was the product in each case? 5. Can the multiplier be concrete? Why? 6. Show by objects that 3 x 4 things of a kind are 12 of that same kind. 7. Give the factors of $21; 35 miles; 18 cases; 49 men. 8. Compare 4 x 5 bu. and 5 x 4 bu. Give each product quickly, stating which factor is multiplicand : — 9. 10. a7 12. 13. 14. 15. $ 800 600 ‘700 rd. 900 800 da. 400 6000 9 BUD ase: ene aby _16 men 13 Prinocrptes. I. Only one factor can be concrete, both may be abstract. Il. The product and the concrete factor are like numbers. III. The order in which factors are used will not affect the product. Give rapidly the products : — 16. 4x $8; 44 x $8; 3x 8%. 195° 3 <1 ORmddacr ee L713 Xp Os x beans OT. 20:u13ix 5 lee: 18. 2x 3874; 5x124; 4x $14. 21°16 XA wy to: 22. Multiply the following by 8; by 9; by 12: — 7 bales, 70 bales, 80rods, 30miles, 90 feet. 23. Multiply by 9 and add 9: — (Cam) orc Paras pee IB AY ees mI Ra Sibson mest (Ini wT EED 24. How could you have got the same result in a shorter way ? Oral REVIEW: MULTIPLICATION 18 1. Give two factors making : — G2 i 2) 40° T008 yO) eet; 1/78 3) 756 days; ) 81 men: 2. Give two factors of 132 sec.; 125 in.; 144¢; 108 hr. 3. Take 4x 7 from 9X 7. Howcan you do this without finding the products of 4x 7 and 9x7? 4. From 9 x 8 take 6 x 8 in the same way. Se Lakemoogo trom 22 x9 16x 7 trom 2ux f, 6. Add 18 x13 to2x13. To do this did you need to know the product of 18 x13? 7. Add 17 x 25 to 13 x 25 in the same way. hae 9. Add at sight 37 x 25 to 3 x 265. 8. ox 6in. +7 xX 6 in. = 10. From 63 x 8 take 13 x 8. 11. 3*means 3 x 3) or 9; 5? means 5% 5 or 25. Find 4*; 67; 77; me Lerten BO 20 eOOs, 7 erin 40-2 0e O00 = O07, 13. Compare 5 and 50. Tell how you would multiply a number by 10. 14. Compare 5 and 500. Tell how you would multiply a number _ by 100. 15. How would you multiply by 1000, 10,000, etc. ? 16. Multiply the following by 10, 100 and 1000: — 59, 28, 147, 261, O15, 140. Principle. Hvery zero annexed to an integer multiplies it by 10. 17. Compare 20 x 5 and 2x10 5; 507 with 5.x 10 x 7. REVIEW: MULTIPLICATION Written 1. Under A what are added to get the B Ga result ? 489 489 2. Explain the position of each product raid ae 3423 34230 and its real value. 3. Show how the same result was got under B without setting down the partial products. 4. Compare the work under B and C, and tell how to multiply by any number of 10’s, 100’s, ete. Find quickly the product : — Deo ATO, 9. 80 x 34,965. 185 80 KZ eo 6. 8 x 4931. 10. 70 x 12,089. 14. 70 x 95,364. 7. 9x 6989. 11. 300 x $42,794 15. 34,000 x 91,000. Sen COUS IL. 12, 30. X11203,900: 16. 28,000 x 75,000. Process Multiplying by any Integer ae 1. In the work at the left read the multi- 3468=6 x 578. plier. 23,120 =40 x 578. 2. What three partial products are used ? 173400’ = 300 573. 3. Explain how each is obtained. 199,989 = 346 x 578. 4. Would the result have been changed if we had multiplied by 300 first? By 40 first? 5. In ordinary work what is omitted from each partial product ? 6. Where is the lowest figure of each partial product written if the zero is omitted ? Give directions for six steps in multiplying : — 1. Arranging the factors. 4. Arranging partial products. 2. Beginning to multiply. 5. Finding the entire product. 3. Setting downandcarrying. 6. Checking the work. Written REVIEW: MULTIPLICATION yaa be 1. What is the cost of 2378 bbl. flour at $7 ? $7 multiplied by 2378 must equal $ 16646 2. Which is the true multiplicand ? for 2378 multiplied by G equals 16646 To shorten the work, why must we use abstract numbers as shown at the right ? Find the product of: — 536 and 846. 8. 3976 and 597. 9. 37 ¢ and 482. 10. 427 and 83 lb. 1 at 329 and 347. ie Find the cost of:— iS: 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 1579 bbl. sugar @ $9.87. 6496 A. of land @ $34.89. 12,865 T. of coal @ $ 4.87. 16,492 lb. cotton (@ $0.09. 24,975 lb. wool @ $0.35. 6425 cords wood (@ $6.98. 47,000 ft. lumber @ $42.25 per M. (per 1000 ft.). 5600 bu. wheat (@ $0.89. 2745 bbl. cement (@ $1.25. 47,892 gal. oil @ $0.08. 135 bbl. pork @ $16.87. 2892 kgs. nails (@ $3.09. Gog andro (6s wis. $627 and 931. 14. 36 doz.and 453. 15. 864 and $45. 16. 387 and $8. 30. . 42,000 lb. coffee @ $ 0.23. . 6425 pairs shoes @ $ 2.25. . 176,000 bricks @ $12.84 84 qt. and 96. 346 and 19 ¢. 8477 and 86. 98 oz. and 43. 17. 304 posts and 97. 7859 bu. potatoes (a) $ 0.79. Pele . 45,761 ft. granite @ $0.75. . 43,628 sq. ft. land @ $0.29. . O79 T. ice @ $4.67. . 47,000 shingles (@ $3 per M. . 87 gal. alcohol @ $2.65. . 375 bu. corn @ $0.87. . 3784 lb. butter @ $0.22. . 967 doz. eggs @ $ 0.18. 42. How many rods in 54,000 miles ? 43. How many square inches in 937 square feet ? 44, How many seconds in 18 days ? 92 MULTIPLICATION; ANALYSIS; STATEMENT OF PROBLEMS Oral and Written Give at sight :— 1. 17x$94+3x$9=2. 4, 125x$9.85—25 x $9.85=2. 2. 87x $7.504+13x$7.50=a. 5. 987x$125+63 x $125=2. 3. 64x $48+36 x $48=2. 6. 157x316 ft.—57 x 316 ft.=2. Use. the same method in finding : — 7. 649 x $12.84 less 145 x $12.84. 8. 317 x $19.83 + 524 x $19.83 + 160 x $19.83. 9. 975 x 846 4+ 973 x 352 — 973 x 198. Analysis and Statement of Problems Much work may be saved in solving problems if a statement is made showing all that is to be done to get the result before any figuring is done. 1. Which is easier to do; to reason about a problem so as to show how it may be solved, or to figure out the result after being told how to solve it? 2. From which do you learn more ? 3. A statement of all that is to be done to get the result written as equal to something denoted (for convenience) by the letter 2, is called an equation. Make a problem in which the equation 4 x 161¢ =a will indicate the work to be done. 4. $2.50 was the expressage on 19 tables at $12.74, and 28 chairs at $2.58; « was the total cost. STATEMENT. $2.50 + 19 x $12.74 + 28 x $2.58 = & (or the total cost). Make an equation showing all that is to be done to find the value of x; then find it: — 5. 130 men at $2 a day, 47 at $1, and 8 at $3.50 receive 2 dollars in one day. 6. A nursery contains 1000 trees; 75 are dead ; Oe rest are to be sold at $2 each. They will bring $a. Oral REVIEW: DIVISION 23 Finding an Unknown Factor .1. The product of two numbers is 48. One is 6. Find the other. 2. How many 9’s in 54? What number is contained 9 times in 63? In 108? Ses fo u5) the multiplicand; how many times is it taken to make the product 84 ? 4. What multiplicand, repeated 12 times, makes the product 108 ? 5. When one factor and the product are known, how is the other factor found ? 6. Illustrate by using « xX $6=$42; and5 x $a7= $45. 7. Why is the process called division ? 8. Show by the examples in Ex. 6 that: — (a) The product becomes the dividend (something to be divided). (b) The known factor becomes the divisor. (c) The unknown factor, when found, becomes the quotient. (d) The quotient shows either, how many times the divisor can be taken out of the dividend, or the size of each of the equal parts into which the dividend is separated. Give the quotients : — 9. 15x «= 30 days. 12. 12x %=144 miles. 10. «x $9 = $108. 13. «x 9 ft. = 1385 ft. Lie Oe a= 120, T4eeo, xtliivd. Sad oleyd: 15. Give the quotients in the following : — 6)120; 96+ 24=%. 360=%; 72: 4=2. a 18 16. Describe the four ways of indicating division shown in the pre- ceding line. 17. Find any two factors :—96 bbl.; 91 days; 168 hr. ; 182¢. 24 DIVISION: PRINCIPLES Oral 1. How many $10 bills make $300? If one factor of 60 yd. is 10 yd., what is the other ? 2. In multiplying two factors to make a product, which factor is always abstract? Which may be concrete ? 3. In division, which term corresponds to the product? Which to the factors ? 4. Can both dividend and divisor be concrete ? 5. Show by the first example on the page that if the dividend is concrete and the divisor like the dividend, the quotient is abstract. 6. If 8 hats cost $40, what will 1 cost ? 7. A rod 90 inches long was cut into 10 equal parts. How long was each part ? 8. Show by the above example that if the dividend is concrete, and the divisor abstract, the quotient shows the size of the equal parts into which the dividend has been divided. Principles. I. A divisor that is like the dividend is one of its equal parts and the quotient shows the number of these parts. II. An abstract divisor of a concrete dividend shows into how many equal parts the dividend is to be divided and the quotient shows the size of one of these parts. 9. When 20 books cost $40, what part of it will one cost ? 10. One factor is 15. The product is 90 yards. What is the other factor ? 11. The divisor is 12, the dividend is 108 bushels; what is the quotient ? . 12. The divisor is 9 ft., the quotient is 11; find the dividend. 13. The dividend is 2100 mi., the divisor 21 mi. ; find the quotient. 14. At17¢ a yard, how many yards can be bought for $1.53 ? Oral REVIEW: THE PROCESS OF DIVISION 25 1. Division is the reverse of and 72 +9= 8. 2. If you had not known that 8 xX 9=72, how might you have found the number of 9’s in 72? ; since 8 x9=72,72+8=—9 3. Find by subtraction the number of 12’s in 60; of 24’s in 96. 4. How many 12’s in 1740 ? 5. Are there 200 12’s in 1740? A Ve 12)1740 oe Are there 100? ‘Subtract them ; what 1200 = 100 12’ remains ? 540 7. How many 12’sin 540? Are there 50 ? 480 = 40 12’s Are there 40 ? 60 ; 8. Subtract 40 12’s; what remains? Ueda es Total = 145 12’s 9. 60=how many 12’s? Subtract them; what remains ? 10. How many 12’s in all have been taken from 1740 by the three . subtractions ? 11. In practice what part of the work might be omitted ? Where may 1 be written to show by its position that it stands for 100? Where may 4 be written to show it stands for 40? In practice the work is written as in the margin. a 145 12. Short Division. Perform aloud the work of C. 12)1740 How does short division differ from long division ? 1 13. If 3465 is divided by 58, in what place me Z ) 48 12)1740 should the first quotient figure be written { a 145 14. 3465+19. How many figures in the 60 quotient ? 15. When should short division be used ? 16. How may division be tested or checked ? 17. How may the process of multiplication be proved correct ? 26 Pe WO WO EH EXERCISE IN DIVISION Written How many 15’s in 4650 ? 27 XxX —— = 40,527. Dividend = 9672; divisor = 372; quotient = ——. 96 and 75 are factors of what dividend ? Product = 33,810; one factor = 245; «=the other. One factor of $475,000 is $250; what is the other ? 84 equal numbers make 6500 yards. Find one of them. 360 miles = multiphcand; 25,520 miles = product; «= multi- . Divisor = $250; dividend = $16,750; «= quotient. . The divisor =197; the quotient = $461; the dividend = 2. . I received $1394 for 17 horses. What was the average price ? . Divide $39,624 into 48 equal parts. . Find J, of 3627. 15. Divide 3962 by 49. . Find 7, of 39,627. 16. In38962 ounces, how many pounds? ' . Divide $82.36 into 32 equal parts. Find the quotients : — 18. 19. 20. 21. 22. 23. 24. 39. 759,470 + 78. 25. 89,175 + 39. 32. 183,974 +94. 624,798 + 48. 26. 284,603 -+- 98. ' $8. 265,371 + 88. 182,347 -- 57. 27. 99,134 + 49. 34. 104,288 + 78. 96,343 + 97. 28. 108,264 - 57. 35. 139,267 + 72. (192,462 ~~ 67. 29. 346,271 + 86. 36. 204,306 + 68. 236,475 + 77. 30. 937,441 +163. 37. 307,961 + 96. 187,931 + 68. 31. 784,267 + 269. 38. 198,001 + 67. Divide each number in Column A, page 9, by the divisor formed by the last 3 figures in the dividend used. Thus: 679,458 + 458. Oral, Written SHORT PROCESS IN MULTIPLICATION 27 1, Compare 5 with 4,2. Compare 5 x 84 with 12 x 84. 2. What does 42 x 84 mean ? cs ; 84 _ what fi 3. Describe a short method of multiplying by 5. 4. Compare 25 with 192. Find 25 x 84 by a short method. Multiply each of the following by 5 and by 25: — ig EE eres AKO 9. 35. 11. 208. Ted92s7 1B. 49: Gr JOS Gun S.6e | i LOASZ¢: 12/,°308: 14.1695. 16. 63. 17. By what must 124 be multiplied to make 100 ? 18. Compare 124 with 19° Multiply 96 by 124 by a short method. 19. By what must you multiply 162 to get 100? {of100 =? 20. Compare 162 with 129. Multiply 42 by 163. 21. 40f100 is what? 331=192, Multiply 45 by 33}. Multiply the following by 124, 162, and 333: — 22. 48. 26. 405. 30. 114. 34, 295. 23. 96. ithe P4se $1. 207. 85. 349. 24. 72. 28. 64. 32. 345, 36. 642. 25. 108. 29. 93. 33. 216. 37. 784. 38. 1of 1000= what? Give a short method of multiplying by 125. 39. Multiply by 125: 168; 256; 976; 856; 1040; 375. 40-55% 64-8 x 6=—13°%)6. Why? 41. 13x 6+418 x 4= what? 42. Find 17 x 18 + 23 x 18+ 60 x 18 by a short method, Find the products : — 43. 25x64. 46. 16x121. 49. 125x912. 52. 163 x 1026. 44. 5x82. 47. 162x144. 50. 121x128. 53. 333 x 126. 45. 14x25. 48. 25 x 256. 51. 25 x488. 54. 84 x 33}. 28 SHORT PROCESSES IN DIVISION Oral, Written 1. Compare the quotients of 24 + 2 and 24 + 6. 2. Compare 380 + 10 and 380 + 5. 3. If 38 is the quotient of some number divided by 10, 2 x 38 is the quotient of the same number divided by what ? 4. Explain: 320+5=2 x 320+ 10 = 64. 5. How do you divide a number by 10? 6. Give a short method of dividing by 5. 7. Compare 375 + 25 with 4 x 375 + 100. 8. Explain: 925+ 25=4 x 925 + 100 = 3700 + 100 = 37. In the same way divide the following by 25:— 9: 1325: 12. 1675. 15. 9675. 10m SOLO: 13974525, 16. 10,275. Lili vo: 14. 17,875. 17. 9625. 18. 96 +162 =? x 96 + 100. 19. Give a rule for dividing by 162. 20. Make a rule for dividing by 334. Use these rules in finding quotients in the following : — 21. 2500 + 121. 25. $15 + $ 0.124. 29. 1675 + 25. 22. 6400 + 33h. 26. $24 + $ 0.25. 30. 1925 + 25. 23. 1300 + 334. 27. $36 + $ 0.162. 31. 8250 -~ 50. 24. 1850 + 162. 28. $42 + $0.50. 32. 1750 + 162. 33. How many will $3 buy at $ 0.25 each ? 34. How many yards will $5 buy at 162 cents a yard? © 35. I paid $8 for tea at 33} cents a pound. How many pounds did I buy ? 36. My milk bill was $8.75 at 25 cents per gallon. How many gal. ? Oral THE USE OF SIGNS | 29 () asin (8+4)x5=85,0or asin3+4 x 5=85, shows that the numbers inclosed or beneath are to be treated as one number. 1. 9+6)x 8—5) =a; 84+7x10—5=«. | 2.3xP4+7=0;4x10+50=a; 27+38—-V4=2. 8. bb —-5 x2=e) 18-8225 =e; 10—4)+6—3=2. Observe the following and tell which process is performed first when xX or + is on one side of a number and + or — on the other : — 4. 344x5=3420=28. 6. 6x12—8+2=72—4=68. 5. 44-16-38 4-2 2—0: T 6x150+24+6=754+4=79. 8. Compare in value 138 —5 X 2+8 and 13—5) x (2+ 8). 9. 86+4—45+9=a%. 11. tof 7 +14) —tof 867—9)=2. 10. 2+5x648+2=¢a. 12. tof 41—16)++2 of d9—T)=2. 13. What expressions are here marked to be treated as one number ? 8x 16+848x 16—9)=V3x9x3 x 108+12—10+1. 14. What is the value of V64+4+8x2? 15. (V100xX4—5) +5432, “18: (29+46—1) x 18=-2. 16. (8?+6) x5+3+45=@. 19. (89+10)x2+6=2a. 17. 4—-4x38+4+10—-—2x5=2. 20. (6?+4—10) x84+9=2. 21. What number is to be divided by 11? (6x54+4x9)+11=[(6+5) x 00 —4)]+11. 22. Show why it was better to use brackets [ ] than curves ( ). 23. x” = (12?-+ 24) x V54—5. 24. (4800 + 100) + 0.01 of 600 =a. 25. w=[6x8—4 x (14—4) +60] = 100. 26. (a) ve] n. 27. (12x446 x 12) +V100=2. 28. x=[(7+3) x2—4 of 39). 29. [(V25)?—V6 xt] 4:4 of 72—2. 30 EQUATIONS Oral a —=12+25 means that w stands for some number which is equal to the sum of 12 + 25, hence x = 37. Find the value of x in the following : — 1. we=12+ 25. 6. 42—¢=—19. 11. 19+-11+-4=—50. 2. 88+12 =>. 7 «+17 =32. 12. 72+2+414= 96. 3. 44—19 =~. 8. 28+2=50. 13. 40+ 20—2=50. 4. e=100—72. 9. $—w=3. 14. $2.75 += $4.50. 5. «—24=48. 10. llb.—w=12 02.15. «—$7.30=92.54. 16. Lam 2 years old; in 8 years my age will be 36 years. (w+ 8 years = 36 years.) 17. After taking $14, $16, and $12 out of a sum of money $3.75 remained. There were $2 at first. «— $14 — $16 — $12 = $3.75. 18. A prize cup contains 23 oz. of gold, 10 oz. of silver, and 2 oz. of alloy. The cup weighs 42 oz. (Make an equation.) 19. 25 gallons run into a tank, and 46 run out. When the faucets were closed, 80 gallons remained. There were «# gallons in the tank when the faucets were opened. (EHquations.) 20. Make a problem about the weather in March to suit this equation: 31 da. =12 da.+ a da. +10 da. 3a means 3x. 3e=15 means that 3 times some number equals 15. Find the value of x in the following and explain your method : — 1. 4x2=60. 6. 87=400. 11. | oo 16. 4¢+-5=21. WO PIS bo 2. 17xd5=—2. 7%. Te=91. 12. =4, 17. 1772—4=80. oe 7 of «=16. 8. 1472=700. Loy | ee Or 18) 18% x10 = 180. 4. 80+a=4. 91/25 0=6275. 914.--=4, 19. 42¢+21=70. le 8/S ble &8| 5. 144+a¢=16. 10. 44”7=45. 15. —=7. 20. 4 of 16%=120. bo aN Oral, Written STATEMENT OF PROBLEMS 31 1. If 16 cords of wood cost $120, 24 cords will cost what? In solving such a problem, which of these suggestions seem most important ? I. What is to be found out? (Cost of 24 cords.) Il. What facts will help to find this ? (16 cords cost $120.) Ill. Comparison of what is given to what is wanted. (24 cords will cost 14 times as much as 16 cords.) IV. Process, briefly set down. (14 x $120 = cost of 24 cords.) V. Work performed. (14 x $120 = $120 + $60 = $180.) VI. Does the result seem reasonable ? 2. Bought 12 lb. tea at 75¢, and 20 lb. coffee at 40¢. How much butter at 30¢ would cost the same ? 12 x $0.75 + 20 x $0.40 _ $0.30 Ctr 3. Exchanged a 60-acre farm worth $ 2400 for 200 acres of wood- land valued at $18.75 an acre. Find the gain. STaTEMENT. 200 x $13.75 — $2400 = x. 4. Gave 3000 sq. ft. of 20% land for a span of horses and $75. What were the horses valued at ? Statement. 38000 x $0.20 —$75=~-2. 5. A man purchased 130 bbl. of flour at $4.50 per barrel, and a number of barrels at $4. He paid $665. How many barrels of the cheaper flour did he buy ? $665 — 130 x $4.50 __,, $4 es 6. Sixty-four men are employed 25 days in digging a sewer. The contract price was $1200. Nothing was gained or lost. What were the men paid each per day ? 7. If 14 lb. cost $2.94, what will 10 lb. cost ? 8. If 17 tons of coal cost $134, what will 51 tons cost ? STATEMENT. STATEMENT. 32 MISCELLANEOUS PROBLEMS Written Applying the suggestions on the preceding page, state the following and explain orally :— 1. I bought a field of 10 acres for $1000. I sold 7 acres of it at $125 an acre, and the remainder at $85 an acre. How much did I gain ? 2. I sold 50 acres of land for $5000. This was a loss of $15 per acre. What did the land cost me? 3. A boat goes 10 miles an hour -downstream and 6 miles an hour upstream. How long does it take to go 30 miles and return ? 4. If 15 men can do a piece of work in 90 days, how long will it take 6 men ? 5. If 14 bbl. of apples are worth $35, what are 21 bbl. worth ? 6. A train runs 280 miles in 11 hours. Seven 3-minute stops are made, and a hot axle makes a detention of 39 minutes. The rate per hour was 2 miles. 7. Six men buy 640 acres at $125, and sell for $95,000. Each man gains w + of ($95,000 — 640 x $125) = each man’s gain. In the statement what represents the cost of the land? The proceeds of the sale? The whole gain ? 8. Bought 39 bbl. of flour at $4.75; sold 15 bbl. at $5, and the remainder at $5.25. Required, my gain. 9. Three 1-pound packages will go by mail each for 1% an ounce plus registration; by express, for 25¢ each. Which way is cheaper? 10. A peck, 2 bushels, and 5 quarts are to be divided equally among 7 persons. Any two receive x quarts. 11. I spent $4485 for cattle and horses, buying the same number of each. If I paid $75 apiece for the horses, and $40 each for the cows, how many of each did I buy ? SucexrstTion. What did 1 horse and 1 cow cost? Then how many times can I buy a horse and a cow with $4485 ? Oral and Written MULTIPLYING BY MIXED NUMBERS 33 1. Explain the process in each of the following: — Do Xd ia K Lo ey OE 2 So. 2. Give the results of the following: — 34x 8; 85 x 10; 54 x 15. 3. What does 2 of 9mean? Find 2 of 9; 3 of 16; 3 of 10. Nore. The sign of multiplication (x) may be used instead of the word ‘‘ of” but it must always be read ‘‘of’’ and not ‘‘ times’’ when the multiplier is a fraction. Thus, 3 x10 means 2 of 10. 4. Give results: — 2x 20; 4x 28; 2x 45; % x 56. 5. Give results:— 8x72; 3x 63; 2 x 36; $x 40. 6. 0.05 of 20 means 4 of 20 or 5 x 1 of 20. 7. zy or 0.1 of 20=—; 1, or 0.1 of 50 = —; 0.1 of 45= 8 10 . Give results : — 0.6 of 20; 0.8 of 60; 0.9 x 70; 0.08 x 400. 9. 25 % =25 or 0. —; 6% =0.—_; 6% of 200 = —. 10. 8% of 400; 15% of 400; 16% of 500; 17 % of 1000. 11. Compare 3} with 10. Show a quick way of multiplying by 33, 12. Compare 831 with 100 and show a quick way of multiplying by 33. 13. Give results: — 34x15; 34x18; 334 x 24; 334 x 36. 576 14. Supply omissions in the multiplication 8% . _at the right. 82 x 576. 12 =% of 576 ee 504 = 4 of 576 15. Give directions for each step in multi- AG08 — . plying by a mixed number. 5112 = 81 ¥ Written Work 16. 93 x 280. 19. 780 x 193%. 22. 13,3, x 280. 17. 183 x 942. 20. 603,35 x 2000. 23. 148 ft. x 784. 18. 11055 x 144. 21. 18% x 1728. 24. 911 lb. x 1080. 34 MISCELLANEOUS EXERCISES Written 1. If the Empire State express runs 115 miles in 108 minutes, what is the rate per hour ? 2. If 71 cu. ft. of water weigh 2 tons, how much will 2414 cu. ft. of water weigh ? 3. I paid $59.22 for potatoes, and sold them for $70.56. What was the gain per bushel, if I paid 94 ¢? 4. My July gas bill was $5.28 for 4400 feet of gas. In August | the gas company raised the price +. How much should my August bill have been, if 1500 feet of gas were consumed ? 5. At the end of August the reading of the meter was 5600. At the end of September the reading was 7400. What is my gas bill for September at $1.25 per 1000 cubic feet ? 6. The mailing clerk in the office of the Herald receives $10.95 for mailing 52,850 papers weekly. How much is that per hundred papers ? 7. The Transvaal gold output in August, 1899, was 459,700 oz., and its value was $9,194,000. The August output in 1902 was valued at only $3,200,000. How many ounces were mined ? g. Find the amount of a bill for the following: — 28% yd. silk @ $1.80. 24 yd. percale @ 0.121. 2% yd. velvet @ 2.25. 8 yd. silk @ 2.90. 9. A merchant purchased 950 barrels of flour at $6.90. He sold 325 of these at $7.20, 460 at $8.10, and the remainder at $9. What was his total gain ? 10. A fruit dealer bought 500 oranges at the rate of 2 for 3¢, and 400 more at the rate of 4 for 5¢. He sold the whole lot at the rate 3 for 5¢. What was his gain ? : Oral COMPARISON OF NUMBERS 35 1. If 5 lb. of cheese cost 80 cents, 10 lb. will cost what ? 2. Why is it needless to find the cost of 1 lb. in Ex. 1? 3. At the rate in Ex. 1, what will 23 Ib. cost? 4. When 21 lb. of steak cost $3.21, what will 7 lb. cost ? 5. Compare the time required by 6 men to do a piece of work with the time required by 2 men. By 12 men. 6. Compare the rent for 5 mo. with the rent for a year. 7. If a house rents for $500 per year, what is the rent for 5 months ? 8. 5 bushels of oats cost $1.70. At this rate, what will 15 bushels cost ? 385 bushels ? 9. 9 for $1 makes 6 cost what? 12 will cost what ? 10. $386 is what part of $108? Of $72? Of $144? 11. What 42 men can do in a week will take 7 men how long? How long will it take 28 men ? 12. Supplies that will maintain a regiment of 1000 for a week will maintain 100 how long? 600 men how long? 13. If 6 men can do a piece of work in 10 days, how long will it take 4 men to de it? 14. If 8 men can build a wall in 3 weeks, how many men will be required to build it in 1 week ? 15. If 60 horses eat 450 bushels of oats in a month, how long will it last 80? How long will it last 90? 16. If 12 dozen eggs are worth $1.80, what are 8 dozen worth at the same rate? What are 60 dozen worth? 17. If oranges sell at 3 for 10 cents, what is that a dozen ? 18. 4 ounces for 25 cents is how much per pound ? 36 PROBLEMS Written 1. Compare the cost of 36 bu. with the cost of 12 bu. The ratio of 36 to 12 is 2. If 12 bu. cost $7.50, what will 36 bu. cost ? Compare the cost of 85 yd. with the cost of 17 yd. If 17 yd. of cloth cost 18.70, what will 85 yd. cost ? Compare the cost of 114 gal. with the cost of 19 gal. 1) gal. of alcohol cost $72, find the cost of 114 gal. Oo oT Fe & ie Compare the time required for 91 men to do a piece of work with the time required for 15 men. 8. If 13 men can pave a street in 42 days, how long will it take 91 men to do it? 9. What will 364 bbl. of apples cost when 52 bbl. cost $175 ? 10. What is the relation (ratio) of ali of anything to 20% of it? When 20% of a crop of beans is 325 bushels, what is the whole crop ? 11. What is the relation (ratio) of all of anything to 162% of it ? If 162% of a certain number is 342, what is the number ? 12. A barrel of flour fills 8 bags and costs $4.50. What is the gain on 3 bbl. sold at $0.621 per bag ? 13. If Hans can haul as much sand in 15 days as Knut can haul in 20 days, which should receive the higher wages ? 14. If Knut receives $60 per 20 days, what should Hans receive per day ? 15. What is the ratio of all of anything to 50% of it? If 50% of a certain distance is 168 miles, what is the whole distance ? What is 25% of the distance ? 16. Compare 9000 lb. with a ton. What should I pay for 9000 Ib. of hay at $12.50 per ton ? Written DENOMINATE NUMBERS 37 Change 1. 3 da. 6 hr. to hours. 12. 13 T. 1600 lb. to pounds. 2. 18 hr. 43 min. to minutes. 13. 9 bu. 8 qt. to quarts. 3. 6 yd. 17. in. to inches: 14. 11 yd. 24 in. to inches. AS sds tt. LOT sq. cin. to 15. 9 hr. 48 min. to minutes. square inches. 16.75 ¢u. ft. 15635" cu. in.’ to 5. 8 cu. yd. 16 cu. ft. to cubic cubic inches. feet. 17. 356 qt. to gallons. 6. 5 gal. 3 pt. to pints. 18. 150 oz. to pounds. 7. 3 mi. 240 rd. to rods. 19. 17,562 cu.ft. to cubic yards. 8. 5 bu. 8 pk. to quarts. 20. 34,628 sec. to hours. 9. 2wk.5 da. 11 hr. to hours. 21. 348 wk. to years. 10. 9 A. 120 sq. rd. to square 22. 1965 qt. to bushels. rods. 23. 3468 cu. ft. to cubic yards. 11. 9 lb. 3 oz. to ounces. 24. 3462 gi. to gallons. 25. Change 120,000 min. to days. 26. Thirteen tons of oatmeal will make how many one-pound packages ? 27. How many pint bottles can be filled from 728 gallons of ex- tract ? 28. A team of strong horses haul 5 tons of granite. How many cubic feet in the load if each cubic foot weighed 165 pounds ? 29. Bought 3 acres of land and cut it into house lots, each con- taining 3267 square feet. How many lots were there ? 30. How many rails 60 feet long will be required to lay 20 miles of double track railroad ? 31. How many 3-ounce packages can be made from a quarter of a ton of pepper ? 32. How many pint bottles of mineral water can be filled from a tank holding 300 gallons of mineral water ? 38 COMPARISON; RATIO Oral 1. 24=2x 12; 12=1 of 24. What is the difference between two ways of comparing 24 and 12 ? In this way compare the following numbers : — pee Ie chivas 5. 20 and 80. 8. 49 and 7. 3. 15 and 60. 6. 25 and 125. 9. 63 and 9. 4. 25 and 75. 7. 30 and 90. 10. 42 and 6. 11. Compare 3 in. and 2 ft. ; 2 yd. and 6 in. 12. What is the relation of 3to9? Of4tol6? Of 5 to15? The relation of one number to another is their ratio. Ratio is ex- pressed as the quotient of the first number divided by the second. Thus the ratio of 6 to 2=3; of 2 to6=4. 13. Read these five ways of expressing ratio: — The ratio of 12 to 24is 4; 12:24=4; 12+24=4; #2=1; 12 is to 24 as 1 is to 2. 14. What is the ratio of 12 to 60? Of 124 to 100? Of 15 to 45? What is the ratio of :— 15. 121 to 25? 18. 81 to 162? 21. 20to 30? 16. 16 to 48? 19. 61 to 25? 22. 60to 90? - 17. 162 to 50? 20. 374 to 75? 23. 108 to 144? 24. What is the ratio of 121 to 183 ? SUGGESTION. 121 is 2 6}’s, and 183 is 8 63’s, hence the ratio is 2 to 3, or 2. What is the ratio of : — 25. 121 to 314? 28. 61 to 25? 31. 162 to 414? 26. 121 to 75? 29. 84 to 162? 32. 3314 to 2662 ? . 27. 374 to 874? 30. 162 to 100 ? 33. 662 to 300? REVIEW: PROBLEMS FOR ANALYSIS 39 If 15 oranges cost 40 cents, 5 will cost what part of 40 cents ? If 3 cost 7 cents, what will 15 cost ? If 64 lb. of honey cost 95 cents, what will 25 lb. cost ? Give two equal factors of 3600, that is, 3600 = what ? Divide 21,000 by 3000. Explain your method. Find the value of ¢ of 2400 — 3 of 1500. Find the sum of 3600 + 120, 3, of 2400, and ;4, of 75,000. 900 V121; V81; 144; 400; 1600; V/2500. 3X 0x T7+5xX6= what? What common factor in both friend and divisor may be dropped ? 10. Uae 12. men. 13. days ? 14. Divide 4 x 6 x 8 by 6; by 8; by 4; by 24. How many 7’s from 910 leave 700 ? Compare the work done by 9 men with the work done by 15 Compare the time required by them to do a piece of work. How long will it take 15 men to do what 9 men can do in 10 At the rate of 8 for 9 cents, what will a dozen cost? What will 24 cost ? 15. 16. VE 18. 2 ounces for 40 cents is how much per pound ? 40 sq. rd. of land for $1000 is how much per acre ? If 6 inches of pipe cost 35%, what will 1 yard cost ? If 18 pounds of sugar is worth $1.00, what will a bag con- taining 63 lb. be worth ? 19. 20. 1000. Divide 360,000 by 4000; 90,000 by 4500. Divide 3600 by 100; by 1000. Divide 280,000 by 100; by 40 DENOMINATE NUMBERS Written 1. Bought 44 barrels of flour at $51 and sold it in 1-bbl. bags at 70¢. Find my total profit. 2. When lessons are $30 per quarter, what is the average cost per week ? 3. How many days in a leap year? How many weeks and days are there ? 4. Bought an 8-peck barrel of cranberries for $8 and retailed them at 2 qt. for $0.25. What was my profit ? 5. A bushel of wheat weighs 60 lb. What is a ton of wheat worth at 86 cents per bushel ? 6. At 60 1b. to a bushel, 3 bushels to the barrel, 9 tons of beans will fill how many barrels ? 7. Bought a field 600 ft. long and 413 ft. wide. What will it cost me to fence it at $1.25 per rod ? 8. A man travels 23 miles per day on every day except Monday, when he goes 6 miles farther, after resting on Sunday. How many weeks will it take him to go 576 miles ? 9. Bought 130 tons of coal by long ton (2240 lb.) at $4.00 per ton, and sold it by short ton (2000 lb.) at $6.50 per ton. How much did I make on the lot ? 10. I paid $300 for an acre of land and sold it for 25¢ per square foot. How much did I gain or lose ? 11. Give the length of a double track railroad laid with 1056 rails 30 ft. long. 12. Ocean steamers sometimes use 300 T. of coal every 24 hours. This is how many pounds per minute ? 13. The velocity of light is 186,337 miles per second. - Light from the sun reaches the earth in 8.3 minutes. What is the sun’s mean distance from us ? TESTS FOR DIVISIBILITY 45 1. Any number is a multiple of 2 if the last digit is zero or a multiple of 2, that is, if it ends in 0, 2, 4, 6, or 8. 2. Any number is a multiple of 3 if the sum of the digits is a multiple of 3. For example, 3 will divide 1278 for it will divide 1+24+7+8 or 18. 3. Any number is a multiple of 4 if the last two figures are zeros or express a number divisible by 4. Thus 38,236 is divisible by 4 for 36 is. 4. Any number is a multiple of 5 if it ends in 5 or 0. 5. Any number is a multiple of 6 if the last digit is even and the sum of the digits is divisible by 8. 6. Any number is a multiple of 9 if the sum of the digits is. Thus 327,654 will contain 9 for 34+24+7+6+5+44 or 27 will contain 9. 7. Any number is a multiple of 10 if it ends in a zero. 8-14. Which of these numbers are multiples of 2? Of3? Of4? GOO 7a O19 te OF 107 360. 6984. 2160 3654 3741 1728 8397 3240 1782 1746 3123 6624 9270 4662 8460 Making use of these tests to find the common factors, change the following to largest units : — 15. 6/428 = 3/21 = Jt. 22. 198, 26. 282. 16. 342, 19. 198, 23. 216, 27. AL. 17. 349. 20. 225. 24, 279, 28. 252. 18. 315, 21, 248, 25. 495, 29, 125. 46 ADDITION AND SUBTRACTION OF FRACTIONS Oral What are ike numbers? Give examples. What isaninteger ? Name some integers and the unit of each. | Whatisafraction ? Name some fractions and the unit of each. Show the difference between an integral and a fractional unit. o FPF WW DO In the folowing numbers name (a) the integral unit, (0) the size and kind of fractional unit, (c) the number of fractional units : — 5 pk.; 75 yr-3 ps3 BS; Zin.; $0.15; 6% of a day. 6. Which of the following fractions have units of the same size ? Of the same kind? Of the same kind and size? . Which are like fractions ? éYT5 FIT Hs Fda; $F; pyd; § mi 7. Why not add ¢ lb. and $4? #2 wk. and 4wk.? Principle. before fractions can be added (or their difference found) their units must be of the same kind and size, that is, the fractions must be like fractions. 8. tet fe tit = ae 11. 35 — $e = ot Ble Sek. OIE ae C 12. 72% —zty=0.48, 10. 0.55 =26 == 01308 13. 162+4+183= 14. 14 da.+ 3 wk. = make before adding ? da.. or wk. What change did you 15. Mention some unlike fractions. Why are they ‘unlike ? 16. 4,3,4,2. Which of these fractions have a common numerator ? A common denominator ? Which are like fractions? Which are unlike ? 17. 6 +2+2475+44+43=2. (Add the like fractions first.) 18. 34+ R= Can 12 be changed to 6ths? Why will it be better to veo L to 6ths rather than + to 18ths ? 19. In Exercise 14, why would it be on to change the 14 days to weeks ? 20. 2+4%=a. Why change 3 to 24ths? Oral and Written ADDITION AND SUBTRACTION OF FRACTIONS 47 1. Which term is common to like fractions ? 9 2. Change 2 and 58, to like fractions, or a common denominator. oO Change to the least common denominator (L. C. D.). .8. In adding 3 and 4% shall we use 4ths or 60ths as the com- mon unit? Will 4 or 60 be the least common denominator ? 4. Why is it easier to use the largest common unit (or the least common denominator) in adding ? 5. Give three steps in adding 7 and §. Method. In adding or subtracting unlike fractions. I. Change to a common unit. Il. Add numerators or find their difference. Ill. Simplify the result. Perform the operation indicated in the following : — By eset ees 9. $422. pee To 4 38. 10. 3+3 13. 24%+4 8. ish i. $4 Me +h LASS rele: aoa wy anion om a 20. ¢t+ayetet+etsa 160 20 + 1 020 + 4, 21, $4184 25491 17. 2841444104 259, SOU USB UNE te Oude 18. $+ yy +39 +46 2. H—$+h+4 19. 18% +.0.12--4 + 2. pte Salle Li'l URS eA Ta 25. +wk.+ 52, yr. +i da.—- mo. + 180 min. = 2. Multiples; Common Multiples; Loast Common Multiples. Show that 36, 60, 72 and 120 are multiples of 12. Show that 50, 60, 80, and 120 are multiples of 10. Show that 60 and 120 are common multiples of 10 and 12. Show that 60 is the least common multiple (L. C. M.) of 10 and 12. Pm Oo WwW 48 LEAST COMMON MULTIPLE Oral and Written 1. Find the least common multiple of 30 and 42, t.e. a multiple containing only such factors as are needed to produce each number separately. WorK a Le): AZ AGO OG 42 x 5 = 210, the least common multiple (L. C. M.). ExpLANnATion. A multiple of 30 must contain all the prime factors of 30, and a multiple of 42 all the prime factors of 42. Now, a number whose factors are 2, 8,5, and 7 will contain 2 x 3 x 5, or 30, and also 2 x 8 x 7, or 42. In practical work we say 380 x 7, or 42 x 5, since 7 is not found in 30, or 5 in 42. 2. Find the L. C.M., i.e. the least number divisible by 60, 72, and 108. Work 60=2x2x3x 5. (22 X20 Xo ro. 108 == 252 x oe OOS. LOS 26 5 == 080 the: iG M3 (a) What prime factor of 60, needed in the L. C. M., is not found in 108 ? (6) What prime factor of 72 is not found in 108, that is, found more times in 72 than 108 ? (c) What is meant by the least common multiple of several numbers ? 3. Find the L.C. M. of 60, 84, and 132. 4. What is the least common multiple of 45, 90, 100, 200 ? Sueerstion. Isa multiple of 90 a multiple of 45 ? Is a multiple of 200 a multiple of 100 ? Notice that we need simply to find the L. C. M. of 90 and 200. Find the L. C. M. of :— 5. 15, 21, 45. 6.1116; 18227, 72: 7. 16, 25, 80, 100. Written FRACTIONS: ADDITION AND SUBTRACTION 1. Add 44 and 54. PROCESS Le 1S ='3'x Bx 2: 18 x 5=90, the L.C.M. 49 EXPLANATION. Since the common fractional unit could not be readily seen, we found the L. C, M. of the denominators. 2. Js+y =o. 7. G++ =2 3. + ise 8. fot t +ib=e 4.ctit=" + 9 ge +4$t+it=a. 5. tHe 10. §} +t =e. ey taias 11. H+ H+ Hae | UTS Pood eoore 74s Ge sis) Add and this result to the sum of the integers. 13. Add 325, 1714, 24,7,, 364 and 212. 15? 14. From 75,4 take 5744 PROCESS EXPLANATION. 90ths, then, is the largest common unit to which both 15ths and 18ths ean be changed. (Reduce each to larger Why ?) the fractions mentally or 42 from 75 and add to the fraction #$. Since the fraction in the minuend How is this tt is smaller than the one in the subtrahend, we take 1 (= S741 = nate = 5 ns = 178 19. 14,9,—6.51, 21. 22. 23. 24. 734 — 1614. 71d — 1948. 8423 1585, 17145 —91.8,. 4815 — 1347 like ‘subtraction of integers when a figure in the sub- trahend is larger than the one above in the minuend ? 20. 25. 114441612 + 8419 — 1633. Can you find some short way of doing this-without changing a fractions to a common unit? 50 FRACTIONS: MULTIPLICATION Oral I. To multiply a fraction by increasing the number of units. 1. 9x7 units = units. 2. In Exercise 1, does it matter whether the units are integral or fractional ? 3. Then it follows that 9 x 7 eighths = 63 eighths ; 9 x 7 = 43 AL ene lan ee 10. 16x 2. 54 Cie toe ae” Vieille GoM ss 9. 15x34. 12. 10x 4. 13. In the preceding exercises did we change the number or the size of the units ? Il. Yo multiply a fraction by increasing the size of the units. 1. Compare }and1; and 4; + and 5. Oune mal 7 2. Compare 3 and 33; 2 and 3,; 7 and 5%. 16> 3. How does dividing the denominator affect the size of the units ? 4. Compare 3 days and 3 weeks. What is the unit in each? Why is 3 weeks 7 times as large as 3 days? 5. AX in ‘. What change in the value of the fraction when i we take { instead of .? Why? Find the product by increasing the size of the units : — 6. 6X 45. 9. 10x 55. 12. 15x 45. 7. 8x28. 10. 12 x 42. 13. 25 x +158). 8. 3x HH. 11. 18x 3. 14. 36 x 43. Principle. A fraction is multiplied by any number either by mul- tiplying its numerator or dividing its denominator by the number. « Oral and Written FRACTIONS: MULTIPLICATION 51 The multiplier a fraction ; the multiplicand an integer. How can you get 9 x 7 by addition ? What does the multiplier show ? 3. Is it proper to say 4 times $6? That is, can $6 be taken 1 of a time as an addend ? 4. In finding + of $6 to be $2, do we multiply or divide ? on! While such expressions as 2 X $9 are called multiplication, the $9 has not really been multiplied, that is, increased. When preceded by a fraction the sign x must be read “of.” The expression means 3 of $9, that is, 2 times one of the 3 equal parts of $9, or 2 x $3. 5. 3 x 24 hr.=3 x G of 24 hr.) or 3X3 hr. or 9 hr. In practice 3 we say ; Neel Chigpet a4 NH Se aah eal bh 10. § x 20. 14. 438 x 105. Tae Xo 11. 5% x 50. 15. 47 x 144. 8. 14 x 84. 120030 100. 16. 47 x 226. 9. = x 60. 13. 33 x 100. 17. 4% x 840. II. One factor a mixed number. 18. At $0.95 a pound what will PROCESS & 0.95 742 pounds of tea cost ? TAL 19. 182 cords of wood at $8.75? 9)$6.65 =7 x$0.95 20. 15 yards at $0.18? 18 9 Wes oh rie 21. 278 yards at $6.25? coll ES AM BOO ee TON 0.05 22. 73 months’ rent at $28? $ 71.038 = 747 x $0.95 23. 32 tons at $8.75? When the final result includes 24. 93 bushels at $3.40? ; ve a fraction of + cent or more, it is 25. 400 dozen at $ 0.162? customary to count the fraction as another cent, 26. 125 feet at $0.347 ? 52 PROBLEMS IN FRACTIONS - Written 1. 4,2,and 4+of a number=110. Is the unknown number larger or smaller ? Hon do you know ? 2. A barrel is 2 full. Draw off 4 of a barrel and 2 of a barrel. What part remains ? 3. A stone wall cost $1 a rod. What costs 3 days’ work, or 62 rd., 53 rd., and 72 rd. ? 4. A chimney contains 132 courses of brick. 4 are under ground, 24 roofed in; how many courses are exposed ? 5. How BED cords of wood in 2§ cd. sawed by hand, 4 ed. by machine, and ;% cd. chopped ? 6. Two pumps contribute 4 and 4 toward filling a reservoir, springs contribute 4 and i, surface’ water the rest. How much more do the pumps yield than other sources ? 7. Inan11-acre marsh lot three men cut 4, 5%, and 2 of the whole. What part remains for a fourth man to cut? How ies acres ? 8. If you invest ;2; of what you have in one way and 3 in another, what remains ? 9. At $3 a day what is due a man for working half a day, 2 da., $da., and 214 da. ? 10. What remains of a 49-yard piece of cloth after selling of it, 2 of the rest, and 4 yards? What is the remnant worth at 85¢ per yard ? 11. Another piece of 47 yards is damaged. One half sold at 7¢. Of the other half 23 yards were unsalable, but the rest went at 5 ¢. Give the total receipts. 12. At the rate of $13 a day, figure a board bill in dollars and cents for 3 months from August 1. 13. Ifa glacier moves uniformly a hundred feet a year, how far does it go in 181 days? Oral and Written FRACTIONS 53 Finding the Fractional Part of a Fraction 1. 2 of 6 things (apples, collars, fourths) = x things. 2. 2 of 10 twelfths = x twelfths; 2 of 18 = 4 ori. 3. 2 of 12 5. 8 of 27. 7. 73; of 82 4. 7 of 18 6. 3% of 42 8 1 of To multiply 2 by 2 is to find 2 of 2: that is, to divide 3 into 5 equal parts and find the value of 5 of these parts. 9. How does increasing the denominator affect the size of the fractional unit ? 10. If we take a denominator 5 times as large, how is the size of the fraction changed ? 11. Then fof }=4. If{ ofa fraction = 5%, 3 of the same frac- tion will be how many times 3, ? 12. Make arule for finding the product of two fractions, i.e. for finding one or more of the equal parts of a fraction. 13. Find 19 x 24. PROCESS 10 24 10x24 240 4 (a) Of what use is cancellation? 12 “ 35 12x35 420 7 (0) On what principle is it based ? Ste) (c) Which is easier, to change the ae 10 ¥ 74 _4 product to lowest terms or to cancel 12 pp 7 first ? 7 14. 3% x #. 17. 23 x 2 20. fy X 23. 15. 4 x $4. 18. 85% x +4 21. 5 x $3. 16. 33 x 33, 19. 96% x #3 22. 44 x Pf. 23. 83 x 68 = 32 x =a 24. 2x Th 28. 4 of 3 of 34 32. 16% of 54 25. 22 x 735. 29. 22 x 44 xX Tehq- 33. 84% of 355 26. 4% x 153. 30. $x 14x f. 34. 72% of 32% 27. 63 x 246 31. 12 x 54 x 1,545 35. 19% of 182 54 FRACTIONS Sight Products Multiply each fraction in the table by the number at the end of its line or column. Change any fraction in the product to a smaller denomina- tion when possible. Thus: — 4x3 yd. = 2° yd.=22 yd.=2 yd. 8 in. 12 4 6 10 15 2 3 qt 3 yd. 3‘ hr. 7 sq. yd. 121% 12 3 2 yd Z gal 41 da. 33; min. 162% 1 4 5 pk 33; lb qe ft 41 sq. ft 334% 10 5 2 ft. +5 yr. 43 T. ay Sec. 621% 9 6 2 wk. 3% in 3 ed. zy hr. 374% 8 fl 8 9 12 7 Written Work. Problems Make out bills in full for : — ) ibs es 17 doz. at $ 1.624. 152 doz. 173 yd. at 10¥. 13 yd. 3. Find the cost of 92 tons of coal at $7.41. 4. Twenty pounds of sugar bought at 4,%,¢ are sold for $1.25. At this rate what is gained on a barrel of 200 lb. ? 5. Oil is bought at $3.50 for a 42-gal. barrel and retailed at 121. The gain is what part of the cost ? 6. Oranges bought at 3 for 5¢ are sold at 4 for 9%. What is gained on a box of 9 doz., 1 in 12 of which are worthless ? 7. I can buy blank books of one dealer at the rate of $1.25 a hundred; of another at $1.60 a gross. How much is one offer better than the other ? 8. Find the cost of seven 50-gal. barrels of oil at three for $16.71. 9. Supposing an empty barrel to be worth $1.25, what is the oil worth per gallon ? at $ 1.00. at 621¢. Oral DIVISION OF FRACTIONS 55 I. When the divisor is an integer 1. To divide ? into 5 equal parts, that is, ?+5=+4 of =~. CG 15 2. enn are ies tana Observe that a fraction may be divided by a number either by divid- ing the numerator or by multiplying the denominator by that number. 3. Observing Exercise 2, tell when you use one method and when the other, and why. Which is shorter ? 4. 5 +7. 7. 4+12. 10. = + 6. 5. 4+8. 8. 78 + 12. 11. 48% +16. 6. 24-8. 9. 74 + 20. 12. 7335 +17. Il. When the divisor is a fraction 13. 2ft.+38in.=«. (Since dividend and divisor must represent like units, we have 24 in. + 3 in.) © 14. 3+2, Are the units of the same size ? rey a 15. 3+2=9+35;9+8=14. 16. al ieee Cie 17. What is the first step in dividing days by hours? Feet by inches? 4ths by 5ths? One fraction by another ? 18. 2+ 2. 21. 75+. 24. 27+ 5, 19. 14+ 3. 22. y+. 25. 18+ 3 20. $+ 2. 23. +4. 26. ge+2 27. How many 4thsin1? In2? Ind? 28. How many 3dsin1? In4? In10? 29. How many dthsin1? In10? In 20? 30. 31. | 4 = + ~ OO SS) = | ~y I Ph pL bt | a ole ple I x) oo ASS jb | co|- Oo|- | Secs es ey) ~3 fs | bol, S| | 56 DIVISION OF FRACTIONS: THE DIVISOR INVERTED Oral 8. l+i=-2@ 6 1 + 53. 9. 1 + 35. Notice that in each of the preceding exercises the quotient was the same as that obtained from dividing the denominator by the numerator. 10. ied We 12. 13. Why was this ? Compare 6 + 2 and 12 + 2. What effect on the quotient if the dividend is increased ? How does 2+2 compare with 1+2? Compare 8 + 2 with 4 + 2. 14. Compare ++ 2 with 1+ 2. We, Sees 4 ys SuecEstion. Since 1 + 2 is 3, 3 + 2 is 3 of $ or 2 ie ee ye aaeetods 10. See EXPLANATION. Since 1 + 3 is 3, { + is 7 as great as 1 + 3, or is 7 of 8. 17. The two methods compared : — First Mretuop Seconp MetTuHop (a) What advantage has the second method over the first ? (6) What disadvantage may it possibly have ? Notr. Cancellation may be used in division of fractions as in multiplication. Apply the shorter process and explain why it is shorter : — 18. 2+ 3. 22. $+ 5. 26. 27+ 5. 19. +4. 23. $+ 45. 27. ~p +. 20. §+ 8%. 24. 34-3, 28. 0.32 + 18 21. $§+41, 25. $+ 5. 29. 0.042 + 5%. Written FRACTIONS.: PROBLEMS 59 1. An heir gets 4 of an estate, then loses 3 of his share. What part of the estate does he keep ? 2. I buy at 20% discount. What is the total cost to me of goods sold regularly for $ 1.42, $ 3.98, $57, $ 0.162, and 9 pieces at $ 0.311? 3. If 81 T. of coal cost $487, what is the cost of 68 T.? 4, Property which cost $5000 is rented for $431 a month; what is the annual income to the owner after paying a tax of $15 ona thousand ? 5. Three cheeses, weighing respectively 343, 423, and 474 lb., were sold for $ 20.60. What was the price per pound ? 6. J. Ff. Sampson bought 721 bu. potatoes at 624 ¢ a bushel, and sold 2 at 642 g, the remainder at 75¢. What did he gain? 7. An electric launch was sold for $ 285, or 42 of the cost. Find $2 of the cost, or the whole cost. Compare #2 with 32. 2) 8. 2 of a ton of hay at $ 20 pays for 14'T. of coal at how much a ton ? 9. 161 ft. of 2-in. pipe at 64¢, and 1020 ft. of 1-in. pipe at 41¢, are exchanged for 120 lb. of tubing at 111f, and 134 ft. at 9%. What is the difference in value ? 10. Two trains start together in the same direction. How far apart will they be in an hour if one goes a mile in 1,3, min. and the other in 85 sec.? . 11. An express train runs 240 mi. in 54 hr. How far will it run in 3ihr.? (Compare 34 with 51.) 12. A tank holds 168 gal. and is 2 full. 23 of the quantity is drawn off. How many gallons will fill the tank ? 13. An automobile started at 10.45 a.m., and at 3.20 p.m. had cov- ered 641 mi. What was the rate per hour? 60 FRACTIONS.: PROBLEMS Written 1. A man had in a bank a certain sum of money. He withdrew 2 of 2 of it, and gave his son 2 of 4% of that. What part of the whole amount did his son receive ? 2. A man walked 29% mi. one day, and 213 mi. the next day. On the third he walked the difference between the two distances. How far did he walk in the three days ? 3. A owned a farm as follows: 33% A. of wood land; 633 A. of meadow; 254 A. of good grass land; and 192% A. producing fruit. He sold 2 of the whole farm to a speculator. This was a A. 4. If 3 bbl. of flour will supply 30 people 134 wk., how many barrels will it take to supply them 46 wk. ? 5. A man traveled # of his journey the first month, and 2 the next month, when he found that he still had 660 mi. to go. How far had he traveled ? 6. A boy bought 23 bu. grain and sold me ;4 of it. Another boy bought 3 bu. and sold me an amount equal to what I bought from the first boy. What fractional part of the second boy’s grain did I buy ? 7. Bought 28'of a barrel of flour of one man, } bbl. of another, and +5 of the third. What was the cost at the rate of $6 a barrel ? 8. I purchased 360 A. of land at $75 per acre, and sold $ of it at $85 per acre. After paying for the land, I had left # A. and §$ y. 9. If 132 bu. of rice cost $11.75, what will 37% bu. cost? - 10. A man bought 100 yd. of carpeting, and sold 374 yd. to one man and + of the remainder to another. How many yards had he left ? 11. How much larger is 24% + 7 than 24% x 7? Oral NUMBERS COMPARED: RATIO 61 Compare with 100 :— 1. 50, 25, 75, 20, 40, 10, 80,70. 2. 5, 15, 4,12, 16,2, 6, 8. Repeat rapidly until thoroughly learned parts of 100:— the values of the following cee ae be 8 St ge a eR iy ns A fe 0 ce oy Es ca en ko2 | OF. See? 02 50 SL Ote a2 lesen 02 O08 0 OF HL UO: Are TEI Be Le eS Bet ana Oe Oe aS ad teat G2 .62" 6! S262 5S 10), P23 216202 02925240 02) 7- § (ie a koe 81) GUL IN Es Ma da an 9 3 pe DL NE be ee Uta aed O75 s 0%no D2. OU 129 16? 409 809 169 16° Gaye ee rg Tee pee Se OL 1 p50 42.62.6773 7. 897 6: erg? 16) 1 2s 407 1G) 1 3° 11. A nurseryman sells 2500 strawberry plants at $6 a hundred. They cost him 2 as much to raise, and he gives an agent 4 of the profit. How much does he gain ? Compare one number with the other in each of the following columns. Thus : — 1. (a) The ratio of 8 to 24 is 4; (0) 8 is 4, or 334%, of 24. 2. (a) The ratio of $ to 4 is 5 to 2, or 24; (6) 2 is 24 times, or 250%, of 1. tt, 8, 24 60, 12 24, 60 48, 72 162, 662 #4 375, 25 1, 100 2.75, 5.50 1, 1000 a iy 89 4 72, 60 37}, 64 6 8 5 2 69 3 10%, 20% $0.75, $ 1.25 874, 874 III. 90, 18 OZ Lal. 2 gal., 3 qt. ~ 30, 50 4, 0.16 0.68, 0.51 | IV. 1 da., 1 hr. 1 wk.,1 da. 5 min., 25 sec. 144, 148 rio aH 4X yor 8 X oo 0.93, 0.31 200, 162 $1, $1.50 1.834, 1.00 62 NUMBERS COMPARED Oral Compare the following and give the ratio in per cent : — Thus, to compare 5 with 20, say 5 1s 25% of 20. i: 2. 16 with 80, 6 with 42. 3. 4. 45 with 15, 48 with 36. 5 2 with 28, 15 with 40. . 385 with 105, 3 da. with 1 wk. . 4in. with 1 yd., 2 mo. with 1 yr. 6 f 25 with 150, 24 with 42. 8. 1 oz. with 1 lb.,1 hr. with 1 min. 9 . 1 1b. with 1 0z., 100 lb. with 1 T. . 18 with 54,18 with 24. 10. 2 qt. with 4 gal., 1¢ with $1. Rib: What is the relation of the whole of anything to 2 of it, i.e. the relation of 2 to 2? 12. If 12 is 2 of a number, what is the number? SUGGESTION. 13. with 2 z* Since all, or 3, is 24 times 2, the whole number is —— x 12. If 28 is 2 of a number, what is the number? (Compare 3 times as large ?) Is the number larger or smaller than 28? How many In this way find the whole when a part is given : — 14. 15. 16. 17. 18. 19. 20. 28. 29. 30. 31; 12 is 4 of what ? 21. 16 is 4 of what ? 22. 24 is 1 of what ? 23. 19 is 2 of what? 24. 28 is 2 of what ? 25. 36 1s 2 of what ? 26. 72 is 4 of what ? 27. 100 is 42 of what ? 450 is 43 of what ? 175 is 25 of what? What part of 23 is 3? 32 is 4 of what ? 123 is 8 of what? 142 is 2 of what ? What part of 2 is 33,? (Change to like units.) 183 is what part of 621? (3 x 64 and 10 x 61.) A farmer sold 200 bu. of beans. This was 4 of his crop. How many had he? 20, or 2 of a farmer’s sheep, are black. How many sheep has he? Oral NUMBERS COMPARED: PER CENT 63 1. What does the phrase per cent mean? How many per cent in the whole of anything ? 2. The sign % takes the place of what denominator ? 3. What is the unit in 8% ? 4. What is the ratio of all, or 100%, of anything to 50% of it? 5. If 50% of a crop is 400 bu. what is the whole crop ? Find the whole when the specified part is known : — 6. 16 is 4, or 50%, of what? 10. 56 is 4, or 874%, of what ? 7. 241s 3, or 75%, of what? 11. 20 is 25% of what? 8. 32 is 2, or 662%, of what? 12. 15 is 121% of what? 9. 40 is 8, or 831%, of what? 13. 80 is 371% of what? PAVE eT is'10%) of == 17. V25 is 5% of —. 15. 31 is 831% of —. 18. 36 is 18% of —. Tomer ist 20 peat ete 119, 9? is 27% of ——-. 20. A teacher pays $6 per week for board and room. This is 40% of her salary. What is her salary for a school year of 40 weeks? (Her salary is how many times 40% of it?) 21. A man’s expenses are $7.50 per week. This is 84% of his income. What is his income? (His whole income is how many times 81% of it ?) 22. How much have I if $1.20 is 20% of my money ? 23. Mary Smith’s salary is $750 per year. This is 331% of her father’s income. What does Mr. Smith receive annually ? 24. After spending $5, James had 60% of his money remaining. How many per cent did he spend and how much had he at first ? 25. I sold 60 of my flock of sheep. If this was 20% of my whole flock, how many had I left? Hint. How many per cent left? How many times 20% is the per cent left? 64. PROBLEMS FOR ANALYSIS Oral EHzxplain exactly how you get each result : — 10 = 12 of what number? 10 = 42 of what number? Give 4 and ;4, their least common denominators. # of a hill is dug away. How many times as much remain ? 10+, =what? 10+ 4= what? A train goes a mile in12min. How far willit goin an hour? At 1 mi. in 90 sec., how much in an hour? 50 mi. an hour = how much a minute ? At 48 mi. an hour, how long does it take to go 1 mi.? co aoant OO oT FF WO WD A mile in 124 i min. is the same as 60 mi. in 10. 34% of certain telegraph lines are under ground. What per cent are above ground ? 11. A foundry uses 100 T. of Swedish iron to 50 T. from other sources. What part or per cent of each class is used ? 12. $160 was s&, or 16%, of the profits. What were the profits ? 13. 23% of a certain stock was glassware, 69% was china. The rest was in brass goods, which were «% of the whole? 14. The 5,000,000 sq. mi. of the Arctic Ocean are what per cent of the area of the Pacific, which is 16 times as large? 15. I gain 100% on 4 my goods and sell the rest at cost. How much do I gain on $100 invested ? 16. Ina36-column newspaper what part of the whole space would be filled by 20 columns of advertisements ? What per cent ? 17. I lose half that I have, and 25% of the rest. What I keep is what part of what I lose ? 18. The board of a horse is $ 20, shoeing $ 1.25, harness repairs $ 0.25, use of carriage $3, new whip $0.50. Each item is what part of the whole? Give per cents when you can. ADB es, OLD". 20. 2 = 2% of 87. Oral EXERCISES IN FRACTIONS 65 1. Change to lowest terms 343; 74%. 2. Compare results: 224.7 yd. +7; 224.4 yd.+7 yd. 8. fof 1W=a. Zofl0=y. 4, 5% league = what part of a league? 5. 2A.=what partofi2A.? Of1,A.? 6. 21s contained in +2 how many times? 7. Cancel mentally: $ of § x 44 of #2 =a, 8. By getting a uate of 2 I pay only $3.33. What is the regular price ? 9. 478+ 5 =-%. 10. What is the least that will pay for 1 article, when the price per dozen is — $1.05? $1.10? $1.15? $1.25? $1.30? $1.35? $140? $1.50? $1.75? $2.00? $2.25? $2.50? $2.75? $3.00? $5.00? 11. 3rd. at $1.25. 13. 34 in 25 a times. 12. 57% |b. at $1.28. Lae = 164 te a, 15. kes + 874 4+.624 + 874 +1124 4121; 662 — 162 —81. 16. By a Fahrenheit thermometer what is the temperature when the top of the mercury column is ;°; of the distance in degrees from zero to the freezing point ? 17. After gaining ;5, or 10%, I have $99. What had I at first ? 18. Find 21x90. Whatis 74, of it? 6% of it? 19. What part of 100 is 94,? Give = of 100; #3 wb 20. How much for a dozen at 16 for a quarter? At 4 for 5¢? At 20 for a quarter? At 3 for 10¢? At3 for 5¢? 21. A newspaper weighing 4 0z. may be mailed for 1%. What will it cost to send — Llb.? 10 oz. ? 5 oz. ? Al oz. ? $ oz. ? 66 | EXERCISES IN FRACTIONS Written 1. Paid at different times 4, 4, and 54 of a debt. The balance was $1170.61. What was the whole debt ? 2. 4 of an estate is divided equally among 14 persons, another 4 among 9 persons. One of each of these shares is to be given to the heir of the remaining third. What part of the whole does he receive ? 3. From a life-saving station to the end of the northern beat is 2, mi. How many full steps will a surfman take in going and returning, if his steps average 2,2, ft. each ? 4. The Minot’s Ledge light revolves twice a minute. It is lighted from sunset to sunrise. How many revolutions does it make between 5.52 p.m. and 5.52 a.m. ? 5. Four equal farms, all adjoining, are offered for house lots. Parts of each are sold as follows: %, 5, 7%, 7%. Add the fractions and tell what the sum shows. 6. What part of a mile is covered by 22 revolutions of a wheel 18 ft. round ? 7. Divide 231 cu. in. into 3 equal integral parts ; into 83. How else can it be exactly divided ? 8. Just when does + of a common year end? + of a leap year ? 9. A bushel of potatoes is commonly 60 lb. A thousand 56-lb. bushels are what part by weight of 1000 60-lb. bushels ? 10. A barrel of 42 gal. will fill how many cans containing 1% pt. ? 8 11. When $80 are earned in a month and 5, of it spent, the saving of 2,4, yr. at that rate would be how much ? 12. A chest contained 74 lb. of tea at 55 ¢, 134 Ib. at 35 ¢, and 98 lb. at 27 ¢, The mixture is worth wf? a pound, and $10 would buy y lb., with 2% remaining. Written FRACTIONS: BUSINESS TRANSACTIONS 67 Find the cost of the following purchases : — 1. 3h yd. silk at $1.374; 84 doz. buttons at 15¢; 74 sticks braid at 75%; 23 yd. ribbon at 162¢. 2. 6470 ft. fencing at 9,5,¢; 3 lots land, 10,280, 7595, 8122 sq. ft. at 54¢; 3400 bricks at 8. 15 per M. 3. 3450 Ib. coal at $8.25 a ton; 324 cu. ft. wood at $5.75 per . cord; 5460 lb. coke at $6.40 per tort 4. Go over the computations in the following bill or invoice to find the errors it contains. Cuicaco, Aug. 1, 1904. Mr. Henry D. WARREN Bought of Joun V. FaARweitu & Co. May 18 | 23 yd. Brussels Carpeting @ $1.50 34 | 50 June 22 | 161 yd. Black Silk @ 1.75 18 | 98 July 6 | 388 yd. Wamsutta Cotton @ 0.123 4 | 75 hale 23 Less 35 82. 41 Cr. June 2 By Cash $ 25.00 By Cash 20.00 —00- Received payment, 41 JOHN V. FARWELL & Co., By Smith. 5. Make out bill in proper form. Supply dates and names. 13 tons Franklin coal at $ 7.25 64 tons Lackawanna at 5.50 1 Cord Hard Wood at 11.00 31 bbl. Cement ee Ree 68 FRACTIONS: BUSINESS TRANSACTIONS Written 1. Friday, Jan. 1, 1904, Sam’l Chase had $32.76 to his credit in a bank. If he deposited $25 every week day during the month, and $100 extra every Saturday, what amount could he draw against HeboLa 2. In buying 785 music books @ 85%, a discount of 1 or 20% is allowed on cash payments. The net cost is what ? 3. Bill 62 lb. Formosa Oolong Tea @ 60¢ 30 lb. Maracaibo Coffee @ 241 ¢ 2 bbl. “ Bridal Veil” Flour @ $5.25 Discount 2% [for cash]. 4. Bill 374 yd. Dwight Cotton — (@11¢ 421 yd. Scotch Gingham (@ 23¢ 112 yd. India Silk (@ $1.75 Credit mdse. returned, $8.75. 5. Invoice 33 gro. No. 514 Eagle Pencils @ $4.20 54 gro. No. 404 Gillott’s Pens (@ 0.374 41 gro. 4to Blank Books (@ 3.66 6. Colonel 8., a Kansas farmer, harvested 4000 acres of wheat in 1903. He estimates the cost per acre as follows: for plowing, $1.00; for drilling, 25¢; for seed, 3 pecks to the acre at 60¢ a bushel; for heading and stacking, $1.25; for threshing, 20 bushels per acre at 6¢; for hauling 4¢ a bushel. The wheat was sold at 60¢ a bushel, and the use of fields for grazing during the winter is worth $2500. Estimate the profits. Rule paper for an account that you keep with John Holmes. It will show that he is debtor for all that is sold to by and creditor for all that is paid to by as below. (See p. 15.) Joun Houimes Dr. Or. 19— 3 May 1 | To 8 Shares Mill Stock ‘} 812 | 00 By 4 River Pasture (3 A.) |} 46° | 00 Written BUSINESS TRANSACTIONS 69 1. Complete Holmes’s account, found begun on page 68, from the following data: 34 days repairing Holmes’s fence at $2.50. Credit him for use of his oxen same time at $2. Sold him 23 bbl. apples at $1.40. Bought of him 3 hogs (732 Ib.) at 1119, and 11 tons hay at $15. Sold him 15 young maples at $1.061, and 3 hoops contain- ing 35 ft. strap iron at 23¢. Holmes paid cash $100. What does he now owe me? 2. January 1 my gas meter read 67,500; March 31 it read 91,500. At $1.60 per thousand my quarter’s gas bill is $a. 3. A week’s sales of wheat in bushels: 2137 , 0476, 972, 3041, 6782, 1849. Valued at 623¢. What are the gross proceeds ? 4. What did my house cost me as shown by these items : — Cellar, 18 days @ $14.75; mason’s contract, $4575.86 ; carpenter, 137 days @ $2.15 and 96 at $3; materials, $576.84; painting, etc., $ 397.68 ? 5. At an auction sale of land the following prices were obtained: 3648 ft. at 237; 2894 at 314¢; 7642 at 194¢; 8641 at 25¢. The auctioneer’s Commission was o¢ on the dollar, and puvauuiatne; etc., cost $37.50. Required the net proceeds. 6. Bought a 100-acre wood lot for $800. Paid 23 men $517.50 for 18 days’ work at cutting. Sold 175 cords at $2.37, 215 at $4.25, and the remainder with the lot for $800. What did I gain? 7. A farmer wintered 17 horses from December 1 to April 1 at $12amonth. He paid $23 a ton for 22 tons of hay, and 42 cents each for 280 bushels of oats. He had $14 worth of provender left. He made $a a month. 8. I can buy of one firm 732 tons of coal at $4.20 and 75 cords of wood at $8.16. Another firm bids $4.16 for the coal and $8.35 for the wood. Shall I buy of the first or of the second, and save what ? . 9. There are 2741 operatives on a corporation. 12 overseers get $3.50 a day, 25 second hands get $2.50, 4305 earn $1.50, 215 men and 731 women earn $1.25 each, and the remainder on the average receive 96 cents. What is the weekly pay roll? 70 FRACTIONAL MEASURES Written 290 ft. hemlock boards at 13 ¢ will cost how much ? 3.25 bu. of beans at 2 qt. for 121¢ will sell for what ? 19 of a 235-lb. barrel of sugar at 4;4,¢ is worth « dollars. zy T. of cream of tartar fills how many 4-oz. boxes ? 322 yd. lace billed at $87 costs how much a yard ? 6. When hay is $13? a ton, what fraction of a ton is worth $13? How many pounds ? ot Ff WW WO 7. When 115 votes are in favor of a project and 46 are against it, what part of the whole are opposed ? 8. Find the gain in 2250 lb. of wool bought at 161¢ and sold at 163 ¢. 9. A man sold 2 of his interest in a mill for $30. If his share amounted to 3 of the whole property, what part of the whole did he sell? What was the value of the whole property at this rate ? 10. Broadway Park measures + of a mile wide and 3 of a mile long. What will a walk 8 feet wide around the park cost at 621¢ per running foot? (Draw a diagram. Do not leave the corners without a walk.) 11. If = of a piece of work can be done in 3 of a day, how long will it take to complete the work ? 12. At an auction one buyer bids + of the cost, another 4. The difference was $75. What did each bid ? 13. If a glass jar contains a hundred thousand fish eggs, how many jars will hold 8,000,000? If 13 are hatched, how many on an average are lost from each jar ? 14. The catch of shad for a certain period is valued at $145,000. What part of this is $4000, the cost of hatching the eggs and stock- ing the waters ? 15. When oysters yield 11 gal. to the bushel, a 25-gal. barrel can be filled from x bushels in the shell. wo wn 4. DECIMAL FRACTIONS at (Review pp. 5 and 6) By a decimal system we mean what ? Compare the values of the 2’s in 222. Compare the values of the 3’s in 33; in 3.3; in 0.33; in 0.033. Which of the preceding numbers are integers? Decimals ? Mixed decimals ? 5. 6. T. S: 2: 10. Li The value of a figure depends upon what two things ? 7654321 .234567. What figure stands for tenths ? Hundredths? Thousandths ? Compare the position and value of the 3’s. Of the 5’s. Of what orders are the 6’s? The 7’s? What is the use of the decimal point ? How is the denominator of a decimal fraction determined ? Write the following with denominators: 0.24, 0.08, 0.175, 0.036, 0.0025, 0.0001, 0.00017. 12. 13. 14. 8 8 8 ] j ? Are 38, 730) ayy Common or decimal fractions ‘ How would you write them decimally ? Compare the number of places each takes up with the number of zeros in the denominator. 15. 16. ies What part of 140.040 should be read first ? Where is “and” used in reading mixed decimals ? Which is the easier way of finding the denominator of 0.040, (a) by counting from the decimal point, — “tenths,” “ hundredths,” “‘thousandths,”’ — or (0) by imagining 1 and three zeros annexed ? Read : — 18. 0.507, 0.0307. 21-5, O-7105'0.0071: 19. 330.03, 0.303. 22. 64.0019, 6400.019. 20. 3003.075, 3.00375. 23. 6000.006, 0.6006. (4 DECIMALS: READING AND WRITING Oral Since 84 per cent, or 84%, means 84 hundredths, or 0.84, in what two ways might the following be read ? 1. 0.06, 334%, 0.163. 3. 0.871, 871%, 0.004. 2. 0.14, 0.374, 413%. 4. 0.00%, 2%, 0.064. 5. 34 thousandths is written 0.0034. To what order of units does the 4 belong? 6. 1% is written 0.004. Write 2%, 14%, 24%. A number made up of decimal and common fractions is a complex decimal. 7. Give other complex decimals. Writing Decimals 1. If the denominator contains three zeros, how far from the decimal point must the numerator end? j ; 1 F 845 12 S17 1264 2. Write decimally: 345, 24%, 24> ith: 3. If the numerator contains but one figure and the denominator three, where is the numerator written? What is written in the other two places ? W 1 . 3 1 9 5 19 4. rite: T0002 T0000? T00? T00009 10002 10000° Wri ° 3 20 1 1 a 24 5. rite: 1 100? 2 Orb Sido Ortiv: 6. Write as common fractions or mixed numbers: 1.003, 20.020, 19.03, 0.0013, 125%, 250%. Write from dictation or at sight : — 7. 8 thousandths. 12. 804 hundred thousandths. 8. 17 tenths. 13. 400 and 4 ten-thousandths. 9. Threeandafifth %. 14. Forty thousand forty millionths. 10. 3075 millionths. 15. Seven hundred six thousandths. 11. 4a hundredth. 16. Two million 71 and 404 millionths. Oral DECIMALS: ANNEXING ZEROS 13 1. Change 12 to larger units. What principle applies? 2. Annex a zero to 0.8. Write 0.8 and 0.80 as common fractions and compare their values. 3. Compare the numerators, the denominators, and the value of 0.90 and 0.9. 4. What effect on the number of units has annexing a zero toa decimal ? What effect on their size ? 5. Omit the zeros at the right of 0.860 and 0.400. How is the numerator affected? The denominator? The value? Read as printed; then in smallest decimal terms, that is, in largest decimal units : — 6. 0.0400, 6.6450. 8. 8.0500, 0.050. 10. 9.400, 0.100. 7. 7.0900, 0.0600. 9. 10.010, 8.450. 11. 0.50, 0.050. 12. What is the effect upon the value of a number when the deci- mal point is moved one place to the deft? One place to the right ? 13. If a zero is annexed to 8 is the decimal point moved? Which way ? 14. To 1.2 annex a zero. Is the decimal moved, that is, have ones and tenths been changed ? 15. Tell how each figure is changed in value when a zero is an- nexed to 135. To 13.5. 16. Explain any change in value made by annexing a zero to 8; to 120; to 0.3; 0.03. Read the following as printed; then with one or more zeros annexed ; — : bie Odie 7.9- "0.07? 0.10 0.008; 245.6; 36.008. 18. Read and announce the change when the decimal point is moved one place to the left. 19. When the decimal point is moved one place to the right. 74 DECIMALS CHANGED TO COMMON FRACTIONS Oral or Written 1. 0.75=745,=%. Give directions for the two steps in this process. Change to common fractions. Give each step. 2. 0.2, 0.4, 0.5. 5. 0.125, 0.480, 0.375. 3. 0.80, 0.25, 0.12. 6. 0.000125, 0.0625. 4. 0.50, 0.75, 9.70. 7. 0.00375, 0.001728. 2 100 e100 1 e100 meee 9. Multiplying both terms of 7 by 4 has what effect on the 00 value? On the form? Upon what principle does this depend ? Aare nT Oe | 10. Explain the whole process of reducing a complex decimal to a common fraction. Change to common fractions : — 11. 331%, 0.162, 662%. 14. 0.031, 41%, 0.061. 12. 0.834, 0.081, 584%. 15. 62%, 311%. 13. 121%, 0.37h, 874%. 16. 412%, 0.621, 912%. Change the following per cents to common fractions : — 17-26. 43¢%, 504%, 623%, 682%, 814%; 413%, 933%, 228%, 291%. 27-36. Subtract each of the preceding from 100% and change the remainder to a common fraction. 37. Change ¢ to 100ths or %. gig LOO P00 = Bote * 800 800+8 100 PRACTICAL CALCULATION 28 = 0.874 = 871%. dare ae a Sits 41, 27, 44, 15, ee Boo , p. 145. 8) 7.000 39s, AQT S.. Cte 0.875 = 874%. 40. 50: 43. si. 46. 5. Written ADDITION AND SUBTRACTION OF DECIMALS Tits) 1. Without copying, write the sum of each column and of each line. whe 2. 3. 4. 5. 6. 96.475 + 186.52 + 0.4875 + 0.64985 + 396.47. 7. 83.8 + 62.379 +293 +3.207 + 82.379. 8. 542 + 48 +8479 + 0.0439 4+ 64. 9. 16.783 + 9.54 +653 +9.642 + 180.09. 10. 4.09 + 72.683 + 2.946 + 8.78514 + 34.769. Il. Find the difference, first explaining whether the denominators must be alike: = ie 2. 3. 4. 5. 6. 7. 3.64 — 1.878. 9. 41. —13.074. 11. 6389 — 0.497. 8. 19 — 0.5694. 10. 9.87 — 4.3. 12. 2.641 — 0.0994. 13. Take seventeen hundred eight ten thousandths from twenty- four and six thousandths. 14. From eighty-six tenths take forty-three thousandths. III. Rewrite as integers and decimals; then add each column and each row : — | Ds 2. Os 4, 5. 6. 7. 16.372 85 79.42 144 863 21.054 8. 2161 B41 POTS) Wo rm OTe Wah 1 6:25 Oeeeb te 0.8470 wy 964: 20 omen 621%, 10. 3 871% 0.758 931% 4 1219, Lise OS 2.08 15 695 116% 6.4837 IV. Find the difference between : — 1. 17.38 and 200. 4. 94 and 7969. 7. 1 and 0.833. 2. 2.0875 and 3. 5. $5 and 24. 8. 0.1 and 0.0833. 3. 44 and 7.011. 6. 2 and 64%. 9. 10 and 8.31. 76 MULTIPLICATION OF DECIMALS Oral 1. Compare 3 and 0.3. Moving the decimal point one place to the left has what effect ? 2. Move the point one place to the left in 18.4 and 0.15, and give the result and the effect on the number, 3. Compare 0.1 of 18.4, 4 of 18.4, and 18.4 + 10. 4. Find 0.1 of the following: 3.4, 0.2, 0.05, 12.5. 5. Compare 400 and 4.00. Moving the point two places to the left has what effect ? 6. Read the quotients after dividing these by 10. By 100. 24.5, 8.65, 42.1, 0.04, 0.875, 625, 0.264, 1.82. 7. Instead of reading the preceding as divided by 10 or 100, we may read them as or of the number. Give at sight : — 8. +1, of 2.46. 10. aby Of 37.6. 12. 0.0001 of 3500. 9. 0.010f 32, 11. 0.001 x 0.9. 13. 0.001 x 25. 14. Having found 0.1 or 0.01 by moving the point, how would you find 0.3 or 0.05, ete. ? 15. Compare 0.1 and 0.8; 0.01 and 0.15; 0.01 and 0.08. 16. What is 0.01 of 300? 0.03 of 300? 17. 0.01 x 32.45 = what? How many decimal places in the result ? 18. 0.09 x 0.03 = what? 19. 0.06 x 300. 0.01 x 0.03 = 0.0008 (why ?) 20. 0.15 x 0.6. 0.09 x 0.03 =9 x 0.0003 (why?) 21. 0.05 x 0.005. 22. Compare the number of decimal places in the product with the number in both factors. The product contains as many decimal places as there are in both its Jactors. Oral MULTIPLICATION OF DECIMALS 17 Wee O.0 Of oUies OC Seve 2. 334% of 60 =4 of 60. Why? Give ao a - WwW lat 12. 13. 14. 16. 2.008 3.46 results and show what process you use :— . 0.06 of 200. 5% of 500. Te) x 0,0: 2.5 X 0.30. . $90 x 0.9. 12% of 1000. 8. 80% of $400. 16% of 40. GOL O00 Vator st, 9. 334% of 360. 0.2 x 0.2 x 0.2. . 0.08 x 0.5. 20% of 60 yr. 10s cewek, 200 x 0.003. Had you first to multiply or divide in these examples ? A man of 50, spending 30% of his life abroad, is at home w@ yr. 2% of $5000 being counterfeit, the rest or $a is good money. 23 X 375 = 8625. 15. 0.23:x 37.5=y. Bo Oe =a, 20 KO. D =z. w X 3.75 = 0.8625. 230 Xu =8625. How would you write the product of 0.02 x 0.004 ? Written 3.46 x 2.008. WorK 2.008 ExpiLanatTion. The first partial product is 0.06 x 3.46 2.008 or 0.12048, which may be written in either of 12048 12048 the ways given. The important thing to remember 8052 6.024 8032 is how to determine the number of decimals in the 6.024 product and to put the decimal point where it belongs. 6.94768 6.94768 (a) Why are there 5 decimals in the first partial product, 4 in the second, and 3 in the last ? 2. 3. 4. Ly 0.84 x 5.076. 5. 0.0874 x 12.50. 8. 8% x $456. 8.47 x 9.432. 6. 1.8% x 0.360. 9. 0.24 x $9.60. 0.84x $9.60. 7. 0.86x36x36. 10. 17% x 34.6. A man’s salary is $1200. If reduced 121%, what will it be? 72 DIVISION OF DECIMALS Oral and Written 0.48 + 8, or 4 of 0.48 = —— hundredths or 0.—. 0.48 + 0.06. Have these a common unit? What is it? What is 4 of 0.81? £o0f0.072? 1 of 0.063? Whatis6.5+7? 81+9? 105+7? Compare 0.8+0.4 with 8+4. 20+4 with 10+ 2. Does multiplying both dividend and divisor affect the quotient ? Compare 0.049 + 0.07 with 4.9+7. How many places to the Jai was the point moved in each? By what did this multiply each? Pe CFO een Vat es 8. When any number of units (dollars, feet, tenths, hundredths) is divided into any integral number of parts, what is the unit of each part ? 9. Then if the divisor is an integer as in 4.9+7, the quotient . will be expressed in the same units as the Hence 4.9 + 7, or AY tenths, +7 =0.7. To divide 0.144 by 0.09. 10. In the example at the left how is the divisor changed from 0.09 to 9.? For what purpose ? To divide 63.44 by 25.6. 11. How and why must the dividend Process also be changed? State the principle. 2.47 + 12. After dividing 14 by 9, how 25.6)63.44 256.) 634.4 many tenths in all remain to be divided? PROCESS 0.09) 0.144 =9.)14.4 1.6 512. 13. Explain the process shown in 122.4 the second example. When there is a 102.4 remainder, how do you continue the 20.00 division ? Le 14. How may you always have an eu. integral divisor ? 15. 38.7 + 4.34. 16. 3.485 + 0.95. 17. 24.6 + 0.17. Principle. Multiplying or dividing both dividend and divisor by the same number does not affect the quotient. Written DIVISION OF DECIMALS 79 Give directions for division of decimals in five steps: — I. Setting down. II. Pointing. III. Dividing. IV. Placing and pointing the quotient. V. Managing the remainder. Divide, noticing whether the quotient will be larger or smaller than the dividend : — 1. 21.6 + 0.006. 6. 102.01 + 1.01 +12.5% of 100. 2. 0.4913 + 1.7. 7. $8.281 + 23, — 6.25% of $3.00. 3. 2.1952 + 0.028. 8. 4.096 + 0.0064 + 0.82369 -+ 28.7. 7 yee) 9. 67.24 x 82% — 67.24% + 82. 5. 0.6345 + 0.009. 10. 400 + 0.662 + 876.16 + 0.296. Common Fractions Changed to Decimals Since 3 + 8 or ¢ of 3 = 3, a fraction may be changed to a decimal by considering it a problem in division of decimals; thus, — & = 8)3.000 0.375 Change to complex decimals of three places: — es Lees 19k 4e) 23.85. 12. 5%. 16. 35. 20. 83. 24. 48. VS tear. Lyd Dod ae 25. 16. 14 he 18. 23. a2 0443) PAO Ee 9 Change to incomplete decimals of four places : — yin nae. 2970 ao, Vie ey B3rete 28. 2. 30. At. 32. gt. 34. +35. 35. A long ton is 224° of a short ton. Express the relation as a mixed decimal 36. 9 is what part of 144? Express the relation as a complex decimal of two places. As a per cent. 37. What per cent of $6000 is $4680 ? 80 REVIEW EXERCISES Oral 1: 88.025 is to be divided into 195 parts. Will each part be more or less than 4+? 2. How many places would there be in the quotient if the division were exact ? 3. A camel goes 3.5 miles per hour. How far will it goin 10 hr. ? 4. Of what two equal numbers less than 1 is 0.25 the product ? 5. Find the square root of 0.0081. 6. 12 is what per cent of 50? Of 25? 7. Find the balance of this account by inspection. Which party is described as Dr. and Cr. ? New Yorks, Jan. 30, 1904. Mr. J. Q. ADAMRB. . In acct. with Joun ReEynoups & Co. Dr. Cr. Jan. 2 | To Mdse. as by bill || $114 | 81 Jan. 1 | By bill for services || $150 | 00 9 | Freight prepaid 2 | 13 5 | By goods returned 14; 81 Storage of barrels 50 By allowance for 16 | Cash on acct. 50 | 00 damages 2 | 00 8. Multiply the sum of 11.507 and 4.493 by the difference between 10.85 and 103. Read rapidly and change to or from decimal forms : — 425%, 0.0006, sf . 20.2020, 202.020, 2020.20. 0.12, $10.012, $412. 35 1000, 3 $) EVO) B00? 15 ee i 900 100 . 0.314, 0.0374, 3.74. 6 2 7 3. 68%, £%, AL, LH. 8. 1 4 9 5 19104 40800 _985 100M MO LOKOrOis . 60,004, 0.048, 6.666-+. 10.) gate eee vat On Written REVIEW EXERCISES 81 1. A city contains 40,000 persons, 26% in the first ward, 32% in the second, 21% in the third, and the remainder in the fourth. How many in each? 2. 21% of a gang of 200 workmen receive $48.30. The wages of the rest are 20% higher. What does a workman of each class receive per day ? 3. The binding of 390 books cost $54.60. What was the entire . cost of each copy if the binding was 14% of it? 4. 164 ft.=1 rd. One girl lives 370 rd. from school; another 142.35 rd. on the same road. ‘Their houses are how many feet apart ? 5. When the cost of transporting coal is 2 ¢ per ton for each mile the freight on 400 tons is $800. What is the distance ? 6. At 1.25 cu. ft. to the bushel, compute the value at 571¢ a bushel of a bin of corn containing 4000 cu. ft. 7. Find the profit on 274 bbl. of flour at $4.114, and 128 bbl. at $ 3.963, if sold at 3¢ per pound. (196 lb. to the barrel.) 8. After melting 2 of a sheet of metal, and later 4, there was 44 of a square foot left. How many square inches in the part first melted ? 9. £1 English money = $4.8665. ind the value of £23,738. 10. $81,271.08 is to be divided among 7 heirs. 65 of them share equally; the others receive each a double portion. What is the amount of a 2% tax on 1 of the 5 equal shares ? 11. A lot of cord Mont is =8, beech, 0.21875 birch, 0.1875 maple, 35 ash, 10 cords oak, 4 poplar, and 34% pine. How many cords in all? 12. When 5.20 francs = $1, how many dollars will 3302 francs equal ? 82 INTEREST: GENERAL METHOD Oral 1. I live in a hired house worth $6000. For the use of the house for a year I pay ;45, or 10%, of its value. What do I pay per year ? 2. If I had used the money which the house cost, $6000, for a year at the same rate, 10%, the annual interest would have been what? The interest for 6 mo. would have been what ? 3. The value of a house used was $3000; rate of rent, 5%. Find a year’s rent.- A month’s rent. 4. Money used, $3000; rate of interest, 5%. Find a year’s interest. A month’s interest. 5. What is the difference between rent and interest ? Interest 1s an allowance to the owner for the use of his money. The principal is the money used. The amount is the sum of the interest and the principal. The rate of interest is the number of hundredths of the principal paid for a year’s use of it. 6. The principal is $200. The rate is 6%. Give the interest for- 1 yr.; 2 yr.3 3.yr.; 4 yr. 35 yr. 3 the interest for 1omos soe 3$mo.;4.mo.;°5 mo.; 6° moe.;/7 mo.; S$ mo. 9imo0., 10 m0 eee 7. What is a year’s interest of $300 at 2%? 3%? 4%? 5%? | Find the interest : — What shall I pay for the use: — 8. Of $300 at 4% for 2 yr. 11. Of $1000 for 2 yr. at 10% ? 9. Of $500 at 6% for 3 yr. 12. Of $600 for 2 yr. at 10% ? 10. Of $800 at 7% for 4 yr. 13. Of $800 for # yr. at 4%? 14. In most business transactions 30 da. make a month. If a month’s interest is $60, what is the interest for 1 da.? For 10 da. ? For 20 da. ? 15. At 6% the interest of $300 for 1 yr. is ——. For 1 da. it is For 1 mo. it is Written INTEREST: GENERAL METHOD 83 1. Find the interest of $240 for 2 yr. 5 mo. at 5%. B A $ 240 incipal a = principa 29 x+y.» yx $949 = 29 05° = rate # ‘00 : $12.00 = int. for 1 yr. 20 2,5, = time in years (a) In B what represents a $5.00 = int. for 7 yr. year’s interest ? 24.00 = int. for 2 ree (6) A month’s interest ? $ 29.00 = int. for 25% yr. (c) How is the process short- ened ? What is the interest of : — 2. $840 forlyr.9mo.at10%? 5. $1000 for 12% yr. at 8% ? 3. $360 for4 yr.10mo. at5%? 6. $400 for 2 yr.5mo.atT% ? 4. $960 for 1 yr. 8 mo. at 4% ? 7. Find the interest of $500 for 1 yr. 5 mo. 6 da. at 5%. A B $ 500 ae 05 eon $215, $ 25.00 =int. for 1 yr. PIP x a * 700 x ae ~——— = $ 35.83 10.412 = int. for 5 mo. 12 Al¢ = int. for'6 da. $35.83 =int.forlyr.5mo.6da. 6 (a) Explain the process in B. (6) What is the advantage of this method ? Find the interest of : — 8. $600 for 60 da. at 4%. 12. $336 for 8 mo. 10 da. at 4%. 9. $250for1mo.1dda.at6%. 13. $1728 for2mo.17 da.at9%. 10. $120 for 80 da. at 7%. 14. $2800 for 9 mo. at 41%. 11. $372 for 36 da. at 10%. 15. $1000 for 93 da. at 44%. To Teacuer. (If drill in interest is desired at this time, see page 156.) 84 ae MEASUREMENTS: LINES Oral Beginning with the shortest, name five units used in measur- ing lengths or distances. Give the table of length measures. 2. oO UWO oT PR w 9. Compare an inch with a foot. An inch with a yard. Compare a foot with a yard. Witharod. With a mile. What part of a mile is a rod ? A yard is what part of arod? Ofa mile? 72; miles = rods. Q1 tn, sees in. 320 x 54 x 3 ft. =1 mi. = —— ft. Learn in some way, as measuring and counting your steps, the distance from your home to school in feet; in yards; in rods. 10. Estimate in feet the dimensions of your schoolroom. ‘Test your estimates by measuring. ak Estimate and test longer distances as the dimensions of your playground in rods; estimate in yards. 12. 13. At 1¢ per foot, what is the cost of 4 yards of picture wire ? At 12¢ a yard, picture molding for a room 25 feet long and 20 feet wide will cost what ? 14. 15. aS: 19. 20. Zl. wee long ? 23. 4rd. = —— ft. 16.1000 .0d. ) base=40 ity, alt==18 ft; 7. 194 ft. and 24 yd. 4. Base==60 ft., alt.=25 ft. 8. 38rd. and 221 ft. Simbase—~o tt, alt.==9 in. 9. 64 yd. and 13 ft. 6. Base—41d., alt, =7 it. 10. 34rd. and 6 yd. 11. Draw or cut out a trapezium, or a quadrilateral no two of whose sides are parallel. 12. Separate it into two triangles along one of its diagonals, as AB. 13. Find the dimensions of each triangle and its area. 14. What will the area of the trape- zium be ? 15. The diagonal of a trapezium is 24 inches; the altitudes per- pendicular to it are 18 inches and 15 inches respectively. What is the area ? 16. The diagonals of a given trapezium cross at right angles. The point of intersection is 50 feet from the upper end of each diagonal. One diagonal is 100 feet long, the other 150 feet. Find the area. (Draw a diagram.) 17. In right triangles the two sides including the right angle, are called the legs. If one leg is the base, the other is the altitude. Why ? 18. The legs of a right triangle are 10 and 15 ft. Find the area. 19. The diagonal of a trapezium is 14 in. The altitudes perpen- dicular to it measure 6 in. and 8in. What is its area? 100 MEASUREMENT OF PARALLELOGRAMS AND TRAPEZOIDS Oral and Written 1. By cutting, find the relation between the two triangles into which any parallelogram is divided by a diagonal. 2. OM, the altitude of the triangle, P C is also the altitude of the parallelo- gram, and AB is the base of each. 3. Compare a parallelogram with a triangle having the same altitude and base. A M B 4. What kind of parallelogram is shown in the figure ? 5. Compare a rhomboid with a rectangle whose dimensions are the same as the base and altitude of the rhomboid. 6. How is the area of a rectangle found ? 7. How, then, may the area of a rhomboid be found when its base and altitude are given? Find areas of rhomboids with : — | 8. Base 121 ft., altitude 74 ft. 10. Bes 2hiyd, A418 im 9. Base 16 rd., altitude 40 ft. 11. A= 84 ft. B= opm 12. To find the area of a trapezoid whose two parallel sides are 24 in. and 16 in. and whose altitude is 12 in. If the trapezoid is divided as in the figure, what are es iC the dimensions of triangle ABO? Its area? If DC is taken as the base of triangle ACD, what is the altitude? The area? What, then, is the area of the trapezoid ? N me A 24 B Sraremennr, 4 of 24 x 12 + 4 of 16 x 12, or $ of 40 x 12 = 20 x 12, or 240 13. Notice that since the altitude is the same in each triangle, time may be saved by adding the two bases before multiplying by half the altitude. State the rule, then, for finding the area of a trapezoid. Written MENSURATION: REVIEW 101 1. Name 2 quadrilaterals that are not parallelograms. 2. The base and the altitude of arhombus are each 16in. Area? 3. The altitude of a rhomboid is 2 of its 2-ft. base. Area is 2. 4. One angle of a parallelogram measures 90°. What do the other angles measure ? 5. The parallel sides of a trapezoid are 25 and 37 ft., respectively, and the distance between them is 15 ft. Required the area. 6. The base of a triangle is 174 in., its altitude 8} in. What is its area ? 7. A lot of land has a frontage of 50 ft. The parallel sides are perpendicular to this frontage. One of its parallel sides measures 10 rd. and the other 80 ft. What is its value at 45¢ a square foot ? 8. A triangular park measures 600 ft. on one side, 300 ft. on the other which is perpendicular to it; how many square rods in the park ? 9. Compare the area of a rhombus having a base of 16 ft., and an altitude of 12 ft., with that of a 16-ft. square. 10. The diagonal of a trapezium is 42 in., and the perpendiculars dropped to it from the angles are 16 in. and 18 in. respectively. Required the area. Find the areas of rhomboids : — 11. Base 138 ft., altitude $ ft. 13. Base 3i yd., altitude 20 in. 12. Base 20 rd., altitude 50 ft. 14. Base 2640 ft., altitude 4 imi. 15. The diagonal of a trapezium is 7 yd. The altitudes perpen- dicular to it are 24 ft. and 124 ft. What is its area? 16. What is the area of one of the triangles into which a diagonal of a field 10 rd. square divides it ? , 17. What will it cost to fence a yard in a shape of an equilateral triangle 36 ft. long with 4 lines of wire weighing one pound to every 20 ft. and costing 3i¢ a pound? ‘There are 12 posts, costing 15¢ each. 102 CIRCLES, DIAMETERS, RADII, ARCS, ETC. Oral 1. An area bounded by a curved line, all points of which are equally distant from a point within called the center, is a circle. 2. The distance from the center to the curve D is the radius, and the curve 1s called the circum- E ference. In the figure how many radii are drawn? Name them. Ve 3. Any straight line through the center ter- i - minating in the circumference is a diameter. Is a diameter shown in the figure? Which lines are diameters ? C 4. Into how many equal parts does a diameter divide a circle? One half of a cirele is called a semicircle. 5. Into how many equal parts do two perpendicular diameters divide a circle? Such parts are called quadrants. 6. Any part of a circumference is called an arc. Any part of a circle bounded by two radii and an arc is a sector. Name some sectors in the figure. Is a quadrant a sector ? 7. For convenience in measuring arcs, every circumference, whether large or small, is divided into 360 equal parts called degrees (360°). How many degrees in a semicircumference? In a quadrant ? 8. Each degree is divided into 60 minutes (60'), and each minute into 60 seconds (60"). 15° =a’. 300! = 2°. An are of 30° contains 2’. 10° =a", 600" =a’. of a circumference = 2°. {oo 9. Cut circles of different sizes from stiff cardboard or bring to the class several circular objects: plates, rings, covers, wheels, or coins. Measure very accurately the diameter and circumference of each. Nore. To get an accurate measurement of a circumference two pupils can work together to an advantage. Take two rulers, one standing edgewise on the other as a guide for the circle. Mark a point on the circumference and roll through one complete revolution, noticing the distance passed over on the bottom ruler, and holding against the second ruler to get a straight path. Oral, Written | 103 RATIO OF CIRCUMFERENCE TO DIAMETER 10. In each case divide the circumference by the diameter, carry- ing the result to several decimal places. 11. Compare your results. Take all the results that are about alike and find the average. 12. If you have measured and divided accurately, the quotient will be 3.1416 nearly. What does this show ? 13. If the diameter of a circle is 10 ft., what is the circumference? 14. 3.1416 is the ratio of the circumference to the diameter. It is represented by the Greek letter z (pi). 15. If D=diameter, A =radius, C= circumference, read the following : — Cee ODD = CO-+ 7, or ean C= 2 Rr, or 27fR. T Find the diameter or circumference or radius. Forecast the result. Goes 20185 Glam. Pome = Auf el) oer tT ee UD tb se) sag, 20eE) = LOAN Ca. LS ere edie Ca, Ql pts =a 2) Vasa tet ye = we. To Finp THE AREA OF A CIRCLE (_) AAAnn nnn 1. Draw a circle on stiff cardboard. Draw two perpendicular diameters. Divide each quadrant into two or more equal parts. 2. Arrange the parts as in B. What figures that you have studied do they most resemble? If these sectors were triangles, how would you find the area of each? Of all? Compare the sum of the bases of all the sectors with the circumference of the circle. 104 CIRCLES: AREA Witien 3. Ifthese sectors were triangles, the area of all would equal the atea of one triangle with a base equal to and an altitude equal to the OR , or Area = 4. While the sectors are not triangles, yet the smaller they are made, the more like triangles they seem, and it 1s proven in geometry that the area of a circle is actually the same as you have found by supposing the sectors to be triangles, that is, The area of a circle is the same as that of a triangle having a base equal to the circumference and an altitude equal to the radius ; Cece or times the unit of measure. 5. Find the area of a circle whose diameter is 10 ft. SoLturion. C=10 x 3.1416;= 31.416. &=43 of 10 = 5 2 ) 157.080 78.54 = number of square feet. Notre. We must use Cand R as abstract numbers representing the number of units. The product will then be an abstract number showing the number of units in the area. Find the area of circles : — 6. R= 6 ft., C= —. 8. C=100 ft., D= ——. 7. D=12 ft. C=——_. O54 C =o¥ DOT, tT aes Sj —- CG x R i) [ ° Inge =4'== Sane and C=2x Kx, we may multiply by 2xXxmxX RYE ») fed generally used when the radius or diameter is known. 2X 7X # instead of C, then A= = 7h’, the formula. Find the area of circles : — | 10. dt =" 15 ft. 13. R22) 1d.) 16. C108 Tt) 1S Seer 1 aaa 1A, Vig— 164 Cle ek 7 (ee 20. = 96 41 12. C400 mm. 915. Doe 80 Gd) (18s) aeshird, aL. Da cee Drawing and Measuring THE CIRCLE 105 1. Draw a 2-in. circle. Draw two perpendicular diameters. 2. Compare the four arcs. Connect the points where the diame- ters cut the circumference as in the figure. G How do these lines compare ? 3. Lines joining the extremities of arcs as AB are called chords. Chords whose arcs p B are equal are also equal. 4. Measure the angles of the quadrilateral ABCD. r 5. What are the two parts into which AC divides the circle called ? 6. The angle ABC is made, or inscribed, in a semicircle, when the vertex B is in the circumference and the sides pass through the ends A and C of the diameter. Measure the angle B with a pro- tractor. How large is it ? Any angle inscribed in a semicircle is a right angle. 7. What kind of figure is ABCD? Why? .- 8. If you consider these figures made up of two triangles ABC and CDA, what is the length of the base of each? What is the alti- tude of each? Then what is the area of each? Of the whole square ? 9. What is the area of the circle ? 10. How much larger is the circle than the square ? 11. Find the ratio of the square to the circle (4 decimal places). 12. Observe Problem 8, and give the area of the largest square that can be drawn in a 3-in. circle. In a 4-in. circle. 13. What will be the diameter of the largest circle you can draw in a 6-in. square ? 14. The top of a 4-ft. round table is what part of a 4-ft. square table ? 106 DRAWING AND MEASURING FIGURES 1. Draw a 3-in. square. On the same base draw a 3-in. rhombus. Which has the greater altitude? The greater area? G 2. Draw a rhombus. Draw the diagonals. Measure the angles made by the diagonals. Compare the lengths into which each diagonal divides the other. A B The diagonals of a rhombus bisect each other at right angles. 3. If the diagonal AO—12 in. and DB=8 in., find the area by considering the figure two triangles. 4. The diagonals of arhombus are 10 in.and 16in. What is its area ? 5. Using 1-in. lines, make a rhombus with an altitude of 4 in. What is the area? 6. Using 1-in. lines, make a rhombus with an altitude of 1 in. What is its area ? 7. Using 1-in. lines, make as small a figure as you can. What is its shape ? : 8. The area of a rhombus, when the length of the sides is fixed, depends upon what ? 9. Draw a trapezoid, the parallel sides being 14 in. and 2 in. One of the other sides 1 in. long is perpendicular to these. Find the area by dividing into a rectangle and a triangle; then find the area in the usual way. 10. Draw a rectangle. With the same sides draw a figure whose angles are not right angles. What is this figure called? How does it compare in size with the rectangle ? 11. What dimensions would you need to know to find the area of the rhomboid you drew in Exercise 10 ? 12. Draw atrapezium. What dimensions must you know to find the area ? Oral Review MEASUREMENTS 107 1. What objects before you are nearest in length to a yard? To afoot? Toarod? 2. How many degrees measure a|_ (right angle)? Can you find as you look about you any but [’s ? 3. After going 4 round a circle, how many degrees complete the circle ? 4. Since the sum of all the angles of a triangle is equal to 2 |’s, each of the three equal angles of an equilateral triangle measures how many degrees ? 5. How may the area of a triangular park be found ? 6. How can you find the area of any surface bounded by straight lines ? 7. How much of an 8-in. square is not covered by a 7-in square ? How much not covered by a 6-in. square ? 8. An area containing 3 sq. yd. contains how many square feet ? 9. Which is larger, a square or a rhombus with the same perimeter ? 10. What part of a square yard is a mat 4 yd. square? 11. A room is 2 as wide as it is long. If it is 30 ft. long, how many square fee in the floor ? 12. The area of a trapezoid is 350 sq. ft. The parallel sides are 30 ft. and 40 ft. What is the distance between them ? 13. The area of a triangle is 25 sq.ft. The altitude is 5 ft. What is the base ? 14. 7, or 3.1416, is nearly 31 or 22. Using this value, what is the diameter of a tree if it takes a string 44 in. long to reach around it? 15. The diameter of a circle is 103 in. What is the circum- ference ? 108 MEASURING CIRCLES: PROBLEMS Written 1. A button is 4.7124 in. round. How long a buttonhole is required ? 2. Find the circumference of the base of a lamp chimney that is 23 in. across. 3. A circus ring is 414,% ft. round. Find the distance to the center in rods. 4. A hogshead is a little over 121 ft. round the middle. Will it go through a doorway that is 3 ft. 10 in. wide ? 5. If a mountain is 10 mi. round, what distance might be saved by tunnelling ? | 6. A pie is cut accurately into 6 pieces. What is the ratio of the curved edge to the straight one ? 7. The hubs of two wheels are alike, but the spokes of one are 3 in. longer. How much greater is its circumference ? 8. Ifa barrel is 8 in. over the chine, how much strap iron will be required to make 100 end hoops with 3-in. laps? Make a statement. 9. In a lawn 100 ft. square the circular basin of a fountain is 40 ft. from each side. Draw a figure and find the area of the green- sward. 10. A square is cut from a circle 12 in. in diameter. What is the size of each of the four equal segments cut from the circle ? 11. In Exercise 10, what per cent of the circle was cut away ? 12. A circular grass plot 2 rd. in diameter is surrounded by a walk 3 ft. wide- How many square feet in the walk ? Hint. Find the areas of the two circles — the grass plot, and the one includ- ing plot and walk. 13. A circle 10 in. in diameter is cut from one 16 in. in diameter. What per cent of the large circle remains ? Oral MEASUREMENT OF SOLIDS 109 1. Lines have one dimension; viz.: 2. Surfaces have two dimensions; viz.: ——- and ——. 3. Solids occupy space and have weight. They have three dimen- Slons ; V1z.: ; , and : 4. Name the three common units of solid measure in the order of their size. 5. ——cuin.=1 cu. ft.; —— cu. ft.=1 cu. yd.; —— cu. ft. = 1 cord. 6. State the method of finding the number of cubic feet in 20,000 cu. in. 7. 10 cu. ft. = cu. in. 9. 10 cords=~2 cu. ft. S720 Cieyd = 2icus tty 1020720 cur ft.==a cu. yd. Rectangular Solids What is a rectangle ? A solid bounded by sia rectangles is a rectangular solid. A solid bounded by six squares is a How many corners and edges has a cube ? Is a cube a rectangular solid? Describe an inch cube, or a cubic inch. Describe a cubic foot; a cubic yard. What is meant by a 2-ft. cube ? 9. What is a9-in. cube? How many cubic inches in a 9-in. cube ? (See the figure.) Along one edge of a cube 9 there is a row of x cu. in.; 9 such rows make a, tier of 9 x x cu. in. or y cu. in.; 9 such tiers contain 9 x y cu. in., or 729 cu. in. SraTEMENT. 9X9 xX9cu. in. =~ cu. in. 110 RECTANGULAR SOLIDS Written Find the contents or volume of : — 10. A 6-in. cube. 12. A 5-ft. cube. 14. A.10-yd. cube. 11. An 8-in. cube. 13. A 12-ft. cube. 15. A 20-in. cube. 16. Show that the volume of each cube is equal to as many units of measure as the product of the number of units in the edge taken as a factor three times. 17. 5*’means 5x5x5. Since this is equal to the number of times 5 is taken as a factor to find the contents of a cube whose edge is five, 5%, or “5 to the third power,” is called “the cube of 5,” or “5 cubed.” TB Tah ox Ogle Ue” Cu ere) OF coals Eee 19. What is one of the three equal factors of 64, or 1/64? 20. »/216? 1728? W512? +/13312? +3729? +/27,000? Prisms 1. Solids whose sides are all rectangles and whose bases are bounded by straight lines are prisms. 2. Prisms are named from their bases: square, rectangular, tri- angular, hexagonal, etc. Name some objects that are square prisms; that are rectangular. 3. Is a square prism also rectangular? What kind of prism is a cube ? Written MEASUREMENT OF PRISMS. PROBLEMS 111 4. To find the contents of a rectangular prism whose dimensions are 4 in., 5 in., and 9 in. —_ Notice (a) the number of cubic inches in one row or square prism 1 in. by 1 in. by 9 in. (6) The number of such rows in one tier or layer 4 in. wide. (c) The number of such layers in the prism, and explain the statement 5 x 4 x 9 cu. in. = @ cu. in. 5. How many cubes may be put into a box 10 in. long, 8 in. wide, and 5 in. deep ? 6. A trunk measures 3 ft. by 20 in. by 18 in. Find its cubical contents. Why multiply by 36 instead of 3? Find the contents of rectangular prisms of these dimensions : — LENGTH WIDTH HEIGHT LENGTH BREADTH DEPTH aeeLO- tte 10 ft. 8 ft. 10. 422 ft. 20:4: 13% ft. Sen Ley d. atts 9 in. 11. 124 yd. 10 ft. 16 in. gedit) 16h Lyd: Tome Oe tee tt ee Glin: 13. How does the number of cubical units (cu. in., cu. ft., etc.) in any layer 1 unit thick compare with the number of units (sq. in., sq. ft., etc.) in the side of the layer ? 14. If the base of a rectangular prism contains 30 sq. in., how many cubic inches in the bottom layer? If the prism is 10 in. high, © how many cubic inches in the prism ? STATEMENT. 10 x 30 cubic inches = z cubic inches. 15. The floor of a cellar contains 36 sq. yd. If the cellar is 8 ft. deep, find its cubical contents. 16. A square prism is 16in. wide and 4 ft. long. Find the volume. 17. A 5-in. cube is cut from the corner of a 20-in. cube. What part remains ? 112 MEASUREMENT OF WOOD Oral and Written 1. Wood for fuel is generally sold in what lengths? In what form is it usually piled ? 2. Give the dimensions of a cord of wood; a half cord; 4 of a cord, or a cord foot. 7 3. What is the unit of measure used in measuring wood ? 4. What kind of solid does a half cord represent ? 5. Explain: 8x4x4 cu. ft.=~2 cu. ft. as applied to cord wood. 6. A pile of 4-ft. wood, 4 ft. high and 8 ft. long, contains a cord. How many cords if 16 ft. long? 24 ft.? 321t.? 96 ft.? 7. A pile of 4-ft. wood of the usual height must be how long to contain 10 cords? 12 cords? 25 cords? 8. How many cords in a pile of 4-ft. wood 4 ft. high and 18 ft. long? Explain the following statement, and show what may be cancelled : — Ax a5 1S: Cun tt 128 cu. ft. 9. Bought a pile of 4-ft. wood 30 ft. long and 8 ft. high at $6 per cord. Find the cost. = the no. of cords. 4x8 x 30 Aga, [age MAP a STATEMENT. In the statement, what represents the number of cubic feet? The number of cords? The cost of all? Find the value of piles of wood as follows : — LENGTH WIDTH HEIGHT PRICE LENGTH WIDTH HEIGHT #£PRICE 10. 24ft. 4ft 6ft $4 13. 24 ft. 4ft. Tift. $3.50 11 40 Thott. Sit.) ob 14. 20ft. 3ft 124 ft. $5.00 12. 60ft. 10ft. 4ft. $6 15. 164ft. 44in. 22 ft. $4.25 16. A pile of tan bark 8 ft. wide and 10 ft. high contains 200 cords. How long is it? Oral and Written MEASUREMENT OF LUMBER 113 1. Timber sawed for building purposes is lumber. What forms can you mention besides boards, planks, joists, and beams ? 2. In measuring lumber no attention is paid to the thickness unless it exceeds an inch. A board 1 ft. square and an inch, or less, thick is called a board foot. 3. A board 12 ft. long, 1 ft. wide, and 1 in. thick contains 12 board feet, or usually spoken of as 12 ft. What would a board 12 ft. long, 10 in. wide and an inch thick contain ? 4. A board 15 ft. long, 8 in. wide, and 3 in. thick contains how many feet ? Suecesstion. If the board were a foot wide it would contain it is but 2 of a foot wide it contains : feet. Since 5. Ten 16-ft. boards averaging 9 in. in width contain how many feet? Explain the statement: 10 x $ x 16 board feet = x. 6. A board 10 ft. long, 1 ft. wide, and 1 in. thick is equal to x board feet. If 14 in. thick it is equal to 14 x & board feet. 7. Find the contents of a 3-in. plank 15 ft. long and 10 in. wide Explain: 3 x 2 x 15 board feet = a. 8. How many feet of lumber in 12 joists, 16 ft. long, and 4 in. square ? 9. Find the number of feet in 8 “3 x 4” joists each 12 ft. long. Note. A ‘3 x 4” joist is one 3 in. thick and 4 in. wide. Find the cats in board feet of lumber measuring as follows : — 10. 6 boards, 16 ft. long, 14 in. thick, and 14 in. wide. 11. Fifteen 3 x 4 joists, 18 ft. long. 12. A stick of timber 18 ft. long and 12 in. square. 13. A board 12 ft. long, 6 in. wide at one end and 10 in. at the other. Hint. Find the average width. 114 THE SURFACES OF RECTANGULAR PRISMS Written 1. How many surfaces has a rectangular prism ? 2. What name is given to a rectangular prism with equal faces ? 3. Find the entire surface of a 5-in. cube. Explain the statement: 6 x 5 x 5 sq. in. = @ sq. in. 4. The entire surface of a cube is 150 sq. in. How long is it ? Find the entire surface of : — How long a cube has : — 5. A 9-in cube. 8. An entire surface of 384 sq. in. ? 6. A cube 10 in. long. 9. An entire surface of 600 sq. ft. ? 7. A 16-in. cube. 10. An entire surface of 294 sq. in. ? 11.. Compare the ends of a square prism with each other. 12. Compare its four sides. 13. Find the entire surface of a square prism 8 in. long and 3 in. wide. (What is the area of each square? How many? Of each rectangle? How many ?) 14. Compare the opposite faces of 2 any rectangular prism. 15. Give the dimensions of each face in the figure at the right. What 4 is its entire surface ? Find the entire surface of prisms : — 6 16. 10 in. by 6 in. by 4 in. 17. 12 ft. long, 9 ft. wide, 6 ft. high. 18. 20 ft. long, 4 ft. wide, 8 ft. high. 19. 16 x18 by 4; 20 by 1 by 1. 20. 12 by 9 by 8; 23 by i by 16. 21. 12 by 12 by 6; 2 by 3 by 74. Oral, Written THE VOLUME OF CYLINDERS hED 1. A solid having ends (or bases) which are equal, parallel circles, and having uniform diameters is a cylinder. Mention some common objects that are cylinders. 2. If the base of a prism is 6 sq. in., how many cubic inches in a section 1 in. thick? How many cubic inches in the prism if its height 18.0 10.3 3. Ifa cylinder is cut as in Fig. A B and arranged as in Fig. B, what does it nearly become ? 4. If the circular end is the base, it has been changed into one nearly lke a rectangle. Has the area of the base changed in B? Has the length of the cylinder changed ? 5. In B, if the base were a rectangle and its area known, how would its volume be found ? 6. If the radius of A is 1 in., what is the area of its base or the base of B? If the length of Bis 8 in., what is the volume ? 7. The volume that we have found by considering that the cylin- der is nearly a prism with an equal base and the same height, is the volume that is found by geometry to be the true one, viz. : — The volume of a cylinder is the same as that of a prism having a base of the same area and having the same height. 8. Find the volume of a cylinder whose radius is 8 in. and whose height is 15 in. Explain the statement: (15 x 3.1416 x 8 x 8) cu. in. = acu. in. 9. A cylindrical pail 6 in. in diameter inside and 12 in. deep contains how many cubic inches ? Explain the statement: (12 x 3.1416 x 3 x 3) cu. in. = weu. in, 116 MEASUREMENT OF CYLINDERS Written 10. A cylindrical tank is 10 ft. deep and 8 ft. in diameter. Find the contents. 11. A gallon contains 251 cu.in. How many gallons will a cistern 8 ft. deep and 4 ft. in diameter hold ? 12. A well is 32 ft. deep and 5 ft.in diameter. Find the contents. 13. To hold a gallon, a pail measuring 33 square inches on the bottom must be how deep ? To find the Surface of a Cylinder s. The rest of 1. In form, the ends of a cylinder are equal the surface is convex surface. 2. Suppose the diameter of a cylinder to be 4 in.; its circumfer- ence = 2, or 3.1416 x D. 3. The circumference of a cylinder is 8 in.; its diameter is x, or ©. 7 4. A cylinder is 20 in. long and 4 in. in diameter. Find the area of its ends. ; 5. Roll an oblong paper to form a cylin- der. Give the length and circumference of the cylinder thus made. . 6. Unroll the paper and give the di- mensions of a rectangle equivalent to the convex surface of the cylinder. Explain the diagram at the right. es 7. The convex surface is equal to CO x I square units, where C= circumference and £ = length. Why is this? 8. A cylinder is 25 in. long, 4 in. in diameter. Its convex surface is x. Ex- plain: (3.1416 x 4) x 25 sq. in. = @ sq. in. 9. A cylinder is 20 in. long, 5 in. in diameter. Find the entire surface, or the convex surface plus the surface of the ends. Oral MEASUREMENTS: REVIEW a ke Ey Nore. In oral work use #2 for 7. 1. An old tree is 22 ft. round; how far is it through ? 2. Give the convex surface of a lead pencil + in. in diameter and 7 in. long. 3. The sides that make the right angle of a triangle are each 10 ft. What is the area ? 4. It is 28 ft. across a pond. How far is it around it ? 5. Which takes more room, a cord of wood or a 5-ft. cube ? 6. 4a circle = a rectangle having the radius for one side and —— for the other. 7. A 20-ft. log averages one square foot in the cross section. The cubic contents are x. 8. How many cubic yards of earth will a bin hold that is pelts OTL. x29 fh.-? 9. About how many cubic yards does your schoolroom contain? 10. A horse tethered by a rope 10 yd. long can graze over how large an area? 11. A cellar 18 ft. by 15 ft. by 74 ft. deep contains how many cubic yards? Written 1. The largest possible circle is cut from a cardboard measuring 16 by 24 in. What area of the cardboard remains ? 2. What will it cost to cement the bottom of a circular cistern 8 ft. in diameter at $ 2.50 per square yard ? 3. The driving wheel of an engine is 6 ft. in diameter. How many revolutions does it make in going a mile if there is no slipping ? 4, A large street roller is 5 ft. in diameter and 7 ft. long. How ereat an area does it cover in one revolution ? 118 MISSING FACTOR FOUND Written 1. o times: 6 = Aster ear 1 So: 2. Multiplicand = 25; product = 400. How is the multiplier found ? 3. 186+e#= 31. 4. Dividend and quotient being given, how is the divisor found ? 5. When the product and multipler are given, how is the multi- plicand found ? 6. What is the area of a rectangle 12 ft. long and 64 ft. wide? 7. A rectangle containing 108 sq. in. is 9 in. wide. How long is hii OR Oe SO aie LOS teqmians) | 8. A lot 200 ft. long contains 24,000 sq. ft. How wide is it? 9. A sidewalk 50 ft. long requires 50 sq. yd. of concrete. How wide is the walk ? 10. One half an acre of land is taken for a new street 40 ft. wide. How long is the street ? 11. The area of a triangle is 325 sq. in.; its base is 25 in. What is its altitude ? | 12. The altitude of an isosceles triangle is 14 ft.; its area is 126 sq. ft. What is its base? 13. At 30 a board foot a mahogany board 1 in. thick and 12 ft. long cost $2.70. How wide was it? 14. The area of a rhomboidal field is 12 A. Its length being 20 rd., what is its altitude ? 15. A square contains 400 sq. in. How long is it? 16. The perimeter of a square is 1000 ft. Its area? 17. A rectangular field 48 rd. wide contains 48 A. What is the other dimension? ‘The perimeter ? 18. A rectangle is 3 times as long as wide and contains 108 sq. ft. What are its dimensions ? Suecrstion. Into how many squares can the rectangle be cut? What will be the size of each? The dimensions of each ? Written MEASUREMENTS 119 CONTENTS OF A SoLIp AND Two Dimensions GIVEN To FIND THE THIRD df LS OPiS ho = POLO: 2. I hired 15 men at $2.50 per day each. At the completion of the work I paid them in all $150. How many days did they work ? 3. A box on my table holds 432 cu. in. It covers 72 sq. in. of the surface of the table. How high is the box ? 4. The area of the floor of your schoolroom is 900 sq. ft. The room contains 10,800 cu. ft. How far is the ceiling from the floor ? 5. A packing box is 48 in. long and 30 in. wide. How deep must it be to hold 10 cu. ft. ? 10 x 1728 48 x 30 Explain the statement, and show a short solution. 6. A closet 8 ft. high and 27 in. deep will contain 72 cu. ft. How wide is it? 7. A pile of 198 ed. of 4-ft. wood covers 16 sq. rd. How long is it? How high is it? Explain the statement : — STATEMENT. = depth. Huong zy sli: 198 x 128 _ ig a Ata ionic 1652723 0 7 8. A cylindrical oil tank holds 10 gal. Standing on the floor it covers 77 sq.in. How high must it be? 9. A bookease holding 32 cu. ft. covers a wall space of 24 sq. ft. How far must it project into the room ? 10. I have room in my stable for a grain bin 8 ft. by 4 ft. How deep shall I make it to have it hold 72 bu. ? 11. A grindstone 4 ft. in diameter contains 6.2832 cu. ft. How thick is it? Explain the statement: 6.2832 + (2? x 3.1416) = a. 12. In digging a trench 3 ft. wide and 44 ft. deep 330 cu. yd. of earth are removed. How long was the trench ? 120 MISCELLANEOUS PROBLEMS Written 1. I buy a corner lot 120 ft. by 50 ft. and use the earth obtained by digging a cellar 60 ft. by 30 ft. by 10 ft. to raise the grade, how many inches ? ; 2. A circular standpipe 75 ft. high is 25 ft. in diameter. When 2 full, how many gallons of water does it contain, reckoning 74 gal. to a cubic foot ? 3. A speculator buys a field 600 ft. long and 500 ft. wide for $2500. He runs a 40-ft. street through the center in each direction at an expense of $425 for labor. He sells the land at 20 cents a square foot. How much does he make or lose ? 4. At $3.75 per square yard what will it cost to pave # of a mile of street 81 ft. wide ? 5. A reservoir supplies a town with 4,573,800 gallons of water daily. If its surface area is 7 acres, how much will the water be lowered in a week, providing 4 as much runs in as runs out? Call 1 cu. ft. equal to 74 gal. 6. From a lot of land 40 rd. square I sold 40 sq. rd. What is the remainder worth at $230 an acre ? 7. The snow fall is 6 in. on the level. How many cubic feet rest on every acre of ground ? 8. A watch chain cost $28. This is 7 of the cost of the watch. Find the cost of the watch. 9. Wood is bought at $3.75 a cord. Transportation adds 15% to the cost, and storage 5% more. It is then sold at a profit of 20% on the total cost. What does it sell for ? 10. A southern county contains 14,700 blacks. 334% of the population are white. What is the population of the county ? 11. A schoolroom 12 ft. high, 30 ft. long, and 28 ft. wide contains 40 pupils. How many square feet of floor space for each pupil ? - How many cubic feet of air for each one? Written MENSURATION: PRACTICAL PROBLEMS 121 1. 64, 8, 44 are the dimensions of my coal bin in feet. Reckon. ing 90 lb. to the cubic foot, what will a bin full cost at $5 per ton ? 2. Quincy granite weighs 165% pounds to the cubic foot. What is the weight of 6 pieces of curbing 8 inches thick, 2 feet wide, and half a rod long ? 3. Find the cost of carpeting a 9-foot hallway 22 feet long with three-quarter carpeting at $0.874. Cut no strip, and allow 14 feet per seam for matching. 4. How many tons of 15-inch ice may be cut to the acre, a cubic foot weighing 574 pounds? Apply your knowledge of cancellation. 5. What is the capacity, in 42-gallon barrels, of a cylindrical oul tank 34 ft. in diameter, 22 feet long? Make a statement and cancel. 3 6. What is the area of a sector of 120°, its radius being 24 inches ? 7. A ball ground 375 feet long and 280 feet wide is inclosed by a tight board fence 8 feet high. What will the boards cost at $24 per M? Add 10% for waste. 8. Bought 12,000 long tons of coal at $4 and sold it at the same price per short ton. What did I gain ? 9. What will it cost to polish the visible portions of a shaft of red granite 6 feet by 2 feet by 22 inches at 62¢ per square inch ? 10. Draw a 6-inch square, a rectangle 9 inches by 4 inches, and one 3 inches by 12 inches. Compare areas and perimeters. What inference do you draw ? 11. A schoolroom 82 feet by 30 feet is lighted by 6 windows, each containing 15 panes of glass 12 by 16 inches. The lighting surface of the room is what per cent of the floor surface ? 12. My garden is 80 ft. by 100 ft. What will a concrete walk around the inside cost at 80¢ a square yard, the width of the walk being 35 feet ? 2? MISCELLANEOUS PROBLEMS Written 1. What decimal of a square prism becomes shavings when the largest possible cylinder is turned from it ? 2. What number subtracted 88 times from 80,005 will leave 13 as a remainder ? 3. A railroad company fences 15 miles of its road at 732 cents a rod. What is the cost ? 4. How many square feet of zinc will line a cubical cistern 5 ft. 10 in. deep ? 5. Bread sells for 10¢ with flour at $5.00. Flour goes up to $6.50. What should bread sell for on this basis ? 6. In a city of 7200 school children there are 2720 cases of tardi- ness in a year, during which there are 400 sessions of the school. The average attendance is 6800. What is the rate of tardiness ? 7. Find the cost of six 8 x 10 sills 18 ft. long at $24.74 per M. 8. In a library every tenth book is a history. If there are 567 other books in the library, how many volumes in all? 9. A schoolroom measuring 32 ft. x 284 ft. x 13 ft. seats 49 pupils. Each one needs 1800 cu. ft. of fresh air an hour. The room full would last the class x minutes. 10. How many feet of wire will it take to fence a square field containing 625 sq. rd. if there are three rows of wire in the fence ? 11. If I buy cloth at $1.20 per yard, how many yards must I sell at a gain of 20% to gain $20.40? 12. How many yards of 3 yd. wide carpeting will be required to carpet a room 12 by 153 feet, allowing 5 inches to each strip for matching? The carpet selected was 85¢ per yard. 13. A cubic foot of mahogany weighs 45 lb. What will be the weight of a piece of mahogany 5 in. thick, 20 in. wide, and 16 ft. long ? ; Oral MISCELLANEOUS EXERCISES 123 1. What does the ga gas of a fraction show? Which is ° the larger unit, 7, or 7? Why? Pe.t WW hiehyis ee 22 or 24? Find this in a short way without changing to a like unit. 3. What is the ratio of 2 to 2? To what unit did you change . both before you could compare them? Why ? 4. Change 28 to 40ths. Explain the process. 5. Explain the process of changing 82 to 28. 6. Explain the process of changing 26 to 82. 7. Compare 164 with 31. 11. $17 is 2 of $a. 8. Add 34, 12, 28. 12. $6 is 3 of $x. 9.. From 7} take 61. 13. 2 of 48 is 80% of «x. 10. 16 x $81 = $z. | 14. 7:3L=11:2 15. What per cent of a 2-foot cube is 3 cubic feet ? 16. A miller takes 4 quarts toll from every bushel. What per cent is this ? 17. I spent $27, or 30%, of my money. How many dollars had I remaining ? Hint. How many per cent remained ? Compare it with 30%. 18. What is the rate of income on an $8000 house rented at $40 per month ? 19. If wages are increased 10%, what will men now receive who received $1.50 before ? 20. Men who now receive $2.75 after a 10% raise in wages received what before the raise ? 21. What is 31% of 500? 51% of 600? 22. $2 is what per cent of $800? Of $80? Of $8? 23. 0.42 + 70 = what? 25. 0.006 + 0.12 = what ? 24. 4.2 + 700 = what? 26. 0.06 + 0.012 = what ? 124. MISCELLANEOUS EXERCISES Written 1. A lot of land cost $2800. This was y% of the cost of a house. The house cost @ dollars. 2. How many bags will be required for 1000 bushels of wheat, if each bag holds 2;°, bushels ? 3. The snowfall is 5 inches on the level. How many cubic feet rest on 24 A. of ground ? 4. The races start at precisely 2.15 p.m. The winner returns to the starting point at 5.36 pw. He has averaged 32 miles an hour. How far did he travel ? 5. Give the day and hour when exactly 3 of the month of March has passed. 6. 2800 mill operatives earn on the average $1.68 a day. If their wages are reduced 10%, what will the weekly saving to the company be? , 7. Inacity with an average attendance of 10,000 in the public schools, which keep 360 sessions during the year, how many tardi- nesses would there have been to each thousand pupils if the whole number of tardinesses for the year had been 3600 ? 8. The interest on a mortgage is payable semiannually at 51%. The face of the mortgage is $2300. How large a check should be remitted to pay the interest as it falls due ? 9. What will it cost to bronze a 2-foot cube at 14 ga square inch ? 10. Hay is bought at $12 aton. Transportation adds 15% to the cost, and storage 5% more. It is then sold at a profit of 20%. What does it sell for ? 11. I can buy alcohol for $2.75. If I import it for school use, I can buy it for 90 ¢. Thisisa saving of 7%. 12. The number of children of school age in a certain city is 14,769. This is 331% of the whole population. How many not school age? Work a short way. Oral MISCELLANEOUS EXERCISES 125 1. A pint of meal is put into a peck of flour. What per cent of the mixture is meal? 2. How many bullets, each weighing } oz., can be molded from 2 lb. of lead ? 3. May was sent to the store for 4 pk. of apples. What did she pay, for them, if they were $3.20 a bushel ? How many pens can I buy for $2, if I buy 2 for a cent ? What numbers between 30 and 50 are perfect squares ? What is the cost of 12 oranges at 3 for 5 cents? 4% of a number is 50. What is the number ? +02. 8 8. The difference between + and 4+ of my money is $30. How much have I? 9. Which is the better bargain, bananas at 20¢ a dozen or 16 for a quarter ? : 10. A mason works 8 hr. for $3.00, and a carpenter 10 hr. for $3.50. How much more does one earn than the other in 100 hr. ? 11. What is 7 months’ rent of a telephone at $50 a year ? 12. How long will it take a girl to earn $5, if she works half the time for 18¢ and half the time for 12¢ an hour? 13. Three boys who are 10, 12, and 14 years old respectively are to share $54 in proportion to their ages. What is the share of each ? 14. Two men agree to cut lumber for $200. One, with 3 men, works 5 days, and the other, with 4 men, works 6 days. How much of the money should each receive ? 15. Paid $21 for insuring my house for 5 yr., at 3%. What is it worth, if it is insured for 7 of its value? 16. A teacher’s salary is $60 a month, If it is raised 84%, what does she then receive ? 126 MISCELLANEOUS EXERCISES Written 1. Find the sum of :— 2496 2. Find the total cost of the following: 33 lb. of but- 3948 ter at 28 cents a pound, 9 lb. 9 oz. ham at 16 cents a eee pound, 8 lb. 10 oz. cheese at 24 cents a pound. 9625 3. A man sold 3 of his farm for $3900. What was 4 a of the farm worth at the same rate ? 9) . 6498 4. A builder bought 6500 brick at $7.50 per thousand, 5936 12,200 ft. of lumber at $16.50 per thousand feet, and 4073 975 lb. nails at $3.80 per hundred pounds. What was Bee the amount of his entire bill? ) 5678 5. What will it cost to carpet a room 54 ft. long and 6935 30 ft. wide with Brussels carpet 2 of a yard wide at $1.24 ee per yard, making no allowance for matching ? 4678 6. A man bought a house for $2500 and sold it for 3979 $1875. What per cent of the cost did he lose ? 8462 9879 7. What is the interest of $320, at 6 per cent for 2 yr. 6432 10 mo. 12 da. ? te 8. A merchant sold goods for $240, thereby losing 4 9346 of the cost. For what amount should he have sold them ange to gain 15% ? 9. During the winter of 1902 and 1903 a ton of coal lasted a family 14 days, average time. What did the coal cost at $5.25 a ton from Oct. 1 until March 31, inclusive? What would the coal cost for the same length of time at an increase of $4.25 a ton ? 10. A house which cost $9600 rents for $48 amonth. This is a% income from the investment, if the yearly expenses amount to $96. 11. A flour merchant sold 240 bbl. of flour for $1582, which was 4 less than he paid for it. What was the whole cost? The cost per barrel ? 12. Ifa cubic foot of granite weighs 165 lb., what will a 6-in. cube weigh ? Oral MISCELLANEOUS EXERCISES 127 1. How many yards of ingrain carpet will be needed for a room Date Oya Loot. 2. A building lot contains 5380 sq. ft. and is 100 ft. long. How Wide is it? 3. Divide 0.6 by 0.015. 4. 4x W256 =o. 5. A house bought for $3200 sold for $3000. What per cent of the cost was lost ? 6. What part of anything is 124% of it? If 121% of my money is $25, how much have I ? 7. What will pay a note of $500 that has been running 2 of a year at 9% interest ? 8. Goods marked at $1.50 per yard were sold at 331% discount. What did 5 yards cost ? 9. Bought 3 for 4¢ and sold 2 for 3%. Did I gain or lose and how much ? 10. Bought for $10 and sold for $2.50. What per cent of the cost was lost ? 11. My weight increased from 150 lb. to 175 lb. What was the per cent of increase ? 12. Read as per cents : — 1 1 Tiegh SOR Vg EY fa NG We a GER YA ray) aoe “ Ae “ 13. Read in largest units : — 123%; 18$%, 314%, 813%, 435%, 564%, 833%. 14. It is 160 rods around a square field. How many acres does it contain ? 15. At15¢ a yard, picture molding for a room 12 ft. by 18 ft. will cost how much ? . 16. Paid $2.40 for 15 lb. of meat, 20% of which was bone. What did the meat really cost per pound ? 128 MISCELLANEOUS EXERCISES Written 1. If 22 yd. are bought for $23.10, what is paid for 15% yd. at the same rate ? 2. 67.24 x 82% — 67.24% + 82 = what? 3. Ina flag 174 ft. long and 2 as wide, how many square yards of bunting, not allowing for seams ? 4. How many cubic yards of earth will be thrown out in digging a cellar 54 ft. long, 2 rd. wide, and 9 ft.-deep ? 5. What per cent of its daily trip has the long hand of a clock accomplished at 3 p.m. ? 6. I insure my house for $2400. How much premium do I pay, the rate being 3% ? 7. What is the interest of $400 for 3 yr. 3 mo. 20 da. at 6% ? 8. If I buy flour at 3; cents a pound and sell at 4,5 cents a pound, what part of the cost is the gain ? 9. What per cent of a floor 16 ft. square is covered by a rug 12 ft. square ? 10. A man sold 374% of his business for $3900. What was ¢ of it worth at that rate ? 11. Aman built a double house at a cost of $4500 on a lot valued at $1500. If he receives $30 a month from each tenant, and pays $100 for taxes and $20 for repairs, etc., what part of his investment will his net income be ? 12. If flour sold for $4.25 a barrel gains 61%, at what price should it be sold to gain 15% ? 13. If 4 of 3 of a ship is worth $8600, what is 2 of it worth ? 14. Change 0.096 to a common fraction whose denominator is 875, 15. How do sixths and twenty-sevenths compare in size ? 16. How many 63ds in 33? TABLES OF MEASURES 129 [FOR REFERENCE | Counting 12 things = 1 dozen (doz.) 12 dozen = 1 gross (gro. ) 12 gross = 1 great gross (G. gr.) 20 things = 1 score 24 sheets (paper) = 1 quire 20 quires or \ = 1 ream 480 sheets Time 60 seconds (sec.) = 1 minute (min.) 60 minutes = 1 hour (hr. } 24 hours = 1 day (da.) 7 days = 1 week (wk.) 2 weeks = 1 fortnight 30 (81, 28, 29) days = 1 month (mo.) 3 months or mee ae 13 weeks } cae ee a 12 months or _ 1 year (yr.) 365 days } ~ (common) 365 da, 5 hr. 48) _ { 1 true or solar min. 49.7 sec. } * year 366 days = 1 leap year 10 years = 1 decade 100 years = 1 century (C.) Value U.S. Money 10 mills = 1 ct. (ct., c., or %) 10 cents = 1 dime (di.) 100 cents or meet \ = 1 dollar ($) 10 dollars = 1 eagle Canadian Money 100 cents = 1 dollar = $1 English Money 12 pence (d.)=1 shilling (s.)=$0.248+ 20 shillings =1 pound (£)=$4.8665 French Money 100 centimes = 1 franc (fr.) = $0.1938 German Money 100 pfennigs = 1 mark (M.) = $0,238 Capacity Liquid Measures 4 gills (gi.) = 1 pint (pt.} 2 pints = 1 quart (qt.) 4 quarts = 1 gallon (gal.) TPegaiton, | 5281 tu..in. Dry Measures (For grain, fruit, ete.) 2 pints = 1 quart 8 quarts = 1 peck (pk.) 4 pecks = 1 bushel (bu.) 10 pecks 21 bushels \ = 1 barrel (bbl.) 1 bushel = 9150.42 cu. in, Weight Avotrdupois Weight 16 ounces (0z. ) = 1 pound (1b.) 1 hundred- pe oats rf eee (cwt. ) 2000 pounds or _ f thtone CD 20 paar eet { (short) 2240 pounds = 1 long ton 130 *60 pounds = 1 bushel { PRON potatoes Sh ac Nt Sih corn or rye H Soe te aril Be ea oats 1 OGiee tS eee barre! flour BOO GSW nie pe beef or pork * In most States Troy Weight (For precious metals, jewels, etc.) . = 1 pennyweight 24 grains { 8 (pwt.) 20 pennyweights = 1 ounce 12 ounces = 1 pound 4371 grains = 1 ounce 7000 1 pond vF 480 (ee 1 OUnes Tro 5760 ‘ =1 pound y Apothecaries’ Weight 20 grains = 1 scruple (sc. or D) 3 scruples = 1 dram (dr. or 3) 8drams = 1 ounce (oz. or 3) 12 ounces 5760 grains \ = 1 pound (lb. or tb) Length 12 inches (in.) = 1 foot (ft. ) 3 feet = 1 yard (yd.) 164 feet or se \ =\1 rod (rd.) 320 rods 5280 feet = 1 mile (mi.) 63,360 inches 4 inches = 1 hand TABLES OF MEASURES 6 feet = 1 fathom 6086.7 feet or 1 knot 1.15 + com- \ 1 nautical mile’ mon miles 1 geographic mile 3 knots = 1 league Circular Measure 60 seconds (/’) = 1 minute (') 60 minutes = Pdegrearts) 360 degrees = 1 circumference 694 miles or 1° of latitude; or 60 ccosraphic} = | 1° of longitude miles on the equator Surface or Square 144 square inches \ nN { 1 square foot (sq. in.) (sq. ft.) 1 square yard 9 square feet = { : (sq. yd.) 304 square yards \ io Ne square rod 2721 square feet i. (sq. rd.) 160 square rods lis 43,560 square feet J ace eee ; BAN eres ia { 1 square mile (sq. mi.) 1 mile square = 1 section 36 square miles = 1 township 1 square 100 square feet = (in roofs, floors, etc.) Solid or Cubic 1728 cubic inches \ ‘ig te cubic foot (cu. in.) (cu, ft.) 27 cubic feet = { 1 fas Wood Measures 16 cubic feet. = 1 cord foot (cd. ft.) 128 cubic feet 8 cord feet jf pcord (oe) THE SOUTHWORTH-STONE ARITHMETIC THIRD BOOK PART II PERCENTAGE Computing by Hundredths 1. One hundred is the common standard of comparison. For example, the merchant may gain $10 on every $100 invested, or 10 per cent (10%). The rate of interest may be 6%, or $6 on every $100 used. 2. What is meant by saying: — 12% of the grain spoiled ? 384% of the month was stormy ? 14% of the pupils were absent ? 38. The phrase per cent (the short form of per centum) means hundredths, or by the hundred. 4. 25% of 400 is 25 times one of the 100 equal parts of 400, or 5. 100 is what part of 400? 1L= 4), or %. 6. If 25% of a number is 100, what is the whole number ? Hint. All of the number or 43° of it =—— x +45 of it, or —— x 100 = ——.. It is seen in the preceding examples that there are three general classes of problems in percentage, viz. : — I. The whole given to find a part. II. A part given to find its relation to the whole. III. A part given and its relation to the whole to find the whole. 151 182 PERCENTAGE Oral The Whole given to find a Part Find : — 1. 121% of 96 lb. 6. 331% of 24 hr. 2. 20% of 90 miles. 7. 30% of 200 acres. 3. 50% of 2000 lb. 8. 14% of 300 pupils. 4. 25% of $ 60. 9. 25% of 1200 votes. 5. 75% of 400 yd. 10. 6% of $3000. A Part given to find its Relation to the Whole 1. Compare 2 and 4; thus, 2 is 4 or 50% of 4; 4 is 2 times or 200% of 2. 2. Compare 3 and 6; 2 and 8; 3 and 12. . 5 is what part of 20? What % of it? %, of 48; $24 is what part of $36? What %? . 12 ounces is what part of a pound? What % ? 3 4. 161s 4, or 5 6 . 800 lb. is what part of a ton? How many 100ths of a ton, or % of it? ! 7. 16=what % of 40? 9. 96 = «7% of 144. 8. 48= what % of 64? 10. 35=%% of 105. A Part given and its Relation to the Whole to find the Whole 1. 4 of my age is 16 years. How old am I? 2. 50%, or 4, of my money is $80. How much have I? 3. ¢ of the price was $36. 4 of the price was 1 of $36; and 4, or the whole price, was x $ AOL 4. 6% of my salary is $72. 1% of my salary is $——, and my whole salary is 5. A whole flock is how many times 25% of it? If 25% ofa flock is 40, the whole flock is x 40, or 6. What is the relation of all of anything to 50% of it? To025% of it? To12}% of it? To 334% of it? To 662% of it? Oral Which are decimals ? it ? PER CENT AS EQUIVALENT TO FRACTIONS 25 = 0.256=4. Which are fractio Which is most anil used ? 1. 25 per cent = 25% = 133 Mise 2. 123% =4 of 25%=4 of f=1. 38. 61% =1 of %=4 of gh Ss 4. 371% = x 124% =——_ xl= 5. 50% =——;, 624% =50%+ %=t+ =—. 6. 15% =3 x I 3 874% = 15% + — = 3 + — = —_. 7. 84% = what fraction ? 12. 143 =F = 1. 8. 163% = —— x 81% = what? 13. 284% = what? 9. 412%= x 84% = — 14. 428% = what? 10. 584% =50%+ = —— 15. 574% = what? 11. 834% =75%+ = —. 16. 712% = what? 17. 912% =100% — 81% =1-—7,=—_. 18. 933% =100% — «2% =—— 19. Find 142% of 280 feet. hn 0 a then 142% of 280 feet = +-of 280 feet = what ? 20. What is 581% of $24? 55 of $24= $2. 21. If 162%, or 1, of my money is $30, how much have I? 22. What is the relation of all, or 100% of anything, to 374% of If 371% of it is 6 ft., what is all, or 100% of the length? The following fractions are used so often that we ought to know at sight that : — $= 50%. = $= 40%. = 124. oy = 5M. $=331%. $=60%. $= 37%. B=4%. $= 663%. 4=80%. $=62%. dy =3h%. $= 25%. $= 162%. f=8T%. B= 2%. $=75%. $=834%. ty =8h%. Hy = 2%. $= 20%. = $= 149%. te = 68%. = = OM. 23. What is 624% (or 8) of 40? Of 64? Of 100? 24. Subtract each per cent in the table from 100%. 134. PERCENTAGE Oral Determine whether you are to find all, a part, or the per cent that a part is of the whole, and then find : — 1. 623% of 72. 5. 121% of 96. 2. 124% of a gross. 6. 142% of 70. 3. 142% of 30. 7. 50% of $37.50. 4. 331% of a sq. yd. 8. 25% of $48.60. 9. 7 is what part of 28? What per cent of 28? What part of 49? What per cent ? 10. 13 doz. are 334% of doz. 14. 161s what % of 20? 1126 qt. are 621% of gal. 15. 15 is what % of 60? 12. 30 pk. are 662% of bu. 16. 20 is what % of 60? 13. $17 are 20% of $—_. 17. 75 is what % of 300? 18. 621% of $49.60 was 4 of A’s indebtedness. He paid 4 of the amount. What is the balance ? 19. A teacher pays $6 per week for board and room, which is 40% of her salary. What is her salary for a school year of 40 weeks ? 20. I pay for rent $750 a year, which is 331% of my income. What do I receive annually ? 21. A mechanic receiving $72 a month spent $60. What per cent did he save ? 7 | : 22. An agent collected a bill of $6000, and received $1500 com- mission. What was the rate per cent ? 23. A farmer sold 64 bu. of apples, 874% of which were of the first quality. How many were of the second quality ? 24. Bought 12 doz. buckets at 25¢ each. I wish to make 331% profit. How much must the marked price be per dozen ? 25. A merchant having $2000 paid $500 for a team. What per cent remained ? 26. A grocer received 60 bbl. of flour, and sold 12 of them the day they arrived. What per cent had he still ? Oral PERCENTAGE 135 What is given? What are you to find ? 1. 45 is 834% of ? 3. 5 oz. are 121% of lb. 2. 3is 2% of ? 4. 6 dimes are 60% of $—_. 5. What per cent of a bushel is 4 quarts ? 6. What per cent of $120 is $10? 7. Dick caught 80 fish in a week, and Tom caught 16. What per cent of the whole did Tom catch ? 8. Paid $75 for a watch, and sold it for $50. I lost #%. 9. A mechanic worked for $3.50 per day. His helper received 142% of $3.50 per day. What is the sum of their wages ? 10. A boy earned $1.50 in a week, 60% of that amount the next week, and 662% of it the third. How much did he receive in all ? 11. A certain locomotive ran 990 miles without repairs. Another ran 662% of this distance farther. How far did the second one go? 12. Find 50% of $6200. 15. Find 1% of 6200. 13. Find 25% of $5200. 16. Find 1% of 5200. 14. Find 75% of $1600. 17. Find 3% of 1600. What is the difference between : — 18. 162% of 480 acres of land, and 1 of 240 acres ? 19. 142% of $105 and 4 of $105? 20. A trader sold a horse for $175, which was 873% of its cost. How much did he gain or lose by the transaction ? 21. A number increased by 20% of itself is 72. What is it? 22. One dollar and twenty cents is 20% of what number ? 23. A merchant sold goods for $54 and lost 10%. The cost? 24. C wrote a check for $40, which was 31% of his bank balance. How much remained after the check was paid ? 25. A man’s expenses are $7.50 per week, which is 81% of his income from a small farm. What are the profits from the farm ? 136 PERCENTAGE Written 1. A farm that cost $5400 was sold for 75% of its value. What was the selling price ? B $ 5400 AY 0.75 3 gen $ 270.00 75% nae $ 5400 = $4050. 3780.0 $ 4050.00 Of the two methods, A and B, which seems preferable? Why? When can method A be used to advantage ? When will it be better to use method B? 5. 434% of my crop of 3290 bushels of wheat has been sold. How many bushels did I sell? (Which method? Why ?) 6. I paid $3400 for a house and sold it for 87% of what I paid. What did I get for it? (Which method? Why ?) 7. Of the $2700 paid for an estate, 121% was in cash and the remainder in notes. What was the cash payment? (Method ?) 8. Of 12,650 bushels of grain, 34% was in corn, 28% in oats, and the remainder in wheat. There were a bu. of corn, y bu. of oats, and z bu. of wheat. Explain the statement : — [100 % — (34% + 28%)] x 12,650 bu. =z bu. ee pe How much is : — Find a discount of : — 9. 25% of 3742 tons ? 12. 15% on 61 yards at $2.50. 10. 74% of 784 miles ? 13. 374% on a $558 piano. 11. 162% of 5733 acres ? 14. 183% on 42 tons at $6.50. 15. Compare ? of $400 with 3% of it. 16. Read: 0.003; 2%; nae Explain : A 17. What is Z of $64,000? 18. Find 2% of $64,000. 19. My property is assessed for $24,800. Tax rate 18%. My tax ? Written PERCENTAGE 137 1. I bought a house for $3600, and sold it at a gain of $540. What % did I gain? 0.15 = 15% EXPLANATION. Comparing the gain with the 360) 54 00 cost, the ratio of the gain to the cost = $4. This fraction eXpressed as a decimal is 0.15, or 15%. 36.00 Observe that the % one number is of another is their 18.00 ratio expressed as hundredths. 18.00 2. 6035 persons bought tickets toa fair. This was what per cent of the 8500 that attended ? 3. 625 pupils belong in the Lincoln school. 600 of them are present, or «% of the whole. 4. 37%, or 11,100 tons, of an ice crop remained unsold. There must have been # tons in the whole crop. 5. The cargo of the Sea Hing was valued at $38,475. The value of the cotton was 162% of the whole, that of the sugar 374%. The miscellaneous part of the cargo was valued at x dollars, or y% of the whole. Take the shortest method. 6. I sold my bicycle for $17. It cost me $25. I must have lost what: per cent of the cost ? 7. If I had lost but 15%, I should have sold it for what ? 8. 192 pages of a book of 432 pages are illustrated. That is what % of the whole ? 9. If a retail dealer has habilities amounting to $1125, and owns property amounting to $675, what per cent will he be able to pay his creditors ? 10. A man owned $175,000 worth of real estate in a certain city, but he has recently sold $145,000 worth of it. What per cent of it does he still own ? 11. A city inereases 24% in 10 years; that is from 37,860 popu- lation to a. 138 PERCENTAGE Written 1. I gained 15%, or $540, when I sold my house. Find the cost. aA x 8 54) = $ 3600. Expianation. The whole cost is 492 of 15% of O00 it, or the ratio of 100% of. et cost to 15% of it is 19 or 15)$ 54000 Hence the house cost 49° x $540. Jf it is not easy 45 to cancel the terms, the aiisiOn may be performed in — 90 the usual way. 90_ 00 2. $260.01 is 27% of $—. 5. $1406.25 is 45% of $—_. 3. $533.40 is 84% of $—. 6. 134.64 qt. = 51% of x gal. 4. $368.93 is 79% of $ —_. 7. 113.49 ft. = 39% of a yd. 8. The Indian population, according to the census of 1900, was about 145,000, which is only 58% of what it was in census of 1890. What was it in 1890, and what was the decrease, in round numbers ? 9. There are 2844 school children ina certain city. This is as of the population. What is the population ? 10. A man sold 20% of his interest in a mill for $38,000. Ashe owned 20% of the mill, the mill was worth $a. 11. About 50,000,000 sq. mi. (or 25%) of the earth’s surface is land and the remainder is water. What is the area of the water ? 12. Ina certain school 19 pupils are post graduates, 391 are regu- lar students, and the remaining 374% are students taking special courses. How many are enrolled ? 13. A man saved $12,597 in ten years, and this amount was 5% of his whole property. What was he worth at the end of that time? 14. A Western ranchman sent 381 cattle to the Chicago market, which was 60% of the number he sent to the Eastern dealers. He found he had sold in all 4 of his drove. How many had he at first ? 15. A man owed $470 in 1900, which was 40% of what he owed in 1898. How much did he owe in 1898 ? Oral and Written PERCENTAGE 1389 1. Which is more profitable, a gain of 3 per dozen, 5 per score, 25 per cent, or 36 per gross? Why? 2. Compare 2 of something with 2% of it. 3. Six wrong out of 24 problems solved is @ wrong out of a hundred, or 7%. 4. The Clevelands won 7 games in their series with the Pittsburg Club, the Pittsburgs won 4, and the one game was a tie. The win- ner’s per cent was z. 5. Thirty-six hits in 80 times is a batting average of 7%. 6. The center fielder has 80 chances, and makes 4 errors. His fielding average is 7%. 7. The crew pulled 36 strokes to the minute at starting, but fell off to 30 at the finish. This was a loss of what per cent ? 8. 9. 10. 2% of 600 = 2. £% of 800 = a. xe = 874% of 128. 2% of x = 2. $% of 2 = 12. “9% of 144 = 120. x% of 1200 = 8. x% of 200 = t. 834% bad and #% good. ie i ~eaky 25% of 52? 19 is %% 57. 25 is 4% of a. 35% of 400 ? 70 is w% 2100. 280 is 14% of a. 81% of 22°? 162 is x% 662. 8 is $% of a. 14. I paid 2% commission to my agent for selling a farm for $1250. How much money did he have left to send me? 15. 21% was paid a collector who earned $22.50 a month in this way. What were his annual collections ? 16. Of a farm of 320 acres 108 acres are given to wheat, 96 acres to oats, and the remainder to corn. What per cent of the farm are the cornfields ? 17. The frost destroyed 27 per cent of a crop of oranges, and only $1660 was realized. What was the loss? 140 PERCENTAGE: TABLE FOR PRACTICE The number of hundredths, or per cent, is sometimes called the The number of which a part is to be found is called the base, and the part of the base required, when found, is called. the percent- For brevity these terms are used in the following table: — rate. age. Find the value of x. a ovo F§ FY FY KF FF FP SYP EF ES oo mo nrt Oo OM Fr wo HO K OC Rate % Oral WRITTEN Base Percentage Percentage Base Rate % $9.30 x 1 32,45 mi. 2674 mi, x x 125 tons 2 170.40 ay 17% 75 yd. 183 yd. 3 36 yr. 130 yr. x x 57 4 184 A. by 454 9000 mi. o 5 4857.6 ft. 5280 ft. cy $0.50 $ 0.314 6 $5.76 ie 4 x 14 da. 7 $ 5400 $ 9600 i 14 tons ~ 8 x 1500 ed. 15 160- 1062 9 | 204 sq. ft. | 5200 sq. ft. | a x 15 sq. mi. | 10 6500 T. % 834 725 x 11 143 da. 365 da. x a century 8 mo. 12 | $13,651.56 $ 75,842 x 608 bu. © 13 5 36 x x 18 bales 14 84 x z 75 rd, 61 rd. 15 a 314 162 $ 12,000 x 16 328.8, a 94 x 35 cords 17 $ 349.06 $ 9006 x 726 gal. 605 gal 18 $18 x 4 $120,000 x 19 13 $8100 fs x 34 Ib 20. x 3 3 Written PERCENTAGE: BUSINESS PROBLEMS 141 1. Sold a house that cost $5000 at a profit of 80%. Proceeds of sale? 2. A merchant’s sales for January amounted to $28,000, but 12% was lost in bad debts. The net proceeds of the sales for the month were w dollars. 3. Gained $12, or 20% in selling a Century Dictionary. It cost me w dollars, and I sold it for y dollars. — 4. A sewing machine cost me $24. I sold it for $32. I gained x . | 32 — 24 Explain the statement: Ea x. Nore. The gain or loss is always reckoned on the cost. 5. A conductor’s wages are $72 amonth. They are reduced to $60. This is a cut down of #%. 72 —60 - A = x%p : 12 6. Cost, $8000; selling price, $6000; loss per cent, a. Explain the equation : 7. Cost, $6000; selling price, $8000; gain per cent, a. 8. Which is more profitable, to buy cloth for $5 and sell it for $9.50, or to buy for $4, and to sell it for $4.80 ? 9. Gas is reduced from $2 to $1.60 per 1000 cu. ft. How much do I save on $45 worth of gas ? : 10. Last winter my coal cost me $6 a ton. This winter, I pay $8. This is an increase of what per cent? 11. A man sold two city lots for $5600 each. On one he gained 1429, on the other he lost 124%. Find the loss or gain. 12. Bought silk at $1.75 a yard. I marked it to sell at a gain of 20%, but sold it at 334% less than the marked price. What per - cent did I gain or lose ? 13. 160 is 10% more than what ? 142 PERCENTAGE: PROBLEMS Written 1. Bought wood at $4 a cord and sold it at a gain of 20%. What did I sell it for ? (4) $44 20% of $4=a. (b) 120% of $4=2a. -. Explain the two statements. Which suggests the shorter solution ? 2. Sold a typewriting machine that cost me $80 at a loss of 10%. What did I receive for it? _ (a) $80 —10% of $80=%. (6) 90% of $80=—z2. Explain the two statements. Which is preferable? 3. Bought a house for $4800 and sold it at a gain of 16%. Find the selling price. Which method ? 4. Sold a dwelling house for $7500 at a profit of 25%. It cost me « dollars. 5. An epidemic decimated a southern village, leaving it with but 639 inhabitants. How many died? (What does decimate mean ?) 6. A farmer who owned 390 acres increased his farm 30% within 2 years. How much did he own at first ? 7. A speculator lost $3000 or 6% of his property. What was it then worth ? 8. A sold a yacht for $800 at a loss of 60%. Required its cost. 9. A piano that cost $450 was sold for $292.50. What was the per cent of loss ? 10. Some people feel that if the seller reduces his price, they are buying at a bargain. I wish to take advantage of their weakness by marking a certain line of goods which cost me $1.20 a yard so that I can fall 10% from the marked price and yet make a profit of aU _ What is the marked price ? 11. How shall I mark goods that cost me $2 so that I can sell at a discount of 20% and yet make 40% on the purchase ? At Sight REVIEW 143 1. In $3 what per cent is the numerator of the denominator ? In 2? 2. What is ;4, of a rod in feet? In inches ? 3. 231 cubic inches=1 gallon. Separate 231 into its prime factors. Give the dimensions of a tin pan that will hold a gallon. 4. What part of a year are the longest three months ? 5. What is 4% of 21,000? 6. 331% of 60% of 4 of the money remained. What part did the thieves take ? 7. My property is assessed for $2500. The rate of taxation is 21%. What is my tax ? 8. What per cent of the surface of a 4-inch cube is on five sides of it? 9. Bought thread for 4 cents a spool and gained 300%. It sold for « cents. | 10. 81% of a yard = 7% of a foot. 11. Three sides of a square = 7% of its perimeter. 12. «% of the day has passed at 9 P.M. 13. (£4+30% + 3) of 64 is 25% of a. 14. 16 is 2 of x and 2% of y. 15. Gave $24 to James and $30 to Lucy. Lucy had #% more than James, and he had y% less than Lucy. 16. Paid the price of a pound for 14 ounces. I thus lost 7%. 17. V9 =2% of V144. 18. In aseries of ball games the Alphas won 40% and the Omegas 50%. Two games were drawn. How many were played ? 19. 4a mile is what per cent of two leagues ? 144 PERCENTAGE: BUSINESS PROBLEMS Oral, Written 1. $12, or 121%, (of cost) is gained; cost = $a. 2. $8, or 162%, is lost; cost = $a. 3. $24=cost; 334% is gained; selling price = $a. 4. $35 = cost; 142% is lost; selling price = $2. 5. $36 = selling price, which includes the cost and a gain of 20% of the cost. $36 = cost + 4 of cost, or $ cost; $36 is $ of Ha. 6. $28 =selling price, which is the cost less a loss of 20%. What part of the cost is the loss? The selling price is of the cost; $28 is - of $y. 7. Bought a bicycle for $80 and sold it for $100. My gain per cent was «. 8. If I had sold it for $60 I should have lost $y, or w per cent of cost. 9. An importer bought silk at $2.50 a yard and sold it to a retailer for $3, who sold it to the wearer for $3.50. What per cent of profit did each make ? 10. Sold a watch for $119 and gained 162%. How much should I have gained or lost if I had sold it for $100 ? 11. A thrifty clerk resolves to live on 60% of his salary. He spends $48 more than he intends, but still saves $300. What was his salary ? 12. I paid $125 for what I thought was 4-foot wood. It proved to be but 45 inches long. What deduction should be made in the settlement ? | 13. Sold telephone stock for $25,000 at an advance of 25% on what I paid for it. What did I gain? 14. I purchased a patent for $8000. The seller lost 84% of its original value. What was its original value ? Wratten PERCENTAGE: BUSINESS, PROBLEMS 145 1. Which is more profitable, buying meat at 16¢ and selling at 19¢, or selling potatoes at 64¢ that cost me 56¢? 2. Butter sold at 28¢ yields no profit. What would be gained on $140 worth sold at 30¢? 3. Milk bought at 20¢ a gallon is sold at 8¢ a quart. What per cent is gained if 25% of the quantity bought spoils ? 4. A 5% increase in wages means $200 more a month for the employer to pay. What was his annual pay roll before and after the increase ? 5. Mr. H. earns $1200 a year selling carriages at 15% commis- sion, all expenses paid. The manufacturer makes a net profit of 142%. If 50 carriages are sold, what is their average cost ? 6. A dishonest dealer buys 50 gallons of alcohol at $2.50 a gallon, adds 14 gallons of water, and sells the mixture at 10% below actual cost. What per cent does he gain ? 7.. I am offered a 10% discount on a suit of clothes marked to sell at $60. I know that even then the dealer will make 121%. I offer $50 and get the suit. What per cent does the dealer gain ? 8. I sell 2 of a lot of land at % the cost and get $200 for the remainder. The original cost being $1200, what is my per cent of loss ? 9. What per cent is gained by selling coal at the rate of 4 of a ton for what 1000 pounds cost ? 10. A farmer’s sheep cost him $200. One out of every seven dies and he sells those that remain for $275. What was the gain per cent, the cost of keeping being $40? 11. A merchant sold a stock of goods for $3042 and gained 17%. What per cent would he have gained or lost had he sold it for $2392 ? } 12. For what should he have sold it to gain 100%? 13. I bought a $6 umbrella at 162% discount. The dealer made 25% profit. What did it cost him ? 146 INTEREST Oral 1. If I pay 6¢ for a year’s use of a borrowed dollar, what is the rate of interest ? 2. What does the expression “6 per cent interest”? mean ? 3. At 6%, what is a year’s interest of $300? 4, What part of a year is 2 months? If the interest for 1 year is $18, what should it be for 2 months ? 5. What is the interest of $400 for 1 year and 6 months at 5%? 6. At 7%, what is the interest of $200 for 2 years, 6 months ? 7. What is the first step in finding the interest of any principal ? The second step ? 8. Explain: 21 x 0.07 x $200 = $ a. This is called the general method of finding interest. It may always be used, but the work may be shortened by some of the special methods. At 6% the interest of any principal for : — 12 months = 6% of it. 2months= 1% of it. (Why?) 20 months = 10% of it. (Why ?) 200 months = 100% of it, or the principal itself. 9. Find the interest at 6% of $380 for 2 yr. 7 mo. (81 mo.). PROCESS Interest for 20 months = $38.00 Observe that the time was so Interest for 10 months = $19.00 separated as to avoid multiplying Interest for 1 month 1.90 by anything except 10. Interest for 31 months = $58.90 10. Into what convenient parts would you separate the time if it were 26 mo.? 387mo.? 3yr.7mo.? Syr.8mo.? 3yr. 11 mo.? lyr.7mo.? 8 yr. 4 mo. ? Written INTEREST: RATE 6% 147 Explain the following process of finding the interest at 6% :— TV.) Of $7 Zosforss yriikl mo. 2. Of $278 for 1 yr. 7 mo. Interest for 47 mo. of $725. Interest for 19 mo. of $278. [ 20 mo. = $72.50 (20mo.= 27.80 |20mo.= 72.50 Int. for { 1mo.=__ 1.39 ah eOr 4 wo: 0, eee Sal (19 mo. = $26.41 | 2mo.= 7.25 [47 mo. = $170.375 Find the interest at 6% :— 3. Of $280 for 2 yr. 8 mo. 7. Of $649 for 7 yr. 8 mo. 4. Of $640 for 3 yr. 7 mo. 8. Of $750 for 8 yr. 4 mo. 5. Of $95 for 4 yr. 11 mo. 9. Of $ 295.75 for 5 yr. 11 mo. 6. Of $73.50 for 1 yr. 4 mo. 10. Of $641.86 for 3 yr. 3 mo. TIME IN DAYS: INTEREST 6% Oral and Written 1. How many days in an interest month? In an interest year? 2. 60 days is what part of a year ? 3. Since at 6% the interest for a year is 6% of the principal, the interest for 60 days 1s #% of the principal. 4. The interest for 6 days is what part of the interest for 60 days ? At 6% the interest of any principal for : — 60 days =1%, or zr, of tt. 6 days = 0.1%, or zp Of tt. 5. Find the interest at 6% of $720 for 75 days. PROCESS Interest for 60 days = $7.20 6. How was 15 days’ in- Interest for 15 days = _ 1.80 terest found from 60 days’ Interest for 75 days = $9.00 interest being known ? 148 INTEREST: BANKERS’ METHOD Written Explain the process of finding the interest at 6% :— 7. Of $196 for 115 days. 8. Of $119 for 89 days. Int. for 115 da. of $196. Int. for 89 da. of $119. ( 60 da. = $1.96 ( 60 da. = $1.19 30 da.= 0.98 20 da. = 0.3966+ int. for-3 20 ‘da, —"— 06538 Int. for 4 6da.= 0.1190 | 5da.= _ 0.1633+ B8da.= 0.0595 [115 da. = $ 3.7566 89 da. = $ 1.7651 What shall I pay at 6% for the useof:— Find the interest at 6% : — 9. $780 for 67 da. ? 14. $94 for 200 da. 10. $640 for 98 da. ? 15. $762 for 5 mo. 14 da. 11. $92 for 3 mo. 12 da. ? 16. $815 for 86 da. 12. $87.50 for 117 da. 7? 17. $924 for 8 mo. 11 da. 13. $106 for 2 mo. 17 da. ? 18. $17.84 for 17 da. In distinction from the general method, this is sometimes known ‘as the bankers’ method. INTEREST AT ANY RATE: BANKERS’ METHOD 1. If the interest at 8% = $18, the interest on the same sum and for the same time at 1% =a. At5d%=5 x $4a=—_. 2. If the interest at 6% = $42, the interest at 7% = $a. To find the interest of $105 for 75 da. at 5%. PRocEsS 3. Show how the in- 6% int. for 75 da. of $105. terest at 6% is found. 6% int. for 60 da. = $1.05 4. At1%; at 5%. 6% int. for 15 da.= 0.2625 5. What if the rate had 6)$1.3125 =6% int. been7%? 10%? 12%? 0.2187+=1% int. $1.0988 =5% int. 2 Written INTEREST 149 Find the interest of : — 6. $640 forl yr. 8mo.at1%. (4 of 6%.) 7. $270 for 3 yr. 10 mo. at14%. (4 of 6%.) 8. $382 for 1 yr. 9 mo. at 2%. (4 of 6%.) 9. $927 for 6 mo. 4da. at 3%. (4 of 6%.) 10. $864 for 2 mo. 7 da. at 4%. (6% — 2%.) _ 11. $318 for 1 mo, 13 da. at 5%. (6% —1%.) 12. $725 for 29 da. at 7%. (6% +1%.) 13. $649 for 67 da. at 74%. (6% + 11%.) 14. $84 for 54 da. at 8%. (6% + 2%.) THE ONE DOLLAR METHOD _1. What two methods of computing interest have previously been resented ? 2. In which one did you first find 6% interest ? Note. 11 3.00. 16:0 027) 2. 10.35. Toe 12. 0.7854. ab oats? oi 3. 6.43. 8. 8.56. 13. 391. 18. 348. 4. 9.8. 9. 9.42. 14. 74. 19. 64.01 Bact. 10. 0.7. 15. 0.360. 20. 36 rs] , 918 APPLICATIONS OF SQUARE ROOT Oral 1. Draw a right triangle with the sides which form the right angle, 3 inches and 4 inches respectively. 2. Measure the length of the other side, or hypotenuse. 3. Draw a square on each of the three sides as base. 4. Compare the square on the hypotenuse with the sum of the squares on the other sides. Pythagoras proved about 500 s.c. that the fact that we find true here is true for any right triangle, viz. that The square on the hypotenuse is equal to the sum of the squares on the other two sides. 5. Carpenters make use of this fact, in laying out the foundation. for a building, when they want to form a right angle. A line 8 feet long is taken in one direction along which the foundation is to be made. Another line 6 feet long is fastened to one extremity of the first line and moved until a 10-foot rod will just reach the outer extremity of the two lines. Draw such a figure, and show that this gives a right triangle. 6. Use the test in 5, and find whether the walls of your school- room are perpendicular to the floor. 7. If the square on the hypotenuse is 100 sq. in. and on one of the sides 36 sq. in., what is the length of each side of the triangle ? Denoting the hypotenuse by H, the base by B, and the perpendicu- lar by P, when these are abstract numbers representing the number of units in the dimensions, we may state from the above principle the following formule : — H=VJVB? +P? 8. Explain the formule. B=V ER? ~ P? 9. 1f Hf asloeand) Pe peer P=VH?— B 10.) iB bands P= Ome eae 11)\ If = 25/ and B= 20a Written APPLICATIONS OF SQUARE ROOT 219 The Right Triangle The truth of the Pythagorean theorem, stated on the preceding page, may be seen by drawing, or cutting from cardboard, figures like the following : — Let ABC be the right triangle. The square on the PRUNE AC is equal to the 4 triangles, 1, 2,3, and 4, and the small square, 5 Now put 1 and 2 in the position of the figure at the right, and ihe figure is equal to a square on AB and one on CB’. 1. The base of a right triangle is 48 feet and the perpendicular is 36 feet. What is the hypotenuse ? 2. The hypotenuse is 85 feet and the perpendicular is 51 feet. What is the base ? 38. The base is 76 feet and the hypotenuse is 95 feet. What is ‘the perpendicular ? 4. What is the diagonal of a rectangle 92 ft. long and 69 ft. wide ? 5. What is the diagonal of a 30-ft. square ? 6. What is the longest line that can be drawn on a sheet of paper 16 inches wide and 20 inches long? 7. What is the diameter of the largest wheel that can be got through a doorway measuring 7 feet by 5? 8. What is the distance between the opposite corners of a field 200 rods long and half as wide ? 220 APPLICATIONS OF SQUARE ROOT Written ISOSCELES TRIANGLE EQUILATERAL TRIANGLE Prove by cutting or measuring that — (1) The altitude of an isosceles triangle bisects the base. (2) The perpendicular from any vertex of an Bar Re triangle to the opposite side bisects it. Since an equilateral triangle is also isosceles whatever side is taken as base, (2) could have been inferred from (1). 1. If the base of an isosceles triangle is 12 and the equal sides 10, what is the altitude ? 2. Find the altitude of a triangle whose sides are each 10 inches. 3. A rectangle measures 22 ft. by 10 ft. How long is its diagonal ? 4. The foot of a 25-foot ladder is 12 ft. from the side of the house against which it leans. How far from the ground is its top? 5. What is the area of a right triangle whose longest side is 20 ft. and its shortest 8 ft. ? 6. Find the altitude of an equilateral triangle whose side measures 24 ft. Find the area. 7. What will it cost to fence a square field containing 5 A. at $1.25 a rod? 8. A regular hexagon is made up of six equi- lateral triangles. Study the figure and discover how to inscribe one in a circle. A REGULAR HEXAGON 9. Find the area of a regular hexagon whose sides are each 10 inches. Written APPLICATIONS OF SQUARE ROOT ZA Remember that we cannot take the square root of a concrete num- ber, such as 25 sq. ft., but of 25. In all the formule that follow, we are to consider areas, lengths, etc., as the nwmber of units, and hence deal with abstract numbers. 1. Since the area of a circle = 77”, or 3.1416 x the square of the radius, eran What is the radius of a circle whose area is T 78.54 sq. in. ? 2. What is the diagonal of a floor 25 feet long and 16 feet wide ? 3. The diagonal of a rectangle 30 feet long measures 42 feet. What is the width of the rectangle ? 4. The perimeter of a rectangle is 36 feet. Its width is half its length. What is its diagonal ? 5. What is the diagonal of a square containing 32 square inches ? 6. The top of a 30-foot ladder, which is placed 16 feet from the side of a house, reaches a window sill in the third story. How far from the ground to the window sill ? 7. Two yachts start together. One sails due north and the other due east, each at the rate of 12 miles an hour. How far apart are they at the end of 4 hours ? % 8. What is the area of a circle drawn with an 18-inch radius ? 9. The area of a circle is 24.3474 sq. in. What is its diameter ? 10. A line reaching from the bank of a stream to the top of a 50-foot pole on the other side is 275 feet long. What is the width of the stream ? , 11. The base of an isosceles triangle is 20 feet and its altitude 15 feet. What is the length of one of the equal sides ? 12. A gable-roof house is 24 feet wide. The distance from the plate to the ridgepole is 12 feet. ‘The rafters project 1 foot over the eaves. How long are they ? 222 APPLICATIONS OF SQUARE ROOT Written 1. What is the length of a square equal in area to a rectangle 24 rd. long and 33 ft. wide? 2. What is the longest straight line that can be drawn on the ceiling of your schoolroom if it measures 32 ft. by 30 ft. ? 3. Compare the perimeter of a rectangle 48 in. by 12 in. with that of a square of equal area. 4. How much do I save by crossing along the diagonal of a square that contains 1296 sq. rd. instead of going around its two sides ? 5. How long is an acre of land in the form of a square ? 6. How long a guy will support a derrick 48 ft. high if fastened 85 ft. from its base ? 7. The hypotenuse of a right triangle measures 90 ft. The other sides are equal.- How long are they ? 8. Two poles are 100 ft. apart. One is 60 ft. high, and the other 80 ft. How long a line will connect their tops ? 9. A rectangle 3 times as long as wide contains 3888 sq. ft. What are the dimensions? (Hint. Divide it into 3 squares.) 10. What is the altitude of an equilateral triangle whose base is 24 feetr? .11. The base of an isosceles triangle is 84 feet, and one of the other sides is 50 feet. What is the altitude ? 12. From the corner of a 12-inch square I cut an isosceles triangle one of whose sides is 4 inches. What is the area of that part of the large square which remains ? 13. My son’s kite is 1500 feet directly above the spot on which I am standing, and my son holds the string 1800 feet away. How much string has he let out? Allow 25 ft. for sagging. 14. How many rods do I save by taking the diagonal of a field 75 rods wide and 200 rods long, instead of going around the corner ? Oral PRISMS AND PYRAMIDS Hadas) C SC | ey a D A RECTANGULAR TRIANGULAR RECTANGULAR TRIANGULAR PRISM PRISM PYRAMID PYRAMID A Pyramid is a solid whose base is a polygon and whose sides or faces are triangles meeting at a common point called the vertex of the pyramid. If the base is a regular polygon, as a square, or ‘an equilateral triangle, and the sides are equal isosceles triangles, the pyramid is a regular pyramid. The distance from the vertex to any side of the base of a regular pyramid is the slant height. 1. The distance from the vertex to the side is the altitude of the triangle, hence it divides the side into two equal parts. Why ? 2. Construct from cardboard a pyra- mid whose base is a 4-inch square, and whose edges AC, etc., are 6 inches. Draw a model, and leave lapels for pasting. Written 3. What is the length CD, or the slant height of this pyramid ? 4. Having found CD, and knowing OD, observe the figure at the top of the page, and find the height. (Observe that the altitude of a regular pyramid meets the base at its center.) 5. How could you have found AO and then the height from OA and AC’? Find it. 994 PRISMS AND PYRAMIDS Oral 1. Make a prism having exactly the same base and altitude as the pyramid you have made. Test the accuracy of your construction by measuring, as in the figure. Make an opening in the base of the pyramid, and fill with dry sand, and fill the prism from this as a measure. 2. What do you find true of their volumes ? 3. Make other prisms and pyramids as your teacher may direct, and test the accuracy of the following : — The volume of a pyramid is 4 of that of a prism having an equal base and an equal altitude. 4. A square pyramid is 12 ft. high and measures 3 ft. along one side of its base. What is its volume ? 5. What part of a square prism is whittled away by a boy who is making the largest pyramid possible out of it ? 6. The volume of a square prism is 86 cubic inches. What is the volume of a square pyramid of the same base and altitude ? 7. The area of the base of a triangular prism is 4 square feet. Its altitude is 5 feet. What is its volume ? 8. The contents of a square prism are 28 cubic feet. Its base covers 4 square feet. What is its altitude ? 9. The contents of an hexagonal prism are 42 cubic inches. The altitude is 6 inches. What is the area of the base ? Oral and Written PRISMS AND PYRAMIDS 995 1. A granite shaft 10 ft. high and 20 in. square is surmounted by a square pyramid 2 ft. in altitude. The contents of both ? 2. Which is more easily measured, the slant height or the alti- tude of a pyramid? Which line of a triangle is the slant height of a regular pyramid ? 3. The slant height of a square pyramid is 15 inches, and the side of the base 10 inches. [ind its contents. 4. The area of the base of a pentagonal prism is 624 sq. in.; its altitude is 24 in.; the contents ? 5. A square prism has a base 2 feet long and an altitude of 10 feet. It is made of granite weighing 165 pounds to the cubic foot. What is the weight of the prism ? 6. How many surfaces has a square prism? What is the shape of each one? 7. A hexagonal pyramid is one having a hexagon as base. How many triangles make its visible surface? Of what kind? 8. What two lines in an isosceles triangle must be known in order to find its area? 9. What is meant by the slant height of a regular pyramid ? The total area of all the faces of a prism is called its convex surface. 10. A square pyramid has a slant height of 12 inches and a base of 4 inches. What is its convex surface ? 11. An octagonal pyramid’s slant is 16 inches, and the perimeter of its base is 48 inches. What is its convex surface ? 12. All four sides of a pyramid are equilateral triangles 6 inches long. Find the slant height and the convex surface. 13. The altitude of a square pyramid is 15 in. and the side of its base is 12 in. Required its contents. 14. Find its slant height and convex surface. 226 CYLINDERS AND CONES Oral A solid having a circle for a base and tapering uniformly to a vertex is a cone. AC is the slant height. While there are other kinds of cones, we shall consider the kind described above. In this kind the altitude from the vertex passes through the center of the base. ) 1. How can you find the altitude when the slant height is known? 2. Make a cone whose base is a circle, whose radius is 2 inches, and whose slant height is 6 inches. Make a model as in the margin. 8. How long is the arc CB? What is its radius ? 4. What will be the height of the cone ? 5. Make a model for a cyl- inder of the same dimensions. 6. What will be the size and shape of the convex surface ? 7. Test the accuracy of your construction by measuring as shown in the figure at the top of the page. 8. Using the cone as a measure, fill the cylinder with dry sand. 9. How do their volumes compare ? 10. Make other sizes and show that — The volume of a cone is equal to 4 of that of a cylinder having an equal base and the same height. 11. How do we find the volume of a cylinder ? Oral and Written CYLINDERS AND CONES 227 1. If a cylinder weighs 3 pounds, what will be the weight of a cone of the same material having the same base and altitude ? 2. The largest possible cone is turned in a lathe out of a cylinder 6 inches long and 3 inches in diameter. What part of the cylinder goes into shavings? How many cubic inches in the cone ? 3. The base of a cone is 6 square inches and its altitude is 12 inches. Find its contents. 4. The contents of a cone are 24 cubic inches. The altitude is 12 inches. What is the area of the base ? 5. The diameter of the base of a cone is 4 feet. Its altitude is 9 feet. What are its contents ? 6. A cylinder of ebony weighs 1 lb. 8 0z. What will an ebony cone of the same base and altitude weigh ? 7. What is meant by a sector? How many lines in its boundary ? 8. Show that the are of a sector multiphed by 4 of the radius will give the area of the sector. Compare the method with that of finding the area of a circle. 9. The convex surface of a cone is a sector, the circumference of the base being the are of the sector, and the slant height of the cone its radius. The circumference of the base is 8 inches, and its slant height 10 inches. Area of its convex surface ? 10. What is the convex surface of a cone whose altitude is 4 feet, and the diameter of whose base is 3 feet ? 11. The radius of the base of a cone is 34 feet. The slant height is 5 feet. Find the entire area. (The area of the base + area of the convex surface. ) 12. What is the convex surface of a cylinder 7 feet in diameter and 10 feet high ? 13. How many yards of cloth will be required to make a conical tent 12 feet in diameter and 15 feet high? Add 5% for seams. 228 SURFACE OF SPHERES Oral 1. Cut away any slice of a sphere, as an apple. What is the form thus exposed ? 2. When a sphere is bisected, that is, divided into two hemispheres, the plane sur- faces thus exposed are great circles of the sphere. Would the diameter and circumfer- ence of one of these circles be the diameter and circumference of the sphere ? 3. Wind the surface of a hemisphere as in figure C with a hard, waxed cord. (Place a small tack in the center, and around this wind the cord.) 4. With the same cord wind a great circle as in figure D. 5. Compare the two lengths, and thus the two surfaces. 6. Using cords of different sizes, and if con- venient several sizes of spheres, show that — The surface of a hemisphere is twice that of a great circle. 7. Compare the surface of a sphere with that of a great circle. 8. What is the area of a great circle of a sphere whose radius is 4 inches? What then is the surface of the sphere ? Remember then that — The surface of a sphere =47m or 7d’. (ry = radius; d = diameter.) 9. Show that 4 7? = qd? 10. What is the surface of a sphere whose diameter is 10 inches ? ‘ f J el if i « | YZ oy) ‘ 11. When the radius is 4 inches, what is the surface of a sphere ? Oral VOLUME OF SPHERES 229 1. Ifa sphere should be dissected as in the accompanying illustra- tion, what solids would its parts most resemble ? | 2. What line in the sphere forms the altitude of each pyramid- like solid ? 3. What forms the base of each ? 4. Taken together, what will the bases of all the pyramid-like solids make ? : 5. If these were perfect pyramids, how would the volume of any one be found ? ‘ . While these solids are not pyramids, for their bases are not plane figures, yet it is proven in geometry that — The volume of a sphere is the same as that of a pyramid whose base is the surface of the sphere and whose height is the radius of the sphere. 6. What should we obtain by multiplying the surface of a sphere by 4 of its radius ? 7. The surface of a sphere is 113 sq. in., and its radius 3 in. What are its contents ? 8. How is the surface of a sphere found? 4 of the radius is what part of the diameter ? 9. Read and explain the following : — 2x 4 REX 314164 BP x 3.1116, or 4 x R'= volume of a sphere; or 2 pat Dae a eA i x 3.1416, or a = volume of a sphere. 230 SURFACE AND VOLUMES OF SPHERES Written 1. What part of a 2-inch cube is a 2-inch sphere ? 2. If a sphere 3 inches in diameter is carefully turned out of a 3-inch cube, what part of the cube will go into shavings, and what | part will remain in the sphere ? | 3. If asphere is 0.5236 of a cube of the same diameter, what will be the contents of a sphere 4 inches in diameter ? Compare 0.5236 Withce 6 4. If a cubic foot of iron weighs 450 pounds, what is the weight of an iron sphere 12 inches in diameter ? 5. A cubic foot of ivory weighs 114 pounds. What is the weight of a set of 4 billiard balls 2 inches in diameter ? 6. How many cubic miles in the moon if we call its diameter 2000 miles ? 7. If we call the diameter of the earth exactly 8000 miles, how many moons will be equal in volume to the earth? Shorten your work by cancellation. 8. Two 4-inch spheres are dropped into a pail even full of water and holding 864 cubic inches. How many cubic inches of water are displaced ? 9. Find the square inches in the surface of a 6-inch sphere. 10. How many square miles in the surface of the moon? Call its diameter 2000 miles. 11. A cubic foot of water weighs 1000 ounces, and gold is about 19 times as heavy. What will a sphere of gold 3 inches in diameter weigh ? ) . 12. If it costs $3 to gild a 3-inch ball, what will it cost to gild a 4-inch ball? Shorten your work by cancellation. 13. If a ball 2 inches in diameter weighs 18 oz., what will a ball of the same material, and 6 inches in diameter, weigh ? Oral MENSURATION: SIMILAR FIGURES VoL 1. What is the ratio of a 2-inch square to a 4-inch square ? 2. What is the ratio of a 2-inch square to a 5-inch square ? 3. What is the ratio of a 2-inch circle to a 4-inch circle? What common factors enter into the areas of any two given circles ? 4. Compare a 3-inch circle with a 5-inch circle. 5. Explain the statement : — A 3-inch circle: a 5-inch circle = (3)? x 3.1416: (8)? x 3.1416, or a 3-inch circle: a 5-inch circle = 37: 5”. 6. Ifa rectangle is 2 by 3, what will be the length of another of the same form or shape that is 4 wide? 6 wide? 12 wide? 7. Two rectangles whose corresponding sides have the same ratio are similar. All surfaces of the same form or shape are similar surfaces. Are all squares similar? All circles? All equi- lateral triangles ? 8. Compare a rectangle 2 by 3 with one 4 by 6. ~ 9. Compare a rectangle 3 by 6 with one 5 by 10. 10. Are the rectangles in Exs. 8 and 9 similar? Why ? 11. Observe that in the squares, circles, and rectangles that you have tried, their areas have the same ratio as the squares of their corresponding lines. Try other figures that you know to be similar, and see whether this is true : — Similar surfaces have the same ratio as the squares of the ratios of their corresponding lines. 12. Compare two equilateral triangles whose sides are respec- tively 5 inches and 5 inches. 13. Are the surfaces of all spheres similar surfaces? If it costs 50 ¢ to gild a 3-inch sphere, what will it cost to gild a 9-inch sphere? 14. Ifa lot 60 ft. square costs $300, what will one 120 ft. square cost at the same rate ? * Lay MEASUREMENTS: SIMILAR SOLIDS Oral 1. What is the volume of a 2-inch cube? Of a 4-inch cube? 2. Compare a 2-inch cube with a 4-inch cube. What is the ratio of 2 to 4? 3. What is the volume of a 4-inch sphere? Explain: ¢ x 2° x 3.1416. 4. What two factors are common to the volume of any two spheres ? 5. Compare a 3-inch with a 5-inch sphere. Explain : — 4x (8)8 x 3.1416: 4 x @)® x 3.1416 = 3°: 5° = Ae. 6. What is the ratio of the diameters in Ex.5? What was the ratio of their volumes ? | 7. Compare a 38-inch cube with a 5-inch cube as to their edges and volumes. 8. Solids that have the same shape are similar solids. Name some similar solids. 9. In the preceding problems, what did you notice to be the relation of volumes of similar solids to their like lines ? 10. Take other solids that you know to be similar, and see whether you find this to be true : — Similar solids have the same ratio as the cubes of the ratios of their corresponding lines. 11. The ratio of a 4-inch cube to a 12-inch cube is (4)’, or a. 12. What is the ratio of a 4-inch sphere to a 12-inch sphere ? 13. Of a 6Linch cube to a 124-inch cube ? 14. Of a 61-inch sphere to a 123-inch sphere? 15. Of a 162-inch sphere to a 331-inch sphere ? 16. Ifa 23-inch sphere weighs 3 lb., what will a 5-inch one of the same material weigh ? 17. How many 14-inch cubes can you put into a box 6 inches each way ? 18. How many }-inch balls can be molded from a 6-inch ball ? Oral and Written MEASUREMENTS 250 Similar Surfaces and Volumes 1. The ratio of the flow of water through two pipes depends upon the area of the cross section of the two pipes. Compare the flow through a 4-inch nozzle and a 2-inch nozzle. 2. If two triangles are similar, their corresponding sides and altitudes all have the same ratio. If the ratio of the altitude of two triangles is as 3 to 9, or 4, what is the ratio of their areas? 3. If a square lot 60 ft. long costs $300, what will one of the same shape 3 times as long cost ? 4, Compare the strength of a rope 2 in. round with that of one 3 in. round. ; 5. If it costs $8.25 to gild a sphere 20 inches in circumference, what will it cost to gild one 30 inches ? 6. A %-inch faucet fills a tank in 20 minutes; a 4-in. faucet will fill it in @ min. 7. The ratio of two similar triangles is 25; the ratio of their altitudes is a. 8. To paint a conical steeple 30 ft. high costs $35; to paint a similar one 45 ft. high costs, at the same rate, $2. 9. A conical tent 16 ft. slant height costs $13.50; one of the same shape measuring 4 ft. more would cost $a. 10. If a 2-inch sphere weighs 1 lb., how much will a 6-inch sphere weigh ? 11. If a rectangular bin 5 ft. long contains 75 bu. of oats, how many bushels will a similar bin 124 ft. long contain ? 12. Itrequires 90 min. to fill a cylindrical tank 3+ ft. in diameter. At the same rate, how many minutes will be required to fill a simi- lar tank 14 ft. in diameter ? 234 SIMILAR TRIANGLES Oral 1. Describe similar surfaces. 2. Make two similar triangles. Are their corresponding angles equal ? 3. Draw atriangle, ABC, as in the figure. C Draw EF parallel to AB. By the use of a protractor compare the angles FHC and BAC. Also angles CFE and CBA. E F 4. Are triangles ABC and EFC simular; that is, do they have the same shape ? 5. Make a triangle in which AC is 6 inches and CB 3. Mark off CE equal to 4 inches and CF 2 inches. Are the two triangles similar ? 6. What is the ratio of CA to CH? Of CB to CF? 7. Cut similar triangles from cardboard. Measure their sides and discover that — B In similar triangles the ratios of the corresponding sides are equal, and the ratio of any two sides of one is equal to the ratio of the corre- sponding sides of the other. . 8. Inaccessible distances may be found by the principle of similar triangles. Sup- pose we are to find the distance AB across a small lake. By measuring from A to C, and from B through O to D, making the ratio of OC to OA the same as of OD to OB, we have similar triangles. If OC =10A, and CD measures 20 rods, what is AB? Note. Make OC any convenient part of OA and then OD the same part of OB. In the figure OC = 3 OA, and OD=3 OB. 9. When a vertical rod 6 feet high casts a shadow 9 feet long, a tree casts a shadow 150 feet long. How high is the tree? Oral SIMILAR TRIANGLES 235 1. A boy, wishing to find the height of a pole CH, made a piece of apparatus which he called his “surveying instrument.” It con- sisted of a right triangle whose two legs were equal. It stood 3 feet from the ground. He moved it along until the point C could just be seen along the hypotenuse of the triangle when the base of : = the triangle AF was parallel with = = : = the ground. A line with a weight (a plumb line) hung from A. If DE was 27 feet, how high was the pole? (Triangles A4F'H and ABC are similar. Why ?) 2. If HF had been twice AF, and DE had been 40 feet, what would CE have been ? 3. Make such an instrument, and find the height of trees, tele- graph poles, ete. 4. Make one with the triangle having one leg twice the other, and measure the same heights. Do your results check ? 5. Two triangles are similar. One has sides 4, 5, and 7 inches respectively. The long side of the other is 21 inches. What are the other sides? What if the short side of the latter were 2 inches ? 6. I wish to find the distance AB. AC is 15 rods and OC is 5. I measure from B through O to D. If BO is 8 rods, what shall Imake OD? Why? I measure DC, and find it to be 74 rods. How far from A to B? 7. In this way measure distances on your school lot. 236 LAND MEASURE: TOWNSHIP AND SECTION Written 1. Government lands are divided by parallels and meridians into townships six miles square, containing 36 sections or square miles. Each section is divided into half sections and quarter sections. How many acres in a section? In a quarter section ? 2. A township is designated by its number north or south of a base line running east and west, and east or west of a principal meridian running north and south. Thus, Cis Township 4 N., Range 3 H. WhatisA? Whatis B? 3. The 36 sections into which a township is divided are numbered as in the accompanying figure. Point out section 15. 4. Half and quarter sections are designated as W. or N. half sections, etc.; and §8.W. or N.E. quarter sections, etc. How many acres in 8. $ 8.W. 4 Sec. 15? Township. Section 15. Surveyors generally use, in measuring land, a steel chain 100 ft. long, divided into foot links, or a steel tape line of the same length graduated to feet and tenths. Sometimes a Gunter’s Chain is used. It contains 100 links, each 7.92 in. long. The chain is 4 rods, or 66 ft., or 792 in. in length. 80 chains, or 320 rods, measure a mile. 5. How many square rods in a square chain ? 6. How many square chains make 160 sq. rds, or 1 acre? 7. How many acres in 8.W. 4 N.E.4 Sec. 27? Draw diagram and locate it. Written MENSURATION: REVIEW Yai 1. Find the cost of plastering the walls and ceiling of a room 18 ft. long, 15 ft. wide, and 9 ft. high, allowing 4 of the area for openings and wood-covered portions. Price 124¢ per square yard. 2. Find the area of a right triangle, base 25 ft., hypotenuse 60 ft. 3. A rhomboidal field contains 5 acres, and measures 50 rods along a straight road. How wide is it? 4. Name the six quadrilaterals and the four parallelograms. Draw a trapezoid and its equivalent rhomboid and rectangle. 5. Required the area of the entire surface of a stick of timber 18 ft. long, 4 in. thick, 8 in. wide at one end, 12 in. at the other. 6. The diagonal of a trapezium is 221 ft., and the perpendiculars lrawn from the vertices of the angles opposite it are 16 ft. and 12 ft. respectively. What is the area of the trapezium? Draw it. 7. Find the cost of 28 “six-by-four” joists, averaging 18 ft. in length at $32 per M. . 8. What is the area of a walk 3 ft. wide, around a semicircular flower bed, the straight edge of which measures 12 ft. ? 9. There is a difference of 6 in. in the diameter of the wheels of a carriage. The fore wheel turns 1000 times in going a certain dis- tance. The hind wheel turns 2 times, and is 4 ft. in diameter. 10. What is the axis of a sphere 4 ft. in circumference ? 11. From a sheet of zinc, weighing 16 lb., and measuring 8 ft. by 4 ft., a square was cut, reducing its weight to 114 lb. How long was the square ? 12. How much ground is covered by 12 cords of 4-ft. wood piled 2 ft. high ? 13. A granite sphere barely clears a gateway 2 feet wide. Find its contents. 238 MENSURATION: REVIEW Written 1. A span of horses draws a load of brick weighing two tons. The brick are of the ordinary size, 8 x 4 x 2, and weigh 100 lb. to the cubic foot. How many bricks in the load ? 2. Find the contents of a cone 6 in. in altitude and 2 in. in diameter at the base. 3. Find the slant height of a square pyramid 12 in. in altitude, base 8 in. on a side. 4. Find the entire area of a hemisphere 15 m. in diameter. . 5. How long is the equator, the equatorial semidiameter of the earth being 3963.296 miles ? 6. Compare the perimeters of a rectangular field 60 rd. by 30 rd., of an equivalent square field, and of a circular field of the same area. 7. The inside dimensions of a cellar are 16 ft., 12 ft., and 8 ft. The wall is to be two feet thick. How many cubic yards of earth will need to be removed ? 8. A cylindrical 8-in. driven well is 76 ft. deep. How many cubic feet of earth, etc., have been taken out ? 9. One fire-engine throws a 2-inch stream, another a 13-inch stream. Compare the quantities of water thrown. 10. One piece of shafting 2 in. in diameter weighs 300 lb. What is the weight of a similar piece 31 in. in diameter ? 11. How deep shall a 12-foot square bin be made to hold 1728 bushels ? 12. How many cords of wood in a section of a giant pine 18 ft. long and 16 ft. in diameter ? 13. Ifa 2-in. rope breaks with a weight of 8000 lb., what weight might break a similar 3 in. rope ? 14. Give area of basin required for a fountain that throws its spray out 15 feet. THE METRIC SYSTEM OF WEIGHTS AND MEASURES) 239 The English system of weights and measures, the system in com- mon use in the United States, lacks a uniform scale of relation. HMOiexa tiles 20 loiter os tbe = liyd. 3 0. yO. = 1 Td.,etc,.and 2 pt. =1 qt.; 4 qt.=1 gal. The United States system of money has a uniform decimal scale, and thus any part of a dollar can be ex- pressed as a decimal fraction. Thus 3 dollars, 5 dimes, and 7 cents can be expressed $3.57. The metric system grew out of an attempt by the French govern- ment to supply a system of weights and measures that would have a uniform decimal scale of relation that would thus facilitate computa- tion by enabling one to change a unit to any other unit by simply moving the decimal point. The committee appointed to report upon a standard unit thought to make this unit, which was to be the unit of length, some part of some well-defined portion of the earth’s circumference. To Tp ap Te 2. Read as fractions in smallest terms: 125%, 183%, 314%, 373%) 433% , 565%) 835%) 683%. 3. Of a mixture 15 gal. of alcohol make 311% of it. How many gallons in the mixture ? 4. Give the cost of :— 8 at $2 per dozen. 9 at $4 a dozen. 7 at $2.50 per dozen. 2 at $5 a dozen. 5 at $1.50 per dozen. 16 at $3 a dozen. 5. Required the semiannual dividend from $18,000 in 31% bonds. 6. What is the value of a ton of soap in 4 oz. cakes at 48 f a dozen ? 7. How many envelopes can I buy for $12 at 75 ¢ per M? 8. Bought pencils at $3 per gross and sold them at 60 cents a dozen. Gain on 5 gross? 9. Paid $x for a pair of $8 shoes less two 10% discounts. 10. Of 300,000 immigrants 40% are illiterate. How many of them can read ? 11. If a half million out of a gift of 3 millions for a hospital is spent for the building, what is the annual income from the remainder at 31% ? 12. Each angle of a triangle measures 60°. What kind of triangle is 1t? 13. x ininutes will take a horse 6 times around a 4 mile track if he trots at a 2:10 gait. 14. It is 160 rods round a square field containing «& acres. 15. What will pay a note of $ 800 that has been running 8 months at 6% ? 16. 121% of 64 is 831% of what? Written REVIEW EXERCISES 251 1. A broker received $75 for purchasing bonds at 1% commis- sion. What was the face value of the bonds ? 2. A man offered to sell his horse for 25% more than it cost him. He afterwards sold it for $190 which was 5% less than he first asked for it. What did the horse cost him ? 3. A house cost $3000, and rents for $25 a month. If the taxes and other expenses amount to $ 50 annually, what per cent does it pay on the investment ? 4, A man desires to settle an annual income of $700 on his son. How much must he invest in U.S. bonds, paying 3£% at 105 to yield that income ? 5. A city map is drawn to a scale of ;4, mile to the inch. What is the length of a line which represents a street } mile long? 6. Mr. Jones kept 75 bushels of cranberries through the winter, but found + of them worthless. He sold the good ones for $3.50 a bushel, receiving $15 more than he would had he sold them in the fall. What was the price per bushel in the fall ? 7. Ifa gas jet burns 4 cubic feet of gas in an hour, and 4 jets are lighted each evening from 6.30 to 10, what will be the gas bill for February at 10 ¢ for 100 cubic feet ? 8. On a certain locomotive the driving wheel 7 feet in diameter turns 24,000 times in going from Boston to Springfield. How many times will a car wheel 30 inches in diameter turn in going the same distance ? 9. 2 of an article sold for the cost of 7 of it. What was the gain per cent ? 10. I send $ 4935 to a broker in Chicago for the purchase of flour at a commission of 5%. If he paid $5 per bbl., how many barrels did I receive ? 11. Sold 1500 lbs. of coal for what a ton cost. Gain = 2%. 22, REVIEW EXERCISES . Written 1. When 5% bonds are quoted at 104, what sum must be invested to have an annual income of $1600? Brokerage 1%. 2. Borrowed $500 at 6% on June 10, 1902. When it was paid it amounted to $546. On-what date was it paid ? 3. If the holder of a $150 note running 4 mo. had it discounted at a bank July 15, 1903, at 5%, what did he receive? Date, June 5, 1903. 4. If the note was unpaid when due, and drew interest at 5% from the time it became due, what would settle it March 27, 1904 ? 5. A young man whose salary is $24.00 a week pays $7.50 a week for board and $8.75 a week for other expenses. In how many weeks can he save enough to pay a debt of $279? 6. Received a consignment of 2000 barrels of flour which sold at $5.50 per barrel. I paid $74 for storage and $27 for carting. How much ought I to remit, after deducting a commission of 4% ? 7. A sewing machine was sold at a discount of 124% on the asking price and at a gain of 40% onthe cost. If the cost was $50, what was the selling price? The asking price? 8. How many feet of wire will be required to fence a square field containing 3136 square feet, if there are three rows of wire in the fence ? 9. A hall 20 feet high has a square floor containing 6561 square feet. A merchant agreed to furnish burlap one yard wide for the walls at 15¢a yard, if the bill was paid on or before Feb. 4, 1903. It was not paid until Dec. 22, 1903, when interest was required at D%. What amount was required to pay the bill? 10. A 90-days’ note for $350, dated Nov. 25, was discounted Dec. 19. What did the payee receive ? 11. How many gallons will a standpipe 15 feet in diameter and 80 feet high contain? One gallon equals 251 cubic inches. Written REVIEW EXERCISES 255 1. A physician received $76 from a collector whose commission was 5%. What was the amount collected ? 2. I pay $13.50 for insuring my furniture at 3%. For how much is it insured ? 3. $16.50 will settle a bill on which the discount is 10% for cash. What difference will it make if I delay the payment? 4. What per cent is made by buying coal at $4.65 a long ton and selling it at $8 a short ton ? 5. After deducting his commission of 4%, how many barrels of apples at $1.50 can an agent buy with a remittance of $2000 ? 6. How much will a broker charge, whose commission is 4 for selling 85 shares of stock, — par value $5 ? 7. Shall I increase or diminish my income by selling 80 shares of 5% stock at 72 and investing in 8% stock at 108}? 8. What is the tax on property assessed for $4780 at $16.30 a thousand ? 9. $360 was paid an agent for buying cotton at a commission of 89. He afterwards sold the cotton at a profit of $4500, deducting his. commission of 11%. What was his commission on the selling price, and what were the proceeds ? 10. An agent receives $2184 with which to buy meat at 163 ¢ per pound. How many pounds can he buy at a commission of 5% ? 11. In a city $275,000 was raised from a 14% tax. What was the assessed valuation of the property ? 12. What is the net price of an article listed at $50 and sold ata discount of 25, 10, and 4% ? 13. The surface of a cube is 576 square inches. Find its volume. 14, Find the exact interest of $250 from July 18, 03, to Jan. 5, 704. 204 REVIEW EXERCISES Written 1. Find the cost of $500 in U.S. bonds bought at 1264, broker- age 3%. 2. What is the per cent of profit if I buy oranges at $1.50 a hundred, lose 10% of them by decay, and sell the remainder at the rate of 3 for 10 cents ? 3. The proceeds of a 90-day note, discounted at a bank at date, were $492.50. What was the face of the note ? ; 4. What premium must be paid on a building valued at $7500 and insured for 2 of its value at 4% ? 5. A man can do a piece of work in 3 days. His brother can do it‘in 4 days. What part of the work does each do in a day ? 6. What part would both do in a day working together? If they work together, how long will it take them to do the work ? 7. A farmer can mow a field in 10 hours. His hired man requires 12 hours to mow the same field. Suppose they work together, how long will it take? g. A can doa piece of work in 6 days, B can do it in 8 days, and C can do it in 12 days. How long will it take the 3 men work- ing together to do the work ? 9. Three men can paint a boat in 4 days. Two of ‘them can do it in6days. How long would it take the third man working alone ? 10. A lot is 200 feet by 90 feet. A house, the main part of which is 24 feet wide and 42 feet long, with an ell 14 by 20 feet, is built upon the lot. 15. 16. S2tanPrh © dD = SOBAD TP ww pe ANSWERS Page 203 Page 205 DOL ep. 1. $1125. . $21.56. 2. 5.13 qt. . $217.01}. 3. $56.47. . $2085.71. 4. 3.278+ %. $ 312.65. 5. Loss $407.69. be Wes las 6. 28,8, %. $ 6185. 7 40857, . $1265.25. 8. $1890 net loss. . $12.09. 9. $4,800,000. . $5.21, 10. Latter $ 27.28. ’ cae Page 207 } ane ‘ 18. 300. (grace); f 4g. 331. $ 87.70. 20. 80. $ 51.14 21-39: (grace) ; 99. 1 . 6° See 93. 8 yd. ; 24. $20. (grace) ; $ 193.95 25.° 75. ints’) 26. 60. Page 204 27. 96. _ $250. 28. 65, . 0.605%. 29. 43.5. $ 150. Oe 15: $ 58.80. 8. $378. Sain Page 208 $941.11. 4 aah 3:4, . $1886.70. “acme WO A150 1 12Iar213 p 8. $33.92 . $1590. 9. 96 . Gain $14.25. | 15 Nah . $1980. i. oe ¢ ° 0 ; rae 12. 56 men. Cerda) 13. $4.80 per bbl. $511.57. Page 209 $ 75. 2. 604 da. $ 500. 3. 2585. a a PO Or CO SO 13. 14. 15. 16. Lis 18. OI AH ow ww ee os Oo $819. # . 8100 times. 112 126 mi.: 251 mi. (depends on rate). . 960 bu. . $3.40. BeBe clei Page 210 . $62.50. 2.7 mi, $2.50. $18. 48 yd. 14 rev. 224 da. . 18 Meise . $126. . Of hr, . 2223 da, . 4285.71 mi. . $133.35, . 24 A, Page 211 35. 96. . 680. 10. 11. 12. 441, 343. 420. Page 213 23.=. 26. 34, 42. 45, 54. a oP oo 290 10. i — POO ONDA Pw DO . 638. FOG: 5 Yess . 84. Page 214 28. 58. 92. ols . dd. 2907: Page 215 582. 547, 636. 746. 869. 2458. (28: 696. 799. . 852. Pata e . 967. . 1074. yivaye . 1594. Page 216 . 0.9682+, © 12, 999. 93, 17. 10. ANSWERS . 64. Page 217 . 1.4142+, 1. 8.94+. . 0.4472+. 2. 3.214. - 0.19864+. 8. 2.53+. PELL 2i43+. An S518 35 28.0178+. 5. 2.68+. . 1856. G4. Lie, . 30.23824+. Wiehe oa . 0.860813+. 8. 2.92+. gun 9. 3.06+. . 0.559+. « 10. 2.64. Moa LI 182: . 2.380+. 12. 0.8862+. . 9,099+. 13...6.25+. . 0,5422+. 14) 8 9 72+. i loee2o ths . 1.4790+. 15. 0.6. 16. 2.39+. Sule 0. 8538+. 17. 8:89+; me 18. 5.87+. 3.7249+, 1004. 19. 8.00+. 3136. 20. 6.058+. 7921. 45.09. Page 219 0.75. 1. 60 ft. 0.96. 2. 68 ft. 6.5. 3.57 ft. 0.8848+. 4. 115 ft. 0.9433+. 5. 42.42+ ft. 4.4121+, 6. 25.61+ in. . 28.7210+. 7. 8.607 ft a 4.12754; 8. 223.60+ rd. 0.8. 0.2529. . 43.9590. Page 220 . 27.9991+. 1338; . 15.08. 9. 8.664, 71420. 3. 24.16+ ft. 8314; 4521-08 ft: 282843 5. 73.82+ sq. ft. 6. a Hw Oo 2 = KP OO MAD AP wD 13 Altitude, 20357 8 ff, : area, 249.36+ sq. ft. . $141.40. . 259.8 sq. in. Page 221 ~ 29.674 It. » 29.64 iti 13.4 ft. 8. . 25.374 rd. 67.8+ mi. . 1017.8784 sq. In. . 5.56+, . 270.4+ ft. 2 18.02/46. » Leer. Page 222 6.9282+ rd. . 43,863 ft. 5:4, . 21.089 rd. = 20387 is ite . 97.616+ ft. . 63.639 ft. . 101.98+ ft. . 108 ft. by 36 ft. . 20.78+ it. . 27.12+ ft. . 137.08 isqe in, . 2368.074 ft. . 69.41 rd. Page 225 . 2917 cu. ft., or 29.629+ cu. ft. 14 8. 471.4 cu. in. 4. 1500 cu. in. 5. 6600 lb. 10. 96 sq. in. 11. 384 sq. in. 12. 62.34+ sq. in. 13.. 720 cu./in. 14. S. H.. 16.15+ in,; ¢. sur- face 887.6+ sq. in. Page 227 habe Poe 1372 ca. in: 1 2 8. 24 cu. in. 4. 6 sq. in. 5 6 9 . 01,6992 cu. TE 00g . 40 sq. in. 10. 20.1219+ sq. ft. 11. 93.4626. 12. 219.912 sq. ft. 13) (1.05 -y a. Page 230 1. 0.5286. 2. 0.4764 ; 0.5236 rem. . 285.62 Ib. . 18.566+ Ib. . 4,188,800,000 cu. mi. . 64. ao oa Pf mi. 11. 14.573 lb. avoir. 12. $74. 13. 486 oz. OID TP wo o . 38.5104 cu. In. D Po 2 7 8. 67.0208 cu. in. 9. 113.0976 sq. in. 10. 12,566,400 sq. ANSWERS Page 233 12. 28.2744, $ 2700. 13. 18,000 lb. res 14. 706.86 sq. ft. $ 18.56. Page 244 195: 1. 847,200 g. ; 5 1867.75 lb. pele 2. 26.43 HI. Be Ee 3. 16.72 sq. mm. : ae 4. 1312.359 yd. . 1171.874 bu. 5, 8395.38 Sq. rd. _ 96 hy. Beane | Page 237 7. $10.63 gain. 8. 960 Ke. | $9.25. 9. 88.9056 Ke. . 681.7925+ sq. 5 Sd: | 10. 2.845184 Mm. ee iis Jas . 425 sq. ft. Ree te 138.38 A. . 315 sq. ft. 13, 264.17 eal . $32.26. a piiea: . 124.686. eel oy 15. $3.21. _ 1.27824 ft. mee aaah OTS bed: 17. 7500 m. ' exe date 18. 1s 220 T. A BBS onittme | Seg ee Page 246 Page 238 aay 3. 12:53:36 p.m. 1080. p 4. 7:15:48 a.m. . 6.2882 cu. in. d 5. 5:53:20 p.m. 12.65- in. 530.146 ec, in. | 8 022282 mil hee eae 7. 6:33:20 a.m. spear tt * |g 6:25:8a.m. ee 9. 11:59:86 a.m. 169.68+ rd. ; 10 oR oe 150.40- rd. se viare ee atees 94.81 11. 12:93:20 p.m. fee 12. 3:5:16 a.m. . 26.529+. PAYOIs Page 251 . 1607.8125 lb. | 1. $60,000. . 14,.9334+ ft. 2. $160. _ eae FOOD DAHA SP wW POG (Soot Sore A ee = OHIATh OV — pt pe On re © _— » 819, . $21,000. 6 in. $2.60. $1.57. 67,200. BEY, . 940 bbl. . 832%, Page 252 . $33,280; $80 brok. . Dec. 22, 1908. $ 148.19. $ 153.58. 36 weeks. . $10,844. $80. . 672 ft. Bab bch . $346.15. . 105,753.6 gal. Page 253 $80. $1800. $1.88. 0.927-. . 1282 + bbl. $0.53. . $24 increase. . $77.91. . $1256.25 ; 99,243.75. . 12,480 lb. . $22,000,000. . $32.40. . 939.84+ cu. in. 28 7 0Sc eT cats bien eeee oe ee Cr OMAR ATP w we —_ wore © —_ Po Sa Ee St oe ho i Page 254 $379.88. . Int. $25. . $647.90. Page 255 . $11,848.80. . 57823 cu. ft. $ 4649.80. 62%. $ 3854.17. $ 2172.50. 43.30 ft. . $816.06. . 25%. . $55.36. . $345. . $905.42. Page 256 . Latter. $ 767.38. $ 2250. 40 ¢, 30 mi. $ 844.95. 162%, $ 59.63. $4704. $43.20. . $42.86 per $ 100 share. ANSWERS 12. 3 mo. 6. $34.45. . 12h yr. 7. $41.48 more. . 6.336 in. 8. $0.133. 9. 461.8152 cu. in. Page 257% | 10. $4936.60, ie ie Page 260 . 280 gal. ; 1. 173%: 200 gal. 2. First, 72% ; 32 ft. better. . 1368 sq. ft. cy Ls . $5.60. ee eG - $1050, 5. $156.08. ition, 6. 63.8: 36.2. . 1143. 7. $0.94. Booms JUsep ie ween ie 9. $3.24. 1348 A. 10. 0.816. - $310. 11. $9.21. $28.80. 12. 11.55. 13. Increase $15. Page 258 14. 608 ; 544, - $595. Page 261 noe 1. 3125. . 49:98=98: | 9 107.603, ah 8. 336.13+. a 4. $12,000. 30 # per yd. 5. Loss $16. See 6. $425. te ’ Fhe 7. $302.25. 3 in, . 10 9. 200,000 Ib. . aa & to . $1279.94. 10. $402. - #819. 50. 11. 9.36+%, 12. $56.53. ea 13. 1963.5 sq. in. 1. 14,9125 %. 14. 140.0616 sq. in. 2. 18849. 3. 7.854 sq. ft. Page 262 4. $0.94. Tees: 5. $17.67. 2. 70.71- mi. OowA sa PE = ee Wore © Oo fF © oO 15 . $206.45 ; $ 208. . $495.08; $494.83 (grace). . $2934.37. - @BH= $16.09 less. ba) tds e251: 588 sq. rd. . L. price $960 ; cost $400. . $1481. . $26.16. . $889.43, . $0.86. . $25,000. . 62%, Page 263 is: 70+ gal. . 389.7+ ft. 384%. 60 %, $ 632.10. . 663 % . $94.50. . $80. . $640.25. . $450. . $84.31, Page 264 . 1.38084, . $518.55. 48 ft. . $555.10. . 420.168+. . $17.92, 16 . $352.837+ - 7 8. $3668.75. 9. $35,000. 10. 87} ¢. 11. $58,500. 12. $2971.25. 13. 56.568+ rd. 14. $344.53. Page 265 $ 40.32. 160 bbl. $10.75. $ 56.77. 16 rd. . $260.40; See Te edie $13.02 com. . §6.56+ rd. . $5.50. . $1930.43. . EY. 18. 5%, - = = woros Page 266 1. $55,000. 2. 44.74- %. 3. 542%. 4. 413,0134. 5eo Ae 6. $2.75. WY 177%. 8, 22,8 4. 9. 4. 10. 1662. 11. 3616.037+ bu. 12. 163%, 13. 93.81 — rd. 14. $22.68. 15. $5190. eet PP CO WO KH © —_ (s) OO AIA TP ww — rOoDO SONIA AP OD OHOWAHAM Pwr ANSWERS Page 267 10. Six hund. six 2, thous. and 12 ft. fifteen tril- 50%. lionths. 80. 11. Nothing. . 154.284+ bu, | 12. 14 §¢. $ 140.623. 13. $625. $ 5.60. | $90.56. Page 270 - 125%, ligbeee . $2.625, 2. $67.20 . $12. 3. $28.80 . $252.58. 4.4% 17.36- . 125 ft. 5. $51,568. - $198. 6. 9929.4. 7%. 62.38376+ min. Page 268 8. 1765.17 gal. . $4,375. 9. 1:10. . 1662 %. 10. $20. 40 ¢. 11. $5157.43. $ 8.27. 12. $4499.25. 4% $ 371.20. Page 271 $ 40. 1. 0.0162. . $105.80. 2. $21. . $391.27. 8. 44.74-9. . $620.82. 4. 2473. - $187.20, 5. 684 ¢. . $220. 6. $93.75. 1.7 33,0803. Page 269 8. 10¢. , 8213. 9. 12 da. 810), 10. $283.50. 8 6150. 11. 32.725 Ib. 165 ft. 12. 136 cu. yd. $ 3761.25. 13. 12 rd. . 9662. Sai Page 272 . 6h ft. Lito $ 5.47. 2. $54181. ee ee oo wo KX © so PDa2Inarronr $94.52. $418.88. . $56,250 5 38% $ 25.92, . 64 men. $8000. $76. . $377.32. . $150. . $421.55. Page 273 Sai 140° . 159,155 cu. ft. 116 ft. . Loses $10. $96.21. . $577.40. $386 ; cost 600 fr. . $12.75. . 0.02078125. . $810. . $5700. Page 274 wae . Seven thou- sand four hundred eighty-five and two ninths ten- thou- sandths. . $43.20. . $82.08. — — wero a OOAWR AP ww yp . $2007.50. 126. $1337. 241 bu. $2.18. . $150. Page 275 22, $8625. $3978.72. 662. $ 125%, $9.45. 8%. . $21.84. $7.48. $1.80. . $4824. . $32.18; $4.95 per thous. Qo Pw dD ANSWERS Page 276 6. 15¢. 1243. 7%. 71.65+1d. . 720.288 +. $720 5. 874, sq. yd. 9. $560. . $7400, 412%. | 10. 1728. $39.65. 11. 55%; $0.75 . $200 income ; per bu. ; $558. 349701, 12. $495.58 $12. (grace) ; . $24.79. $495.83. 80. oe. Page 278 . $53.90. . $48. 1. $18.29. . $640. 2. $36.75. 3. 20 in. Page 277 4. 5¢. +o- SRI By ye 67%. 6. 9 cd. Liat, tb Eph $9; 60%. 8. $26.18. . $1115. 9. $4.50. 10. iv 12. — — — pe wOwreo Oo 14. 15. Sap ketal Nig ed Hine ase 17 $88.36. $ 1562.50. $ 8.66. Page 279 64. . $1546.79. $ 150. 0.853 +. $ 120. 1.002. . 32 shares. 53° . 12.806 + ft. . Variable. . July 2, noon, or July 2, 12 p.m. 75 ¢. 123%. aN he a be wr. ay i oe URBANA ” oO = a | = ts =) 7 > _ “a cc Lhd = = => < os) i=] i) o _ wn v003 STONE ARITHMETIC BOSTON TUS C001 THE SOUTHWORTH (3.0112 017106359