9FAL16~LY~< ECLECTIC EDUCATIONAL SERIES. R A Y' s A RlITHMETICAL KEY: CONTAINING SOLUTIONS TO THE QUESTIONS AND TO SOME OF THE MOST DIFFICULT QUESTIONS IN THE SECOND BOOK: ALSO AN A P P E N DIX OF TEST EXAMPLES FOR THE SLATE OR BLACKBOARI). BY JOSEPH!AY, MI. D., PROFESSOR OF M3ATHEMATICS IN WVOODWARD COLLEGM. NEW AND IMrPROVED EDITION. PUBLISHERS: CLARK, AUSTIN & SMITH, NEW YORK; WVINTHROP B. SM3IT-I & CO., CINCINNATI. M JATHEI IATICAL WORKS. By PROFESSOR JOSEPH RAY. TYPE ENLARGCED-N>EW AND IMPROVED EDITIONS. EachL Boost of Ray's Aritfhmetical Course, also of the Algebraic, is a covmplete work ins itself, and is sold separately. FIRST BOOK, PRJIfARY LESSONS AND TABLES, simple and progressive, for little learlners. SECOlID BOOK. INTELLECTUAL ARITHIMETIP'IC, by Induction and Analysis; a thorough course of meontal exercises. T I D BOOK, i PRACTICAL ARITHMETI[C by Induction and Analysis; a simple and thoroulgh work for schools antd privazte students. KEY TO PibY'S A1RITHMIETIC, THIRD BOOx. IIGHER ARITHMBETIC. Principles of Arithmetic analyzed and practically applied. For advanced students. KEY TO RAY'S HIGHER ARITHM-IIETIC. ELEIENT ARY ALGEBRA. RAY'S ALGEBRA, FIRST Boor, for Common Schools and Academies; a simple, progressive, and thorough treatise. H-IGt E1 ALGEBRA i AY'. S ALGEBRA, SECOND BOOIc, br advanced students in a academies, ancl for colleges; a, lucid and comprehensive work. KEY TO RAY' S ALGEB3RA, FQ.IRST AND SECOND BOOKS; complete. ji ione v:1liasm, l2meo. Entered accordinog to. Act of Congress, in the year Eighteen Iliundred and FortS-firve, 1byev Ws5erifo, B. SrlITr,: in the Clerk's Office of the I)istrict Court of the ULnitedtrates, for the District of Ohio. C. F. O'DI9ISCOLL & CO., STErEOTYPERS, CINCINNATI, O. PREFACE. The leading objects in the preparation of the following work, have been:Fiirst, to aid the teacher, who may not at all times have leisure, in the midst of his labors in the school-room, to furnish the pupil at once with the aid which he requires. For this purpose, the work contains full solutions to all the questions in Third Book, as far as Compound Numbers; after that, the method of working each example is indicated by arithmetical signs, so that any one acquainted with the elementary rules, can readily perform the required operations. Second, to assist young teachers. For this purpose, besides what has been already named, the work contains solutions to all the difficult questions in the author's Mental Arithmetic, (Second Book); remarks designed to illustrate more fully the nature and principles of several parts of arithmetic; with suggestions on the best methods of instruction. The examples for exercise at the blackboard have been inserted at the request of several of the best teachers, who have found it both interesting and:useful to the learner, to propose questions not found in the text-book. It has been a complaint with some respectable teachers, that answers are given to the examples in the Third Book. When the pupil is required to explain, at the blackboard, the method of working out an example, it is desirable that he should rely on the correctness of his own reasoning, and, if he understands the method of obtaining the true result, it is of little moment whether he has seen the answer or not. The experience of many eminent teachers is, that the use of a book without answers has an immoral effect on the pupils.@ A superior pupil will obtain the answers correctly, while other members of the class will procure them privately, either from him or others, who have previously worked them. It is the author's firm conviction, founded on an experience of many years, that the only proper test of a learner's knowledge is, to place him at the blackboard, to solve and explain the questions; and, with this test, it appears to him that the objection to furnishing answers to the questions in the book is removed. A celebrated Boston teacher once said, that an Arithmetic without answers, tended to make the pupils both rogues and liars. o~~~~P _____ __ I -- -C-r. -s r - N OW PUBLIStHED. IAY'S THIGH ER ARITTHMETIC, embracing the principles of the science, analyzed and applied; designed for advanced students and business men. An admirable and highly comimended treatise, peculiar for its freshness, philosophical accuracy and adaptation to the purposes of instruction. Its rapid introduction into schools of the highest order, and the continued preference given to it, over other admired and popular Arithmetics, may be regarded as evidence of decidedly superior excellence. OBSERVE.-That it is especially adapted to those desiring enlarged intelligence with reference to Mercantile, Mechanical and Agricultural pursuits. Just Published.-A KEY to Ray's Higher Arithmetic, embracing full and lucid solutions to the examples contained in that work. (4) KEY TO RAY'S ARITHMETIC, THIRD BOOK. SUGGESTIONS ON TEACHING WRITTEN ARITHMETIC. IN teaching written, or, as it is frequently termed, practical arithmetic, the first step is to make the learner well acquainted with numeration and notation. After this, pupils should be frequently exercised in reading and writing numbers. Similar remarks apply to the numeration and notation of decimals; when these have been properly studied, the subject of decimal fractions presents few difficulties to the student. It is scarcely necessary to mention that every pupil should be able to perform Addition, Subtraction, Multiplication, and Division, with ease and correctnzess, before proceeding further. In order to do this, it is frequently necessary to review these subjects more than once, and to require the pupil to solve examples additional to those contained in the text-book. The instructor and pupil should both feel that no time-is lost in dwelling upon and reviewing the elementary rules, until the pupil is thoroughly master of them; it is the most certain means of insuring his future rapid and successful progress. In explaining the nature and operation of many of the rules, the instructor will find it of great advantage to illustrate the subject by simple mental exercises. Every competent teacher can generally frame such as are suitable, and present them orally to the class. In this way, they mnake a more decided impression on the pupils, than when presented in the text-book. Mental exercises, suitable for the illustration of those subjects, when they can be lised with advantage, will be found in a succeeding part of thlis volume, under their' appropriate heads, and precedinig the examriples for exercise at the black-board. 6 IKEY TO RAY'S THIRD BOOK. tWhen practicable, the pupils should be arranged in classes, clue regard being had to their ages, acquirements, etc. After this, the proceeding in the best schools, is somewhat as follows: A certain number of examples is arrangefl for a lesson; it will, also, frequently be necessary that a. part, or even the whole, of the lesson shall consist of the illustration of principles, or the memorl. izing of definitions or rules. When the class meets for recitation, each pupil passes his slate into the hands of the pupil next above him, except the pupil at the head, who passes his to the foot scholar. The teacher then reads the answer to the first question, while each pupil examines the slate lie holds, to see if the answer is correct and properly obtained. In addition to reading the answer, the teacher, in many cases, such, for example, as proportion, should state the general method of working the question. The pupils mark the answers that are wrong, or obtained improperly. In the same manner, each question is examined and marked. Instead of the teacher reading the ansswers, the pupils in succession may read them. When there is a blackl-board (and there should be at least one in every school-room, 4 or 5 feet wide, and as long as the room will permit), each pupil should be required to work out one or more of the examples, and give the reasons for th[e method of performing the operation. The time required to examine the questions is generally short, while the lhabit of closely scrutinizing each other's work, improves the perceptive faculties of the pupils. The instructor will find the suggestion contained in the Note, Article 138, with reference to the examples having a star, -, orthy of particular attention. in: order to teac]h either intellectual or practical aritlhzetic well, the instructor nmust be Jfaniliar with the subject, so as to be able to explain, pr'omptly and clearly, the solution of every question, as well as the priniciples on uwhich each rule is based. IIe should have, also, a strongy desire to render his pupils fanciliar with the subject: for, it zmay be stated as an accknowledlecl truth in education, that, a teacher who is indifferent to the cadvancemenz t of his pupils, will renzder them indifferent likewise; (land that, on the coati rn'yar, onTe that is deeply interested in the student's I. p'c/e.dsu, wil n c/el/! fareli, to e/nkiznd7le the salice spirit. OF SYMBOLS. 7 O9f SYMBOLS. 1. The sign- is called the sign of equaZity; it denotes that quantities between which it is. placed are equal. Thus, $1 — 100 cents; read, one dollar equals one hundred cents. 2. The sign + is called pluts; it denotes that numbers between which it is placed are to be added together. Thus, 4+3 —7; read, four pzus three equals seven. 3. The sign - is called nminus; it denotes that the number placed at the right of it is to be taken from the number on the left. Thus, 5 —3=2; read, five 7minus1t three equals two. 4. The sign X is called the sign of mzutii2lication; it denotes that the two quantities between which it is placed are to be multiplied together. Thus, 4X3=12; read, four times three equals twelve, 5. The sign --- is called the sign of division; it denotes that the number on the left of it is to be divided by that on the right. Thus, 12-_ 3-4; read, twelve divided by three is equal to four. Division is also denoted thus, Gl, or 3)12; either of which indicates that twelve is to be divided by three. 6, The parenthesis, ( ), when it incloses several numbers, indicates that these are to be regarded as one single numler. Thus, (5 +3)X4-32; read, five plus three multiplied by four is equal to thirty-two, If the parenthesis were wanting, the expression would stand thus, 5 +3X4-17; this indicates that five is to be added to the product of three multiplied by four, and that the sumn is seventeen. 7. A number placed over another number, a little to tile right, is called an expoznent. Thus, 2 and 3 are called the exponents, in the expressions, 492, 5', etc., 43, 53, etc. The exponent of a quantity denotes that the quantity is to be used as a factor as many times as there are units in the exponent. Thus, 42 —=4X4, 2-=2X2X2, etc. The exponent 2 depotes the seconId )pow-i, or squafre.of the number over7 whtihel it is placed. The exponent 2 denotes the 1 8 KEY TO RAY'S THIRD BOOK. thlird power, or cube. Thus, 3'= 3 X 3 -9-the square of three; and 33=2X2X2=8 —the cube of two. 8. The sign -/ denotes that the square or second root of the number before which it is placed, is to be extracted. Thus, 1/16=4; read, the square root of sixteen is equal to four. In a similar i-manner the sign g/ denotes that the cube or third root of the quantity before which it is placed is to be extracted. Thus, g/27-3; this is read, the cube root of twenty-seven is equal to three. 1KU I DRATION. SUGGESTIONS TO TEACIES. —Be careful to impress the mind of the pupil with a clear idea of a unit. Show him that numbers are formed by the successive addition of units. Explain the distinction between units of different orders; this may be done conveniently by a reference to the coins of the United States, stiff pieces of paper being used to represent units of the first order; those of the second order will be represented by cents; those of the third, by dimes; and those of the fourth, by dollars. MIake the pupils well acquainted -with readbing numbers under one thousand before proceeding higher. Until the pupil is familiar with the subject, require him to point the numbers into periods of three places each, before attempting to read them. Let the pupil be made well acquainted with the method of writing numbers under 1000; then, by the use of points, he may soon be taught to write any number with ease and accuracy. TO TEACHERS.-Where the word "ARTICLE" occurs in this volume, it has reference to the corresponding "A' T." in "Ray's Third Book.'" The examples in the Key are numbered, and refer to the corresponding number of example, under the same artiole, in the Arithmetic. ARTICLE 13. 1, Five., Sixty-three. I 6, One hundred anId forty3, Ninety. 4, One hundred,. seven. 5, One hundred and f;ur. 7, Two hundred and eight. &, _e —------------ ~ —-- ----- NUMERATION. 9 8, Two hunldred and eighty. 37, Seven hundred and four 9, Four hundred and three. thousand two hundred and 10, Seven hundred andtwen- eight. ty-nine. 38, Eight hundred thousand 11, Seven hundred and ten. one hundred and forty-one. 12, Nine hundred and one. 39, Nine hundred thousand 13, One thousand. and sixteen. 14, One thousand and five. 40, Six hundred and one 15, One thousand and fifty. thousand and twenty. 16, One thousand and eigh- 41, Seven hundred thousand ty-five. four hundred. 17, One thousand one hun- 42, Eight hundred thousand dred. and two. 18, One thousand one hun- 43, Nine hundred and ten dred and eight. thousand one hundred and 19, Three thousand and three. three. 20, Four thousand and fifty. 44, Seven hundred thousand 21, Three thousand and for- one hundred. ty-five. 45, Twenty thousand and 22, Nine thousand six hun- twenty. dred and ninety-nine. 46, One hundred and six 23, Ten thousand. thousand three hundred and 24, Twelve thousand. seven. 25, Thirteen thousand two 47, Four hundred thousand hundred. and one. 26, Fifty thousand and four. 48, Three hundred and two 27, Sixty-two thousand and thousand four hundred and one. four. 28, Seventy thousand four 49, Eight hundred thousand hundred. and ten. 29, Eighty thousand and 50, Seven hundred thousand ninety. and ten. 30, Ninety-seven thousand 51, One million, thirty thouand ten. sand seven hundred and twen31, Forty thousand three ty-five. hundred and five. 52, Four millions, fifty thou32, Seventy-six thousand sand six hundred and seven. and fifty-two. 53, Six millions, six hun33, Eighty-three thousand dred and one thousand. nine hundred and ninety-one. 54, Seven millions, four hun34, Two hundred thousand. dred and six thousand and 35, Four hundred and fifty- thirty-five. six thousand. 55, Nine millions, seven hun36, Six hundred and eighty- dred and twenty-five thousanld two thousand three hundred. and fourteen. 10 KEY TO RAY'S THIRD BOOK. 56, Three hundred thousand hundred and eighty-nine thouthree hundred. sand six hundred and two. 57, Nine hundred and nine 66, Ninety-nine millions, thousand and ninety. ninety-nine thousand and nine58, Eight millions, six hun- ty-nine. dred thousand and fifty. 67, One hundred and thirty 59, Two millions, one hun- millions, six hundred and sevdred and two thousand one enty thousand nine hundred hundred and two. and twenty-one. 60, Four millions, eighty 68, Six billions, nine hunthousand four hundred. dred millions, seven hundred 61, One million and eleven. and two thousand and nine. 62, Three millions, thirty- 69, Twenty-three billions, six thousand. four millions, ninety thousand 63, Nine millions, one thou- nine hundred and one. sand and three.' 70, Nine hunldred and forty64, Thirty-three millions. two billions, twenty-nine mnilseven thousand eight hundred lions, three hundred and seven and twenty. thousand and twenty-nine. 65, Sixty-one millions, one I ARTI CLE 14. 1. 100. One hundred. 9. 5 in the first order is five, 2. 203. Two hundred and or five units. 4 in the secthree. ond order is forty. 3 in the 3. 3020. Three thousand and sixth order is three hundred thousand. 8 in the 4. 50204. Fifty thousadl two third order is eight hun-.hun50red Fiand ftousra tv cdred. 7 in the fourth orhunre a o. cler is seven thousandcl. 9 5. 9000500. Nine million, five in the seventh order is nin hun7dred. millions. 4 in the fifth or6. 70050. Seventy thousand tder is forty thousand, 2 in and fifty. the eighth order is twenty 7. 600004000. Six hundred millions. 8 in the 6th order million, and four thousand. is eight hundred thousand. S. 20050200. Twenty mil- 4 in the 9thi order is f),u lions. fifty thousand two b hundre'd millio,,s. 5 in the | undtr~u:i'edn., l 10e11h,'(ler1 i: five billion,.l NOTATION-ADDITION. 11 NOT AT ION. A RTICLE 15. 1. 4 13. 65 25. 222 37. 503 49. 45083026 2. 16 14. 97 26. 999 38. 827 50. 870743579 3. 118 15. 87 27. 807 39. 1000 51. 2804 4. 20 16. 104 28. 870 40. 1020 52. 4029 5. 24 17. 120 29. 901 41. 25006 53. 6006 6. 28 18. 130 30. 606 42. 345 54. 22765 7. 30 19. 175 31. 309 43. 760 55. 80201 8. 32 20. 203 32. 190 44. 3406 56. 90001 9. 37 21. 340 33. 202 45. 42030 57. 30030 10. 41 22. 435 34. 101 46. 30000 58. 41.0205 11. 46 23. 502 35. 21 47. 163000 59. 800669 12. 59 24. 625 36. 300 48. 341563 60. 900001 61. 500050 64. 100010001 67. 700010001 62. 100010 65. 91007060 68. 1001000040 63. 909090000 66. 70000004 69. 40000200005 70. 726050001243. ADDITIONI). SUGGESTIONS TO TEAcnERS. —Simple Addition is the method of adding numbers of the same name or kind into one sum. We can not add numbers of diffierent kinds together. If a reason is asked for this, the reply is, that it is self-evident; thus, 2 apples and 3 apples added, make five apples; but, we can not add 2 apples and 3 pears together; the sum would be neither 5 apples nor 5 pears. If it is asked why we write unitst under units, tens under tens, etc., the answer is, since we can only add numbers of the same name together, we write those of the same denomination under each other, merely for the sake of convenience in performing the work. We carry one for every ten, because ten units in any place make a unit of the next higher place or order. If the base of our scale of Notation had been twelve, instead of ten, we should have carried one for every twelve. After the pupil has become acquainted with Compound Addition, he will better untderstand the reason for carrying in Simple Addition. Celerks, wh-o add long columns of figures with great rapid(ity, never call the names of the figures in adding; thev "Yv Ad) Ace fto 12 KEY TO RAY'S THIRD BOOK. look7 at each figure as they proceed, and either mentally or orally name the sum which it makes in connection with those that precede it. In this manner the addition is performed in about half the time that it could have been, if each figure had been named al.ou(d. ARTICLE 20. (Es. 3) sheep. (Ex. 4) dollars. (Ex. 5) 4321 (Ex. 6) 50230 43 210 1254 3105 21 142 3120 423 14 35 - Ans. 8695 Ans. 53758 Ans. 78 Ans. 387 ARTICLE. 22. (Ex. 11) Ans. 21023. (Ex. 12) Ans. 27910. (Ex. 13) Ans. 89569. (Es. 14) Ans. 2499593. (Ex. 15) Ans. 24194086. (Ex. 1.6) 23 (Ex. 17) 45 (Ex. 18) 51 (Ex. 19) 263 41 19 48 104 74 32 76 321 83 74 85 155 16 55 4 — A2S. 843 Ans. 237 Ans. 225 Ans. 264 (Ex. 20) 94753 (Ex. 21) Jan. 31 (Ex. 22) June 30 2847 Feb. 28 July 31 93688 TMarch 31 Aug. 31 9386 April 30 Sept. 30 258 May 31 Oct. 31 3456 Ans. 151 Ans. 153 Ans. 204388 (23) (24) yds. (25) dollars. First five, 1,51 50 245 Next five, 153 65 325 November, 30 42 187 December, 31 89 96 Anis. 365 Ans. 246 Ans. 853 ew —-------— L~m-P — SUBTRACTION OF SIMPLE NUMBERS. 13 (Ex. 26) 1732 (27) 1656 67 11.64 180 Ans. 1799 1004 Ans. 4004 (28) (29) sheep. lambs. barrels. dollars. 65 43 30 285 187 105 18 144 370 243 23 235 416 95 34 408 Ans. 1038 486 Ans. 105 1072 (30) 287 (31) 337 596 49 841 431 607 fourth, 1724 146 Ans. 3448 Ans. 1570 (32) 3005 (33) 275432 (34) 880000889 42627 402030 2002002 105 300005 77436000 307004 872026 206005207 800791 4002347 49003 320600 990019919 Ans. 5851840 Ans. 1474132 Ans. 2155513020 SUBTRACTION. SUGGESTIONS TO TEACRERS. —For the same reasons as those already given with reference to simple addition, we can only find the difference between numbers of the same kind. The order of writing units under units, is adopted for convenience. It brings the individual figures, between which the subtractions are to take place, in the most convenient position with reference to each other. It is a matter of indifference which of the numbers is placed uppermost. The method of placing the number to be subtracted under that from which it is to be taken, is generally supposed to be the more convenient. | 14 IKEY TO RAY'S THIRD BOOK. ARTICLE 25, (2) sheep. (3) dolls. (4) 734 245 875 531 123 751 -Ans. 203 Ans. 122 Ans. 124 (5) 8752 (6) 529 (7) 79484 3421 8 25163 Ans. 5331 Ans. 521 Ans. 54321 APRTICTLE 27. (12) 4444444 (13) 91516171 (14) 63046571 1234567 15161718 34992884 Anls. 3209877 Ans. 76354453 Ans. 28053687 (15) 153425178 (16) 100000000 53845248 10001001 Ans. 99579930 Ans. 89998999 (17) cents. (18) dollars. (19) feet. (20) years. 63 137 75 1837 17 65 37 1492 Ans. 46 Ans. 72 Ans. 38 Ans. 345 (21) dollars. (22) dollars. 1840 10104 475 7426 Ans. 1365 Ans. 2678 (23) dollars. (24) 20010 100000 19019 11 Ans. 991 Ans. 99989 (25) 91201.0 (26) 4000000 (27) 2020930 50082 4004 1009006 Ans. 861928 Ans. 3995996 Ans. 1011924 g M —- M-nmr -m-un ~-Aaar~l-p —----- MULTIPLICATION OF SIMPLE NUMBERS. 15 ii ULTIPLICATION. SUGGESTIONS TO TEACnaiRs.-It is important that the pupil should understand the nature of multiplication, namely, that it is a short method of performing addition, when the numbers to be added together are all equal to each other. The multiplier shows howv many times the multiplicand is to be repeated, hence it is always an abstract number. The multiplicand, however, may be either an abstract or a concrete number; thus, the abstract number 5, repeated three times, makes 15; and the concrete number, 5 apples, repeated three times, makes 15 apples. It is obvious from this, that the product is always of the same name as the multiplicand. Some, who have only examined the subject superficially, suppose that two concrete numbers may be multiplied together, as in the following question: What will 3 yards of cloth cost, at 5 dollars a yard? The answer, it is obvious, is 15 dollars. Now, it is asserted, that in this case we multiply 5 dollars by 8 yards, and that the product is 15 dollars. This view arises from an ignorance of analysis and the elementary principles of numbers. The correct solution is as follows: The price of 1 yard is 5 dollars, acnd 3 yard.s being three timzes as much as 1 yard, the price qwill be three times Cs nmuch, that is, 15 dollars. (See Ray's Arithmetic, Second Book, page 37.) ARTICLE 31. (12) 49 (13) 57 (14) 128 (15) 367 3 4 5 6 Ans. 147 Ans. 228 Ains. 640 Anzs. 2202 (1s) 1427 (17) 19645 (18) 44386 7 8 9 AlAs. 9989 Ans. 157160 Ans. 399474 (19) 708324 (20) 964578 (21) 96432 7 9 10 Ans. 4958268 Ans. 8681202.Ans. 964320 o —. - -- -- 16 KEY TO RAY'S THIRD BOOK. (22) 46782 (23) 86458 (24) 680323 11 12 11 Ans. 514602 Ans. 1037496 Ans. 7483553 ARTICLE 32. (25) 1236839 (29) 235 (30) 346 12 13 19! Anes. 14842068 705 3114 235 346 Ans. 3055 Ans. 6574 (31) 425 (32) 518 (33) 279 29 34 37 3825 2072 1953 850 1554 837 Ans. 12325 Ans. 17612 Ans. 10323 (34) 869 (35) 294 (36) 429 49 57 62 7821 2058 858 3476 1470 2574 Ans. 42581 Ans. 16758 Ans. 26598 (37) 485 (38) 624 (39) 976 76 85 97 2910 3120 6832 3395 4992 8784 Ans. 36860 Ans. 53040 Anls. 94672 (40) 342 (41) 376 (42) 476 364 526 536 1363 2256 2856 2052 752 1428 1026 1880 2380 Ans. 124488 Ans. 197776 Ans. 255136 g~~~ mcRE~~~~ i~~ - -------- --— ~ —---------------------- ~~~~~~~~~~~~~~~~~~~~~~~I-I MULTIPLICATION OF SIMPLE NUMBERS. 17 (43) 2187 (44) 3489 (45) 1646 215 276 3'65 10935 20934 8230 2187 24423 9876 4374 6978 4938 Ans. 470205 Ans. 962964 Ans. 600790 (46) 8432 (47) 6874 (48) 2873 635 829 1823 42160 61866 8619 25296 13748 5746 50592 54992 22984 2873 Ans. 5354320 Ans. 5698546 Ans. 5237479 (49) 4786 (50) 87603 (51) 83457 3497 9865 6835 33502 438015 417285 43074 525618 250371 19144 700824 667656 14358 788427 500742 Anls. 16736642 Ans. 864203595 Ans. 570428595 (52) 31624 (53) 126 (54) 823 7138 6 12 252992 Ans. 756 dollars. Ans. 9876 dollars. 94872 31624 221368 Ans. 225732112 (55) 675 (56) 496 (57) 152 13 24 28 2025 1984 1216 675 992 304 Ans. 877a cents. Ans. 11904 cents. Ans. 4256 miles. Key.-2 18 S:KEY TO RAY'S THIRD BOOK. (58) 1760 (59) 365 (60) 1130 209 24 109 15840 1460 10170 3520 730 1130 Ans. 367840 yards. 8760 hours. Ans. 123170 feet. 8 -Ans. 70080 miles. (61) 2029 (62) 80401 (63) 101032 1007 60007 20001 14203 562807 101032 2029 482406 202064 Ans. 2043203 Ans. 4824622807 Ans. 2020741032 ARTICLZE 33. (24 () 124 (3) 2395 (4) 2873 124 124 8 6 8 ---- 19160 25857 744 or 992 7 6 4 3 - -- Ans. 134120 Ans. 155142 Ans. 2976 2976 (5) 1673 (6) 2874 (7) 8074 9 9 12 15057 25866 96888 7 8 9 Ans. 105399 Ans. 206928 Ans. 871992 ARTICLE, 34. (l) 245 (2) 138 (3) 428 (4) 872 100 1000 10000 100000 Ans. 24500 Ans. 138000 Ans. 4280000 Ans. 87200000 DIVISION OF SIMPLE NUMBERS. 19 ARTICLE 35. (1) 2350 (2) 80300 (3) 10240 (4) 9600 60 450 3200 2400 Ans. 141000 4015 2048 384 3212 3072 192 Ans. 36135000 Ans. 32768000 Ans. 23040000 (5) 18001 (6) 8602 (7) 3007 26000 1030 9100 108006 25806 3007 36002 8602 27063 Ans. 468026000 Ans. 8860060 Ans. 27363700 (8) 80600 (9) 70302 (10) 904000 7002 80300 10200 1612 210906 1808 5642 562416 904 Ans. 564361200 Ans. 5645250600 Ans. 9220800000 DIVISIOiN. SUGGESTIONS TO TEACHERs. —Since division is only a short method of performing many subtractions, when the numbers to be subtracted are all equal to each other; and since we calln only find the difference between numbers of the same kind, it is obvious that the divisor and dividend are always of the same name. Thus, if the dividend is dollars, the divisor must be dollars; if the dividend is an abstract number, the divisor must be the same. Since the quotient shows how many times the same number has been subtracted, or, which is the same thing, how many times the divisor is contained in the dividend, it is always an abstract number. In the question, I-ow many yards of ribbon at 2 cents a yard, can be bought for 6 cents? the answer is evidently 3 yards. It is not, however, to be inferred from this, that the quotient of 6 cents, divided by 2 cents, is 3 yards. The analysis of the question is, Since 2 cents buys 1 yard, and 6 cents is 3 i~ --- _.. —-.. _.......... __ -.. w 20 KEY TO RAY'S THIRD BOOK. times 2 cents, therefore 6 cents will buy three times as many yards as 2 cents, that is, 3 yards. (See Division, in Ray's Arithmetic, Second Book, page 43.) Since we commence on the right in Addition, Subtraction, and Multiplication, it may be asked, why do we commence on the left in Division? The reason is, that remainders are often left in the higher orders, which must necessarily be carried to the lower orders. ARTICLE 42. (4) 3)60 (5) 2)400 (6) 3)6000 (7) 4)80000 Ans. 20 Ans. 200 Ans. 2000 Ans. 20000 ARTICLE 43. (2) 3)693 (3) 4)848 (4) 2)4682 Ans. 231 Ans. 212 Ans. 2341 (5) 4)8408 (6) 3)36936 (7) 2)88468 Ans. 2102 Anis. 12312 Ans. 44234 ARTICLE 44. (4) 3)462 (5) 5)1170 (6) 4)948 Ans. 154 Ains. 234 Ans. 237 ARTICLE 45. (12) 2)8652 (13) 3)406235 (14) 4)675043 Ans. 4326 Ans. 1354112 Ans. 1687603 (15) 5)984275 (16) 6)258703 (17) 6)523408 Ans. 196855 Anis. 43117- Anis. 872344 (18) 7)8643275 (19) 8)6032520 (20) 9)9032706 Ans. 12347534 Ans. 754065 Ans. 1003634 (21) 10)1830024 (22) 11)603251 (23) 12)41674008 Ans. 183002-A4 Ans. 54841 Ans. 3472834 and~~~~ --- ---- -M ——`.~~ 4 —s DIVISION OF SIMPLE NUMBERS. 21 (24) 3)894 (25) 4)140 (26) 4)812 Ans. 298 oranges. Ans. 35 cents. Ans. 203 barl. (27) 9)423 (28) 12)540 (29) 4)321276 A1zs. $47 Ans. 45 years. Ans. 80319 gals. (30) 8)1736 (31) 7)734566 (32) 11)495 Ans. 217 barl. Ans. 104938 weeks. Ans. 45 (33) 9)3582 (34) 2)56 (35) 2)3725 Ans. 398 Ans. 28 Ans. 1862', (36) 3)147 (37) 4)500 (38) 5)1945 Ans. 49 Ans. 125 Ans. 389 (39) 6)4476 (40) 7)2513 (41) 8)5992 Ans. 746 Ans. 359 Ans. 749 (42) 9)8793 (43) 10)1090 (44) 11)4125 Ans. 977 Ans. 109 Ans. 375 (45) 12)5556 (46) 4)144 (47) 5)195 Ans. 463 3)36 each son's share. 3)39 Ans. 12 Ans. 13 ARTICLE 46. (3) (5) (6) 13)4056(312-Ans. 14)11577(826-i Ans. 15)48690(3246 Azs. 39 112 45 15 37 36 13 28 30 26 97 69 26 84 60 13 90 90 22 KEY TO RAY'S THIRD BOOK. (7) (8) 23)1110960(48302: 4- Quotient. 67)122878(1834 Quotient. 92 67 190 558 184 536 69 227 69 201 60 268 46 268. 14 (9) (10o) 53)12412(234'S Quotient. 72)146304(2032 Quotient. 106 144 181 230 159 216 222 144 212 144 10 (11) (12) 54)47100(87214' Quotient. 88)71104(808 Quotient. 432 704 390 704 378 704 120 108 12 (13) (14) 66)43956(666 Quotient. 99)121900(1231i Quotient. 396 99 435 229' 396 198 396 310 396 297 130 99 31 DIVISION OF SIMPLE NUMIBERS. 23 (15s) (16) 112)25312(226 Quotient. 123)381600(3102156A4 Quotient. 224 369 291 126 224 123 672 300 672 246 54 (17) (18) 204)105672)518 Quotient. 1234)6o0000(486.. Quotient. 1020 4936 367 10640 204 9872 1632 7680 1632 7404 276 (19) (20) 4674)47263488(10112 Quot. 2034)26497935(13027,1 Quot. 4674 2034 5234 6157 4674 6102 5608 5593 4674 4068 9348 15255 9348 14238 1017 (21) (22) 9876)48905952(4952 Quotient. 12345)4049160(328 Quotient. 39504 37035 94019 34566 88884 24690 51355 98760 49380 98760 19752 19752 24 KEY TO RAY'S THIRD BOOK. (23) (24) 973)5521G60000(567482s4g Quotient. 15)3465(231 Ads. 4865 30 6566 46 5838 45 7280 15 6811 15 4690 3892 (25) 7980 26)364(14 days. 7784 26 1960 104 1946 104 14 (27) (26) 19)1083(57 dollars. 107)9523(89 bushels. 95 856 133 963 133 963 (29) (28) 63)14868(236 hhd. 365)25000(68 dollars per day. 126 2190 226 3100 189 2920 378 $180 rem. 378 (31) (30) 365)37960($104 1235)6571435(5321 Ans. 365 6175 1460 3964 1.460 3705 2593 2470 1235 1235 ~~-~ —------------— __ — DIVISION OF SIMPLE NUMBERS. 25 (32) (33) 405)1247400(3080 Ans. 1006)10401000(10338ts7 Quotient. 1215 1006 3240 3410 3240 3018 0 3920 3018 9020 8048 972 A RTICLE 47. (3) 9)2583 (4) 4)6976 (5) 4)2744 (6) 6)6145 7)287 8)1744 7)686 7)1024-1 rem. Ans. 41 Ans. 218 Ans. 98 Ans. 146-2 6X2+-1=13 rem. (7) 11)19008 (8) 8)7840 12)1728 8)980 Ans. 144 Ans. 122-4 8X4-=32 rem. (9) 8)14771 (10) 9)10206 9)1846-3 9)1134 Ans. 205-1 Ans. 126 8X1+3 —11 rem. ARTICLE 48. (2) 110)26812 (3) 1100)47100 (4) 1100)372101 Ans. 268 -0 Ans. 47 Ans. 372~-f-o (5) (6) (7) (8) 1100)462150 1100)620134 11000)181003 11000)3751000 Ans. 462-i-ou Ans. 62018 4 Ans. 18 T Ans. 375,~~~~8 K D'C 26 KEY TO RAY'S THIRD BOOK. ARTICLE 49, (3) 41000)731005 (4) 91000)361001 Ans. i.8-0 -- Ans. 4-g 1 (5) 111o000)10781000 (6) 1810)401617(22327 Quotiezt. 36 Ans. 98 41 36 56 54 2 (7) () 21100)9072137(432!TgiG Quot. 64100)3640106(565@S Quot. 84 320 67 440 63 384 42 56 42 (9) 2510000)7654]6037(306~%%Vo7 Quotient. 75 154 150 4 (10) 634100)435637154(687 f 95j1 Qu-otient. 3804 5523 5072 4517 4438 79r DIVISION OF SIMPLE NUMBERS. 27 ARTICLE 50. (1) (2) 7895)434225(55 Ans. 327)106929(327 Ans. 39475 981 39475 882 39475 654 2289 (3) 2289 375)10741125(28643 Ans. 750 (4) 3241 60352)63550656(1053 Ans. 3000 60352 2411 319865 2250 301760 1612 181056 1500 181056 1125 1125 ARTICLE 56. (1) (2) 96 500 243 1St. 243 1265 120 271 61 304 772 55 - - 225 - Ans. 229 304 2d. - Ans. 493 271 79 772 225 3d. (3) (4) 157 428 9503 57068 264 186 586 16967 305 - 4794 97 614 1234 Ans. 40101 123 850 946 16967 614.Ans. 332 28 KEY TO RAY'S THIRD BOOK. (5) $123078 $237 spent in Jan. 538 8706 301 spent in Feb. 10 months left. 131784 538 5380 5918 5380 Ans, $125866 $5918 spent. (6) 72 (7) 4 yards. yards. 73 88 288 w. 9 12 24 657 1056 1152 657 576 Ans. 1713 Ans. 6912 lights. (8) days. (9) 39 vols. (10) 365 774 86)31173(362 Ans. 49 258 156 - 3285 273 537 1460 273 516 17885 30186 pages. 213 99 2 172 160965 60372 columns. 41 160965 67 Ans. 1770615 422604 362232 4044924 lines. (12) 63 gallons. 28 (11) 10 5 - $ 40449240 words. 15)315(21 Ans. 25)1400(56 Ans. 30 125 4044924 lines. 47 15 150 15 150 28314468 15 16179696 190111428 letters...................... EXERCISES IN PRECEDING RULES. 29 (13) 73900 (14) 148 148 70 56 56 214)73830(345 Ans. 204 sum. 92 642 92 diff. 963 408 856 1836 1070 23)18768(816 Ans. 1070 184 36 23 138 138 $ (15) $ (16) 15)360(24=gallons filled 60 15 30 in 1 hour. 8 14 - _- 60 480 60 60 210 15 - ~~- ~ 210)3610 6)690 210 18=gallons emptied in 115 Ans 1 hour. 24 (17) 18 70 6=gallons left in cistern per hour. 2100 $6000 2100 6)360 25)3900(156 acres. 60 hours. Ans. 25 70 acres. 140 226 acres Ans. (18) 222 125 125 150 Ans. 347 150 30 IKEY TO RAY'S THIRD BOOK. (19) 500 (20) 45 (21) 340 (22) 27 135 15 11 98 Ans. 365 225 9)351 125 diff. 45 246 39 675 15 Ans. 371 65 - Ans. 24 Ans. 740 ARTICLE 61.a-CANCELLATION. 4X1$ 41x11 (2) -— =4 Ans. (3) -- 11 Ans. 3 15X 32;4x4 (6) - =30 Ans. (7) — =12 Ans. 37 X 1 3 36X (8) - -111 Anzs. (10) — =12 Ans. 14X9 7X>X$Axx3 (11.) --- =21 Ans. ARTICCLE 61.b (1) 42X25X18 _XX2X$X5X$X6 2 -1 --— =2X5X6=-60 Ans. 21X 15 9,xx O$x 2 1X 23 (2) =2 X 23=46 Ans. 14X15 2XTX3X0 (3) - -=6 Ans. 35 - xo cg d b —------------— ~~ —-------------- COMPOUND NUMBERS.-U. S. MONEY. 31 (4) 21X11X6X26 _XX11XlX ~X3X$X 113..11X3=33 13X3X14X2 OX$X.X XA Ans. 21X 15X 33X 8X14X17 (5) 20X34X 22 X27 _7x ~x x x xx 1x x x x x TXtX x~ 7X$X=X — X=7XX$XAXAX7XfXT 7 =49 AX XXX iXX 1X X$X$X$ XX Ans. (6) 21X95X6 _ x$x$X19X2X$.....= 19 X 2=X 38 Ans. 35X9 tX0X$x $ FEDERAL, OR U. S. $ MONEY. ARTICLE 67.-NUmIERATION. (1) Eighteen dollars, sixty-two cents, five mills. (2) Twenty dollars, thirty-two cents, four mills. (3) Seventy-nine dollars, five cents. (4) Forty-six dollars, three mills. (5) Seventy dollars, one cent, five mills. (6) One hundred dollars, twenty-eight cents, three mills. (7) One hundred and fifty dollars and two mills. (8) One hundred dollars and three mills. (9) Six dollars and twelve cents. (10) Three dollars and six cents. (11) Four dollars and thirty-one cents. (12) Five dollars and forty-three cents. (13) Twenty-nine dollars. (14) One hundred dollars and three cents. (15) Twenty dollars and five cents. (16) Forty dollars and seven mills. ARTICLE 63.-NOTATION. The examples in the Arithmetic are all answered there. It is recommended to the teacher to exercise the pupils with additional examples, should it be found that they are not sufficiently acquainted with the subject, after going through with those in the book. ls -- -- --------— ~ —------- 32 KEY TO RAY'S THIRD BOOK. ARTICLE 71.-RED UCTION. Since the operations in this section consist simply in adding ciphers or removing them, or erasing points or inserting them between the different denominations, it is deemed unnecessary to occupy space, as the whole solution, when presented to the eye, would consist in notling more than writing down the question to be solved, and then placing the answer under it. ARTICLE 72.-ADDITIONo (2) $ cts. (3) $ cts. m. (4) $ cts. 17.15 18.04 1 43.75 23.43 16.31 7 29.18 7.19 100.50 3 17.63 8.37 87.33 8 268.95 12.31 718.07 - Ans. $222.19 9 Ans. $68.45 Ans. $1077.58 (5) $ cts. m. (6) $ cts. m. (7) $ cts. m. 200.00 0 504.06 3 5.07 0 43.87 5 420.19 7 30.20 3 56.93 7 105.50 0 100.00 5 8.50 0 304.00 5 60.02 0 2.31 3 888.45 5 700.01 1 1000.10 0 Ans. $311.62 5 Ans. $2222.22 0 40.00 4 64.58 7 Ans. $2000.00 0 ARTICLE 73e-SUJBT RACTIO N. (2) $ cts. m. (3) $ cts. (4) $ cts, (5) $ cts. 29.34 2 46.28 20.05 3.00 17.26 5 17.75 5.50.03 Ans. $12.07 7 Ans. $28.53 Ans. $14.55 Ans. $2.97 (6) $ cts. m. (7) $ cts. m. (8) $ cts. m. (9) S cts. m. 10.00 0 50.00 0 1000.00 0 1000.43 5 1 50 5 1.01 1 900.68 7 Ans. $9.99 9 Ans. 49.49 5 Ans. $998.98 9 Ans.n 99.74 8,, —----------------- COMPOUND NUMBERS.-U. S. MONEY. 33 ARTICLE 74, —MULTIPLICATION. (2) $ cts. (3) $ cts.m. (4) $ cts. m. (5) $ cts. 7.83 12.09 3 23.01 8 35.14 8 9 16 53 Ans. $62.64 Ans. $108.83 7 138108 10542 23018 17570 Ans. $368.28 8.Ans. $1862.42 (6) $ cts. (7) $ cts. (8) cts. m. (9) $ cts. 125.02 40.04 12 5 3.28 62 102 17 38 25004 8008 875 2624 75012 4004 125 984 Ans. $7751.24 Ans. $4084.08 Ans. $2.12 5 Ans. $124.64 (10) $ cts. (11) $ cts. (12) 465 Here, instead of 1.06 5.75 7 multiplying 7 338 38 - cents by 465, Ans. 32.55 it amounts to 848 4600 the same thing 318 1725 to multiply 465 318 by 7, and is more Ans. $218.50 convenient. Ans. $358.28 (13) cts. (14) $ cts. (5) $ cts. 34 5.67 2.69 89 24 169 306 2268 2421 272 1134 1614 269 Anns. $30.26 iAns. $136.08 Ans. $454.61 Key. — 3 3-4 KEY TO RAY'S THIRD BOOK. (I6) $ cts (17) 73 (18) 281 1.25 63 4 691 -- 219 1124 lb. 125 438 65 1125- - 750 4599 gal. 5620 55 6744 Ans. $863.75 22995 AnSo. $73.050 22995 Ans. $2529.45 (19) 35 (20) 312 (21) 10 11 18is $ cts. 3 1.25 350 yd. 3432 hours. - 54 1 13 Bushels 54 500 Ans. $3.50 10296 625 3432 -Ans. $67.50 Ans. $446.16 (22) cts.rm. (23) 17 cts. n. 10.00 1 51 14 7 150 - 867 1 17 500).050 85 1029 10001 -- 882 867 lb. 1176 Ans. 1500.15 0 Ans. $127.44 9 AP TICLE, 75. —DIV IsION. (4) (5) ets. yd. (6) Bhl. 8)280 25)300(12 Ans. 805)16100(20 Ans. 25 1610 Ans. 35 yd. - 50 0 50 COMPOUND NUMBERS -U. S. MONEY. 35 (7) 75)1200(16 Ans. (8) 1125)234000(208 bu. Ans. 75 2250 450 9000 450 9000 (9) 4)92.250 (10) 8)57.500 (11) $ cis. 16)25.76(1.61 Ans. Ans. $23.062+ Ans. $7.187+ 16 97 96 16 16 (12) cts. m. (13) $ cts. m. 755)328.425(.43 5 Ans. 313)800.000(2.55 5+ Ans. 3020 626 2642 1740 2265 1565 3775 1750 3775 1565 1850 1565 (14) $ cts. m. cts.m. (15) $ cts. m. cts. nm. 133)10000.00 0(75.18 7+-Ans. 154)2705.01 0(17.56 5 Anps. 931 154 690 1.165 665 1078 250 870 133 770 1170 1001 1064 924 1060 770 931 770 I,:-~ r~ ~~ ~ --- 36:KEY TO RAY'S THIRD BOOK. (16) 25 (17) 235 15 8 m n. cts.m. 125 lb. 1880)122200(65=6 5 Ans. 25 11280 ~ ~ cts. cts. lb. 375)6000('16 Ans. 9400 375 9400 2250 2250 ARTICLE 76. (1) $ cts. (2) $ cts. (3) $ Ct8. (4) 47.50 35.25 18.38 $ cts. $ cts. 38.45 23.75 81.62.75 5.00 15.47 -.35 3.10 19.43 59.00 100.00.50 59.00 100.00 1.50 Ans. $1.90 Ans. $120.85 1.00 100.00 - $3.10 Ans. $119.00 Ans $300.00 (5) (6) $ cts. cts. cts. $ cts. 50.00 8.10.75 20.00 30.50 1.65 8'1975 6.00 19.50 4.00 $6.00 Ans..25 cts. 6 $19.75 Ans. $117.00 (7) $cts. (8) $ cts. (9) 143 (10) 435 3.85 37.06.13 45 1.25 200.85 - 2.50 400.00 429 2175 1.50 236.75 143 1740 124.34 9.10 $18.59 $195.75 999.00 12.60 $ cts. 889.25 - 400.00 21.75 - Ans. $5.99 195.75 9.10 Ans. $109.75 $204-25 AnsAns. $12.654.25 Ans. -12.65 COMPOUND NUMBERS.-U. S. MONEY. 37 (11) 365 (12) 21 (13) 19 (14) $ cts..65 3 4 2000.00 163.75 1825 63 76 2190 35 23 5)1836.25 $237.25 315 228 5)367.25 189 152 $400.00 -- -Ans. $73.45 237.25 Ans. $22.05 Ans. $17.48 Ans.$162.75 (15) $ cts. (16) $ cts. (17) $ cts. 4)516.00 410)910.00 22= —2Xl 1000.00 _ls)s ~~.500.00 4)129.00 10)2.25 500.00 cts. 2)1500.00 43)32.25(75 Ans. Ans. cts. 22 5 m. 301 11)750.00 215 Ans. $68.18+ 215 (18) $ cts. (19) $cts. 9 lbs. at 16 cts.........$1.44 8 yards at $1.10........8.80 4 lbs. at $1.25...........5.00 18 yards at 25 cts.......4.50 45 lbs. at 9 cts.......... 405 25 yards at 15 cts.......3.75 17 lbs. at 13 cts.......2.21 12 yards at 35 cts....... 4.20 6 yards at 65 cts.......3.90 Ans. $12.70 Ans. $25.15 (20) $ cts. (21) $ cts. 5 copies at 35 cts.......1.75 27 copies at 19 cts......... 5.13 12 dozen at $10.50....126.00 25 " "27 "......... 6.75 6 M. at $1.60............9.60 8 " " $4.50........ 36.00 5 quires at 25 cts.......1.25 27 " " $2.25.........60.75 3 copies at $4.50......13.50 18 " " $1.50.........27.00 75 " 31cts........23.25 Ans. $152.10 Ans. $158.88 38 KEY TO RAY'S THIRD BOOK. COM iPOUND ITUM iBEIRS. In the remaining part of this work, instead of writing out the solution in full, we have merely indicated the operations to be performed. When a person understands the elementary rules, and the meaning of the signs, the method of solution is just as easily understood, as if written out at length. Pupils, however, should be taught to work the questions, after the manner given in the book. In Reduction, pupils should be frequently required to give the reason for multiplying or dividing by the numbers used: also to point out the name of the remainder, and give the reason for it. ARTICLE 81.-DRiY MEASURE. bu. pt. pk. pt. (11) 2X4XSX2-128 Ans. (12) 12X8X0 —192 Ans. bu. qt. (13) 8X4XS=256 Ans. bt. plc. pl. pk. pt. (14) 1X4+1=-l5: 5><8X2=80 Ans. bu. pk. pk. qt. qt. qt. pt. (15) 2X4-8: 8XS+2-66: 6GX2-132 Ans. bu. pk. pk. pk. qt. qt. qt. pt. (16) 4X4+-2=18: 18X8+1=145: 145X2=290 Ans. bu. pk. pk. pPk. qt. ct. qt. pt. pt. (17) 7X4+3=31: 31X8+7-=255: 255X2 —1=511 Ans. bu. pk. pk. qt. qt. pt. pt. (18) 3X4 —12: 12XS=96: 96X2+=1 93 Ans. pt. qt. qt. pk. pk. bu. (19) 384-. -2192: 1928 —8=24: 24 *. 46 Ans. pt. qt. pt. qt. pk. qt. pk. qt. pt. (a0) 47.e2=-23 1:23 — 8=2 7. Ans. 2 7 1 pt. qt. pt. qt. pk. qt. pk. bu. pk. (21) 95 — 2=47 1: 47 —8=5 7: 5. 4 — 1 1: collecting the different remainders, the Ans. is, 1 bu. 1 pk. 7 qt. 1 pt. ~~ —-- ----------- -- - - - -— ~~~~~~~~~~~~~~~~~~~~~~~~ef REDUCTION. 39 pt qt. pk. qtt. t1. u,. pk. bu.pk.cit. (22) 5-8 * -,2. 234-.-8=-31 (): 31-:-4=7 3. As. 7 3 6 ARTICLE 82. —TrtoY EIGHT. lb. gr. lb. oz. oz. (2) 4X12X20X24=2 — 3040 Anms. (3) 5X12+4=64 Ans. lb. oz. oz, oz. pwt. pwt. (4) 9X1'2+3111: 111X20+5=22 25 Ans. lb. oz. oz. oz. pwt. pwt. (5) 14X12+11-179: 179X204-19 —3599: pwt. gr. gr. 3;599X24- 923-86399 _'ts. lb. oz, Oz, OZ. pAwt. pwt. (6) 8X1-1t't-9=l105: l5X20 15 -L3 — 213 pwt. g. gr. 2113X'24 —+17-=50729 Ans. fgr. pwt, gr. (7) 171 24z-7 3 Anls gr. pwt. gr. pwt. Oz. pwIt. 0o. pwt. gr. (8) 505. -24'21 1: 21 20=1 1. Ans. 1 1 1. gr. pAwt. gr. pwt. oz. pt. oz. lb1) Z. (9) 12530-9-24-522 2: 5'22 20,26 2: 26 12-2 2. lb. oz. pwt. gr. Alns. 2 2 2. pwt, oz. pwt, oz. lb. oz. lb. oz. pwt. (10 805 — 20-4() 5 400 1 —-3 4. Ans. 3 4 5. gr. pwvt. gr. p\wt. oz. pwt. oz. b. oz. (11) 25591-+4 —1066 7: 1066 20 = 53 6: 53 —12=4 5. lb). oz. pwt. gl-. Ans. 4 5 6 7. A.RTICLE 83.-APOTHECARIES, -NVT EIIGHI-'T. lb>~~ ~ gr. (1) 3Xl2X8X3X20= —:'172SO Ans. 40 KEY TO RAY'S THIRD BOOK. lb 3 3 3 3 D (2) 4X12=48: 48X8+5=389: 389X3=1167: 3 gr. gr. 1167X20+2 —23342 Ans. lb 8 8 3 3 D. (3) 7X12=84: 84X8=672: 672X3-+2=-2018 Ans. b i b 3 8 3 3 D 3 (4) 7X12+2-86: 86X8=688: 688X3+1=2065: gr. 2065X20-41300 Ans. (5) 67 —; 12=5 7 Ans. 3 33 3 Tb t1l 3 (6) 431 —-8=53 7: 53l12_4 5. Ans. 4 5 7. D 3 3 8 3 8 ltb 3 3 (7) 975- -3=325: 325 8=40 5: 40.-12-3 4. Ans. 3 4 5. gr. 3 gr. 3 3 3 (8) 6321+e 20=316 1: 316 -3=105 1: 105-8=-13 1: b I is I 3 D gr. 13 -.12-1 1. Ans. 1 1 1 1 1. gr. D gr. D 3 3 3 (9) 30941-.-20=1547 1: 1547 33=515 2: 515-.-8-64 3: 3 lbf i lb 3 3 D3gr. 64__12=5 4. Ans. 5 4 3 2 1. gr. 3 gr. D 3 3 8 3 (10) 29239 -20=1461 19: 1461_. 3=487: 487- 8=60 7: 3 Blb lb3 gr. 60. 12=5. Ans. 5 7 19. ARTICLE 84. —AVO TRDUPOIS WEIGHT. cwt. lb. cwt. qr. qr. qr. lb. (1) 2X4X25=200 Ans. (2) 3X4+3=15: 15X25=375 [Ans. REDUCTION. 41 T. cwt. cwt. cwt. lb. (3) 1X20+2222: 22X4X25=2200 Ans. 1T. cwt. owt. qr. qr. qr. lb. (4) 3X20 —60: 60X4+3=243: 243X25 —6075 Ans. cwt. qr. qr. qr. lb. lb. (5) 4X4+1= —17: 17X25+19 —444 Ans. T. cwtt.. qt. qr. qr. qr. lb. lb. (6) 5X20=-100: 100X4+3-403: 403X25+15=10090 [Ans. cwt. qr. qr. qr. lb. lb. lb. oz. oz. (7) 2X4+3=11: 11X25+2277: 277X16+-12=4444 [Ans. cwt. qr. qr. lb. lb lb. oz. (8) 2X4-=8: SX25+ —17=217: 217X16=3472: oz. dr. dr. 3472X16+3=55555 Ans. T. cwt.cwt. cwt. qr. qr. lb. lb. (9) 1X20-+6-26: 26X4=104: 104X25+4=2604: lb. oz. oz. oz. dr. dr. 2604X16+2=41666: 41666X16+10-666666 Ans. lb. qr. lb. qr. cwt. cwt. lb. (10) 4803- 25=192 3 192 -4=48. Ans. 48 3. lb. qr. qr. cwt. cwt. T. cwt. (11) 22400~'. 25896896:96-4 —224: 224-*-20=11 4 Ans. dr. oz. oz. lb. (12) 2048000-.-16=128000: 128000-. 16=8000: lb. qr. qr. cwt. cwt. T. 8000 —. 25-320: 320 -4-4=80: 80-. 20-4 Ans. dr. oz. dr. oz. lb. oz. (13) 64546 16 —4034 2 4034-. 16=252 2: lb. qr. lb. qr. cwt. qr. cwt.qr. lb. oz. dr. 252 25 —10 2: 10. 4-2 2. Ans. 2 2 2 2 2. oz. lb. oz. lb. qr. qr. cwt. qr. (14) 97203 — 16-6075 3: 6075-. 25-243: 243. 4-60 3: cwt. T. T. qr. oz. 60 -. 20-3. Ans. 3 3 3. 42 KEY TO RAY'S THIRD BOOK. drl. oz. OZ. lb. oz. (15) 544272-. 16 —34017: 34017-:16=-126 1: lb. qr. lb. qr. cwt. qr. cwt. T. cwt. 2126 — 25=85 1:85- 4-=21 1:21 — 2031 1. T. cwt. qr. lb. oz. Ans. 1 1 -1 1 1. lb. lb. lb. qr: lb. qr. cwt.qr. (16) 18X52=936: 936 —. 25-37 11: 37- 4=9 1. cwt. qr. lb. Ans. 9 1 11. A TICLE 85.-LONG ]MEASUREo yd. ft.. ft.. ft. in. i nI (1) 2X3+-2=8: SX12+-7 —103 Ans, yd. ft. ft. in. in. (2) 7X3=21: 21X12l2+11-263 Ans lIii. fur. fur. rd. (3) 12XS=96: 96X40,-3840 Aas. mi. fur. fur. fur. rd. (4) 7X8+6=62': 62X40=2480 Ans. mi. fur. fur. rd. rd. (5) 9X8 —72: 72X40+31=2911 Ans. in. ft. in. ft. yd. ft. yd. ft. in. (6) 133-.-12=11 1: 11 —3=3 2 A-ts. 3 2 1 in. ft. in. ft. yd. yd. in. (7) 181 —12=15 1: 15 —3=5. Ans. 5 1. rd. fur. fur. mi. (8) 2240-:-40=56 56-.-8=7 Ans. rd. fur. fur. mi. fur. (9) 2200 —40=55: 55 -8-=6 7 Ans. ~~ —-- ------ ---- ---- ----— ~~~~~~~~~~~~~~~~~, REDUCTION. 43 ARTICLE 88.-LAND IMEASURE. sq. yd. sq. in. A. P. (1) 8X9X144=10368 Ans. (2) 4X4X40=640 Anzs. sq. mi. P. (3) 1X640X4X40-102400 Ans. sq.yd. sq.ft. sq.ft. sq.ft. sq. in. (4) 2X9+3-21: 21X144=3024 Ans. A R.R. R. P. P. (5) 5X4+-2-2: 22X40+20-900 Ans. P. R. R. A. (6) 960 —40-_24 24-.-4=6 Ams. sq. in. sq. ft. sq. ft. sq. yd. (7) 3888 —144-27: 27 — 9=3 Ans. P. R. P. R. A.R. A.R.P. (8) 243 —40=6 3: 6 — 4=1 2. Ans. 1 2 3. P. R. P. P. A.R. A. R. P. (9) 603-4-40=15 3: 15 —4=-3 3. Ans. 3 -3 3. sq. in. sq. ft. sq. ft. sq. yd. sq. ft. (10) 4176 —144=29: 29-.-9-3 2 Ans. ARTICLE 89. ft.. ft. sq. ft. yd. yd. sq. yd. (2) 16X12-192 Ans. (3) 5X4-=20 Ans. ft. yd. ft. yd. ft. yd. ft. yd. (4) 18 — 3=6: 12 —3 —4: 21 —3=7: 15-.-3=5: sq. yd. sq. yd. sq. yd. 6X4-24: 7X5=35: 24-1-35-59 Ans. ft. ft. sq. ft. sq. ft. sq. yd. (5) 18X14-252: 252 -9=28 Ans. rd. rd. P. P.. R. R. A. (6) 35X32=1120: 1120, —40-28: 28 4 —4-7 An?. 44 KEY TO RAY'S THIRD BOOK. ARTICLE 90. sq. ft. ft. ft. sq.yd. sq. ft. sq. ft. ft. ft. (1) 132~. 11=12 Ans. (2) 30X9=270: 270 —18=15 Ans, A. P. P. rd. (3) 9X4X40=1440: 1440-. 45-32 Ans. A. P. P. rd. (4) 21X4X40-3360: 3360-.-35-96 Anzs. ARTICLE 93.-CUBIC MEASURE. cu. yd. cu. in. C. cu. ft. (1) 2X27X1728=93312 Ans. (2) 28X128=3584 Ans. C. C. ft. C. cu. in. (3) 34X8=272 Ans. (4) 1X128X1728=-221184 Ans. cu. in. cu. ft. cu.t. cu.yd. cu.ft. (5) 63936 1728-37: 37 —27-1 10 Ans. cu. in. cu. ft. cu. ft. tn. cu.ft. (6) 492480 —1728=285: 285 ~.40-7 5 Ans. ft. ft. ft. cu.ft. yd. yd.yd. cu.yd. (7) 8X5X4=160-Ans. (8) 8X5X2=80 Ans. ft. ft. ft. cu.ft. cu.ft. cu.yd. (9) 18X15X7-1890: 1890-. 27=70 Ans. ft. ft. ft. cu.ft. cu. 0f C. (10) 40X12X8-3840: 3840 128=30 Ans. ARTICLE 94.-CLOTHt MEASURE. yd. na. E. F1. na. (1) 19X4X4=304 Ans. (2) 14X3X4-168 Ans. yd. qr. qr. qr. na. na. (3) 5X4+2=22: 22X4+3-=91 Ans. E. En. qr. qr. qr. na. (4) 13X5+1=66: 66X4-264 Ans. REDUCTION. 45 E. Fr. qr. na. na. (5) 23X6+3=141: 141X4+2=566 Ans. la. qr. na. qr. yd. qr. yd. qr. na. (6) 159 —4=39 3: 39 4=9 3. Ans. 9 3 3. na. qr. na. qr. E.Fr. na. EFr. qr. na. (7) 287 4=71 3: 71 6=11 5. Ans. 11 5 3. yd. qr. qr. E.F1. (8) 6X4=24: 24-.~3=8 Ans. yd. qr. qr. qr. E. F1. (9) 9X4+3=39: 39-. 3=13 Ans. yd. qr. qr. qr. E. Fl. qr. (10) 12X4+1=49: 49. 3=16 1 Ans. E.Fl. qr. ar. yd. qr. (11) 37X3=111: 111 —. 4=27 3 Ans. E.En. qr. qr. E.Fr. (12) 36X5=180: 180-6 —=30 Ans. E.En. q q. qr. E.Fr. (13) 22X5+4=-114: 114 -6=19 Ans. E.En. qr. qr. yd. qr. (14) 47X5-235::235. 4 —58 3 Ans. ARTICLE 95,-LIQUID MEASURE. gal. pt. gal. gi. (1) 17X4X2=136 Ans. (2) 13X4X2X4=416 Ans. hhd. pt. (3) 2X63X4X2=1008 Ans. T. gi. (4) 5X4X63X4X2X4=40320 Ans. T. hhd. hhd. hhd. gal. (5) 3X4+3=15: 15X63=945 Ans. 46 KEY TO RAY'S THIRD BOOK. hhd. gal. gal. gal. qt. qt. pt. pt. (6) 1X63+60-123: 123X4=492: 492X2+1 —985 Ans. hhd. gal. gal. gal. qt. qt. qt. gi. (7) 2X63+17=143: 143X4+3-3=575: 575X2X4-4600 [Ans. T. hhd. hhd. gal. gal. gal. pt. pt. (8) 2X4=8: 8X63+62=566: 566X4X2+1=4529: pt. gi. 4529X4=18116 Ans. gi. pt. pt. qt. qt. gal. (9) 96 —4-24: 24. 2=12: 12 — 4=3 Ans, gi. pt. pt. qt. qt. gal. (10) 6048 —4=1512: 1512 —-2_756: 756 —4=189: gal. hlhd. 189 —-63=3 Ans. gi. pt. pt. qt. qt. gal. (11) 32256-4=8064: 8064 —2=4032: 4032-.-41008: gal. hhd. hhd. T. 100 —63 —16: 16 —4=4 Ans. gi. pt. gi. pt. qt. qt. gal. qt. (12) 4050 —4-1012 2: 1012-2=506: 506- -4-126 2: gal. hhd. hhd. qt. gi. 126 * 63-2. Ans. 2 2 2. gi. pt. gi. pt. qt. qt. gal. (13) 30339 —4 —7584 3: 7584 —-2=3792: 3792 -4 —948: gal. hhd. gal. hhd. T. hhd. T. hhd. gal. gi. 948-1-63=15 3: 15 4 —=3 3. Ans. 3 3 3 3. gi. Pt. gi. pt. gi. pt. qt. pt. qt. gal. qt. (14) 10125+ 4-2531 1: 2531-.-21265 1: 1265 4=316 1: gal. lhh. gal. hhd. T. hMd. T. hhd. gal. qt. pt. gi. 316-)'63: —z5 I: 5-i-4=1 1.:InsS. 1 1 1 1 1 1..~~ — I —6 REDUCTION. 47 pn. gi. (15) 3X84X4X2X4-=8064 Ans. p. qt. (1G) 5X126X4=2520 Ans. pt. qt. qt. gal. gal. tr. (17) 5712 —— 2856: 2856 4=714 714-42=17 Ans. A R T I L 96.-BEER MEASURE. hhcld. pt. (1) 4X54X 4X2'-1728 Ans. hhd. gal. gal. qt. q't. qt. pt. (2) 7X54=378: 37SX44-3=1515: 1515X2 —3030 Ans. pt. qt. qt gal. gal. bl. gal. (3) 1000 —-=500: 500 —4 —125: 125 —-36=3 17. An1s. p-t. qt. pt. qt. gal. qt. gal. hhcl. gal. (4) 443. —2-221 1: 221-*4=55 1: 55-.54=1 1. lhhd. gal. qt. pt. Alns. 1 1 1 1. ARTICLE 97.-TIMIE. hr. SCG. da. min. (1) 2X6OX60=7200 Ans. (2) 7X24X60-10080 Ans. da. hr. hr. hr. min. m. min. sec. sec. (3) 1X24+3=27: 27X60+44=1664: 1664X60+-3=99843 [A.zs. wk. da. da. da. hr. hr. (4) 9X7+6-69: 69X24+10=-1666: hr. min. min. 1666X60-+40=100000 Ans. mon. wk. wk. da,. da. da. hr. (5) 1X4=4: 4X7+-3=31: 31X,-4=744: hr. mlin. n in. 744X60+4=446)44 Ars.I. -.-. M 48 KEY TO RAY'S THIRD BOOK. yr. da. da. da. hr. hr. hr. min. (6) lX365+20=385: 385X24+19=9259: 9259X60+15 min. min. sec. sec. -=555555: 555555X60+33=33333333 Ans. da. hr. hr. hr. min. min. (7) 365X24 —5=8765: 8765X60-148=-525948: min. sec. sec. 525948X60+48=-31556928 Ans. sec. min. min. hr. (8) 10800-.:60=180: 180 60=3 Ans. sec. min. min. hr. hr. da. (9) 432000-. 60=-7200: 7200 —60=120: 120 — 24-5 Ans. sec. nin. sec. min. hr. min. hr. min. sec. (10) 73227. 60=122 2: 122-. 60=2 2. Ans. 2 2 2. rlin. hr. min. hr. da. da. min. (11) 4323-60=72 3:72. 24= —3. Ans. 3 3. mnin. hr. hr. da. hr. da. -wk. wk. hr. (12) 20280 -60=338: 338 — 24=14 2:14-7=2. Ans. 2 2. min. hr. min. hr. da. da. wk. da. (13) 41761 —60-696 1: 696 —24=29: 29-~ 7=4 1; wk. mon. mon. da. min. 4 -4=1. Ans. 1 1 1. ARTICLE 98.-CIRCULAR MEASURE. (1) 50X60+3'=303' Ans. (2) 8oX60+41l=-521': 521'X60+45"-31305"/ Ans. (3) 3sX30=900: 90~X60 —25=-5425' Ans. (4) 1CX360X60X60=1296000/ Ans. (5) 244// —. 60=4' 4i" Ans. (6) 915/' 60-15~ 15' Ans. (7) 1861-. 60-310 1/: 31~-.-30-1s 1~. Ans. 1s 1~ 1/. REDUCTION. 49 ARTICLE 100. bu. pt. cts. cts. $ (1) 2X4XSX2 —128: 5X128=640=6.40 Ans. bu. pl. plc. cts. ets. $ (2) 3X4 —2 —14: 50X14=700=7.00 Anzs. pk. qt. qt. qt. pt. cts. $ (3) 3X8+-3=-27 27X2-54: 3X54=-1.62 As. $' cts. ts. cts. Ptc. pkl. bLi. (4) 3=-300: 300 —15=20: 20 —4. 5 A'zis. cts. cts, cts. qt. (5) 1.66166: 166-1-4=41 tl d 2 cts. over, which will l,uy qt. plk. qt. pk.' bu. pk. 1 pt. at 4 cts. a qt. 41-.-8=5 I: 5-.-4=1 1. bu. pl. qt. pt. A2,i. 1 1 1. Or' thus: 4 cts. a quart is 2 cts. cts. cts. pt. bu. pkl. qt. pt. a pt.; and 166 -2=83 —1 1 1 1. lAns. bu. plc. lk. p1c. pk. plk. bags. (6) 3 2 —14: 91=364:36 4 —14_=26 Ans. lb. pwt. gr. cts. cts. S (7) 1 3=5832: 3X5832=17496=174.96 Ans. lb. oz. pwt. pwt. oz. pwt. pwt. pwt. pwt. (8) 2 5 5=585: 2 5=45: 58-. 45=13 Ais. lb. oz. pwt. gr. gr. pwt. gr. gr. gr. gr. (9) 1 8 1S 1=10033: 7 —127: 10033 -127=79 Ans. lb 3 gr. gr. gr. dose. cts. cts. nm. (10) 1 4=6000: 6000 — 15=400: 12 5=125 m..111 125X400 —. 50000=50 Ans. lb 3 3 9 cts. cts. $ (11) 1 1 1-315: 4X315 —160=12.60 Anis. cwvt. fctr. lb. cts. cts. $ (12) 6 1=625: 625X3 5=575=18.75 Aiis. Key. — 4 50 KEY TO RAY'S THIRD BOOK. T. cwt. cwt. qr. $cts. $ (13) 1 l121=84: 2.25)(84=189,00 Ans. lb. oz. oz. cts. cts. $ (14) 7 8=120: 5X120=600=-6 Ans. gr. gr. lb 3 3 D (15) 20X316=6320-1 1 1 1 Ans. lb. oz. dr. dr. dr. dr. (lV) 15 9 12-3996: 3996 —. 4=999 Ans. cwt. qr. lb. oz. lb. oz. oz. oz. oz. (L7) 44 2 2=71232 9 15=159: 71232 159=448 Anss. cwt. qr. lb. lb. lb. lb. (18) 14 1 3=1428:1428-84=17 Ans. cw;t.qr. lb. lb. lb. lb. (19) 7 2 6=756 756- 12=63 Arts. cwt. lb. oz. lb. oz. oz. oz; oz. (20) 6 10=9760: 3 13-61: 9760-. 61=160 An. A. R. P. P. cts. cts. $ (21) 2 3 5=445: 20X445=8900()89 Ans. sq.yd. sq.ft. sq.ft. sq.in, cts. cts. $ (22) 2 2 = 20=2880: 2880>)5=14400=144 Ans. sq.yd. sq.in. in. in. sqin. sq.in. sq.in. (23) 1=1296: 3X2=6: 1296 —. 6=216 Ans. A. R. P. P. A. R. P. P. P. P. (24) 16 1 13=2613: 1 1 1=201: 2613 — 201=13 Ans. cu. f. cu.ft. cu.in. (25) 2X2X2=8 8X1728=13824 Ans. oz. oz. lb. oz. (26) 1000X5=5000=312 8 Ans. oz. oz. T. (27) 1000X128=128000=4 Ans. C. cu.ft. o oz. T. cwt. (23) 2X128=256: 950X256=243200=7 12 Ans. --- - -- ------ ~~~~~~~~~~~~- — ~~~u REDUCTION. 51 yd. qr. na. na. yd na. na. na. pieces. (29) 13 3 2=222: 666=10656: 10656 —222=48 Ans. yd. qr. qr. yd. qr. qr. qr. suits. (30) 5 1=21: 147=588: 588-.21=28,Ans. yd. qr. na. na. yd. na. na. na. (31) 2 1 1=37: 37 —592: 592 — 37-16 Ans. yd. qr. qr. E.FI. qr. qr. qr. suits. (32) 3 2=14: 70=210: 210. 14=15 Ans. blid. gi. cts. cts. $ (33) 1=2016: 5X2016=10080=100.80 Ans. gal. qt.. qt. qt. qt. cts. cts. $ (34) 31 2=126: 126X5 —630: 10)X630 —6300=63 Ans. $ cts. cts cts. cts. pt. gal. (35) 2=200: 200 —5-40 40=5 Ans. $. cts.m. m. m. gi. gal. (36) 12=12000: 1 5=15: 12000 —15-800=25 Ans. hhd. pt. qt. pt. pt. pt. pt. (37) 1=504: 3 1=7: 7X12=84 in 1 doz. bottles: pt. pt. doz. 504 — 84=6 Ans. gal. qt. pt. pt.:hhd. pt. pt. pt. pt. (38) 4 3 1=39: 1-=504: 504 -39=12 and 36 left: pt. gal. pt. gal. qt. 36=4 2. Ans. 12; and 4 2 left. gal. pt. gal. qt. pt. pt. pt. pt. (39) 165=1320 1 1 1=11: 1320 —. 11=120 Ans. hhd. qt. cts. cts. $ (40) 1=216: 3X216=648=6.48 Ans. da. min. beats. beats. (41) 1=1440: 70X1440=100800 Ans. Qa —---, is - --- 52 -- KEY TO RAY'S THIRD BOOK. (42) 1840 is a leap year, because it is exactly divisible by 4; hence, February has 29 days: 29 days=2505600 seconds. Ans. (43) 1839 is not a leap year, not being exactly divisible by 4; hence, February contains 28 days. January has 31 days, which is 3 days more: 3 days=72 hours. Ans. (44) By dividing by 4, it will be found that 1,838 and 1839 are not leap years, and that 1840 is one; by adding 4 to this, we find that 1844 and 1848 are both leap years. da. hr. min. sec. sec. sec. sec. (45) 365 5 48 48-31556928: 1609403328-. 31556928 yr. — 51 Ans. yr. da. da. min. min. da. hr. (46) 60X365=21900: 30X21900-657000=456 6 Ans. wk. da. hr. hr. mi. 1mi. (47) 3 2 3=555: 8X555=4440 Ans. da. (48) 2s 40=64~: 64~ —. 2~=32 Ans. min. (49) 90~-324000'// 43/ 12"=2592": 324000" /-2592"'/125 hr. min. =2 5 Ans. cts. Cts. $ (50) 2X20=40 quires: 20X40=800=8 Ans. cts. cts. $ (51) 3X24-72 sheets: 2X72=144=1.44 Ans. doz. cts. cts. (52) 3-36 apples: 2X36=72 Ans. (53) 1 dozen dozen-12X1 2=144, and 144X4=576 oranges: cts. cts. $ 3X576=1728=17.28 Anzs. (54) 1 gross =12 dozen, and 5 gross =60 dozen: cts. cts. $ 5X60=300=3 Aiis. fa ------------- -- ----- REDUCTION. 53 cts. cts. $ (55) 4X196 —784=-7.84 Ans. m1. mi. $ (56) 125X200=-25000-25 Ans. (57) 1 dozen dozen =144: 144X6=864 eggs in all: half a dozen dozen -72: 864-72-792 eggs left: et. ets. $ lX792 —792-7.92 Ans. NoTE.-In the solutions to several of the preceding examrples, it may appear that we have not adhered to the principle of always regarding the multiplier as an abstract number; for instance, in solution to question 22, it might be supposed that we multiply 2880 by 5 cents. In this case, however, the 2880 is to be regarded as the multiplier, and the 5 cents as the multiplicand. It is an established principle, that where two abstract numbers are to be multiplied together, either of them may be made the multiplier. The same principle applies where one of the numbers is concrete: thus, in example 22, the 5 cents may be taken 2880 times, or 2880 cents may be taken 5 times; the result, in either case, being 14400 cents. For convenience, the smaller number is generally made the multiplier. For the sake of occupying the least possible space with the solutions, we have frequently represented the quotient of one concrete number divided by another, as a concrete number; though, in all cases, it is really abstract, as stated in the remarks on Division, page 19. Thus, in the solution to question 6, preceding, pecks divided by pecks are represented as giving bags for a quotient; the analysis of the question, however, shows, that 14 pecks is contained in 364 pecks 26 times, and as it requires 1 bag to hold 14 pecks, it will require 26 bags to hold 364 pecks; hence, the answer to the question is 26 bags. A similar method of analysis applies to several of the other examples. The quotient in all cases of division is an abstract number, and the analysis of the question under consideration always determines what name or denomination is to be given to it. 54 KEY TO RAY'S THIRD BOOK. ADDITIONi OF COMPOUND NUMBERS. SUGGESTIONS TO TEACHEiRs.-The attention of the pupil should be particularly called to the nature of compound numbers, and the analogy that exists between them and simple numbers. In the latter, the number of each order of units required to form a unit of the next higher order is uniform, ten units in any order always making one unit of the next higher order. In compound numbers the rate of increase is not tenfold, nor is it the same for the same table. Owing to this peculiarity, in compound addition, instead of carrying one for every ten, as in simple addition, we sometimes carry one for every two; again, we carry one for every three, etc. The reason for carrying in compound addition is easily understood by every pupil, and advantage may be taken of this, to explain more clearly the reason for carrying in simple addition. ANSWERS TO THE QUESTIONS IN COMPOUND ADDITION, TO WHICH THERE ARE NO ANSWERS IN THE BOOK. ARTICLE 101. (4) 871b. 93oz. 17pwt. 22 gr. (5) 191I 11 3 3 23 1 gr. (6) 79 T. 15 cwt. 3 qr. 20 lb. (7) 94 cwt. 3 qr. 5 lb. 5 oz. 5 dr. (8) 38 mi. 5 fur. 19 rd. (9) 12 yd. 4 in. (10) 299 A. 3 R. 30 P. (11) 51 sq. yd. 4 sq. ft. 73 sq. in. (12) 49 C. 58 cu. ft. 519 cu. in. (13) 143 cu. yd. 2 cu. ft. 990 cu. in. (14) 26 yd. 3 qr. 2 na. (15) 37 E. Fl. 2 qr. 3 na. (16) 46 E. En. 2 na. (17) 26 T. 34 gal. 2 qt. (18) 128 gal. 3 qt. 1 pt. 3 gi. (19) 8 mon. 4 da. 8 hr. 49 min. 35 sec. (20) 3290 5' 52" g~~Dt~ I~PISQ. god~P~L-R~ —~~-~C COMPOUND NUMBERS. 55 (21) bu. pk. qt. pt. (22) bu. pkt. Pt. (23)ib. oz. pwt. gr. 21 3 2 1 2003 2 10 15 21 14 0 5 0 143 0 2 1 ] 1 16 14 23 2 0 1 400 3 0 0 2 8 5 12 18 I 0 1 255 1 5 0 Ans.6 8173 As 71. 1000 0 0 0 Ans. 100 0 0 0 (24) lb 3 3 9 gr. (25) cwt. qr. lb. (26) lb. oz. dr. 3 4 1 0 8 1 11 13 11 15 4 3 2 0 4 2 14 17 13 11 4 18 5 0 19 14 14 0 6 5 2 18 7 3 O 1G 0 10 Ans. 1 3 2 O 16 7 3 9 19 7 12 Angs. 33 3 3 A-ns. 99 9 9 (27) mi. fur. rd. (28) A. R. P. (29) sq. yd. sq. ft. scin. 104 1 10 186 3 14 17 3 119 95 6 30 286 0 17 18 0 141 jAls. 200 0 0 113 2 9 23 7 00 nAns. 586200 2 0 29 5 116 Ans. 586 2 0 Ans. 88 8 88 (31) yd. qr. na. (30) C. cu.ft. 17 3 2 (32) hhd. gal. qt. pt. 7 78 13 2 1 73 43 3 1 16 24 23 0 2 27 3 O 0 35 127 27 1 2 15 0 3 1 29 10 29 1 2 162 0 1 0 Ans. 88 111 Ans. 111 1 1 Ans. 277 48 0 0 (33) bl. gal. qt. pt. (34) Jan. 31 (35) Jan. 31 1 28 0 0 Feb. 28 Feb. 29 1 17 0 0 March 31.March 31 5 2 0 0 April 30 April 30 1 2 1 May 1 layy 31. 7 2 1 June 30!8 3 0 Ans. 121. July 4 33 0 0 Ans. 186 An's. 10 0 0 0 (36) Cape HIorn, 55~ 58' 30/" S. New York, 40 42 40 NAns. 96 41 10 ~ —--- - ~ ~ ---- ~ —-- --- 56 KE(~Y TO RAY'S THIRD BOOK. I SUBI3TR AC T IO N. SUGGESTIONS TO TEACHERIS.-After the pupil clearly understands the method of borrowing when the lower denomination is larger than the upper, his attention should be directed to the method of borrowing in simple subtraction, as he will then be better prepared to understand the rule there given. ANSWERS TO QLUESTIONS IN COMPOUND SUBTIRACTION, TO WHICI THIERE ARE NO ANSWEIRS IN THME BOOK ARTI C LE 02. (4) 8lb. 1 oz. ls13pwt. 7gr. (5) 4T. 2cwt 3 r 5 b. 15. S oz. 3 dr. (6) 7 mii. 38 rd. (7) 1 ydcl. 2 ft. 11 in. (8) 250 A. 38 P. (9) 2 sq. yd. 8 sq. ft, 104 sq. in. (10) 55 da. a5 hr. 55 min. 55 sec. (11) 90 59' 56" 1bu. pk. qt. bu. plk. qt. Pt. lb. oz. pwt. gr. (12) 4 0 (13) 100 0 0 (1s) 19 9 0 0 2 1 1 24 0 0 1 9 9 10 10 Ans. 1 2 7 Ans. 75 3 7 1.Ans. 9 11 9 14 lb. oz. pwt,. gr. b g 9 3 gr. lb. oz.dr. (15) 8 2 11 0 (16) 3 3 1 1 12 (17) 46 9 0 2 4 7 16 1 7 0 2 18 19 0 4 Ans. a 10 3 8 Ans. I 8 1V 14 Ans?. 27 8 12 cwt. qr. lb. mi. fur. rd. A. R. P. (18) 32 2 16 (19) 25000 0 0 (20) 146 2 0 8 2 17 100 1 1 86 2 14 AnIs. 23 3 24 Ans. 24899 6 39 Ans. 59 3 26 COMPOUND NUMBERS. 57 C. cu. ft. E.Fi. qr. na. yd. qr.na. (21) 8 50 (22) 25 0 0 (23) 11 0 0 3 75 14 1 3 3 2 2 Ans. 4 103 Ans. 10 1 1 Ans. 7 1 2 gal. qt. pt. gi. da. hr. min. see. yr. mon.da. (24) 63 0 0 0 (25)5 10 27 15 (27) 1857 3 5 51 1 0 2 2 4 13 29 1776 7 4 Ans. 11 2 1 2 Ans. 3 6 13 46 Ans. 80 8 1 yr. mon. cda. yr. mon. da. (28) 1848 8 1 (29) 1832 6 28 (30) 55~ 58' 30" 1]822 6 24 1807 11 25 33 55 15 Ans. 26 1 7 Ans. 24 7 3 Ans. 22 3 15 quires. sheets. (31) 71~ 10' (Y' (32) 500 0' (Y' (33) 20 0 36 6 30 10 25 48 5 11 Ans. 35 3 30 Ans. 39 34 12 Avns. 14 13 (34) 1 dozen dozen - 12X12_144 6 dozen dozen =144X 6 —864. a dozen dozen -144 — 2- 72 Ans. 792 AR,TICLE 104.- IJULTIPLICAT ION. bu. pk. qt. pt. lb.oz.pwt.gr. lb. oz. pwt. gr. (4) 4 3 3 1 (5) 2 11 6 (6) 10 10 10 12 6 10 Ans. 58 1 2 0 1 3 7 12 8 9 4 4 58 KEY TO RAY'S THIRD BOOK. Bt 33 D gr. T.cwt. qr. lb. T. cwt. qr. lb. (7) 4 o 2 15 (8) 8 2 (9) 10 3 14 8 9 7 Ans. 2 8 7 1 0 Ans. 3 17 2 8 Ans. 3 16 0 23 mni. fur. rd. mi. fur. rd. A. R. P. (10) 208 4 16 (11) 30 4 10 (12) 130 3 30 15 12 4 Ans. 3128 2 0 Ans. 366 3 0 Ans. 523 3 0 cu.yd.cu.ft.cu.in. yd. qr. na. gal. qt.pt. (13) 23 9 228 (14) 26 2 2 (15) 62 1 1 12 6 8 Ans. 280 1 1008 Ans. 159 3 0 Ans. 499 0 0 gal,. qt. pt. da. hr.min.sec. (16) 123 2 1 (317) 365 5 48 48 (18)s 40 11/ 15// 5 12 8 Ans. 618 0:1 Ans. 4382 21 45 36 Ans. 1 3 30 0 doz. bu.pk.qt. m;.fr. r A. A. P. (19) 144 (2a1)2 3 5 (22)3 5 16 (23)6 3 30 12 4 6 8 1728 11 2 4 22 0 16 55 2 0 3 6 10 7 Ans. 5184 Ans. 69 3 0 Ans. 220 4 0 Ans. 388 2 0 T. cwt.qr. lb. oz. dr. hhd.gal. qt. pt. gi. (24) 8 2 14 12 13 (25) 3 2 1 1 11 8 95 0 12 12 15 29 1 0 0 2 2 Ans.9 10 I 0 9 10 1 0 9 14 58 2 0 0.6 3 07 Ans. 6 31 2 0 0 COMPOUND NUMIBERS. 59 T. cwt. qr. lb. yd. qr. na. hhd.gal. qt.pt. (26) 16 2 24 (27) 37 3 2 (28) 47 3 1 119 89 59 Ans. 99 12 0 6 Ans. 3370 3 2 Ans. 44 52 2 1 (29) mi. fLr. rd. (30) C. cu. ft. 27 3 35 7 98 31 17 Ans. 852 0 5 Azs. 132 2 ARTICL 106.-] D IVISIO N. (4) (5) (6) bu. pk. qt. pt. lb. oz. pwt. gr. oz. pwt. gr. 5)67 3 4 1 8)14 8 16 16 10)8 12 12 Ans. 13 2 2 1 Ans. I 10 2 2 Ans. 17 6 (7) (s) c-wt. qr. lb. oz. dr. Imni. fur. rd. 11)35 1 17 3 7 7)39 7 8 Ans. 3 0 22 0 5 Ans. a 5 24 (9) (11) (12) yd. qr. na. bu. pk. qt. lb. oz. dr. 5)62 0 3 10)490 2 4 5)266 9 10 Ans. 12 1 3 10)49 0 2 10)53 5 2 Ans. 4 3 5 Ans. a5 5 (13) (i) lb. oz. pwt. gr. da. hr. min. see. 6)339'7 9 18 6)114 22 45 35 7)56 7 4 23 9)19 3 47 35 5 rem. Ans. 8 1 0 17 Ans. 2 3 5 17 2 rem. 2)X6+5=17 sec. rem. ~~ ~ —pl~~~opl.~~~- c~~i~,~~-~~ c~i-~r~~~l~~-~ —spm*~lu ~~~~~P~~-* —'Ic 60 KEY TO RAY'S THIRD BOOK. (16) (17) lb. oz. dr. lb. oz. dr. lb. oz. pwt. gr. lb.oz.pwt.gr. 23)1027 1 8(44 10 8 Ans. 245)17 7 6 6 (0 0 17 6 92 12 [Ans. 107 245)211(0 oz. 92 20 15 245)4226(17 pwt 16 245 91 1776 15 1715 23)241(10 oz. 61 23 24 11 250 16 122 74 245)1470 (6 gr. 11 1470 23)184(8 dr. 184 (18) (19) bu. pk. qt. hu. pk. qt. imi. fur. 1rd. mi. fur. rd. 78)309 2 2(3 3 7 Ans. 319)788 4 9(2 3 31lAns. 234 633 75 150 4 8 78)302(3 pk. 319)1204(3 fur. 234 957 68 247 8 40 78)546(7 qt. 319)9889(31 rd. 546 957 319 319 RE-MARK.-For cases involving the division of a compound number by a fraction; also, one compound number by another, see " Ray's Ihigher Arithmetic." Ob —-. —-.- ------ -------- COMPOUND NUMBERS. 61 ARItTICLE 107. (1) (2) (3) lb 3 D gr. oz. p-t. gr. A. RP. P. 1 0 0 0 0 8 10 12 104 2 37 4 2 0 13 9 3 5 87 1 38 8 9 7 7 5 2 7 192 0 35 3 1 2 14 26 3 0 40 1 0 5 12 > Ans. 4 3 2 13 12 3)151 3 35 Or, add the two q 3an1i- 1 As. 50 2 25 ties together, and take 5)8 12 12 their sum from 1 lb. As. 1 14 12 (4) (5) (6) mi. fur. rd. yr. mon. da. 350 30/ 0// 3 2 5 1845 2 3 Sub. 20 35 0 4 1843 6 27 14 55 0....___ i ______- 14 55 0 1 6 2u 1 7 6 Add 14 20 0 30 5 29 15 0 Ans. 54 3 0 Ans. 8 0 0 Sub. 25 4 30 4 10 30 Add 6 19 20 Ans. 10 29 50 ARTICLE 108. (1) 15)18~ 25' 30// (2) 30~. 15=2. Ans. 2 hr. Ans. 1 hr. 13 min. 42, see. (3) 15)71~ 4' 0"// (4) 15)10~ 35' 0// Anls. 4 hr. 44 min. 16 sec. Ans. 0 hr. 42 min. 20 sec. min. sec. hr. min. sec. hr. min. sec. (5) 37 20 (6) 1 4 56 (7) 5 8 4 15 15 15 Al.s. 9 20' 0// Ans..16 114' 0'" Ans. 770 1/ 0// 62 KEY TO RAY'S THIRD BOOK. A.RTICLE 109. (8) hr.min. sec. (9) hr.min.sec. Time at C. 12 0 0 Time at N. Y. 11 0 0 A. M. Add diff. 37 20 30~- 2 0 0 to be added. Ans. 12 37 20 Azs. I 0 0 P. M. (10) hr.min.sec. (11) hr.min.sec. Time at Ph. 12 0 0 Time at N. Y. 11 0 0 Subtr. diff. 37 20 Subtract diff 1 4 56 Ans. 11 22 40 A. M. Ans. 9 55 4 A. M. (12) 124~-80~ 42'-43 18': 43~ 18' — 15:= 2 hr. 53 mJin. 12 sec. lir.min.see. N o T E. — In performing the subTime at W. 1 0 0 traction we can not take 3 hr. from Subtr. diff. 2 53 12 1 hr., but 1 P. M. is the 13th hour Ans. 10 6 48 A. 5M. from midnight, from which, after taking 3 hr., the remainder is the 10th hour from midnight, or 10 A. iI. ARTICLE 113.-FACTORING. N o TE.-Most pupils in a short time may be taught to give the prime factors of all the composite numbers, from 4 to 100, mentally, or by inspection. The great simplicity of the first 21 examples, renders it unnecessary to present their solution here. (22) 105. -3-35: 35-+-5.7. Anis, 3, 5, 7. (23) 168-. 284: 84.2-42: 42 — 221: 21-+ 37. Atns. 2, 2, 2, 3, 7. (24) 216- -2-108: 108- -2 —54: 54 -22=27: 27 — 3-=9: 9 —3=3. Ans. 2, 2, 2, 3, 3, 3. (25) 330 — 2=165: 165-. — =55: 55 —5511. An.s. 2, 3, 5, 11. (26) 110=2X5Xll: 210z-2X3X5X7. Ans. 2, 5. (27) 105-3X5X7: 231=-3X7Xl1. Ans. 3, 7. (28) 330-2X3X5X11: 390=2X3X5X13. Ans. 2, 3, 5. (29) 231-3X7X11: 330-2X3X5X11. Ans. 3, 11. ~~4 Ib~~~ —- ----------------- ------— ~ —--------------- ~~~~~~~~~~~~~~~~~~~~~~~~~1.1 GREATEST COMMON DIVISOR. 63 1 AraTIcLE 114. (1) 42=2X3X7: hence the divisors are, 2, 3, 7; and 2X3=6; 2X7 —14; and 3X7=21. Ans. (2) 105=3X5X7: hence the divisors are, 3, 5, 7; and 3X5 —15; 3X7=21; and 5X7 —35. Ats. (3) 20O2X2X5: hence the divisors are, 2, 5; and 2X2-4; and 2X5l10. Ans. (4) 24-2X2X2X3: hence the divisors are, 2, 3; and 2X2 =4; 2X3=6; 2X2X2=8; and 2X2X3=12. Ars. A RTICLE 116.-RULE I. (4) 42-2X3X7: 54-2X3X3X3: hence 2X3=6 gr. c. d. (5) 70-2X5X7: lO= —2XX:l hence 2X5-10 gr. c. d. (6) 105=3X5X7: 165=3X5Xll: hence 3X5=15 g. c. d. (7) 60=2X2X3'X5: 90-2X3X3X5: hence 2X3X5 =30 g. c. d. (8) 140=2X2X5X7: 210-2X3X5X7: hence 2X5X7 _70 g. c. d. (9) 66-2X3Xl: 154= —-2X7X11: hence 2Xll-=22 g. c. d. (10) 154-2X7XlI: 2980=2X2X2X5X7: hence 2X7 -14 g. c d. (11) 231-3X7Xll: 273=-3X7X13: hence 3X7=21 g. c. d. (12) 30=2X3X5: 42 —2X3X7: 66=2X3Xll: hence 2X3=6 g. c. d. (13) 60=2X2X3X5: 90=2X3X3X5:150-=2X3X5X5: hence 2X3X5=30 g. c. d. ARTICLE 117.-RULE II. (4) 495 — 42=11+33 r.: 42 —33=1+9 r.: 33 — 9=3+6 r.: 9-6=13 r.: 36 —3=2. Atzs. 3. (5) 323 —247=1+-76 r.: 247-+-76-3+19 r.: 76 -19=4. Ans. 19. (6) 465. 285=1+180 r.: 285 —. 1801+105 r.: 180 —105 -=1+75 r.: 105_75=1+30 r.: 75 o. 30-2+15 r: 30. — 15=2. Ans. 15. 64 KEY TO RAY'S THIRD BOOK. (7) 1274-. 532=-2-210 r.: 532 —210=2q-112 r.: 210~-112 =1+98 r.: 112 —98-1+14 1r.: 98 —. 14=7. Anls. 14. (8) 759,337=2+85 r.: 337 — 85-3+82 r.: 85o 82 -1+3 r.: 82 —-3 7-1 r.: 3 — 1==3. Ans. 1. (9) 69087 — 9873=6+-9849 r.: 9873 —. 9849 —=1+24 r. 9849 9 — 24=410+9 r. 24 — 9=2+6 r.: 9 —6= —1+3 r. 6 3-3=2. Ans. 3. (10) 1814 —. 1776138 1 r.: 1776 —38-46+28 r.: 38 —. 28 -d1-10r.: 28 — 10=2+8 r.: 10 -8-1+2 r.: 8 — 24. AIns. 2. (11) 1815 —693 —24'29 r.: 693 —429=1-1-2641r. 429+264 -1+-165 r.: 264- 165=1*-99 1r.: 165 —99- 1+66 r.: 99 -66=11-33 r. 66 _ —33-2. Ans. 33. (12) 3471 — 2145 —1-1326 r.: 2145 -1326=1+819 r.: 1326 — 819- 1-+507 r.: 819 — -507_-1+312 r.: 5073-.312 z=1-.195 r.: 312 —1951=1+117 r.: 195 — 117=1d-78 r.: 117 -78=1 —39 r.: 784-39-22. 1A1.. 39. (13) 1701l7 —840-20217 r.: 840-.-217=3+189 r.: 217 —189 — 1+28 r.: 189 —28-=6+21 r.: 28 —21=1+7 r.: 21 —7 — 3. Ands. 7. (14) 153452.66284-2+20884 r.: 66284 —20884 =3+3632 r.: 20884 —3632=5+2724 r.: 3632 —2724 1-f-90S r.: 2724 —908 —3. Ans. 908. (15) 55-4-40=1+15 r.: 40 —-15-2+10 r.: 15i —10=1+5 r. 10-.-5=2. Again, 105 —-5=21. Ans 5. Or, by resolving the numbers into their prime factors, and takingl, the pr(oduct of those factors that are common. 40X —2X2X2X5. 55=5X11. 105-3X7X5. Since 5 is the only factor common to each of the numbers, it is their only common divisor. (16) 154 —70-2+14 r.: 70- 14-5. Again, 819 —14 -58+-7 r.: 14 —7-2..Ans. 7. (17) 168 —120=1q+48 r.: 120 —48=24-24 r.: 48 —-242. Aain, 1768 —24-73+16 r.: 24 —16-1+48 r.: 16- 8-. A s.. OS. 0, i20(-2X2X2X3X5. 168-=22X2X3X7. 1768-2X>2X2Xl3X17, hence, 2X2X2=-z, is the greatest common divisor. d. — - ------------------— g LEAST COMMON MULTIPLE. 65 ARTICLE 119o- RULE I. (3) 12=2X2X3 (4) 6=2X3 30= X $5 10=-x5 2X2X3X5=60 Ane. 18=-s=x X3 2 X 3 X X3=90 Ans. (5) 6=2X3 () 6-=2 A 3 8=gX2X2 15=-$x5 9 —=X3 35= x7 2X3X2X2X3 —72 Anv. 2X,3X5X7=-210 Ans. (7) 10=2X5 (8) 9=3X3 12= X2X3 15=-X5 t15=,x 18=2XtX$ 2X5X2X33=60 Ans. 24 — X2vX2X$ 3X3X5X2X2X 2=360 Ans. (9) 8= —2X2X (1o) 14-='X 15 3X5 21=-3X7! 12-xx 320=' X$XS 30o=xox 35=0xI 22X2X 3X5=120 Ans. 2X7X3X5=210 AnEs. REMIARI.-When the humbers are small, or when they are fewt Rule 1st is preferable; but in other cases Rule 2d is the several of the same exsamples by both rules. ARTICLE 120.-Ru-E, II. (2) 2)6 10 (3) 3)15 21 35 (4) 3)9 12 3 5 5)5 7 35 3 4 2X3X5=30 Ans. 7)1 /> 4=3.6 A1.s 7)1 7 7 1. 32X5><7=1-05 An.. (5) 7)14 21 (6) 3)6 9 15 (7) 2)4 14 3)5 2 3 2 3 5 7)2 7 35 7X2X3 —42 Ans. 3X2X3X5=-90 Ans. 2X7X2X5 —140 Ans. Ke.. - -5 _ 1 66 KEY -O RAY'S THIRD BOOK. (8) 2)6 - ) 1O 1' (9) These numbers contain no common factor, there3 5 1 9 -ore their least common g 5 )1 t) tj 3 llmultiple is their product. 1 1 1 3 7X1X13X3:3003 Ans. X 2X 3X3 - 90 Ans. (10) 2)G312 SQ 7 (11) 3)54 81 63 2)63 6 42 7 3)18 27 21 3)63 3 21 7 3)6 9 7 7)21 1 7 7 3 7 3 1 1 1 s3X3X3X2X3X7=-1134 Ans. 2X2X3X7X3z:252; Ais. (12) 2)8 12 20 24 25 (13) 2)9 10 24 25 32 43 2)4 6 10 12 25 2)9 5 12 25 16 45 2)2 3 5 6 25 2)9 5 6 25 S 45 3)1 3 5 3 25 3)9 5 3 25 4 45 5)1 1 5 1 25 3)3 5 1 25 4 15 1 i I 1 5 5)1 5 1 25 4 5 2><2<5 20 1X3X4 12 1X5X7 35 (5) > (6) (7) 2>X4X5 40 2X3X~4 244 2X5X7 70 3X2X5 30 1X2X4 8 3X2>(7 42 4X2X5 40 3X2X4 24 5X2X7 70 4X2X4 32 1X2>(3 6 6X2X5 60 5X2X4 40 4X2X3 24 7X2X5 70 70 KEY TO RAY'S TIIIRD BOOK. 1X3 X4X5 60 2X5X7X8 560 ( 2s 3X4X5 12a 3X5X7XS 840 2X2X4X5 80 2X3X7X8 336 3X2X4X5 120 5X3X7X8S 840 3X2X3X5 90 3X3X5X8 360 4X2X3X5 120 7X3X5X8 840 4X2X3X4 96 5X3X5X7 525 5X2X3X4 120 8X3X5X7 840 6X7X13 546 (11) - of s-. (12) i- of -=5. 1 iX 7 X13 1001 7,, 3v 74 XllX13 572 3X3X1_ 9 5X7 35 7 X11X13 1001 SX3X1 24 SX7 56 8 X11X 7 616 7X8X1 56 IXS s 13X. 1X 7 1001 3XSX1 24 7X8 56 3X8X3 72 I.XS><3 24 ARTICLE 145, In solution of questions in this Article, the multiplier of both terms of each fraction is supposed to be obtained mentally, by dividing the largest denominator by the denominator of the fraction to be reduced. 1X2 1X4 l X2 1X5, (2) (3) (3) () 3X2 (5) 2X5 4 —a 1X<2 1X3 2X2 4X2 2X3 5X2 1_d 553 5- c~, —--— ~ —---------------— ~ —6 COMMON FRACTIONS. 71 ( X4 (- X = (3) 3X'4 6X4 o 5/2;~. 5 >\2 X3 6X2- 7/2 32- 42.2 —-4' - J'7 91 o 11 I II I T-2 —T7~ J 4 — 14 1 r I 6 -' 6 ARTICLE 146. —CASE VI. (4) Least common multiple of 2, 3, and 4=!2. (Art. 119, 1920.)?6, iS =4, 4 4 1 6- \46,1X- 3X1-3 As. _4 3. (5) Least common multiple of 3, 6, and 9 —18. (Art. 119, 1.'20.) 18, 11-, U 182 1\/6-69 /13=3, 1\/2=2, AnTs..5 (6) Least common multiple of 2, 4, and 5:20. 20:0o, 4~=5,?50_4. 1X><10=10, 3X5=5, 4X=16G. Als.,,. (7) Least comm1ou multiple of 8, 5, and 10= -40 4 0-5 4o=', 4o__.; 3X5=15, 4X8=32, 9X4-36. Ans. -i 4.3. (8) Least common multiple of 3, 4, and 8 —24.'2 4=g Z-'-6, =r __3; 2X8 —46, 3X1 =8, 7X=21 Ans., li (9) Least commnon multiple of 4, 8, and 9-72. 74z 18,s9,: $;S 3X18=54, 5X9-45, 5X8-40. Ans. 4, -, 4-0j (10). Least common multiple of 7, 14, 7, and 2S —28. v -4 28_, ~ t8=4, Is-=; 2X4 —8, 5X/=10, 3X4-12, 11><1 —11..Z!1. 8 I 0 1 2 1 I Ins..-,,. (11-) Least common multiple of 2, 3, 4, 5, 6-60.` 390, 6;~- 20, 5O0-1 5, 1 -120 60-2 10: 1X30=30, 2/20: 40, 3X15-45, 4X12=4s, 5/Xo=50o. rins 0 40 45 4, 580 ~5Xl~O-50. An"s. ~fGap, Ug, 40 0 9o, 0U-U. 72 KEY TO:RAY'S THIRD BOOK. (2 ) 1 20 5.f2 2 ]2 Thle fractions then are, 9, 9, -, and (I' Least common multiple of 4, 8, 6, and 5=120. 12 4 12 ___0 120=30, 120=15, 2=20, 1~:o241; 3X30=90, 7)\15=-105, 5)(20-=100, 12)>24-288. 9a0s. 90, 1 o 100 288 SUGGESTION, —In working the questions, until the pupils are well acquainted with the operation, they should be required to putdclown the work after the manner of Ex. 1, Art. 146, Arith., as this presents the reason why the values of the fractions are not altered. A.:RTICLE 147, -ADDITION. (2) Ar,. =(3) A ns. 902 (4) lls. 7 2. (s) As. 16-2. (6) Ans. t-1. (7) s. 37 T IcLE 148. (5) 4+17+29+-48+76= 174; hence, sun-l s71=2 A2j AS. (6) 11 + 9+29+101=160; hence, sumn- 1 4 - ~- 9 A. (7) 999-1-888+777=2664; hnce, su,- -8-4 o=a — 2853 1.,. (8) ~+'-, 1,, o _9,= 24 1 _ 1 I (9) I A+fS. (3zO) 2 1 + 2'~2 4a (10) 8+1 _ 12. -, 88! 1424 24 1S. (12) 8 1 +4 3 3 1 n Ans. a aI 1.4 ('V i +-1Y - + jo= 1o4 +5 + 4n 4+1 G 37 i 1= 629 -5 4&' An1s ( 15 ) 1 +S.=3+1 5 9 4 COMMON FRACTIONS. 73 (:L8) I+- 4 off'; _-o8=15;_ 5315+2 lo~ -- Ans. ( )i-4 1 + _ _ 1304i = 3 4-_; 16 —12+8+2-38; 38+2-1=4 0 2'LD Ants. I I 3 __ 3 0 1 5 (l9) -of 3 1- 6 - 4 O1f 5 1I+- 3 -4 -3 Ans. (l9) -D of 1'= of 3- 3 o f2-of 5-= of of ~1, 3 5 + 1 8 10+6 _ +3 6 =2 3 1 -0 Ans. q 1_ rl__ 37 3+ -3 0= 1 8O ~0 + gA 1 3 - I 6 8 T 332o o -- &B0 0l Ans. ARTICLE 149.-SUBTRACTION.: - 1 o I RT TICLE 150. (5) * —_-5 - 1 ALns. (6) i A -- -.- t (3) 3I__= 19 4 7 AnlS. (9) 2- 9 8 - Ans. (10) -1 1 = _ _4 Ans. (11) 8 3 -8- 4 -4' — 49 A 3. (13) 45`10 -—'- A s — s. (18) 13 - =4n 2 ns. (19) 4 T__ 1= 9 - 9 =1_ A s. 4) o8 3 I — 9 - 7 1 -~:Ans. (12) 57 -1-2-54 - 91-'1 Ans. 2139) 5' 141 54-4=12 A2s. (21) 1 -__1 ____q 2 3 ns. 74 KEY TO RA.Y'S TIHIRD BOOK. (22) s8 -- — 7 q 74t Anls. (23) li -4 99 41_ 58=6 Ans. (24) 8-33=713 3 t4-n0 Als. (25) of ~-O-6;'x of Q =4; 6-4=2 Ans. of 3_ 3 f*.3 5 2 O 7 O -7 4Aui7s. sz 6 6i GT9i (27) -2of fI 9 —8 41 4i7 57 38-171 152-19 =-1 Ans. -- T'-2 l ~ 74-f (28), ~ 1; 5 -- (28) of 21e3 of 72o7-l1h3; 1 35 63=7 =143 Ans. 1U L T I PLI CATI 0 N. SUGGESTIONS TO TEACIdEnS.-When the pupil comes to the multiplication of fractions, he will probably be surprisedl, at first, to find that the product is firequently less than the multiplicand, while in the multiplication of whole numbers, it is always greater. The pupil should be reminded that the multiplier shows how zmany timzes the multiplicand is to be taken. Thus, if the multiplier is 2, the multiplicand is to be taken two times; if 1, it is to be taken one time; if, it is to be taken half a time, etc. Hence, if the multiplier be less than 1, the product will be less than the multiplicandc also, the less the value of the fraction by which we multiply, the less the product: when the multiplier becomes infinitely small, the product becomes infinitely small, and wh!n the multiplier is 0, the product is also 0. It should also be noticed, that to multiply by 2,'-, etc., is the same as to divide by 2, 3, etc.; also, that to multiply by 2, is the same as to multiply by 2, and divide by 3, etc. Indeed, it is a general rule, that when an2y result is obtained by dividcig one nLmber by another, the same'esutlt may be obtained by mzultiplying the given number b7y the divisoa svrittenz in a fractional form, aCnd inzverted. ARTICLE 151. —CASE I. (14) X11-12=- 132 7 37 Axns. (15) \9,<7- 3 Ails. T i 3~ 17` 172 rps ------------------------------ 9I COMMON FRACTIONS. 75 o 25 (16) WX9 -=1Z Ans. (17) -5=32 Ans. 36_ 9X21 9X3 27 An,(18 ) -__X21....... 14 2 (19) { X10 X 1 2= — 23-71 Ans. 15 3 (20) 5X9=5X95 Ans. 9 (22) 18-X8= — X8+18X8=6+144=150 Ans. (23) 16'2X3=- X3+16X3 —2+48=50 Ans. (2a4) 10-X7-X71X7 7:4,9 -70-75-, Anzs. (25) 10-X9 —-X99-10X9=.-1490 =97'- Ans, ARTICLE 152,-CASE II. (12) 2SX= 4 =4X4=16 Ans. 7 (13) 36X7=36X7=4X7=28 Ans. 9 (14) 50X —0 XO-5X9-45 Ans. 10 (15) 31X X3 2 lX==202 Ans. 3 3 (16) 29Xl 29X_3 7_ =2 n As. 437 (17) 37X4 -- 37X4'=209 Ans. (19) 25X8 o=X 453=5X43=215 Ans. (20) 45X64-45x='- 12 " 981 AsS. (21) 55X93= —X6-6=30 ~5184 Ans. (22) 64X8=-'6-X7~'l=78X71=568 Ans. 76 IKEY TO RAY'S TItIRD BOOK. ARTICLE 153.a-CA-sE III. 13X2 (9) 8- 35x 1- 47-= 4 s. (:O 10-) - L- T%-1X 9 Xl9 171 =15 As ~11 I 11 (JJ) 25 2X-/ t6~4 Ans. (12) 161 33 33Xv3 —_1089 —2721.Ans. 2X5 (13) -1 of 4-; &- of 5 —5;x A — s. 3X3 (14) 7of? o 3 of13A, o; 3 o 7=7 3 T3 z —2 3 —; 6X-7=13 -4 Ans. (15) g of 9 -- of 37= 1; i of 17 1l\/h34 o3f17 6 29=785 Als. 16 ~ __3X17 __ 3Xll a186 —1 a_1of? (136) - of 1 1 0 of -; I, of -r 9 5 a 3 5 10 1 5 -O 17 Ans. (l1) 7X <0 X X 9 AQ. (19) X X l X - X~2 14Ans. 10 2 (7) 5 -- 10 "-i - X=,- o $6'2 Ans. ( — ~ 0 Xt 3= —; 3X( 1) 5 Ta7-= 1 X 3=4 -3 —=425 mi. Ans. (8 5',E 74 4 78 KEY TO RAY'S THIRD BOOK. 2 (9) W Of 3 X $=x Ans. 11 5 $ 5 $$: 2' 7 ~ 5x x5 97X4 ARTICz LE 154_.-CAsOf I. tient will always be less than the dividend, as in the division of whole numbers; thus, to divide ~- by 4, we really find what part of a time 4 is contained in 3; we also find the -- of ~, since to divide by 4, is the same thing as to multiply by I. (10) P, -5it s (i 51 -1 1 >'I;lS In the solution of the following question s we have not marked the extra space occupied by this kind of type: in op rations performed on the slate, or at the blackboard, the canceled factors should always be maiked in owder to avoid mistakes. (s~) 1.49.34 4 (7) y. A4 is. (8) -tt a =l-7 Aby.I (9) -4 -t4 Ans. (10) - -T0 (12) 3 A ns, 5 5 A. 5 —~S(13).As. 1.: 7-=15 As (14') caneld 9= Ars. (15) Le-2o4 24 8eo Ac S. (16) 5. A=3oss.. (17) 12 - 8q; - -3-'.i? AAs. o 4o (19) 475- - t-; a_152 7=; _-_ _-3, 4. Ars. 392 C' 4- Ans. (1) 5195 (20) A3O ns. (n) 44Xyi8 4= Ai3s. (2C) 313 9. l;3 99'. T-33 9 -' 5 -i I A X K~~~~1,_ ~ COMMON FRACTIONS. 79 ARTICLE, 155.-C0ASE IIL (4) 4 2 10 Ans. (5) 16 -16X4 4=211 As. 24 3 (6) 8 8Xl54X15-60 A8 2 (7) 6 -=6X=2X77=14 Ans. (8) 13 - 3 -13X 5-212 Ains (9) 21..11.X =3X1=33 Ans. (3-) 2 12; 6, X= 5 X= = 2 yd. Ans. 12 (12) 34 30 — -2X 4-S lb. -Ans. 15 (13.) 4_ 31; 50 — 31__350-X7-30~ —i I A _7s. 31 49 - -PupRTICE ofte t.-CA losSE. y i RE1IAA. RTupiCs are often at -a loss to undlerstandl why it is that the quotient of one proper fraction, dividel by another, is sometimes a whole number, or greater than unity. The teacher should be careful to explain this subject, by means of familiar examples, such as may be found in Ray's Arithmetic, 2d Book., Section xxi. It should also be shown, that if we take any dividend, and divide it by different numbers, that as the divisor becomes less, the quotient becomes greater; so that, by making the divisor sufficiently small, the quotient may be made as large as 80 KEY TO RAY'S THIRD BOOK. we please. Thus, the quotient of I divided by 1, is 2; by,, is 4; by T1Oj is 8; by ~0-oUUU1 is 2000000, etc. It is on this principle, that mathnematicians, say, that the quotient of any number, divided by 0, is infinitely large. ~(5)-4- 43 *l. 1(6) 2 X /= —X 2 AtS (, ) ~ ~=-' X2= A.ls. (8) _.,2,,=5X3. A4_. (O) 7 27, L5. ) X1+ —40 Ais, (3La4) 9.- 4j 958 8; a93 3 2 - 519X2 i 41 4:!.3 ) 2-4 —9; 7 -12 4.19. a =4 1 5-= _ ___-__o___- 41 (3.4) e * \/o= U=1=25 Ans.. (s:.6) 51.f 4_1 v1 4 3_ 3 1- T 4 01 5 ARTICLE 167. (2) i 2 4 1. I__v Ans._'_ ( ) -- 2;1 5'V - I -1 4 A. (3) 8-.7v 457X1 =X34 _4 An. (i) 8X. o Ans. (6) 2* 4. 3 fv 3r- v 3 pa=- Ants. (7) 12,=; 4j-" 7X-J X4='9!3, A??,,. Tv — ~XT G 3 3 - 3 (8) 6 S'O 5 In solving this example e we fincl 12-,l3 i I - f ~, 1 andc 8- o45. When the latter is;jjx:$ S ine ted, 41 becomes a divisor, and, therefore, is placed on the left. 6 =5.Ans.. 6 COMMON FRACT.)I.. ARTICLE 158.a 2 -. —- A s Ans. 1 I ($) "' 97.S, 7747 t 2(9) 7 =; 97=; 9 = o; 1- 113; ) 2 5Io~ g 3 3 19 47 I 33 (p (3 2 -3 33=].1 Ans. 47 19X2 — A'.s A R. T ICLE 15g. (2) 2, 93 37 l. An. (3) 2- 4 Je/3 X. 6 _ 1 1 i-\/ - ~ (4) 4-2 - 1 1 33 >< 1-3_ _1/- yd. As. 1 45 3 lf,.-,x 82 KEY TO RAY'S THIRD BOOK. (5) 10. — - 11v8_8-o 26' Ans. (6) 3 of 1 Of =.4; 3= o- 24=; b24' 924-,, 4- 61~ -n7 1 4/\3 2 1 3 3 3- (7)'A4' of 27= - 5,of5=10; 1 of 21-1=3G Of 8 \1j=_51; 10. 11X- 8_=8 129 Ans. (8)'. — X3 2 2_= 12 X-I2X =72 2!~2. RTICLE 59. (1) (2) $ cts. cts. 16.06-4 9.12-1 9.122 4.4335*434i 1)O97.l4 5.435~ ~ 0.370 Ans. 833.43- Avf s. 615.56-j (3) (4) (5), cts. $ cts. $ cts. 50.25 32.31J 5.81' 8 4 (10) 1.8 11. 129 1 3.i9 Als.`$209.061- As. AL7.18 Ans. $4.621 (6) 121X93-10 \8 +4 t 112 cts.=4$1.1254 Ans. (7) 61X21=26+511=1-1- cts.=-$1.31 4 zs. (8) $3.18:3X15:47.70-. 114-$47.81-~ An.s. 1(9) 6,X5 1='.. 1.1 o375- 343-3 cts.-=$3.433? Ans. (13) 17>? ---:1., -:1)X 6 -- cs$2.342,. (3.0) IS 3-5 ~ 13 2 — T7 X >..-5 25X9- -225 cts_=$2.425 Ans. (:12) 8$3.37v'',s4 S 7 5-I=3459- ts.=$34.59(13) 17 3g =3: 375X =125X53=6625 cts.=-$66.25 A4ns. 3,> 3 3 COMMON FRACTIONS. 83 (14) 225 -18: X — — 3-X4 12 yd. lns. (15) 5S 4l-, 137 1_2 3 25 5 3X1-X —-=154 bu. Ans. (16) $11.56}-~-5-$2.31k- Ans. (17) $31.0614 — 7=4.4s3 Ans. mi. fur. fur. rd. rd. ft. (18) 5X8=40: 40X40=1600: 1600X16-=26400 ft. in. 26400X12=316800 Ans. mi. fur. fur. rd. rd. rd. ft. ft. (19) 2X8=16: 16X40+2=642: 642X161+2=10595 Ans. ft. rd. ft. rd. fur. rd. fur. mi. (21) 15875 —.16=-962 2: 962 —. 40-24 2: 24 8=3. Ans. 3 mi. 2 rd. 2 ft. in. ft. in. ft. yd. (22) 142634 —12=11886 2: 11886 —. 3=3962: yd. rd. yd. rd. fur. fur. mi. fur. 3962 —51 —720 2: 720 —. 4018: 18 — 8-2 2. Anjs. 2 mi. 2 fur. 2 yd. 2 in. mi. in. f. in. in. in. in. (23) 2=126720: 2 8=32: 126720 — 32=3960 Ans. mi. in. ft. in. in. in. in. (24) 65=4118400: 9 2=110: 4118400-. 110=37440 Ans. A. R. R.R. P. P. P. sq.yd. sq.yd. (25) 1X4+3=7: 7X40+16=296: 296X301+25=8979 Ans. sq.yd. P. sq.yd. P. R. P. (26) 7506-.301=248 4: 248-.+40=6 8: R. A. R A. R. P.sq.yd. 6 —4=1 2. Ans. 1 2 8 4 ch. 1. 1. in. in. in. in. in. (2 7)5 15=515 7-9 oo =T7f9: 19X515=47OCso8 0=40784 Ans. 84 KEY TO RAY'S THIRD BOOK. P. P. (28) 40z8=%1 ~: 1X32=81X16=1296: 1296-. 40 R.... A. A.P. -32 16: 32-.-4=8. Ans. 8 16. da. da. da. hr. (29) 3651-X4=1461: 1461X24=35064 Ans. hr. da. hr. da. yr. da. (30) 914092 —.24-38087 4: 38087. 365 —104 101: yr. cen. yr. cen. yr. da. hr. 104- I100=1 4. Ans. 1 4 101 4. da. (31) 238545 31 —7695: 7695 —3651-30780 —1461=21 yr., and 99 quarter days remaining, which reduced to days, by div-iding by 4, makes 24- days. Ans. 21 yr. 243 cda. ARTICLE 160.-CASE I. lb. OZ. lb. oz. (3) 1* Xl6=-=vg AAns. ( 1) Xi12-=-= Ans. yd. qr. qr. na. (5) 1-AX4i=: -X4=4 Ans. A. R. R. P. (6) T -6IX4d —= 32UX40-= Ans. ct. (7) 3-5XloO=-.~-3 Ans. da. hr. hr. min. (8) Tk4 X245='1 _: o-X60=6=l Ans. bu. pk. pk. qt. qt. pt. (9) y3UX4 —: oX -3 X2 Ans. ARTICLE 161.-CASE II. na. qr. qr. yd. (3) 4 4-4: - 4-.i -' Ans. gr. pwt. pwt. oz. oz. lb. ( -) ~24+=Q: * o=20z: ~ 1+12=-gI Anls. --------— 6-. -gou ----- — g ---— U 9 ---- COTMMION FRACTIONS. 85 D 3 3 lb (5), 3 — 4 4 8- - 12=B:~-g- Als. 9 -~, — 2 I 1;,1' - -~ 4 Ans. pt. qt. qt. pk. pk. bu. (6) 3- 2=- 8= —' 4 — 3 Ans. oz. lb. lb. qr. qr.. ct. (7) 4 16=1 25- 4-1- As. in. in. na. na. qr. qr. E.En. (8) 5. 2 X3 4 __ 4 4 a 1 I 5 AS. min. h'. hr. da. (9) I 60-= 2 -2 a. 24= ~ -- Ans. dr. oz. oz. lb. lb. qr. (10) -7 T 16=41 7 I T 1= T-25- F -o- Aus. ARTICLE 162.-CASE III. cts. cts. (3) s of 100 3~~=60 Ans. fur. fur. fur. rd. rd. fur. rd. (4) - of 8='1 —3 =5 of 40=8. Ans. 3 8. oz. oz. oz. pwt. pwt. oz. pwt. (5)4 of 123-=:-9: ~ 4 of 20=-~ —=12. Ans. 9 12. oz. oz. oz. oz. dr. dr. oz. dr, (6) 4 of 16=?74=9, of-' 16=1-6=2. Ans. 9 42 R. R. R. P. P. R. P. (7) 4 of 4=4=21: 4= of 40=-20. Ans. 2 20. hhd. hhd. hhd. gal. gal. gal. qt. qt. (8) of 4-=3: of 63=31: = — of 4=2. Ans. 3 hhd. 31 gal. 2 qt. ARTICLE 163. —CASE IV. hr. min. min. da. min. (5) 13 30=810: 1=1440:'s=-40J Ans. fur. rd. rd. m. rd. (6) 3 25=145: 1=320: - -45 —49 As. -- --- - I- I -3 - -- 86 KEY TO RAY'S THIRD BOOK. ft. in. in. yd. in. (7) 2 8 —32: 1-36: 3-= Ans. (8) 49G_- Ans., that is, 96 pages are - of 432 pages. mni. fur. rd. rd. mi. fur. rd. rd. (9) 15 3 3-4923: 35 7 7=11487: -4t9 2g993 AZs. A. P. P. A. R.P. P. (10) 168 28==26908 37 2 14=6014: Ans. oz. dr. dr. lb. dr. (11) 7 17- 10124: 1=4A,3O04 -024__ Ans. (ll-9 79 2 3: 4 —qt. pt. pt. bu. qt. pt. pt. (12) 2 1 3_136 1 1 I 3203 Ans. 12-. yd. ft. in. in. yd. ft. in. in. (13) 1 1. 54 838G 3 2 I8 -9sG6 10846 f1 —- f - 1 -3I-a 3836 191 8 Ans. AR TICLE 164. (2) (3) (4.) hr. min. da. hr. mlin. da. hr. mlin. sec. da.=16 0 Q wk.=l 18 0 wk. - 4 16 0 0 - hr.= 45 da. = 6 0,da. = 13 20 0 Ans. 1645 41 hr. = — 15 - hr. - 40 0 Ans. 2 0 15 3 min.= 40 Ans. 5 6 00 40 (5) gal. qt. pt. gi. (6) A. R. P. gal. = 3 1 1 2 P. 9 2 34q 3 hhd.- 5 1 0 0 - A.- 2 262 Ans. 6 0 1 3 28q Ans. 2 1 91(7) (8) (9) hr. min. sec. cts. pwt. gr., da.=18 40 0 $ —6 oz. -12 0 Ts hr.== 3 20 4 d..= 7- -- pwt.= 21 Anis. 18 36 40 Ails. 55 Ans. 11 3 __ ~~~~~ CO-IMMON FRACTIONS. 87 1 (lo) (11) (12) fur.rcl. ft. in. hr. min.sec. qr. na. mni.-l 31 1 10 - da.=20 34 17- - E.E.=4 121 7Tfur.= 25 7 6 - hr.- 40 0;I yd. =1 0q Ans. 1 5 104 4 Ans. 19 54 174 Alns. 3 01 - Or, 1 5 10 10 ARTICLE 165. —PROMISCUOUS EXAMPLES. 299 8 92999X9/ 1 Ans. 29999X19 (2) _ 4+ 8 =27 +43 - 36 9 3 6;. — a 3 -: + =3 — 5, 1 3 + 9 24 14 (3) 43- 2 1 2 1 ~ -1 1 8 3- 3 4 2 T.5_{)_ 2Ma 6 - 3 —=9 of V-=- of 23 6 6 127 3 9 8 3 o 3 _7 29 7 87 28__ 5 9 21 AnS. (4) of 7 - of - 7 ~ -+7 _. - o_. % ( s 6. Ans. (5) 52 %1-. 31a 1 Xt 8 4 5; 4 + 3 5- + 8 - 4 T _2 - 83- A s. ( W) 1{(_ -_, —Go ~- - - - (r71 —0o0 16 Ails. (8) If 4 of any thing are taken from it, there will be 4 left; if 16 is 4-sevenths, 1-fourth of 16, w-hich is 4, is 1-seventh, and if 4 is 1-seventh, 7X4-28, is the number. (9) If to any thing there be added 3 of it, the sum will be +, — 10; if 20 is 10-sevenths, 1-tenth of 20, which is 2, is 1-seventh; and if 2 is 1-seventh, 7 times 2, whicllh re 14, is the number. (10) - of = 45, and 4 -_4_ part' left.. Or, the part left may be found thus. If he sell -4 of his share, he has - of it left, and 2 of 8 ~-4.1-.t i52 of $900-4, —2=$375 Ans. 7,h.... - = -...............h -- D 88 KEY TO RAY'S THIRD BOOK. (:L) j of = 19441 7 o 3 T —-~ _iU _ 9, 7 Having 7-thirty-sixths, if we divide by 7, we get 1-thirtysixth; then, multiplying this by 36, we get the value of the whole. Hence the answer will be found by multiplying the g6* of the value by 3-60 1 —00-X3a 6 = $10000 Anis. (2) - of 2=-4; 3-=: Ans. (13)..= ASS..| 16X23 (14') I 1 L9~t'. L 333 148 312 -793 2 7 9 -3 I 8 00 793 _ 1007 A23. (15) 4j% -;E 1 - of __ of 4 — =13; 1 31 T4 T2 TT-1T140 i T I 0 2X4X2 [Ans. 5 4_ x7 1 4 5. I=056-9\ 1 1 - 1. I 4 1 1_i24 1 C 59 (1'7) If a man walk 2044 rods in 7-fifteenths of an hour, lhe will walk 1-seventh of 2044 rods in 1-fifteenth of an hour, and 15 times this distance in an hour; this, multiplied by I14, will give the distance walked in 1 1 4 hours. Thus, - of 2044=292, alnd 292X!5 — 4380-rods walked in an hour; and 4380X1- 4-1 80X i — 8468 rods Ans. O0 0-44y1X > 2044X29 292X29-8468 Ans. Or0, ~.o44x] _? _8468 Ans. 7 1 (18) 11 feet- 15 inche 3- 0 0 2 Ans. (19)1 3~'~:: — ~~c~I ~ 3 3 4 5 _o -9%) 3 1 _L 11 + 5 Ans. — 3 4 and $ $ $ (20) 8 of 2400=5>300 —1500: 1500+500=2000: If { of B's money is $2000, ~ is; of $2000, which is $400, and the whole will be 4 times $400, which is $1600 Ans. (21) Since 2200 is five-twelfths of what the elder son received, one-fifth of this will be one-twelfth of what he received; and twelve times this last sum will be the whole of the elder son's share: DECIMAL FRACTIONS. 89 of 2-2 —440, and 440Xi125280-elder son's share; for a similar reason, dividing this last sum by 16, and multiplying by 35, will give the whole estate: y U of 5280=330, and 330X35=11550, whole estate: Youngest son's share 2200 11550 Eldest son's share.. 5280 7480 7480 3)4070 Eaach daughter's share-$13562 DECIMAL FRACTIONS. SUGGESTIONS TO TEACHERS.-AS it is impossible for pupils to understand the-primary rules without first becoming well acquainted with Numeration and Notation, so in studying decimals it is equally important to become familiarl with their numeration and notation; pupils should, therefore, be drilled at the black-board until they can read and write decimal fractions with ease and accuracy. A R T ICE 1 75.-N UM1E RIATION. To uE REAnxD.-lst Colulmn. (1) Five tenths. (2) Six huncdredths. (3) Three thouZsandths. (4) Twenty-eight thlousanZdths. (5) Thre[ hundred ancd forty-one thousandths. 2d Column. (1) Three ten-thousandths. (2) Six hundred and twenty-five ten-thousac.ndths. (3) Two thousand three hundred and seventy-four ten-thousandclths. (4) Two thousand and six ten-thouszndclths. (5) One hundred and four ten-thousandths. 3d Column. (1) Four hundred-thousandths. (2) One hundred and thirty'-seven hundred-thousandths. (3) Two thousand three hundred and seventy-six hundred-thousandths. (4) Thirty-one thousand four hundred and fifty-six hu7ndred-thousandths. (5) One thousand and seven hundred-thousandths. 4th Column. (1) Seven smilliolzths. (2) One hundred and thirty-three millionths. (3) One thousand seven hundred and sixty-eight millionths. (4) Forty thousand and thirty-five szillionths. (5) Three hundred and sixty thousand and four soillionths. - b. -- do 90 KEY TO RAY'S THIRD BOOK. 5th Column. (1) Eight ten-millionths. (2) Nine hundred-nillionths. (3) One million ten thousand one hundred and one ten-mnillionths. (4) One hundred thousand three hundred and four hundred-nmillionths. (5) Forty thousand and five hundredmillionths. 6th Column. (1) Six units and five tenths. (2) Sixty units and four hundredths. (3) One hundred and eighty-four units and one hundred and seventy-three thousandths. 7th Column. (1) Six tenths, or Six million ten-mzillionths. (2) Eight thousandths, or Eighty thousand ten-millionths. (3) One hundred and two thousandths, or One million and twenty thousand ten-nzillionths. 8th Column. (1) Forty tlousand five hundred and four units, and one thousand and thirty-seven hun.sclred-thousandths. (2) Fifty-four million units, and fifty-four millionths. (3) Thirty million seven hundred and one thousand units, and one million thirty-seven thousand and twenty-five ten-suillionzths. ARTICLE 176. -() 1T0jj5; 0ThO=D; -~o oo-6-.013n5; - 30 0o=.002036 T aoa --—.0004 006 (2) 137,-6-13.7; 2410o=24.OS; 30T o-o-30.025; 16~00004'6.0004; 8oW 0UU —S.000105 (3).4 (9).4125 (15).300004 (21).00000907 (4).26 (10).020'5 (16).000203 (22).20020003 (5).35 (11).0008 (17).000007 (23).01010001 (6).08 (12).20304 (18).0000024 (24) 20.25 (7).005 (13).00605 (19).0080006 (25) 106.037 ($).304 (14).00009 (20).00000002 (26) 1000.001 (27) 200.025 (30) 200.0000000002 (28) 29.000029 (31) 65.006005 (29).001000005 ARTICLE 177. (32) 14.3 (33) 32.04 (34) 1.0502 (o! -. DECIMAL FRACTIONS. 91 ARTICLE 178.-ADDITION. (2) 3.25 (3) 21.611 (4) 6.61 (5) 4.8 6.4 6888.32 636.1 43.31.35 3.4167 6516.14 74.019 67.1234 11.204 Ans. 10.00 Ans. 6913.3477 1 ul3. 133.333 5 3Ans. 133.333 Ans. 7231.0967 (6) 45.019 (7) 432.432 (8) 16.041 7.00071 61.0793 9.000094 93.4327 100.07794 33.27 6.0401 6.009 8.969 Ans~. 151.49251 1000.1001 32.719906 Anls. 1599.69834 Ans. 100.000000 (9) 204.0009 (10).0035 103.00000009.00035 42.009099.000035 430.99.0000035 220.0000009 A -ls..0038885 Ans. 999.9999999 ARTICLE 179.- SIBTRACTION. (2) 5.03 (3) 24.0042 (4) 170.0035 (5).0142 2.115 13.7013 68.00181.005 Ans. 2.915 Ans. 10.3029 Ans. 102.00169 Ans..0092 (6).05 (7) 13.5 (8) 3.00000 (9) 29.0029.0024 8.037.00003 19.003 Ans..0476 Ans. 5.463 Ans. 2.99997 Ans. 9.9999 (10) 5.000 (11) 10000.0000 (12) 1.000000 (13).025.125.0001.000001.000025 Ans. 4.875 Ans. 9999.9999 Ans..999999 Ans..024975 ARTICLE 181.-MULTIPLICATION. (2.).a (5).15X.7= _ o —~ -.105 Ans. 7 Ans. 2.45 (8) 1.035X17- -17.595 Ans. Ans. 2.45 92 IKEY TO RAY'S THIRD BOOK. (9) 19 (10) 4.5 (11).625 (12) 61.76.125 4 64.0071 95 Ans. 18.0 2500 6176 38 3750 43232 19 Ans. 40.000 ens..438496 Ans. 2.375 (13) 1.325 (14) 79000 (15).1 (16) 1..0716.079.01.0001 7950 711000 Ans..001 Ans..0001 1325 553000 9275.....9275 Ans. 6241.000 Ans..0948700 (17).043 (18) 40000 (19).09375.0021.000001.1000064 43 Ans..040000 37500 86 56250 93'5 Ans..0000903 Ans..09375600000 DIVISION. SUGGESTIONS TO TEAcHERS. —The division of decimals is generally a troublesome subject to pupils; this arises from a want of attention to the rule. Should the pupil be at a loss to understand why, in some cases, when the divisor and dividend are both decimals, the quotient should be a whole number, let him read the remarks on the division of fractions, page 79. When the divisor contains more decimal places than the dividend, it is best, before commencing the division, to reduce them both to the same denomination, that is, to make the number of decimal places the same in both; the quotient will then be a whole number. ARTICLE 183. 1.125 2 2.000 (5) =-37.5 Ans. (6) -— _ — =250 Anls..03.008.008 DECIMAL FRACTIONS. 93 37.2 37.20 86.075 (7) - =7.44 Ans. (8) -- 3.13 A1S. 5. 5. 27.5 24.73704 206.166492 (9) 24'77 7.191 Ans. (130) -50.004 Ans. 3.44 4.123 100.8788.000343 (11) 100 2222 Ans. (32) 0003 0001 Ans. 454 3~43 9811.0047 9811.004700 (13) -..90300 Ans..108649.108649.21318 (14) = 1,.122 Ans..19 102048 102048.0000 (15) 320000 Ans..3189.3189.102048 9.9 9.9000 (16) 124 000032 Ans. (17) *- 440 Ans. 3189.0225.0225 (18) 10).10 (19).1)1.0 (20).01)10.00 Ans..01 Ans. 10. Anis. 1000 1.7 1.7000000 (21) - =.-0265625 Ans. 64 64 (22) 80).080 (23) 1000).001000 (24).001).001 Ans..001 Ans..000001 Ans. 1. (25).0000001).ooo00ooo1000 (26).000004).010000 Ans. 1000. Ans. 2500. (27) 1-5000000=2.142857+.7.7 AL RTICLEr 185.-CASsE I. ( 1.000 A9.0 Ans. (2) ~='~~~=.125 -Ans. (3)'~ 8 5 -----— ~ —--------------------- 0 94 KEY TO RAY'S THIRD BOOK. 4 4.00 3.000 (4) 24' =.16 Ans. (5) -43 300. 075I. (6)> Z 6 —.9375s16 (n)s.- 07=5 An0As. 25 40 1 15.0000 1.0000 (6) 59375Ans. (7)T-.1 0008 Ans. 16 1250 9.0000 (8) 6-j= —.0225 Ans. 400 1.00000000 (9).~.00390625 Ans. 256 5.0000 (10) 8- -..8333+ Ans. 4.000000 (12) - 1.000000 090909+ Ans. 11 ARTICLE 186.-CASE II. (2).= As. (3) 5.2-= 4T=q Ans. (4) As.s. (7).033=5 4 bu.s AnAns. (8).5625=6 Ans. (9).34375=Ans. (10).1484375j,4s4 s A19n. ARTI0CLE 187. —CASE III. (3).125 bu. —.125X4-.5 pk. Ans. (4).7 pk. -.7 —-4=.175 bu. Ans. (5).0625 lb.X12-.75 oz. Ans. (6).05 yd.X4X4=.8 na. (7).00546875 A.X4X40=.875 P. Ans. Q - _ _ _- - - --— d DECIMAL FRACTIONS. 95 (8).0004375 mi.X8X40-.14 rd. Ans. (9).25 pt. — 2-.125 qt.:.125 qt. —— 4=.03125 gal. Ans. (10).6 pt.- -2-.3 qt.:.3 qt. —.8=.0375 pk.:.0375 pk. — 4 —.009375 bu. Ans. (11).3 min. —*60-.005 hr.:.005 hr. —24-.0002083 da. — Ans. (12).7 rd.-~-40 —.0175 fur.:.0175 fur. —8.-.0021875 mli. Ans. ARTICLE 188.-CASE IV. (2).875 ycd.X4-3.5 qr.:.5 qr.X4=-2. na. Ans. 3 qr. 2 na. (3).7 lb.X12-8.4 oz.:.4 oz.X20=8. pwt. Ans. 8 oz. 8 pwt. (4).8125 bu.X4=3.25 pk.:.25 pk.X8=2. qt. Ans. 3 ph. 2 qt. (5).3375 A.X4-1.35 R.:.35 R. X40-14. P. Ans. 1 R. 14 P. (6).04318 mi.XS=.34544 fur.:.34544 fur. X40 =13.8176 rd.:.8176 rd.X5-=4.4968 yd.:.4968 yd.X3 -1.4904 Rft..4904 ft.X12-5.8848 in. Ans. 13 rd. 4 yd. 1 ft. 5.8848 in. (7).33625 cwt.X4 —=1.345 qr.:.345 qr.X25=8.625 lb.:.G25 lb.X16=10 oz. Ans. 1 qr. 8 lb. 10 oz. ARTIcLE 189. —CASE V. (2) 2 pk. 4 qt.=20 qt.: 1 bu.=32 qt.: o0-5 025 AIs. (3) 13 hr. 30 min.=810 min.: 1 da.=1440 min. -1 - L -9 -.5625 Anzs. (4) I lb.-=256 dr.: -,=&.03515625 Ans..028 (5) 1 A.-160 P. 02 —.000175 Ans. 160 (6) 1 da. —-1440 min. y 7a-U=.0048611+ Ans. (7) 4 gal. 1 qt. 1.28 pt.=35.28 pt.: 1 hhd.=504 pt. 35.28 5 =. 07 Ans. 504 _- -- -- --- --- - - 96 KEY TO RAY'S THIRD BOOK. (8) 3 pk. 7 qt. 1 pt.=63 pt.: 2 hbu. 2 pk. 4 qt.-i68 pt. -?-=-=. 375 A>7s. (9) 99 pages= —9, of 512 pages: -99=.193359375 A, s. (10) 55 A. 2 R. 17 P. —-8897 P. ss 97 175 229 A. 2 R. 16 P. —36736 P. 3673GARTICLE 190. (1) $0o.4X9=$3.6: $0.1875X12=$2.25 $3.6+$2.25=$5.85 Ans. (2) $0.45X2.3=$1.035: $0.375X1.5=4.5625: $1.035+$.5625=$1.5975 Ans. (3) $2.6875X16=$43: $2.6875X16 —=$43.671875 Ans. (4) $35.25 — $0.75=47 bu. Ans. (5) $98.4 —$2.5625 —38.4 yd. Ans. (6) 6 cwt. 2 qr.=6.5 cwt.: $3.25X6.5=$21.125 Ans. (7) 7 hhd. 23 gal.=7.365079-+ $49X7.365079=$360.8888+- Ans. (8) 343 yd. 3 qr.=343.75 yd.: $0.16X343.75-$55 Ans. (9) 14 bu. 3 pk. 4 qt.=14.875 bu. $.625X14.857=-9.296875 Ans. (10) 13 A. 2 R. 35 P.=13.71875 A. $17.28X13.71875-$237.06 Ans. (11) $19.065, $1.24-15.375 yd.=15 yd. 1 qr. 2 na. Ans. (12) $9.296875 —.3125=29.75 bu.=29 bu. 3 pk. Ans. (13) $59.265 —. $4.32=13.71875 A. —13 A. 2 R. 35 P. Ans. (14).34 yd.X4-1.36 qr.: 1.36 qr.+.325 cjr.=1.685 qr.:.685 qr.X4 —2.74 na.: 2.74 na.+.4 na.-3.14 na. Ans. I qr. 3,14 na yd. qr. na. (15) 1.53 yd.-l 2 0.4S 1.32 qr. — 1 1.28 Ans 1 0 3.20 DECIMAL FRACTIONS. 97 (16) 365.25X.05=18.2625 da..2625 da.X24=6.3 hr. 6.3 hr. —.5 hr.=5.8 hr..8 hr.X60=48. min. Ans. 18 da. 5 hr. 48 min. (17).41 da.-9.84 hr.: 9.84 hr. —.16 hr.=9.68 hr.:.68 hr.X60=40.8 min.:.8 min.X60-48. sec. Ans. 9 hr. 40 min. 48 sec. (18).4 T.X4=1.6 hhd.: 1.6 hhd.+.3 hhd.=1.9 hhd.: 1.9X63=119.7 gal.: 119.7 gal.-.8 gal.=120.5 gal.: 120.5 gal.X4=482 qt.=964 pt Ans. (19) 365.25 da.X.3=109.575 da.:.575 da.X24-13.8 hr..8 hr.X60=48 min. Ans. 109 da. 13 hr. 48 min. (20) 365.25 da.X.005X24X6OX60-=157788 sec. Ans. (21) 1 cord=128X1728=221184 cu. in. Adi z=.000004+ Ans. (22) 17 mi. 3 fur. 15 rd.=17.421875 mi.: $690.35X17.421875-$12027. 19140625 Ans. REfMARnS.-Pupils must have a thorough knowledge of common fractions, before they can understand fully the reason of the rules in decimals. When a pupil is in doubt with regard to the accuracy of the result in any operation involving decimals, let him convert the decimals into common fractions, and then perform the work; the results, in both cases, ought to be the same. It is a useful exercise to perform the same operations in equivalent common and decimal fractions. Thus, they may be required to perform the operations indicated in the following examples, by the rules for common fractions; then to convert the common fractions into decimals, and work by the rules for decimals. EXAMPLES. By common fractions. By decimals. (1) 4 + 1. +-=+ 3= A-ns. 4=.25 -— =.375 -z=.625 Ans. Key.-7 98 KEY TO RAY'S THIRD BOOK. (2) + —. Ans. -5-.3125 (3) 1 AMS a375 (4G)!X.Ig AnES.,3L-.09375 ( -- 7. 4. Ans. 1:-=1.4 (6) 9 — 3 Ans. T 9 —.072 The great advantage of decimals is, that the operations are performed with the same facility as in whole numbers; and, that the most complicated arithmetical expressions may be reduced to equivalent decimals that are comparatively simple. PROPORTION. SUGGEasTIONS TO TEACHERS.-Simple proportion is justly regarded as one of the most important subjects in Arithmetic; particular care should be taken, therefore, to render the pupil well acquainted with its principles. First, let him thoroughly understand the meaning of ratio; if the questions and exercises in the maly deem necessary. It should be kept before the pupil that there is no such thing as ratio between quantities of different kinds, anid hence, that the first and second terms in every proportion must be of the same kind. In performing the work necessary to obtain the fourth term, it may be asked, do we not use a concrete number as a multiplier, and, also, as a divisor? The answer is, we do not; we multiply and divide by abstract numbers that represent the ratio of the first. term to the second. Thus, the question, if 2 yards of cloth cost 5 dollars, how many dollars will 6 yards cost? being stated, is, yd. yd. $ As 2: 6:: 5 to the answer. In performing the operation, we multiply the 5 dollars by 6, and divide by 2; because, the ratio of 2 to 6, is the same as that of 2 yards to 6 yards. The same method of reasoning will apply to every case, both in simple and compound proportion. ARTICLE 203. (8) As 4 hats: 10 hats:: $14: $35 Ans. PROPORTION. 99 (9) As 3 caps: 11 caps:: 69 cts.: $2.53 Ans. (10) As 4 yd.: 9 yd. $7: $15.75 Ans. (11) As 8 yd.: 12 yd.:: $32: $48 Ans. (12) As 12 yd.: 8 yd.:: $48: $32 Ans. (13) As $32: $48:: 8 yd.: 12 yd. Ans. (14) As $48: $32:: 12 yd.: 8 yd. Ans. (16) As 12 men: 8 men:: 24 da..: 16 da. Ans. (17) As 2 da.: 8 da.:: 60 men: 240 men.Ans. (19) As 6 gal.: 2 hhd.:: 65 cts. Ans. Or, 6 gal.: 2X63 gal.:: 65 cts. I gal.: 21 gal.:: 65 cts.. $13.65 Ans. (20) As 5 cwt. 3 qr. 10 lb.: 35 cwt. 1 q.:: $21.06: $126.90 Ans. (21) As 1 yd. 2. qr.: 1 qr. 2 na.:: $2.50: $0.625 Ans. (22) As 90 bu. 450 bu.:: 6 da: 30 da. Ans. (23) As 5 da.: 15 da.:: 6 men: 18 men Ans. (24) As 60 bu. p.: 140 bu. p.:: 15 bu. c.: 35 bu. Ans. (25) As 3 cwt. 1 qr.: 16 cwt. 1 qr.:: $22.60: $113.00 Ans. (26) As 4 ft. 6 in. in shadow: 180 ft. shadow:: 3 ft.: 120 ft. Ans. (27) As 12 hr.: 9 hr.:: 60 da.: 45 da. Ans. (28) As $1: $2200:: $.60: $1320 A's, Ans. As $1: $1800:: $.60: $1080 B's, Ans. (29) $800.30+$250+375.10+$500+$115=$2040.40 As $2040.40: $612.12:: $1: $.30 Ans. 30 cts. (30) As $6: $8:: 9 oz.: 12 oz. Ans. (31) As $300: $250:: 6 mon.: 5 mon. Ans. (32) 27X7= —189, 36-27=9. As 9 mi.: 189 mi.:: I da: 21 da. Ans. (33) 9 hr.: 12 hr.:: $152: $20.881, one month's services when he works 12 hr. a day. $20.88'X4_=$91.91' Ans. 2 t 100 KEY TO RAY'S THIRD BOOK. (34) As 5 lb.: lb.:::' j Ans. (35) As 6 yd. 74 yd.: $53~ $6' Ans. (36) As bu. bu.: e$8 * - A. YXX=9 (37) As 14 yd.: 2 yd.. 74. A ~ 4X — ~.X (38) As $294: 314:: 59. yd.: Ans. TX4'X1 19=1 25-=62' yd. Ans. (39) As.85 gal.:.25 gal.:: $1.36: $0.40 Ans. (40) As 61.3 lb.: 1.07 lb.:: $44.9942: $0.78538 Ans. (41) As yd. E. En. $ Ans. q qr.: 45 qr.':: $4: Ans. oX, 9X84 Ans. (42) As y yd.: 17! yd.:: $44: Ans. -7X' 9Xq=$ —8178.38~ Ans. cogs. cogs. rev. rev. (43) As 26: 35:: 1: 1.e. Hence, the smaller wheel gains 09 of a revolution in each revolution of the larger wheel. rev. rev. rev. of larger. Then,.:!: 10: 1: 28 =revolution of larger, Ans. (44) 1 gal. =32 gills, 32-1=31. As 32: 31:: 100 gal.: 968 gal. Ans. (45) As 70 p.: 20 p.:: 60 sec.: 174 sec. Ans. 1142X174=19577' ft.=3 mi. 5 fur. 145 yd. 2- ft. Ans. (46) As 25 ft.: 25 ft. 5.25 in.:: 643 ft. 8 in. 654 ft. 11.17 in. Ans. (47) As 60 lb.: 2136 lb.:: 75 cts.: $26.70 Ans. (48) As 60 lb.: 1225 lb.:: 81 cts.: $16.53+ Ans. (49) As 56 lb.: 1000 lb.:: 63 cts.: $11.25 Ans. (50) As 32 lb.: 3000 lb.:: 24 ets.: 822.50 Ans. ARTICLE 205. (2) 2 da.: 10 da.: 24 mi.: 240 mi. Ans. 4 hr,. 8 hr. SUGorESTIoN To TEACHERS.-In stating questions in Compound Proportion, it will be found both interesting and instructive for PROPORTION. 101 the pupil to give the reason for the arrangement of each pair of terms, as in the following statements. NOTE. —For convenience, the third term is placed below the second. (4) As 18 rd.: 72 rd. The more rods, the more men. 8 da.: 12 da. The less days, the more men.:: 16 men. Ans. 96 men. This statement is read thus: As 18 rods are to' 72 rods, and as 8 days are to 12 days; so are 16 men to the required number of men. (5) As 6 p.: 15 p. The more persons, the more dollars. 8 mon.: 20 mon. The more months, the more dollars.:: $150. Ans. $937.50. (6) As 7 da.: 9 da. The more days, the more miles. 6 hr.: 11 hr. The more hours, the more miles.:: 217 mi. Ans. 5111 mi. (7) As $100: $75. The less dollars, the less interest. 12 mon.: 9 mon. The less months, the less interest.::$6. Ans. $3.375 (8) As 10100 lb.; 100 lb. The more lb., the less miles. 20 cts.: $60.60 The more money, the more mi.: 20 mi. Ans. 60 mi. (9) As 12 cwt. 3 qr.: 10 T. The more weight, the more money. 400 mi.: 75 mi. The less miles, the less money.:: $57.12 Ans. $168. (10) As 20 men: 18 men. The more men, the less days. 40 rd.1.: 87 rd. 1. The more length, the more days. 5 ft. h.: 8 ft. h. The more hight, the more days. 4 ft. t.: 5 ft. t. The more thickness, the more da.:: 15 days. Ais. 58'" days. (11) As 100 men: 180 men. The less men, the more days. 200 yd. 1.: 180 yd. 1. The less length, the less days. 3 yd.w.: 4 yd.w. The more width, the more days. 2 yd.d.: 3 yd.d. The more depth, the more days. 8 hr.: 10 hr. The less hours, the more days.::6 days. Ans. 24.3 days. CPI_ 102 KEY TO RAY'S THIRD BOOK. ALIQQUOTS OR PlRACTiCE. ArTICLE 207.-(CASE I. (3) $24-cost of 24 yd. at,$1 (5) $173 50 =- l2i=cost of 24 yd. at 50 cts. 25-1i,$ 43.25 lAns. 12 —-t 3=: " " at l121 ctls. $15 Ans. (6) $45 (7) $54 25 _' 11.25 25 -' 13.50 61=l 2.81k 12,=A 6.75 $14.06~ Ans. 0 = 3.37nk $23.62- Anls. (8) $32 (10) 80 (11) 3B (,21 s so=1 162 8 25 =1 8 160 288 12=1 4 25= 20 50o = 18 61=k s_=2 121 10 25 = 9 i29I-1 4.50 Ans. e30 Ans. $190 2 Ans. $319.50 (12) 77 (13) 175 (14) 224 1 14 3 77 700 672 50 O = 38.50 175 25=- 56 12_= a 9.620 1 2 2 8 Ans. $125.12-. 25- 43.75 64-= 14 12, —z 21.87,1 Ans. $770 Ans. $2515.62r (15) 462 (16) 185 1 1 462 185 61 —' | 28.87~ 33 1 —{ 61.66Ans. $490.87~ Ans. $246.66~ —------- ---- -------------------— ~ —""=- ~: PRACTICE. 103 (17) 150 (18) 24 3 1 450 24 9339 1- 50 62= 4 3 33=3 50 Ans. $28 Ans. $550 ARTICLE 208.-CAsE II, (2) $10 (3) $12.50 (4) $12 14 28 5 140 10000 60 2.=1 5 2500 1.= 3.00 Ans. $145 350.00 10 P 4 5 2 R. — 6.25 Ans. $63.75 I R.g 3.121 Ans. $359.37-y (5) $18 (6) $12.50 (7) $1.75 12 14 3 216 5000 5.25 1 R.=4 4.50 1250 2qr —'.871 1o P.~= 1.121l 175.00 Ans. $6.12, Ans. $221.621 2 R.; 6.25 1 R.= 3.12' 20 P.=. 1.5614 5 P.-4 39T1 Ans. 186.32o-{(8) $1.50 (9) $17.25 (10) $6 4 56 83 6.00 10350 498 2 qr. —.75 8625 2 pk.-= 3.00 1 qr.-.37 966.00 1 pk.= 1.50 Ans. $7.12' 2 qr.=- 8.622 2 2 t.4 377 1 qr.-= 4.311 Ans. $502.87~ Ans. $978.93,b...................... _.... 6 104 rKEY TO RAY'S THIRD BOOK. (l1) $4 (12) $3.20 (13) $2.36 24 40 17 96 128.00 1792 2 pk.- 2.00 2 pk.=~ 1.60 256 1 pk.='- 1.00 1 pk.=4.80 43.52 4 qt.=-.50 4 qt;.=.40 1pk -i.64 2 qt.=..25 2 qt..20 08 I qt. —.12' 1 qct.-.10 I pt.=.04 Ans. $99.87 1 --— 05 Ans. 844.28 Ans. $131.15 (14) $.24 (15) $2.56 (16) $3.20 5 3 17 1.20 7.68 2240 8 oz.-=1.12 8 oz.=g 1.28 320 2 oz.= 1.03 4 oz.==.64 54.40 Zo=.-.011 1 oz.==.o16 2Ro.=1 1.60 Ins. $1.36~ 8 dr.=z-.08 1R.i. 80 4 dr.==.04 20 R.-I.40 2 dr.=-.02 10 P.-.20 1 d.=4.01 s p.==.10 Ans.;9.91 4 P.-'.08 Ans. $57.58 ARTICLE 209.-PERCENTAGE. 7,%o.07: 4, I-o=.045: 5- %-=.0525: 10 %=.105: 37-;%=.375: 120 %=1.20 ARTICLE 210.-CASE I. (2) 7 o=-.07 $175X.07 -$12.25 Ans. (4) 6;o=.06 $250X.06 — $15.00 Ans. (5) 7 %o=.07: $162X.07 =$11.34 Ans. (6) 5 % —.05: $11SX.05 =$ 5.90 Ans. (7) 8 %o=.08: $ 11X.08 =$ 0.88 Ans. (8) 1 =.01: 278X.01 -$ 2.78 Ans (9) 2-{ I-=.02 —: $ 68X.02 —$ 1.53 Ans.: —-----— ~I4 4 4 PERCENTAGE. 105 (10) 4, 2-=.045: $220.50X.045-$9.922+ Ans. (11) 71 7-.075: $115.42X.075=$8.656+ Ans. (12) 54 yo=.o05: $243.16X.05V=$13.981+ Ans. (13) 34.034 $1250X.03=-$40.625 Ans. (14) 25 =.25: $25X.25=$6.25 Ans. (15) 101 % =1.015: $2002X1.015=$2032.03 Ans. (16) 208 %o=2.08: $650X2.08-$1352. Ans. (17) 1000 %= —10: $24.75X10=$247.50 Ans. (18)' %o=.001: $400X.001-$0.40 Ans. (19) %-o.008: $464X.00=$81.74 Ans (20) Yo fr=.00.: 81950X.00-1-=$1.624 Ans. (21) 8 %=.081': s72X.08=$ 6.12 ~ $72 —6.12 -=65.88 Ans. (22) 36 Yo=-.36: $800X.36-$288: $800-$288-$512 Ans. (23) $56X.133=8$7.35: $51X.14 —-$7.48: $7.48-7.35 -$0.13 Ans. (24) $1764X.23g=$405.72: $1764X.31X$546.84: $1764 —.09-8158.76: $1764X.034 —57.33. Byadding together the amounts expended, we find their sum is 41168.65, which, being deducted from 81764, the remainder is $595.35, the amount paid for muslin. Or, by adding the several per cents together, and deductingtheir sum from 100 %o (the whole), the remainder is 33 2~ to be expended in muslin: and $1764X.33=$-595.35 (25) 75 lb.X4=300 lb.: 300X.12 —37 lb. 300 lb.-37 lb.-263 lb., and 14 cts.X263-$36.82 Ans. (26) 160X.35-56: 160 sheep+56 sheep-216 sheep Ans. (27) 320X.05 —16: 320-16=304: 304X.25= —76: 304-76-228 Ans. (28) 18+15+23+12 —68: 100-68=32'o left: $800X.32= —256 Ans. 106 KEY TO RAY'S THIRD BO3OK. ARTICLE 211.-CASE iI. (2),15X.01 —.15: $3.00 — $.15=20 Ans. (3) $50X.01=$.5: 6. $.5-12 Ans. (4) $75X.01 —.75: $4.50-:-$.75=6 Ans. (5) -1.331 —33 3-1 70 Ais. (6) $25X.01-8.25: $0.25 —.25-1 Ans. (7) 8142.60X.01=1.426: 67.13 —1.426=5 Ans. (8) 89X.01=-.09: $9 —*$.09=100 Ans. (9) $9X.01=$.09: $13.50 —. $.09=150 Ans. (10) $243X.01=$2.43: $8.505 — $2.43 —3.5=-3 Ans. (11) $2X.01=.02:.002-.02=.1= —jL Ans. (12) 63532X. 01=835.32: $13.245 — $35.32-.375= Alns. (13) $300X.01-83 25-o-.83=8 A7ns. (14) $500X.01=-5: $75 — 8515 Ans. (15) z=.60=60 % Ans. (16) 5 bu.-160 qt. To-g-m-(o=.03 —— o3 % ins. ARTICLE 213.-COMMISSION. (1) $240X.05-=12.00 Ans. (2) $460X.021= $11.50 Ans. (3) $180X.04=$7.20: $119X.03= — $3.57: $7.20+$3.578$10.77 Ans. (4) $240X.03=$7.20: $225X.05=$11.25: $7.20+$11.25=$818.45 Ans. (5) $48OX.02=-$10.80: $275X.03=-$8.80: $216X.02N=$ 5.76 $10.80-+-$8.80+$5.76 — 25.36 Ans. (6) $275X.03=$8.25: $341X.15=$51.15: $964X.25-$241.00: $217X.02-=4.8825: 68.25+$51.15+$241.00+$4.8825=$305.2825 Ans., —----- -- ----- ----- -- - PERCENTAGE. 107 (7) $13X25=$325: ~$325X.02-=-8.12. Agent receives: $325-$8.12!$316.87- Owner receives. (8) 1114 lb. X15=16710 lb.: 8 cts. X16710-$1336.80: $1336.80X.034 —$43.446 commission: $1336.80-$43.446=$$1293.354 Ans. (9) $15X250=$3750: $7X175=-$1225: 0.25X1456 =$364.00: $3750+$1225+$364=$5339: $5339X.03 =$160.17 com.: $5339-$160.17=$5178.83 Ans. ARTICLE 214. (11) $1.00+.04=$1.04 $312 —. $1.04=$300 Ans. (12) $1.00Q-.08=$1.08: $1323.54 — $1.08=$1225.50: $1323.54-$1225.50-$98.04 Ans. ARTICLE 215.-INSURANCE. (1) $2250X.01 —$33.75: $33.75+81.00=$34.75 Ans. (2) $12600+$14400-$27000: 2 of $27000-_$18000: $18000><.042-$810.00: $810+$1 — $811 Ans. (3) 3 of $5000-$,3750: $3750X.00-=$18.75: $18.75+$1.50= —20.25 Ans. (4) 4 of 21000-$12000: $12000X.121-$1500: $1200X.Ol-l$18.00: $1500+$18+81+$1-=$1520 Ans. ARTICLE 216. (6) 1-.01.99: $2475 -.99-$2500 Ans. (7) 1 —.125=.875: $13125 —.875=$15000 A7ns. (8) 1-.01 =1-.0175=.9825: $2358 -.9825=$2400 Ans. (9) 1-.00-=:1-.00375-.99625: $2287.39-.99625 -$2296:.2296 —$2287.39=$8.61 Ans. Or, 1 —.00-.99-= 7'" 7: $ 2287.39 —. 7' 7=$2296, etc. As —--— Or, a 8 -9. 108 IKEY TO RAY'S THIRD BOOK. ARTICLE: 217. —STOCKs. (1) $1400X.04=$56 $1400+$56=S1456 Ans. Or, 104 -o'1.04: $1400X1.04=$1456 Ans. (2) $1400X.96=$1344.00 Ans. (3) $50X11=$550 $550X.05=$27.50: $550+$27.50 -=$577.50 Ans. (4) $75X15=$1125: $1125X.10=$112.50: $1125- 112.50-=1012.50 An~s. (5) 108 — 104-=-3: $1500X.03$=$57.50 Ans. Or, $1500X1.04 =81567.50: $1500X1. 08=r81625.00: $1625.00-$1567.50=857.50 Ans. (6) 3+5-S=8: $1300X.08=-$110.50 Ans. Or, $1300X.97 —$1261.00 cost: 81300X1.05' — 1371.50 sold for: $1371.50 —1261.00 =$110.50 Ans. (7) 4+21=6,: $860X.061=$55.90 Ans. Or, 8860X1.04=$894.40 cost: $860X.97- =$838.50 sold for: $894.40-$838.50=855.90 Ans. ARTICL E 218.-BRO KERAGE. (l) $600X.01-$9.00 Ans. (2) $289X.01O -83.61- Ans. (3) $360X.003 -$1.35: 8360 —$1.35=8358.65 Ans. (4) $134X.02=y-3.35: $134-$3.35=$130.65 Ans. (5) $200X.00-$-1.25: $200+$1.25z=$201.25 Ans. (6) 1.+-=-. $500X.00O-$3.75 Ans. Or, 8500X.00-=$2.50 gain in buying. $500X.00' =81.25 gain in selling. $2.50+$1.25==$3.75 Ans. ARTICLE 220.-CASE I. (3) $65 X.05==$3.25: 83.25X4-$13.00 Ans. (4) $200X.08=$16.00 Ans. (5) $150X.05=$7.50 Ans. PERCENTAGE. 109 (6) $300X.06=$18.00: $18.00X2=$36.00 Ans. (7) $275X.06-$16.50: $16.50X3=849.50 Ans. (8) $187.50X.05=-9.375: $9.375X4=$37.50 Ans. (9) $233.80X.06=$14.028: $14.028XO1=$140.28 Ans. (l1L) $215.00X.06=$12.90: $215+$12.90=-$,227.90 Ans. (12) $45.00X.08=$3.60: $3.60X2-$7.20: $45+$7.20=$52.20 Ans. (13) $80.OOX.07=$5.60: $5.60X4=$22.40: $80+$22.40-$102.40 Ans. (14) $420.00X.05~ =$22.40: $420.00+$22.40=$442.40 Ans. (15) $237.16X.033=$8.8935: $8.8935X2=$17.787: $237.16+$17.787=$254.947 Ans. (16) $74.75X.04-$2.99: $2.99X5 —14.95: $74.75+-14.95=$89.70 Ans. (:17) $85.45X.06=$5.127: $5.127X4 —$20.508: $85.45+$20.508=-$105.958 Arns. (18) $325.00X.059=-$17.55: 17.55X3=852.65 $325-+52.65 —$377.65 Ans. (19) $129.36X.04 =-$5.6595: $5.6595X4422.638: $129.36+$22.638=:$151.998 Anzs. ARTICLE 221.-CASE II. (2) $240X.08-=$19.20~ 2 mon.-x —1 yr.: 6 of $19.20=$3.20 Ars. (4) 8360X.06=$21.60: -Z of $21.60o=$1.80: 2 da =,.r=- r mon. of $1.80=$.12 Ans. (6) $86X.06=$5.16: 3 mon.-I~ yr.: { of $5.16=$1.29 Anrs. (7) $50X.08=$4.00: 4mon.=- yr.:' of $4.00=$1.33- Ans. (8) $150.25X.08-$12.02: 6 mon.= yr.: { of $12.02=$6.01 Ans. (9) $360X.05=818.00: 7 mon.=-7 yr.: > of $18.00=$10.50 Ans. ~~a- - - -- -- - --- ---- - - -- ---- ---— ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~- ---''P 110 KEY TO RAY'S THIRD BOOK. (10) $204X.07=$14.28 11 mon.= -- yr.: } of $14.28=$13.09 Ans. (11) $726X.06=-$43.56: ~1, of $43.56=$3.63: 10 da.=~ mon.: 3 of $3.63=$1.21 Ans. (12) $1200X.06=$72.00: Ti' of $72.00=$6.00: 15 da.=- mon.: ~ of $6.00 —$3.00 Ans. (13) $180X.08=$14.40: T of $14.40-$1.20: 19 da.=-m-o mon.: -s-3 of $1.20=$0.76 Ans. (14) $240X.07-$16.80: -1l of $16.80=$1.40: 27 da, -— 1 — mon. TU9 of $1.40-=$1.26 Ans. (15) $100.80X.05-$5.04: ~-' of $5.04=$.42: 28 dla. -1 4 mon.: o of $0.42 —$0.392 Ans. (16) $228X.06= 13.68: 9 mon.=-9= yr.: -, of $13.68-$10.26: $2284-+$10.26=-$238.26 Ans. (17) $137.50X.06-$8.25: 8 mon.= -1-= yr.: a of $8.25-$5.50: $137.50-+$5 50 —$143.00 Ans. (18) $150X.05=-7.50: -' of $7.50=$.625: 18 ~dx.=- =5 mon.:, of $.625=$.375: $150+$.375=$150.375 Ans. (19) $360X.06=-$21s.6: ~' of $21.60=$1.80: 11 da.=i4o mon.: 3i 0 81.80 —8.66 8360-q -$,360.66 Ans. (20) $264X.06-$15.84: -- of J15.84 —1.32: 9 da. =, -,3o mon.: y-i0- of $1.32 —$.396: $264+$.396_=264.396 Ans. ARTICLE 222. (2) $350 (3).150 (4) 375.40.04.06.06 14.00 9.00 22.5240 7 4 6 mon.=: 11.262 98.00 6.00 2 mon~ - 3'754 3 mon.=-I 3.50 2 mon.- - 1.50 Ans. $37.540 Ans. 4101.50 Ans. $37.50 PERCENTAGE. 111 (5) $92.75 (6) $500 (7) $560.06.06.08 5.5650 30.00 44.80 3 1 mon.=,l 2.50 2 16.695 15 da. = 1125 89.60 4 mon.=~ 1.855 3 da. = 25 4mon.-= 14.93: 1 mon.={.463- Ans. $34.00 15 da.= { 1.86:_ Ans. $19.013+ Ans. $106.40 (8) $750 (9) $456 (10) $216.06.05.10 $45.00 $22.80 21.60 4 3 5 180.00 68.40 108.00 3 mon.= 11.25 4 ron. —- 7.60 6 mon.=jl- 10.80 6 dca. 15 0.75 1 mon.= z 1.90 1 mon. — 1.80 Ans. $192.00 15 da. =$.95 15 da. =.90 3 da. =-.19 10 da. =.60 Ans. $79.04 2 da. 5.12 Ans. $122.22 (11) $380 (12) $300 (13) $250.15.06.06 1900 $18.00 15.00 380 3 6 mon.=- 7.50 57.00oo 5400 1 mon.= 1.25 3 6 jmon.= — 9.00 Int., $23.75 171.00 2 mon. 3.00 250.00 6 mon.= - 28.50 Int., $66.00 Ans. $273.75 3 mon.= l 14.25 300.00 9 da. = — 1.425 Azs. $366.00 Ans. $215.175,a. —--— ~ ------------- 112 KEYr TO PAY'S THIRD BOOK. (14) (15) (16) $205.25 $150.62 $210.25.06.05.07 $12.3150 $7.5310 814.7175 2 3 2 24.630 22.593 29.4350 6mon.=-1 6.1575 4 mon.-=l 2.510+6 mon.=- 7.3587+ 2mon.=-/ 2.0525 1 mon.=.627 1 mon.=zl 1.2264 15 da. = —.5131 12 da. --.251 20 da. =.8176 Int., $33.3531+ Int., $25.981+ Int., $38.8377+ 205.25 150.62 210.25 Ans. $238.6031+- Ans. 8176.601+ Ans. $249.0877+ (17) (18) (19) $57.85 yr. mon. da. yr. mon.da..05 1849 4 19 1849 4 27 $2.8925 _1847 1 9 1848 2 15 2 2 3 10 1 2 12 5.7850 $150 $240 3 mon. -,7231+.06.08 15 da. =f1 -1205 15 do 1205G 9.00 19.20 d.0482 2 2 mon.= 320 2 d. 0.0160 6.2 18.00 12 da, --.64 Int., $6.6928+ 3 ou.on.-l 2.25 Ans. 823.04 57.85 42 20 10 da..25 Ais. 64542 Ans. $20.50 PERCENTAGE. 113 (20) (21 (22) yr. mon. da. mon. da. mon. da. 1845 8 28 11 27 8 28 1.843 5 14 7 3 3 1 2 3 14 4 24 5 27 $180 $137.50 $125.40.07.09.o___ 12.60 12.3750 100320 2 6270 2 mon.-. 4.125 125.20 20 da. =.6875 $1 0.6590 3 mion.-= 3.15 4 dc.- -.1375 4 monI. —' 3.553 10 da. — Aim 84.9500 1P on - o88* 2 cda. =.07 2 dal. -.07 Od 0.la..296,Ai.s,,s28.84 ^ 050 Int. $5.240-.-. 125.40 (23) As. 130.64+ — (23) yr. mon. da.. (24.) 1848 3 9 yr. mon. da. 1847 8 2 1847 7 24 1846 10 25 7 7 8 29 $234.60 8 29 3.05o0 $153.80.05 11.7300 5865 7.6900 12.3165 6 mon. — 3.845.2 -' 1,2816+6 mon.= — 6.15825 6 Cnl 6.1.5825 20 da. 42 72 1 mon.= 1.02637+ - 5 da.' 1068 6 da. =,.20527 4 1 da.:[.03421 int., $5.7460 TInt., $7.42410 153.80 234.60 |Ans. S159,546+Ans. $242.024 A 114 H1KEY TO tAY'S THIRD BOOK. AR TICLE 223. (3) 16 mon. 13 mon. 4 mon. 18 da. 7 mon. 12 da. Ai7.S.,C Ais. $.065 Ans. $.023 $.035 Ans. $.037 10 moen. 13 do. 5 mon. 17 da. 11 mon. iAns..052- $.025 iAns. $.055 Aas. 0, 0277'2 mon. i do. 9 Imn. 3 da. 14 mon. 4 da. i.ts. (.0101 As...0070-!I7 mon. 27 do. 33 mon. 20 da..os085 $165 44 33 Ans. 8.0891- Ans..168 ARITICLE 224. (2),.04><65 —$2.60 Ans. (3) $.o055X36=$1.98 Ans. (4) $.0828=-$2,24 Ans. (5) $.125X500=862.50 Ans. (6) $.022X75T=$1.65 Ans. (7) $.048.X186=$9.083 Ans. (8) M.072X125-$9.00 Ans. (9) 1.204X210.25-$402.891 Ans. (10) $.087-X134.45-$,11.764+ Aits. (11) $,138~X144-.24$119.929+ Ans. A:RT'IC E 226. (2) Principal............. $200.00 Interest to st payment, (12'mon.,)... 12.00 Amount duo, Jan. 1st, 1846 212.00 1st payment, (greater than int.,) to be deducted 70.00 Balance due Jan. 1st, 1846. 142.00 interest for 1 year..... 8.52 Ans. Amount due Jan. 1st, 1847. $150.52 x ~"U"~~"~~.~~""m`w PERCENTAGE. 115 (3) Principal.... 300.00 Interest to 1st payment, (6 mon.,) 9.00 Amount due Jan. 1st, 1844. 309.00 1st payment, (greater than int.,) to be deducted 109.00 Balance due Jan. 1st, 1844,. 200.00 Interest to 2d payment, July 1st, 1844 (6 mon.,) 6.00 $206.00 2d payment, (greater than int.,) to be deducted 100.00 Balance due July 1st, 1844....... 106.00 Interest to Jan. 1st, 1845, (6 mon.,) 3.18 Ans. Amount due Jan. 1st, 1845. $109.18 (4) Principal $150.00 Interest to 1st padyment (i yr. 4 mon.,).. 12.00 Amount due Sept. 10th, 1851.162.00 1st payment (greater than int.,) to be deducted 32.00 Balance due Sept. 10th, 1851.130.00 Sept. 10th, 1852, paid a sum less than the interest then due $6.80 Interest from Sept. 10th, 1851, to Nov. 10th, 1852, (1 yr. 2 mon.,)........... 9.10 139.10 Deduct 2d payment....... 6.80 Ans. Amount due Nov. 10th, 1852. $132.30 (5) Principal.... $200.00 June 5th, 1842, paid a sum less than the interest then due........ $20.00 Dec. 5th, 1842, paid, (making in all a sum greater than the interest).. 50.50 Interest from Mar. 5th, 1841, to Dec. 5th, 1842, (1 yr. 9 mon.,)....... 35.00 $23.o00 Deduct the sun of the 1st and 2d payments. 70.50 Balance due Dec. 5th, 1842,.164.50 Interest on balance to settlement (1 yr. 6 mon.,) 24.675 Ans. Amount due June 5th, 1844. $189.175 116 KEY TO RAY'S THIRD BOOK. (6) Principal.. $250.00 June 1st, 1845, paid a sum less than int., then due.$....6.00 Jan. 1st, 1846, paid (making in all a sum greater than int. then due)... 21.50 Interest to 2d payment, (12 mon.,).... 17.50 267.50 Deduct the sum of the 1st and 2d payments. 27.50 Balance due Jan. 1st, 1846... 240.00 Interest to July 1st, 1846, (6 mon.,)... 40 Ans. Amount due July 1st, 1846. $248.40 (7) Principal.......... $180.00 Interest to ist payment, (6 mon.).... 5.40 Amount due Feb. 1st, 1845.... 185.40 1st payment, (greater than interest,).... 25.40 Balance due Feb. 1st, 1845 160.00 2d payment, less than interest then due $4.30 3d payment.30.00 Int. on bal. to Jan. 1st, 1846, (11 mon.,) 8.80 168.80 Deduct sum of 2d and 3d payments... 34.30 Balance due Jan. 1st, 1846... $134.50:Interest to July 1st, 1846, (6 mon.,).. 4.035 Ans. Amount due July 1st, 1846. $138.535 PERCENTAGE. 117 (8) Principal.. $400.00 Sept. 1st, 1845. Paid, a sum less than interest then due.$10.00 Jan. 1st, 1846. Paid in all, greater than interest then due. 30.00 Interest to Jan. 1st, 1846, (10 mon.,).. 20.00 $420.00 Deduct sum of 1st and 2d payments... 40.00 Balance due Jan. 1st, 1846... $380.00 July 1st, 1846. Paid a sum less than interest then due. $11.00 Sept. 1st, 1846. Paid in all, greater than interest then due.80.00 Interest to Sept. 1st, 1846, (8 mon.,)... 15.20 $395.20 Deduct sum of 3cd and 4th payments. 91.00 304.20 Interest to March Ist, 1847, (6 mon.,)... 9.126 Ans. Amount due March 1st, 1847. $313.326 (9) Principal. $450.00 ist payment, Jan. 1st, 1847, less than interest then due... $20.00 2d payment, April ist, 1847,'less than interest then due.14.00 3d payment, July 16th, 1847, in all, more than interest then due... 31.00 Interest to July 16th, 1847, (15 mon.,).. 45.00 495.00 Deduct sum of 1st three payments.... 65.00 Balance due July 16th, 1847..$430.00 4th payment, Dec. 25th, 1847, less than interest then due. $10.00 5th payment, July 4th, 1848, less than interest then due. 18.00 Interest to June 1st, 1849, (22 mon. 15 da.,). 64.50 494.50 Deduct sum of 4th and 5th payments... 28.00 Ans. Balance due June 1st, 1849. $466.50 _~ —----------------- _s 118 1KEY TO RAY'S THIRD BOOK. (10) Principal.......... $1000.00 1st payment, May 1st, 1840, less than interest then due. $18.00 2d payment, Sept. 4th, 1840, less in all than interest then due...... 20.00 3d payment, Dec. 16th, 1840, less in all than interest then due...... 15.00 4th payment, Ap. 10th, 1841, less in all than interest then due...... 21.00 5th payment, July 13th, 1841, more in all than interest then due... 118.00 Interest to July 13th, 1841, (18 mon. 12 da.,). 92.00 $1092.00 Deduct sum of 1st five payments..... 192.00 Balance due July 13th, 1841,..900.00 Interest to Dec. 23, 1841, (5 mon. 10 da.,). 24.00 Amount due Dec. 23, 1841......... 924.00 6th payment, (greater than int.,)..... 324.00 Balance due Dec. 23, 1841,.600.00 Interest to Oct. 1st, 1843, (1 yr. 9 mon. 8 da.,) 63.80 i4ns. Amount due Oct. 1st, 1843. $663.80 REMARK. —In accordance with suggestion preceding Example 2d, in Arith., in all the preceding solutions the computation of interest has been deferred, until the period when a payment, or the sum of two or more payments, exceeds the interest due. ARTICLE 227. (1) Principal...........$...... $320.00 Interest to Jan. 1st, 1847, (1 yr.)...... 19.20 339.20 Ist payment, May Ist, 1.846..... $50.00 Interest to Jan. 1st, 1847, (8 mon.,). 2.00 2d payment, Nov. 16th, 1846... 100.00 Int. to Jan. 1st, 1847, (1 mon. 15 da.,).75 $152.75 152.75 Ans. Balance due Jan. 1st, 1847. $186.45'I,-"~~~l~" —----— ~cl PERCE NTAGE. 119 (2) Principal........... 540.00 Interest to Jan. 1st, 1848, (10 mon.,)... 36.00 $576.00 1st payment, May 1st, 1847..... $90.00 Int. to Jan. 1st, 1848, (8 mon.,). 4.80 2d payment, July 1st, 1847... 100.00 Int. to Jan. 1st, 1848, (6 mon.,)... 4.00 3d payment, Aug. 1st, 1847.. 150.00 Int. to Jan. 1st, 1848, (5 mon.,).. 5.00 4th payment, Oct. 11th, 1847... 180.00 Tnt. to Jan. 1st, 1848, (2 mon. 20 da.,) 3.20 $537.00 537.00 Ans. Balance due Jan. 1st, 1848 $39.00 A TICLE 228. (l) Principal of the note.......... 875.000 Interest to Aug. 10, 1834, (43 mon.)... 188.1125 Amount............... $1063.125 1st payment deduct...... 260.000 Due Aug. 10, 1834, forming a new principal, $803.125 Interest to Dec. 16, 1835, (16 mon. 6 da.,). 65.053 $868.178 2d payment deduct........... 300.000 Due Dec. 16, 1835, forming a new principal, $568.178 Interest to Dec. 16, 1836, the time between the payments being less than 1 year.. 34.090 8602.268 Amount of $50 fromn March 1, 1836, to Dec. 16, 1836, (9 mon. 15 da.,) to be deducted. 52.375 Due Dec. 16, 1836, forming a new principal, $549.893 Interest from Dec. 16, 1836, to Dec. 16, 1837 32.993 $582.886 Amount of $150 from July 1, 1837, to Dec. 16, 1837, (5 mon. 15 da.,).154.125 Due Dec. 16, 1837, forming a new principal, $428.761 Interest to Sept. 1, 1838, (8 mon. 15 da.,). 18.222 Ans. Due on settlement. A446.983+ - 0- — ~ —-- --- 120 KEY TO RAYN'S THIRD BOOK. ARTICLE 230. (2) $ 6OX.05-$ 3.00: $12 —$ 3=4 yr. Ans. (3) 8375X.08=$3.000: $90. — 30=3 yr. Ans. (4) 8225X.04=-$ 9.00: 66 —$ 9=71 yr. 7-~ yr.=7 yr. 4 mon. AJIS. (5) 2500X.06=$15.00: $34.50-. $15 =2.3 yr.=2 yr. 3 mon. 18 da. Ans. (6) 860X.06-$3.60: 813.77. $3.6 -3.825 yr.=3 yr. 9 mon. 27 da. Ans. (7) $800-$500=8300: $500X.10 — $50: $300-. $50=-6 yr. Ants. (8) $7938 — 600-$198: $600X.09=$154: $198_ $54=3- yr.=3 yr. 8 mon. Anrs. (9) $200X.06-$12: $200-.-$12=162 yr.=16 yr. 8 mon. Ans. (10) $100-.$4=25 yr. Arts. (11) $100-.o$5=20 yr. Ants. (12) $100- - 7=14- yr. Aits. (13) 100 *.$8S=121 yr. An-s. (14) $100 r$ 10-10 yr. Ains. (15) $100.$12-=8 yr. Ans. (16) 4110* —$16=64 yr. Ans. (17) $100X-3=300: $300-('100=$200: $200. +815 -131 yr. Ans. $100X4=$ 400: $400 — 100= 300: 8300-s-$15 =20 yr. Anls. ARTICLE 231. (2) $80\,.01=$.80: $.80X5=$4.00: $28 $4=7 % Ants. (3) $300X.01=83: 3X1~=$5: $3085-56 % Ans. (4) $1000X.01=10: $10X2~-=$25: $200-.-$25=S c ArZs. (5) 8260X.01:$2.60: $2.60X2=5.20: $23.40 — 85.2=4.5=4- % Ans. (6) $640X.01=$6.40: 86.40X6=$38.40: $110.40-.-$38.40-2.875-=2 7 Ans. _a —- _- ---- -- -- - -~~~l~r""~ PERCENTAGE. 121 ARTICLE 232. (2) $.05X3=$.15: $8.25 — $.15=$55 Ans. (3) 8.05X3=$.15: $341.25 -$.15=-2275 Ans. (4) Int. of $1.00 for 16 mon., at 6 % (Art. 223)=-$.08: 92.26 —. $.08=$28.25 Ans. (5) $1023.75-.-$.05=$20475 Ans. (6) Int. of $1 for 12 yr. 3 mon. 20 da. at 9 %o (Art. 222) -$1.1075: $525.398 —. 1.1075=$474.40 Ans. ARTICLE 233. (2) $200X.08=16: $200+$16-$216, 1st yr. $216X.08=$17.28: $216+$17.28=-$233.28, 2d yr. $233.28-$200-$33.28 Ans. (3) $500X.06 —$30: $500$-830=$530, 1st yr. $530X.06=$31.80: $530+$31.80=$561.80, 2d yr. $561.80X.06-8$33.70: $561.80+-833.708 —$595.508 Ans. (4) $800X.06 —:$48.00: $48+$800=8848, 1st yr. $84SX.06-= 50.88: $50.88+$848=$898.88, 2d yr. $898.88X.06=$53.9328: $53.9328+-$898.88 -=952.8128, 3d yr. $952.8128X.06=$57.168d-: $57.168+952.8128 =1009.98+ Ans. (5) $250X.06=$15: $15+$250=$265, 1st yr. $265X.0615.90: $15.90$: $15 9265-=280.90, 2d yr. $280.90X.06-$16.854: $16.854+$280.90 =$297.754, 3d yr. $297.754-$250 —$47.754 Ans. (6) $300X.05 —$15: $15+$300 —$915, 1st yr. $315X.05=$15.75: $15.75+$315=-$330.75, 2d yr. $330.75X.05-,$16.5375: $16.5375+$330.75 =$347.2875, 3d yr. $347.2875X.05= —$l7.364+-: $17.364+$347.287 =-364.65+, 4th yr. $364.65-$300=64.65- Ans. ~ —---- ----- - ------- - - --- 122 KEY TO RAY'S THIRD BOOK. (7) Int. of $1.00 for 6 mon. at 6 % (Art. 233) =.a03 $.03X200=$6: $6+$200 —-206, 1st 6 mon. $.03X206=-$6.18: $6.18+$206=$212.18, 2d 6 mon. $.03X212.18:$6.3654: $6.3654+$212.18=$218.5454, 3d 6 mon. $.03X218.5454- =$6.5563+: $6.5563+$218.5454 — $225.101+, 4th 6 mon. $225.101-$200-$25.101 Ans. (8) 20 ro per annum is equal to 5 %o quarterly, the question then is the same as to find the amount of $500 for 8 years, at 5 % per annum. $500X.05 —$25: $25-$500=$525, 1st quarter. $525X.05=$26.25: $26.25+-$525:$551.25, 2d quar. $551.25X.05:$27.5625: $27.5625+$551.25 =$578.8125, 3d qr. $578.8125X.05-$28.9406+: $28.9406+$578.8125 =-$607.7531, 4th qr. $607.7531X.05-$30.3876+: $30.3876+$607.7531 -$638.1407, 5th qr. $638.1407X.O5 —31.907 -: $31.907+$638.1407 =$670.0477, 6th qr. $670.0477X.05-$33. 5023+ $33.5023+$670.0477 =8703.55, 7th qr. $703.55X.05=35.177+: $35.177+8703.55 =8$738.727+ Ans. (9) $300X.06=$18.00: 818.00+$300=318, Ist yr. $318X.06=$19.08: $19.08+$318=$337.08, 2d yr. $337.08X.06=$20.2248: $20.2248- -2=$10.1124: $10.1124+- 337.08 —347.1924: $347.1924 —$300 =$47.1924 Ans. (10) $100OX.06-= 60: 860+$10O00-$1060, Ist yr. $1060X.06=$63.60: $63.60+$1060 —$1123.60, 2d yr. Int. of $1123.60 for 8 mon. 15 da. (Art. 232) -=$47.753: $47.753+$1123.60=$1171o 353: $1171.353-$1000 =$171.353 Ans. PERCENTAGE. 123 ARTICLE 234. (12) Tab. num. 1.790848X345-$617.84+ Ans. (14) 30 yr.=20 yr.+10 yr. Tab. num. for 20 yr.-=3.207135: Tab. num. for 10 yr.==1.790848. 3.207135X890 =2854.35015: 2854.35015X1.790848-$5111.70d$5111.70-$890-$4221.70+ Ans. (15) 70 yr.-25 yr.+25 yr.+20 yr. Tab. num. for 25 yr. -3.386355: Tab. num. for 20 yr.=2.653298. 3.386355X200-677.271: 3.386355X677.271 -2293.48: 2.653298X2293.48 -$6085.28+: $6085.28-$200.=$5885.28-'- Ans. ARTICLE 236.-DISCOUNT. (2) $.04X2=8.08: $1+-$.08=$1.08: $81 — $1.08 -$75 Ans. (3) $.06X5 — $.30: $11q-$.30-=$1.30: 1300 —.1.30 =$1000 Ans. (4) $.06X3=$.18: $1$.18 —$1.18: $4720-.- $1.18 — $4000 Ans. (5) $1+$.06=$1.06: $257.50-$1.06=$242.924+ Ans. (6) Int. of $1 for 1 yr. 10 mon. (Art. 223)-%.11: $1-l4-.11=$1.11: $199.80 —$1.11-$180.00 Ans. (7) Int. of $1 for 5 yr. 10 mon. (Art. 223) -$.35: $1-+$.35=8$1.35 $6.75 —-1.35= — 00 Ans. (8) Int. of $1 for 5 mon. (Art. 223) =$.025 $1-.$.025 —$1.025: $307.50 — $1.025 —$300.00 Anzs. (9) Int. of $1 for 7 yr. 9 mon. 20 da. (Art. 223) =$.468: $1+$46 1.4 —$1 68~: $493.20 —$1.4683 -$335.89+ Ans. (10) $.06X4-$.24: $1+.24=1.24 24: $496 —$1.24 — $400, present worth. $496-$400-$96 Ans. (11) $.06X2=$.12: $1+$.12=$1.12: $276.64 —$1.12-$247, present worth. $276.64-$247.00- -$29.64 Ans. 124 KEY TO RAY'S THIRD BOOK. (12) Int. of $I for 3 yr. 4 mon. (Art. 223)=$.20: $1+$.20=$1.20: $330 —.$1.20-$275 present worth. $330 —$275=$55 Ans. (13) Int. of $1 for 8 mon. (Art. 223)=$.04: $1+$.04=$1.04: $260 —$1.04=$250 Ans, (14) $1+$.12=-1.12: $235.20 —. $1.12=$210 Ans. (15) $1+$.07 —=1.07: $401.25-$1.07=$375 present worth. $401.25-$375=$26.25 Ans. (16) $1080X.06-$64.80: $64.80X10=$648=interest. $.06X10-=$.60: $1+$.60=$1.60: $1080 —$1.60 =$675. $1080-$675= —$405 discount. $648-$405 = —$243 Ans. (17) 1o of $.12-$.10: $1+$.10=$1.10: $1221+. $1.10 =$1110. $1122 —$1110=$12 Ans. ARTICLE 237. (18) $956.34 — 3 —$318.78, each payment. $.05+$1 —=$1.05: $318.78 —. $1.05$303.60, present worth 1st payment. $.05X2-.10: $1-K.10=$1.10: $318.78-i-$1.10 =$289.80, present worth 2d payment. $.05X3=$.15: $+1+.15=~1.15: $318.75+, $1.15 =$277.20, present worth 3d payment. $303.60+$289.80+$277.20:$870.60 Ans. (19) $1440.2 —=$720, 1st payment: $1440 — 3=$480, 2d payment: 81440-($720+$480)=$240, last payment. Int. of $1 for 3, 6, and 9 mon. (Art. 223) is $.015, $.03, $.045 $1+$.015=$1.015: $720 — $1.015=$709.3596+,-, present worth 1st payment. $1+$.03=$1.03: $480. — $1.03=$466.0194+, present worth 2d payment. $1+$.045=$1.045: $240 —. $1.045=$229.665+, present worth 3d payment. Adding the present worth of the three payments together, their sum is $1405.044+ Anzs. ~~9- -- -- -- -- --— ~~~~~~~~~~A PERCENTAGE. 125 (20) $700-($60+$180+$-260)-=$200, last payment. Int. of $1 for 6, 12, 18, and 24 mon. (Art. 223) is $.03, $.06, $.09, $.12 $1+.03=$1.03: $60 — $1.03-=58.2524+, present worth 1st payment. $1+$.06=$1.06: $180 —. $1.06=$169.8113+, present worth 2d payment. $1+$.09=$1.09: $260. $1.09=$238.5321+, present worth 3d payment. $1+$1.12=$1.12 $200.$1.12=$178.5714+, present worth 4th payment. Sum of payments $645.167+ Ans. ARTICLE 238. (1) $.06X5=$.30: $1+$.30= $1.30: $650.-$1.30 =$500 Ans. (2) $.05X9=$.45: $1+$.45=$1.45: $725 —$1.45 -$500 Ans. (3) $.05X4=$.20: $1+$.20=$1.20: $571.20- $1.20 =$476, principal. $571.20-$476=$95.20, interest. Ans. (4) $.06X2==$.15: $1+$.15=$1.15: $690 — $1.15 =$600, principal. $690 —$600=$90, interest. Ans. ARTICLE 240. —BANIK DISCOUNT. (2) $137X 93 — 60=212+. Int.-$2.12+Ans. (3) $1780X93 -60=89X31=2759. Int.=$27.59 A8ns. (4) $375X 33 -60=75Xl —. 4=2061. Int.==$2.06~ Ans. (5) $165X 63- 60=33X21-.-4=1731. Int.=$1.731 Ans, (6) $140X123 -60= 7X41=287. Int.=$2.87 Ans. (7) $ 80X183 —60- 4X61=244. Int.=$2.44 Ans. (8) $180X 33 60= 3X33=99. Int.=-99 cts. 8180 —$.99=$179.01 lAns. ~o~................. —------ - --- - ---— g 126 KEY TO RAY'S THIRD BOOK. (9) $960X33 * 60=16X33-528. Int.= —5.28 $960-$5.28=-954.72 Ans. (10) $875X93 —. 60-175X31-.4-1356-. Int,=$13.561 $875-$13.561z= $861.443 Ans. (11) $3900X63 —. 60=-65X63=4095. Int.=8$40.95 $3900-$40.95-=3859.05 Ans. (12) $2580X103 —. 60=43X103=4429. Int. at 6 %. =$44.29 Int. at 5 %. (Art. 225, Rem. 3.,) =- of $44.29-$36.908-. $2580 —$36.908 =$2543.09+ Ans. (13) $3.50X225-$787.50, purchase price. $4X225-=900: $900X183-60 —15X 1832745. Int.=$27.45 $900-$27.45-$872.55, proceeds of note. $872.55-$787.50=$85.05 gain Ans. (14) $1+$.07=$1.07: $535$. $1.07=-$500: $535-$500 =-$35, true discount. $535X.07-$37.45, bank discount. $37.45-$35-$2.45 Ans. (15) $.06X4=$.24: $1+$.24=$1.24: $1209+- $1.24 =8975: $1209- 975==$234, true discount. $1209X.06-$72.54: $72.54X4=$290.16, bank discount. $290.16-$234=-$56.16 Ans. ARTICLE 241. (2) Int. of $1 for 3 mon. 3 da. (Art. 223) =-.0155 $1-$.0155=-$.9845: $393.80 —-$.9845=$400 Ans. (3) Int. of $1 for 63 d,. (Art. 223) )-$.0105: $1-$-.0105 —.9895: $500 —$.9895=$505.305+ Ans. REMARK.-For a full discussion of the difference between Bank and true discount, see Ray's Higher Arithmnetic. ARTICLE 243.-CASE I. PROFIT AND Loss. (3) $150X.10=$15 Ans. (4) $874X.25-$218.50 Ans. (5) $500X.12=860: $500+$60=$560 Ans. (6) 8382.50X.04-$15.30: $382.50-$15.3.0-,367.20 Ans. PERCENTAGE. 127 ARTICLE 244.-CASE II. (2).x90X./ 25=22 90 cts.-+22'-=81.12' Ans. $.90X.10=.09: 90 cts. —9 cts.=81 cts. $.81 Ans. (3) $4.37>X(.33=1$1.458+: $4.375+$1.458=$5.83+Ans. $4.37!X.20=.877: $4. 371,-$.871 —$3.50 Ans. (4) $8.50X.20=$1.70: $8.50+$1.70=$10.20: and $8.50-41.70=$6.80 Ans. (5) $300 -40=$7.50 $7.50X.20 —$1.50: $7.50+-1.50 -$9.00 Ans.': $7.50-$1.50=$6.00 Ans. ARTICLn 245. (6) 10cts.+ y' of lOcts.-=llets. 15cts.+-i of 15cts.=16-cts. 20cts.+- 1- of 20cts.=22cts. 30cts+-l of 30cts.=33 cts. 40cts.+-T1 of 40cts.=44cts. 50cts.-+-l of 50cts.=55 cts. 60cts. -i —, of 60cts. —66cts. (7) Scts.-+ of ScS.-t= ts. 12cts.+ ~ of 12cts.=13ects. 16cts.+ of Icts.=18Sts. 20cts.+ 8 of 20cts.= —22cts. (8) lOcts -4- of l0cts.-12cts. 15cts.+- of 15cts.=18cts. 25cts.+- of 25cts.=30ct-. (9) *$11+- of $1-$1.25 8$1.20+-1 of $1.20=$1.50 $1.50+- of $1.50=.$1.87- $2+f of $2=$2.50 $3+-! of $3=$3.75 $4+- of $4-$5.00 $6+-4 of $6=$7.50 (10) 25cts.+-1- of 25cts.=33~cts. 50cts.+-~ of 50cts.-66hcts. (11) $2+- of $2=$3.00. $3 — of $3=$4.50 $4++ of $4=$6. $5+- of $5=$7.50 $7+1 of $7. =$10.50 ARTICLE 246.-CASE III. (2) $105 —75=$30 ~: -=.40-40 %o Ans. (3) $ 40-$30=$10 ~='=.33!=33-1 2 Ans. (4) 30 cts.-25 cts.=5 cts.:,5 =.20=20 o A s. Q~~~~~~~~~~~~~~~~, 5 — 128 KEY TO RAY'S THIRD BOOK. (5) 21 cts.-20 cts.=l ct.: — =.05=5 %o Ans. (6) $9-$8$1: =-.12-=-12 o Ans. (7) 30 cts.-24 cts.=6 cts.: --— =.20-=20 % Ans. (8) $40X40=$1600:.-f664= -=.44-44. Ans. ARTICLE 247 —CASE IV. (3) $1+$.20 =-$1.20: $6- -$1.20-$5 Ans. (4) $1-$.25 =$8.75:83-..75-$4 Ans. (5) $1+$-.25 =$1.25:75 —. $1.25=$60 Ans. (6) $1 —$.25 =8.75: 875 —8.75=$100 Ans. (7) 81 —tI.10 =$1.10:.22-$1.10-=$.20 Ans. (83) 81+-8.1281.1. 125 S X 8.10 1.125=$7.20 Ans. (9) $1 —8.15 $=.85: 1.19 —$.85=$1.40 Ans. ARTICLE 248. -PR.OBISCUOUS EXA IPLES. (1) 6 cts.X30-$1.80: $1.80X.25-$.45 Ans. (2) $250 __40=-6.25, purchase price per yard.,$6.25X.20 -81.25: $6.25+$1.25 —7.50 Ans. (3) 1 hhd.=252 qt.: $.40X252-$100.80: 8100.80 —$75 =$25.80: 2580o=.342=342 a Ans. (4) 2428921-1918604-510317 ~ 1. 3o,-7c-.265+ =26.5+ % Ans. (5) 1519467-937903 581564: 9 o -A=. 62+ =6G2+ % Ans. (6) 81+$8.10=$.10: $.42~. 81.10=$8.38-ll, purchase price. $.51-$.3_8_ —.1= 2 -- 1T -=- 4O- T14=334 %O Ans. o.S =~ —=T7 s (7) 81+$.25-=$1.25: $1- 1.25=$.80, purchase price of rye. $1X.25=,.25: 81-$.25 —$.75, selling price of wheat. $.80 —.75=$.05 Ans. b —--------- - - Q g PERCENTAGE. 1 2'9 (8) $9.60X14 —$134.40, purchase price: $134.40X.20$26.88: $134.40+$26&.88,$161.28, what the whole must be sold for. 814.40X5=$72.00: $12X4-$48.00: 14-(5+4) — 5 pieces left. $72.00+-848.00=$120.00: $161.28-$120-841.28, what the remaining 5 pieces must sell for. =$41.28-.-5-$8.256 A1S. (9) $1-$.14=$.86: $1.29-~-..86 —1.50, purchase price, $1.50X.14=-$.21: $1.50+$.21= —1.71 Ans. (10) $1-$.15=$.85: $1.36 —.$.85-$1.60: $1.856 —81.60 =$.256:.~256=.16=16 O Ans. (11) $1+$-.12-$1.12: $1.96. $1.12 —$1.75, purchase price. $1.75X.16=$8.28: $1.75-$.28-$1.47 Anls. (12) 8$1+-.16= $1.16: 8,1.682_ _$1.16=$1.45, purchase price. $1.$45-$1.247=.203:?i. ~ -—.14=-14 %o,Ans. (13) X.2,5-81.25: $100 -$1.25 —80 yd. A1s. (14) $2.50X.20$.50: $2.50-$.50=83.00, selling price of each cask. $3.00-.$.05-60 lb. Ams. The number of casks has nothing to do with the solution. (15) 812-8 $15=1.253: $1 —$1.25=$. 80-purchase price per bu. $.80X.35=$.28: $.80+$.28=$1.08=selling price per bu. $59.40 — $1.08=55 bu. Ans. A R T I C LE 250.-A SS ESSMENT o T Xs. (1) $500 — $125000=.004, or 4 mills on the dollar.,1650X.004=$6.60 Aizs. (2) $1.25X1258-$-1572.50, aimount raised on polls. $9057.60-$-1572.50=$7485.10, amount to be raisecl on property. 87485.10 —$534650=.014 =i et. 4 m1. on $1?Ans. 130 KEY TO PAY'S THIRD BOOK. (4) $ 00,) is taxed,14.00 (5) $3000 is taxed $42.00 5().) 2.8. 0 500 7.00 40. 50 80. so 1.12 3..'6.042 0..126 3 polls.. 3.75 4 polls 5.00 Aas. $21.152 Ans. $55.246 ARTICLJE 251.-AMIERICAN DUTIES. (1) 100+182+264-546: Draft=1-2-3=6 11). 546 —-=540 1b.: 540X.15=81 lb.: 540-81=459 1b. 2 cts.X459-$9.18 Ans. (2) 400 lb.-4 lb. draft-396 lb.: 396X.05 —19.8 lb. tare: 396-19.=8-376.2, net weight of each bale. 376.2X40=150S 1b1). net. $.25Xl15048-$3762. $376209X.20= — 752.40 Ans. (3) 196 lb.-2 lb. draft= —194 lb. 194X.15=29.1 lb. tare: 194 —29.=164.9 net: 164.9X75=-12367.5 lb. $.05X 12367.5 —$618.375 Ans. A R,l T II GL E 252. -— P ARTNEtSl I P. (2) i -['-8-'- of M232 —$87, A's share, 1,% 0, g of -;2322 —-- 145, B's share. (3) $0i0-1t,0l0 —$0 -$300, whole stock, o~ —o'-3 7 of $120-$28L, A's share.'2-=$500 (5) $100X 6-=600 500X5 —2500 75X 8 — 600 750(X8=6000 125X12=1500 $1500) 9000(6 mon. Ans. $300) 2700(9 mon. Ans. (6) - of,200=$40.00 - of 200= 80.00:40X 0= 0 O8X 5-400 80X 10-800 -4200) 1200(6 mron. Ans. ARTI CLE 258. (2) Counting from April 2d, it is 90 days to the first payment, and 150 days to the second. $200XA 90-=$18000 $300X150- 45000 $500) 63000(126 da. from April 2d, i-which will give Aug. 6th Ans. ARtTICLE 259. (8) $103-.-$1.03 —100 $106 —$1.06 —$100 $103+ 1006-$209 $209 —,$200$ 9 $200X.06=,9$ 12 $9- -$12=.75 yr. —9 mon. Ans. ALLIGATIO1N MEDIAL. ARTICLE 260. (2) 6 lb. at 3 cts.=18 cts. (3) 25 lb. at 12 cts.-$3.00 4,1b. at 8 cts.=32 cts. 25 lb. at 18 cts.= 4.50 I0 lb. cost 40 lb. at 25 cts.= —10.00 10 lb. cost 50 cts. 50 cts. — 10=5 cts. Ans. 90 lb. cost $17.50 $17.50 —*90-$.19 4 Ans. - ------ ----- -- - - --- I-I-9 134 Y T'O RAY'S THIRD BOOK. (4) gal. co"st $0.00 (5) 10 at $3-$30.00 12 gal. at 50- 6.00 12 at 4 — 48.00 - at 90 7:2.00 15 gal. cost $6.00 $6.00 —. 15=$.40 Ans. 30 worth $150.00 $150- -30 —$5 An.., (6) 6 to 10= 4 hr.: 63~X 4=252~ 10 to 1- 3' 70~X 3 —-210~ 1 to 3= 2 75 X 2 —150~ 3 to 7 — 4 73~X 4=292~ 7 to 6=11 hr. 550X11-605~ 24)1509~(62"~ Als. ANALYS IS. REMAaRxs. —The subject of analysis is one of the most important in Arithmetic. With a thorowuh knowledge of the elementary rules, fractions, and analysis, nearly all the operations that are purely arithmetical may be solved without the aid of special rules; because these rules are mostly derived from analytical solutions of particular questions. The pupil, however, who desires to understand analysis fully, and apply it readily, must first make himself well acquainted with Mental Arithmetic. See Ray's Arithmretic, 2d Book, ARTI3CLE, 261. (2) $36 (3) $24 (5) 7bl). (6) 10 lb. (7) lO yd. (9) 7' y. (10) 4. (i4L) 43. (13) 2: 20. (14) 28: 27. (16) 0. (17) s. (19) 18. (20) 28. (23) 16 cts. (24) If i-fifth of a bushel cost 14 cents, 1 bushel will cost 5 times 14 cents, which are 70 cents; and 9T of a bushel will cost y% of 70 cents, which are 63 cents. Ans. (25) i-eighth will cost i-third of 3-eighths; if 8 cost 21 cents, -} will cost - of 21 cents, which is 7 cents, and if S cost 7 cents, 1 yard, or 8-eighths, will cost 8 times 7 cents, which are 56 cents; then q of a yard will cost -. of 56 cents, which are 48 cents. Ans. ANALYSIS. 135 ARTICLE 263. (2) $2.16 —24 —9 cts.: 9 cts.X>23-$2. 07 Anl (3) $32.50-: 13=-2.50: 02.50X1i48-35 Ans. (4) $3-:.4=~$2: 24 — $2 — 12 yd. Ans. (5) $20.67-0 159 —$.13: 34.71~-L.13-267 yd. Ain s. (6) $147. 0341=S —---: $41X9- 838 —" Ans.| (8) 25 da.X17=425 da.: 425 da. — 10 42'- da. Ans. (9) If 6 men can build a wall in 10 days, it will require O1 times- 6-60 men to build it in 1 day, and 1-fifteenth of 60 men-u4 men, to build it in 15 days. tAns. (10) If 5 men eat a barrel of flour in 12 days, 1 man will eat a barrel in 5 times 12 days, which are 60 days, and 4 men Awill eat a bahrel in i-fourth of the time of 1 man, that is, in - of 60 days —15 da. Ans. (11) 10 hir.X7=70 hr. to do the work: 70 hr. — 12 hr.=5o da. of 12 hr. each. Ains. (12) If 18 men reap 72 acres in 7 day}s, 1 man would reap it in 7X1S —126 days, and 8 men in - of 126 —15:3 da. Ans. (13) Each 9 sheep are worth 5 hogs: in 54 sheep there are 6 times 9 sheep; then 54 sheep are iworth 6 timies 5 hogs, which are 30 hogs. Als. (14) In 42 gal. of wine there are 14 times 3 gal. of wine; hence 42 gal. of -Wine are worth 14 times 7 gal. of cider -98 gal. Atns. (15) If a 3 cent loaf weigh 8 oz., when flour is $4 a barrel, if the flour were $1 a barrel it should weigh 8 oz.X4 =-32 oz.; and when flour is $5 a barrel, it should weigh 1-fifth of 3. oz. —64 oz. Ans. (16) If 3 stacks of hay keep 12 horses 5 mon., they would keep 1 horse 5 monr.X12=60 mon., and 20 horses.l-tentieth of 60 inon. —3 non. Ans. (17) If o of a yard cost 40 cts., - of a yard will cost - of 40 cts.=20 cts., and w will cost 3 times 20 cts. —-60 cts. Asis. 136 i:KEY TO RAY'S THIRD BOOK. (18) If 3 yd. cost $2, one yard will cost 3 times -2= $6, and 3 yd. will cost 4 of $6-$4.50. Ans. 44 1 of (19) If A of a tun cost $4.;25; one-seventh will cost of $4.25 —$.85, and a tun will cost 7 times $.85 —= 5.95: then —' of a tun will cost - of $5.95 —3.85 Ans. (20) If T9 of a cord cost $2.52, one-eleventh will cost -a of S2.52$ —.28, and a cord will cost 11 times $.28 =3.08 then -1 of a cord will cost'- of $3.08 —-2.86 Acs. 14'j' 14 (21) If a lb. cost e$, one-seventh will cost - of $> —-, and a lb. will cost 7 times $'Y —4; then 83 Ib. will cost, of: —,44 ~A's. ARTICLE 264. (23) If 2 men can earn $16 in 4 days, 1 man will earn ) of $16=88 in 4 days, and in 1 day he will earn l of $8 =$2: then if 1 tman earn $2 in 1 day, 7 men will earn 7 times $2=$14, in 1 day; and in 3 days they will earn 3 times $14=$42 Ans. (24) If 2 men can build 12 rd. in 9 days, 1. man can build 4 of 12 rd.=6 rd. in 9 days, and in 1 day he can build W - of 6 rd. —= rld.: then if 1 man can build 3 rd. in 1 day, 7 men can build 7 times' rd. —4 rd. in I day, and in 6 days, they can build 6 times 42 rd. —28 rd. Ants. (25) If 11 A. can be plowed in 5 da. with 15 oxen, it would require 15X5=75 oxen to plow 11 A. in one day; and, since 33 A.-3 times 11 A., it would require 75 oxen/3 — 225 oxen to plow 33 A. in 1 day; and to plow 33 A. in 9 days, it would take 4 of 225 oxen —25 oxen Arms. ARTICLE 265. (26) Since the first pipe fJills 41 gal. in 1 hr., and the second emp)ties 6 gal. in the same time, therefore there are left in, 41 —6-35 gal. per hour; it will therefore require 75o _05 0 hr.=:7 hr. 8 min. 342 sec. to fill the cistern. (27) 600 —. 8-75 gal. filled by A in I hr.: 600 —. 12 —50 gal. filled by B in 1 hr.: then 75+50-125 gal. filled by both in 1 hr.; and fi00_.24_444 hr. Ans. c~- ------------ ~ —4~ hr..Ans. —---- ANALYSIS. 137 (28) 900 —: 1090 gal1. filled by A in i hr. 900 — 12-75 gal. eMlnlied by B in 1. hr.: then 90-75=15 gal. left in per hr., and 900 — 15 —=0 hr. to fill the cistern. (30) A does.,10 B 1 and C -1 in 1 day, hence in 1 day they all do I —'r a- - )+6-d- 0 in one day, and G 3 d -6-ay therefore all do it in 1 -- das. Ans. (3)1) Reasoning as in the preceding example, A does y, B 1', C l-, and D1) in 1 day, and in 1 day they all do I- 1 J _L 1l] + 40 Ef3Bo >2 0o o- 1 5%3; hence it will require 1-s. — 8 —4t4 days to do the whole. (32).9- - -- s_7, and 1 —I 1-4- 4 da. Ans. Or, suppose the meadow to consist of 72 parts, then A mows I of 72=9 parts'in one day, and B I of 72=8 parts in one day 9' —8=17 parts both do in one day; 72 —: 17=4f7 da. Ans. (33) Since A and B together, mow it in 12 days, they mow TAin one day, and since A can mow it in 20 days, he can mow ~6 in one day, hence B can mow -3 8- -3-0 in one day; hence he dan mow it in 1 —. 31 — 30 da. Ans. (34) Since both eat it in 6 days, they eat ~ in 1 day, and since the woman eats it in 15 days, she eats jl in 1 day; hence the man eats ~ —a - in 1 day, therefore he will eat in 1 —T-= —10 da. Ans. (35) The first will empty 1, the second 4j and the third 1 in 1 minute, hence they will all empty a-l + 1 - oe 3 A -boo-__3 in 1. minute; and 1 ~-o o3fo1 mi n. 154- sec. Ans. (36) Since A can mow 5 in 6 days, he can mow -- in - of 6 days, which is 2 days, and since he can mow I in 2 days, he can mow the whole in 5 times 2 days=10 days; therefore he can mow 1 in 1 day. Again, since B can mow - in 10 days, he can mow q in a of 10 days, which is 2 days, and since he can mow -} in 2 days, he can rev —------------ ---- - -- - --- - - ~ 138:K'EY' TO RAY'S THIRD BOOK. mow the whole in 7 times 2 days=14 days; therefore he can mow ~- in 1 dty. da,-L 7 L h — =-e -, the part both mow in one ece 5 days, the time in which both can 10o it. (38) $3000-$5500 —S2500: $2500-_ 2= —1250, less share, and 4(1250-+$500-=$1750, greater share. (39) The wife received %3$00 more than the son, and the son $300 more than the daughter; hence, the wife received $800+-f$300=81100 moloe than the daughter. $300+$1100 —81400, whlat the wife and son received more than twice the daughlter's. $3500 —$1400-$2100:.$2100 —3-$700- daughter's share: $;700I-$300-$1000, son's share: $1000+-$800 -=S1800, wife's share. (41) In this case the minute hand mlust gain 25 min. on the hour hand. NowT, according to the solution of Example 40, in the Arithmetic, }- of this 25 mil. is the distance over whicih the minute hand must move to overtake the hour hand. of mi. Ans. (42) When the hour hand is at VIII, the minute hand is at XII. If the hour hand did not move, the minute hand would have to pass over 10 minutes to be opposite to the hour hand, but since the hour hand does move, the minute hand must gaia 10 minutes before it is opposite to the 1,iour hand. It is then required to find what distance the minute hand must move to gain 10 minutes on the hour hand. Reasoning as in the Solution to Example 40, we find T? of 10-10 0 min. Ans. (43),s of $3276=$1365 Since -L of what A put in, was equal to 6 of what B put in,. of 81365=$273, is -' of what B put in; hence $273X9:2457, iwh-at B put in. ] of $45i7$1001; $3276 — $1001 —:$2275, w hich is.5 of what C put in; hence of 9975 —$45 1 of what C put ill; hence C put in $4oo55X'X 4- 80. Since A received a sum the o' cif w-ich was equal to the amount --------------- ------- - - J ANALYSIS. 139 he put in,. of the amount he put in, multiplied by 5, loust equal his slhare; of $3276=,$546, and $546X5 =$2730, A's share. 6 of 1820 —$2184, C's gain. $7000- ($2730-F$21 84) — $2086, Bs gain. ARTICLE 266. (45) By taking 1 lb. at 4 cts. I gain 1 ct.; and by taking 1 lb. at 8 cts., I lose 3 cts.; hence, in order to equalize the gains and losses, as often as 11). at 8 cts. is taken, there must be taken 3 lb. at 4 cts. (46) By taking 1 lb. at 3 cts. there is a gain of 3 cts., and by taking 1 lb. at 8 cts. there is a loss of 2 ets.; hence, to equalize the gains and losses, there must be taken 2 lb. at 3 cts., which will give a gain of 6 ets., and 3 lb. at 8 cts., which will give a loss of 6 cts. (48) On each lb. at 11 cts. there is a loss of I ct., hence on 10 lb. there will be a loss of 1 ct.><10-10 cts. On each lb. at 8 cts. there is a gain of 2 cts.; hence, if 1 lb. gains 2 cts., it is merely required to find how many lb. miust be taken to gain 10 cts., which is eviden tly as many as 2 cts. is contained times in 10 cts., that is, 5 lb. Alns. ARTICE, 26{7. (51) 1 hence -; of the n1umber is 8, and the number is 12 times 8-96 Ans. (52) 7 ——' I'; hence -r' of the number is 7, and the number is 10 times 7-70 Anis. (53) o:o —-, and 1 —8=, the part of his money left. Then -7- of his money is $35, and - is - of 35, which is,,5n, and 15-fifteentihs, or the whole, is 15 timeos $5, which is 87 Ans. (54) 1I.-o +I 3o__...; hen ce 3 —73, and.o is v3 of 73, which is 1, and 3 o or the whole, is 30 times 1 —30 As. (55) 1 4 -, part remaining; lihe then had left 20+15=35 cts.; - if is 35 cts., wh of 35 cts., hich is ts is, and the whole is 9 times 7 cts. —63 cts. 4A1s. I140 ICKEY TO RAY'S THIRD BOOK. (5r6) i —; 43a of -1-:. 1_ l2 part left; then 10 cts.; and a of 10 cts., which is 5 cts, is 1'1; hence 5is or the whole, is 15 times 5 cts.-75 cts. Ans. (57) Of each 4 days he worked 3, earning $6, and idled 1 day, forfeiting $1; hence, for each 3 days he worked he received 85, and since he received $25, which are 5 times $5, he worked 5 times 3 da.=15 da. A7ns. (58) -1 I~ ]__ 0l _ 44 o 1 9 I = part bearing pears; hence o is 4, and 4O-, or the whole, are 20 tinmes 4=80 Anls. (59) As many, (1,) as many more, (1,) half as many more, ({-,) and one-fourth as many more, (1-,) are equal to 1+ 1i1+ -- 4 - --; then since 1 of the number are 110, one-fourth will be ~w of 110, which is 10, and the whole, or 4-fourths, will be 4 times 10=40 Ans. (60) 1-1 —,-:=part left at close of 1st day. I of 4-13; 4- 9 1- — part left at close of 2d clay. 4 196o6-94; 9 6 7part lef at close of 3d day. 4 of 674'2 7 27_ _ -Part leftatcloseof 4th day. Then since 5 —$1.62, I-eighty-first part of $1.62, which is 2 cts., is part of the whole; hence 256, o1 the whole, are 256 times 2 cts.=$5.12 Ans. (61) i +-3:4part of capital at close of 1st yr. -- Of ( —; f+- 1 2 — partof capital at close of 2d yr. { of 12 3; 15'- 3a=9part of capital at close of 3d yr. Then since 4=-$3375, one-ninth of $3375, which is $375, will be one-fifth of the capital, and s, or the whole, will be 5 times $375-81875 Ans. EXCHANGE OF CURiRENCIES. ARTI C L E 269. (1)',r5X20+,3 os.=103 s.: 103 s.X12+2 d.-1238 d.: 1238 d.X4+1 far.=4953 far..Ans. (2) 8675 far.-;-4=21683 d.: 2168s d. -12-2180 s. 8. d. 180 s. — 20 —9 Ans. ~9 8- d. X -I —----— c —--- — o EXCHANGE OF CURRENCIES. 141 (3) z s. d. (4) ~ s. d.., 0 64 17 6 5 5 10 1 15 5 1 2S II-) 4 Ants. 9 0 5'. Arzs. 1.1 G 0 (5) z S. d. (6) z d. 3 12 2 6)2s o o1.0 Ans. 4 3 54 Ans. 28 17 8 (7) ~.625X\20-12.5 s. ~.5 s.X12-6 d. Ans. 12 s. 6 d. (8) ~.796875X20-15.9375 s.:.9375 s.X12 —11.25 d.:.25 d.X4=1 far. — d. Ans. 15 s. 11'- d. (9) 7s. 6d.=-90d,: ~~1-240d.: 90o= -~.375 Ans. (10) 8 s. 9d.=105 d.: ~1=240 d. ~: I0=1~..43 75 Anls. ARTICLE 270. (11).~75 10 s.=~75.5: ~75.5X.04=~3.02: ~3.02X2=~47.55: ~.55X20-1 s. Ans. ~7 11 s. (12) 12 s. 6d.=~.625: ~85.625X.065=~: ~5.1375X 1- (1 yr. 9 mon.) —~8.990625: ~.990625X20 =19.8125 s.:.8125 s.X2 —=9.75 d.:.75 d.X4=3 far., or 4 d. Ans. ~8 19 s. 93 d. ARTICLE 271. (4) ~25X4.84-$121 Ans. (5) 8 s.=~.4. ~15.4X4.84=$74.536 Ans. (6) 15 s. 9 d.=~.7875. ~36.7875X4.84=$178.05+ Anzs. (7) $179.08 —.$4.84=~37 Ans. (8) $124.388- -$4.84=.~25.7=~25 14s. Ans. (9) $1000 *. 4.84 —~206.61157+=-~206 12 s. 2{ cl. + A~ns. e~ ~~ —-— ~ —--- -- t) —----— ~ 142 KEY TO RAY'S THIRD BOOK. ARTICLE 272. (2) ~150OX4-=$ 6664: $GC66X.08S —$3.3 — $53- prem. $6i66-+ 853'3 — 87o An7s. (3) 10s.=~.5 S~0.5X44 —$357.777+: $357.777X.09.=$33.988+ prem.: 8357.777+$33.988 $391.76 — Aizs. ARTICLE 273. (1) $120.50X-o-3=~36.15_ —~36 3s. Ans. (2) $ 75.25X =-~30.1 =-~30 2 s. Ams. (3),l9SX —~36.75=~36 15 s. Anzs. (4) ~30 115 s.=9~30.75: ~.30.75 5 —. 1 —-102.50 Ans. (5) ~25 17s.-~25.85 5- 64.625 A~-s. (6) ~29 8 s.-=~20.4 ~29.4 -7-4-812G.00 Ans.:LrnTICLE 274. Ex. 1.166 fr.- -5.30 fr.=$220 LIEns. ART ICL E 276.-DU D ECI ALS. (2) ft. in. (3) ft. in. (4) ft. in. 53 5 7 8 6 1 5 1 10 7 3 5 3 //7 5 59 11 3 2 23 4 7 10 2 1 8 3 Ans. 7 5 3 Ans. 10 2 10 Ans. 62 0 1.1 3 (5) ft. in. " (6) ft. in. "/ 8 4 6 4 5 6 2 7 4 2 3 5 16 9 0 "' 8 11 0 4 10 7 1 1 4 //// 2 9 6 1 10 3 6 Ans. 21 10 5 0 A'ns. 10 2 2 9 6 _2 ~ ~ ~ ---.,,~ — M ------- - INVOLUTION. 143 ARTICLE 2979 -INVOLUTION. (1) 1-4-9-16-25-36-49-64 —81-1100 121. (2) 1-8-27 —64. (3) 4 4 1 (4) -1-8 27 (5) 16 —32-81. - (6) 65X665=4225 Ans. (7) 25X25X25z-15625 Alw. 8) 2 = 34 -2.231 X,3-4-2 72 4 1S. (9) 12.. 2 5 X 25 X 1, -L5S,_a-19O38 Alw. (9) l,2 — ~ 41953~ Ans. (10) 13X13X13X1313-28561 Ans. (12) 9X9X9X9X9X99-531441 Ans. (13).025X.025X.025X.025-.0000003906250 Ans. (14) 14 3 — 14X 4X14=2744 Ans. (15) 194=19X19X19X19=130321 Ans. (16) 2 1 X- 7 X 7X 7 X7 -80-69 4j Ans. (17).09 6Q).09X.09X.09X.09X.09X.09=.000000531441 Ans. ARTICLE 286. (2) 529(20+3-23 Ans. 400 20 129 2 J29 40 3 43J ARTICLE 287. (6) 625(25 Anls. (7) 656i(81 Ans. (8) 30625(625 Ans. 4 64 36 45)225 161)161 122)306 225 161 244 1245)6225 6225 144 KEY TO RAY'S TilIRD BOOK. (9) 1679616(1296 Ans. (10) 5764801(2401 Ans. 1 4 22)67 44)176 44 176 249)2396 4801)4801 2241 4801 2586)15516 15516 (12) 987606329(31427 lAns. 9 (11) 4;3046'721(6561 Ans. 61)87 36 61 1125)704 6)24)2665 625 2496 1 306)7967 628.2)16963 7836 12564 13121)13121. 62847)439929 13121 439929 (13) 289442169(17013 Ans. (14) 234.09(15.3 Ans. 1 1 27)189 25)134 189 125 3401)4421 303)909 3401 909 34023)102069 102069 ~~~ -----— ~~~~~~~~~~~ —---------- -------- EVOLUTION. 145 (15) 145.2025(12.05 Adns. (16).0196(.14 Ans. 22)45 24)96 ~~~~44 96 2405)12025 12025 (1.7) 1.008016(1.004 Ans. (18).00822649(.0907 Ans. i 81 2004)008016 1807)12649 8016 12649 (19) 1/2535I, 1/729=.27, V., s A nes. (2 0) — 3111 169-13, As I. (21) 3O... —-", Ans. (22) 10(3.162277+ Ans. (23) 2(1.41421+ A.ns. 9 1 61)100 24)100 61 96 626)3900 281)400 3756 281 6322)14400 2824)11900 12644 11296 63242)175600 28282)60400 126484 56564 632447)4911600 282841)383600 4427129 282841 6324547)48447100 44271829 Key. —10 1 146 KEY TO RAY'S THIRD BOOK. (24)) ~2-.666666+(.S1649+Ans. (25) 6=6.4(2.5298+- Ans. 64 4 61i)266 -45)240 161 225 1626)10566 502)1500 9756 1004 16324)81066 5049)49600 65296 45441 163289)1577066 50588)415900 1469601 404704 (26) 1/6241=79 Ana. ARTICLE 288. (l) 16=2X2X2X2: V16=2X2= 4 Ans. (2) 36=2X2X3X3: V/36=2X3= 6 Ans. (3) 100=2X2X5X5: /100=2X5=10 Ans. (4) 225=3X3X5X5: /22-5=3X5-=!15 Ans. (5) V1/(16X25)= 4X5 —20 Ans. (6) V(36X49)= 6X7-=42 Ans. (7) /(64XS1)= SX9=72 Ans. ($) /(121X25)=1]lX5=55 AnS. ARTIC LE 290. (1) 30-= 900 (2) 812= 6561 402=1600 1082=11664 I/2500-50.Ans. 1/18225 —135 Alns. (3) 100-=_10000 (4) 452=2025 60-2= 3600 602=3600 1/6400= —80 Ans. 1/5625=75 Ans. (5) 36 -1296 242= 576 1/720=26.83+ Ans. EVOLUTION. 147 (6) 60;=3600 37'2=1369 1/2231 —47.2334+=wiidth of street from foot of ladder on one side. 60,2=3600 232= 529 V3071-55.4166+=- width of street firom foot of ladlAs. 1.02.65 der on the other side. (7) 6002%360000 140'.= 19600 V/340400=583.43+ 100 —2= 50. Ans. 533.43+ (8) 202-400 The square root of 656, will give the 162=256 length of the diagonal line joining 656 opposite corners of the floor of the room: this is the base of the triSquare of base=656 angle, of which the hypotenuse is 122 —per.2=144 required. 800 1/800=28.28+ Ans. ARTICLE 291. (1) 1/4096=64 Ans. (2) 4X4=16: 16X9-=144: i/144=12 Ans. Or, /(16X9)=4X3 —12 Ans. (3) 1/43560=208.71+ ft., side of acre. 43560 —. 2=21780: 1/21780-147.58+ ft., side of half A. 43560. 4=10890: 1/10890-104.35+ ft., side of qr. A. (4) 10 A.=1600 P. 2 A.-=2000 1/3600=60 Ans. 148 KEY TO RAY'S THIRD BOOK. ARTICLE 295. 1728)10+2=12 root. 1000 10X0OX3=300 728 10X 2X3= 60 2X 2 = 4 364 728 ARTICLE 296. (5) 91125(45 Ans. (6) 195112(58 64 125 [Ans. 27125 70112 4X4X300=4800 5X5X300=7500 4X5X 30= 600 5X8X 30-1200 5X5 = 25 sX8 = 64 5425 27125 8764 70112 (7) 912673(97 Ans. (8) 1225043(107 729 1 LAns. 183673 1X 1X300= 300 225 9X9X300-24300 10X1OX300=30000 225043 9X7X 30= 1890 10X 7X 30= 2100 7X7 = 49 7X 7 = 49 26239183673 32149 225043 EVOLUTION. 149 (9) 13312053(237 Ans. 8 2X2X300=1200 5312 2X3X 30- 180 3X3 = 9 138914167 23X23X300- 158700 1145053 23X 7X 30- 48301 7X7 = 49 1635791145053 (10) 102503232(468 Ans. 64 4X4X300=4800 38503 4X6X 30= 720 6X6 = 36 1 33336 46X46X300-634800 5167232 46X 8X 30- 11040 8X 8 = 64 645904 5167232 (11) 529475129(809 A-is. 512 SX 8X300= 19200 17475 80X80X300=1920000 17475129 80X 9X 30= 21600 9X- 9 -- 81 1941681 17475129 -~~~~~~~~~~~~~~~~... 150 KEY TO RAY'S THIRD BOOK. (12) 958585256(986 Ans. 729 99XX300=24300 229585 9X 8X30= 2160 8X8 - 64 26524 212192 98X98X300-2881200 17393256 98X 6X 30= 17640 6X 6 - 36 2898876 17393256 (13) 14760213677(2453 Ans. 8 2X2X300o=1200 6760 2X4X 30- 240 4X4 = 16 1456 5824 24X24X300-172800 936213 24X 5X 30= 3600 5X 5 - 25 t76425 882125 215X245X300 —18007500 54088677 215X 3X 30- 22050 3X 3 - 9 18029559 54088677 EVOLUTION. 151 (14) 128100283921(5041 Ans. 125 5X 5X300- 7500 3100 50X50X300-750000 3100283 50X 4X 30= 6000 4X 4 = 16 756016 3024064 504X504X300=76204800 76219921 504X 1X 30= 15120 1X 1 = 1 76219921 76219921 (15) 53.157376)3.76 Ans. 27 3X3X300=2700 26157 3X7X 30= 630 7X7 -- 49 3379123653 37X37X300=410700 2504376 37X 6X 30= 6660 6X 6 = 36 417396- 2504376 (16).i99i76704(.584 Ans. 125 5X5X300=7500 741]76 5X8X 30=1200 8X8 = 64 8764 70112 58X58X300=1009200 4064704 58X 4X 30= 6960 4>( 4 = 16 1016176 4064704 152 KEY TO RAY'S THIRD BOOK. (17) V3216- 6. A (18) U/2744-14. it.,4 Ans3.-7, AAs. T V]343=7. Is /6859-19 s. 48778 24389 V/24389-29, 118638 59319' V59319-39. (20) Alo4n. /72 99 Ans - 5-T-T'2, &/125=5 (21) 2(1.25992+ Ans. 1 1XiX300=300 1000 IX2X 30:= 60 2X2 4 364 728 12X12X300=43200 272000 12X 5X 30= 1800 5X 5 -- 25 45025 225125 125X125X300=4687500 46875000 125X 9X 30= 33750 qX 9 = 81 4721331 42491979 1259X1259X300=475524300 4383021000 1259X 9X 30- 339930 9X 9 = 81 475864311 4282778799 12599X12599X300-47620440300 100242201000 12599X 2X 30-= 755940 2X 2 4= 47621196244 95242392488 EVOLUTION. 1 53 (22) 9(2.08008+ Ans. 8 2X 2X\300- 12001000 20X20X300=120000 1000000 20X 8X 30= 4800 8X8 - 64 124864 998912 208X208X300=-1 2979200 1088000 2080X2080X300=1297920000 1088000000 20800X20800X300=129792000000 1088000000000 20800X 8X 30= 4992000 8X 8 = 64 129796992064 1038375936512 (23) 200(5.84803+ Ans. 125 5X5X300=7500 75000 5X8X 30-1200 8X 8= 64 8764 70112 58X58X300=1o009200 4888000 58X 4X 30= 6960 4X 4- 16 1016176 4064704 584X584X 300-1 02316800 823296000 584X 8X 30= 140160 8X 8 — 64 102457024 819656192 5848X 5848X300- 10259731200 3639808000 58480X58480X300=1025973120000 3639808000000 58480X 3X 30- 5263200 3X 3- 9 1025978383209 3077935149627 154 KEY TO RAY'S THIRD BOOK. (24) 9 =9.166666+(2.092+ Ans. 8 2X 2X300- 12001166 20X20X300-120000 1166666 20X 9X 30=- 5400 9X 9- 81 125481 1129329 209X209X300=13104300 37337666 209X 2X 30- 12540 2X 2-= 4 13116844 26233688 (25) U1953.125 —12.5 ft. Ans. (26) 3X3X3=27=cu. in. in a cube, of which each side is 3 in. 1728 —27=64 Ans. (27) 2X2X2=8= cubic inches in the cubical box. XA4X=g14-=- solidity of a cubical block, each side of which is ~ of an inch. 8 — - — 512 Ans. (28) A solid half foot is a cube, each side of which is 6 in.; it contains 6X6X6=216 cu. in.; half a cubic foot=- of 1728=864 cu. in.; 864-216=648 cu. in. Ans. cu. ft. cu. ft. ft. (29) 288X216X48:2985984, Vj2985984=144 Ans. (30) 1728X3=5184,.J5184-17.306+ inches Ans. ARTICLE 297. (31) 63-216; 3 —27. As 216: 27:: 32 lb.: 4 lb. Ans. (32) 11Xl1X11=1331 Ans. ARTICLE 298. (1) 216=2X2X2X3X3X3: V216=2X3=6 Ans. (2) VJ (27X 64)=3X4=12 Ans. (3) V (125X343)=5X7=35 Ans. ARITHMETICAL PROGRESSION. 155 ARITPIHME TICAL PRO GRE SSION. ARTICLE 31. —CASE I. (3) 50-1-49; 49X 3+2-2149 Ans. (4) 100-1 —99; 99X5+9-504 cts.=-$5.04 Ans. (5) 54 —1=53; 53X2=106; 140-106-34 Ans. (6) 20 —1-19; common difference=$216-$200=-$16; $16X);19+$216-$520 Ans. (7) 99 —1=98; 98X~=854; 329-85-3=243~ Ans. A R TICLE 302. —CASE II. (2) 300-3=297; 10 —1=9; 297-9 —-33 Az2s. (3) 50-5= 45; 10-1 —9; 45-*-9- 5 Ans. ARTICLE 303. (1) 49 —544; 44-.-4=-11; 11-1=12 Ans. ARTICLE 304.-CASE III. (2) 50+ 2-52; 52X24-1248; 1248-.2=624 Ans. (3) 1+12=13; 13X12= 156; 156 —-2= 78 Ans. (4) 10000 —19999; 9999X1+1=10000, last term; l10000=1000l; 100001X0000=10001000100010000; 100010000 - 2 —50005000 Ans. (5) The number of terms is evidently 100. The boy travels 6 yards to put the first apple in the basket, 12 the second, and so on; hence, the first term is 6, and the common difference 6. 100-1=99; 99X6+6=600, last term. 6+600=606; 606X100-60600; 60600- 2=30300 yd. 30300 yd. —17 mi. 380 yd. Ans. (6) 30 —1=29; 29X-1-7Z-1; 30 —7-224 mi. distance traveled last day. 30+22z=-,52!-; 521X30 —15821 mi.; 1582_.2 —=7917- mi. Ans. >4 _ _ __ - - -- --- - - 156 KEY TO RAY'S THIRD BOOK. (7) Common difference=193X2=386 in. 60-1=59; 386X59+193=22967 inches, distance fallen in the last second. 193+22967-23160; 23160X60=1389600; 1389600- -2 =694800 in. —57900 ft. Ans. GEOM(ETRICAL PROGRESSION. ARTICLE 307.-CASE I. (3) 212= 4096; 4096X2=8192 Ans. (4) 48=65536; 262144 —.65536=4 Ans. (5) Ratio=3; 39-19683; 19683X 10=196830 Ans. (6) 234-17179869184; 17179869184X1=17179869184 Ans. (7) 34 —16677181699666569, which, multiplied by 1, gives the same number for the 35th term. ARTICLE 308.-CASE II. (2) 36=729; 729X10=7290, last term. 7290X3=21870; 21870 —10=21860; 21860-+-2-10930 Ans. (3) 211=2048, 2048X1=2048=last term. 2048X2=4096, 4096-1=4095, and 4095 —1=4095 Ans. (4) 41=4194304, 4194304X1=4194304, last term. 4194304X4=16777216, 16777216-1=16777215, 4 —1=3, 16777215 —3=5592405 cts.=$55924.05 Ans. (5) Ratio=2,239=549755813888, this multiplied by 10, gives 5497558138880, for the last term. 5497558138880X2=10995116277760; from this subtract 10, and divide by 2-1-1, and we get for the Ans. $109951162777.50 ~4 —--- -- - - ----- GEOMETRICAL PROGRESSION. 157 (6) Ratio=3, 33'=617673396283947, 617673396283947X3=1853020188851841 3-1=2) 1853020188851840 $9265100944259.20 Ans. (7).3X10-3; 10-1=9; 3 —9 — Ans. (8) 100X1.04=104; 1.04-1=.04, 104_..04-2600 Ans. (9) Ratio=3, -IX3=1; 3-1=2; 1. 2=- Ans. (10) Ratio=2, IX2=1; 2-1=1; 1. — =1 Ans. ARTICLE 309. —PEIUTATIO N. (1) 1X2X3X4X5=120 Ats (2) 1X2X3X4X5X6X7X8X9XlOX11X12 =479001600 Ans. (3) 1X2X3X4X5X6X7X8 —-40320 days =110 yr. 170 da. Ans. (4) 1X2X3X4X5X6X7XSX9XlOX11X12Xl3Xl4X15 16X17X18X19X20X21X22X23X24X25X26 =403291461126605635584000000 Ans. The subject of permutations and combinations properly belongs to Algebra. (See Ray's Algebra, Part 2d.) ARTICLE 311.- MENSURATION. (1) 17X15=255 sqc. ft. Ans. (2) 2 ft. 3 in.=2.25 ft.. 12 ft. 6 in.=12.5 ft., 2.25X12.5 =28.125=28- sq. ft. Ans. (3) 15 in.=14 ft.; 1G ft.Xl1 ft.=20 sq. ft. Anas. 4 4 --------— ~ 158 KEY TO RAY'S THIRD BOOK. (4) ft. in. 12 6 1 2 Ans. 14-7 sq. ft.=14 sq. ft. 84 sq. in. 12 6 Or, 1 ft. 2 in.=14 in., 12 ft. 6 in.=150 in. 2 1 150X14-=2100 sq. in.==14 sq. ft. 84 sq. in. Ans. 14 7 (5) ft. in. $ cts. 5 7 1.50 1 10 10 5 7 15.00 4 7 10 2 in.=-.25 10 2 10 6" =[.0625 4// =/.0416+ Ans. $15.354+ (6) 120rd.X 84rd.=10080 P.=63 A. Ans. (7) 65 rd.X 65 rd.= 4225 P.=26 A. 1 R. 25 P. Ans. (8) 35rd.X 16rd.= 560 P.= 3 A. 2 R. Ans. (9) 693X693=480249 sq. ft.=1764 P.=11 A. 4 P. Ans. (10) 15X 15=225, 225+225=450 sq. ft., 30X30=900 sq. ft. 900-450=450 sq. ft. Ans. (11) 80- 10=8 ft. Ans. (12) 10X144=1440 sq. in., 1440.10=144 in.=12 ft. Ans. (13) 6>< 5=30sq. yd., 30 —1. —=20yd. Ans. (14) 21 ft. 3 in.=21.25 ft., 13 ft. 6 in.=13.5 ft., 21.25X13.5= 286.875 sq. ft., 1 yd. —3=3.75 ft., 286.875~. 3.75=76.5 ft.=25.5 yd.=25. yd. (15) 3X1=4 —- sq. yd., 4'~:3=9>K-4=6 yd. Ans. 0 MENSURATION. 159 ARTICLE 312. (1) ft. in. (2) ft. in. 25 9 22 7 21 3 13 1.1 540 9 293 7 6 5 3 20 8 5 9)547 2 3 9)314 3 5.Ans. 60 sq. yd. 7sq. ft.+ 34 8 3 5.20 6.80.17777+ =cost 8sq. ft..00555+- -" 3'.00077+- " 5/" Ans. $6.9841+ (3) ft. in. ft. in. ft. in. 20 6 20 6 6 3 20 6 16 3 4 2 16 3 16 3 328 0 25 0 - 5 1 6 1 0 6 73 6=circum — - 10 4 ference 333 1 6=ceiling. 26 0 6=-fire735 -0 [ofroom. [place. 24 6 Deductions. ft. in. 6 26 0 6 4 2 759 6 —walls. 29 2 7 333 1 6 333 69 0 29 2=door. 1092 7 6 94 2 6=deduc- ft. in. 9)998 5 tions. 3 6 110 sq. yd. 8 sq. ft. and 5/-=5 gsq. ft. 19 6 [Ans. 2 39 0-windows. 4e; —---------------- - 160 KEY TO RAY'S THIRD BOOK. (4) ft. in. ft. in. 20 0 4 4 20 0 4 14 6 14 6 17 4=fireplace. 69 O=circumference (5) ft. in. 10. 4 [ofroom. ft. in. 35 4 32 83 690 0 32 8 23. 282 8 19 8 10 713 0 19 55 4=deductions. 17 4 9)291 9)657 8 55 4 32 sq. yd. 3sq.ft. 18 [6t 73 sq. yd. 8' 27 256 32 511 32 146 1576 1619.461 3 sq. ft._-! 6 s8/=2-2l 2 6' = 1 1 Ans. $19.73 Ans. $5.83 yd. (6) 21 15 5 ft. 3 in.S — f t.I — yd. 105 21 21>\.58.4375; /8.4375 -2.9047375-.+ sq. ft. A1ns. A P RT I CLE 314. (1) 35-26=61; 61X11 —-671; 671 -2=335~ sq. in. Ans. (2) 19-25=44; 44X1 32= —i 408; 1408 -2=704 P. — 4 A. 1 R. 2P. Anms. A rTIC LE, 315. (1) 48oX3.1416=150 7968 ft. Ans. (2) 4X301416-12.5664 ft.-12 ft. 6.7968 in. Ans. (3) 791o2.4>3.116-=4857o59584 mi. Aas. ARTkICLE 316. (!) 15 —3.-141i —4.7746+ft. —4 ft. 9.905-in. An2s. (2) 12 ft. 5 in.=12.416G6+; 12.4166- 3.1416=3.952338 ft. =3 -ft. 11.428-.in. _ALs. ARTICLE 3'17. (3) 42-> -.2=2!; 21\ 921-41; 3.!416X4X41 =1385.4456 sq. f.. Ans (2) 7 rods( 7.s; 7.5<7.5-56.25;; 3.1416X56.25 =176.715 P.-l1 A. 16.715 P. Anls, c_~~~ —------------------ X MENSURATION. 163 ARTICLE 318, (1) 962.115 —.7854=1225; V1225=35=diameter; 35X3.1416=109.956 Ans. (2) 160'..7854=203.71785077+; i/203.71785077 =14.2729+; 14.2729 —2 —7.1364+ rods; or 117 ft. 9+in. Ans. ARTICLE 320. (1) 3 ft. 3 in.=3~ ft.; 4 ft. 4 in.=4~ ft.; 12X3vX4' =169 cu. ft. Ans. (2) 2 ft. 6 in.=2.- ft.; 1 ft. 9 in.=l- ft.; 6X2AXI -264 cu. ft. Ans. (3) 3 ft. 2 in.=3"; 2 ft. 8 in.=2-; 2 ft. 6 in.-=2 ft. 3VX2 —X21-= 1 X19SX =- -'211 cu. ft. 21!-X81 cts.=$17.10 cts. Ans. (4) ft. in. 4 10 2 11 9 8 4 5 2 14 1 2 2 2 28 2 4/' 2 4 2 4///"' Ans. 30 6/ 6// 4/// ARTICLE 321. (1) 97 ft. 5 in.==97.416+ft.; 18 ft. 3 in.=18.25 ft.; 2 ft. 3 in. =2.25 ft.; 97.416X18.25X2.25=4000.1445 cu. ft. 4000.1445. 24.75=161.6+P. Anls. (2) 53 ft. 6 in.-53.5 ft.; 12 ft. 3 in.-12.25 ft., 53.5 X1.25X2=1310.75 cu. ft.=52.95+P. Ans. ~ —-— _- __ M —— ~ —--- 164 KEY TO RAY'S THIRD BOOK. (3) 53 ft. 6 in.-53.5 ft.; 12 ft. 6 in.=12.5 ft.; 53.5X12.5X2 =1.337.5 cu. ft.=54.0404+ P., 54.0404X$2.25 8-$121.59+ A4ns. 6 1_ 4x 5X 2 S_ X'4l S. (i) 48~X16'Xl' —-. X2 — =1196- cu. ft. 1196 X20=23925 bricks.Ans. (5) 15 in. —14 ft.; 32OX6XlI=2400 cu. ft.; 2400X20 =48000 bricks; 48000X$5.875-. 1000 —S282 Anls. (6) 120X8Xl —1440 cu. ft. —248S320 cu. in. in wall. SX4X2.25=72 cu. in. in each brick; 2488320-. 72 =34560 bricks iArs. (7) 240X6X3=4320 cu. ft. =7464960 cu. in. in wall. 9X4X2=-72 cu. in. in each brick, 7464960 — 72=103680 bricks. 103680-.-1000=103.68; $3.25X103.68 =$336.96 Ans. ARTICLE 3822. (1) 2+2+2=6; 6- -2=3; 3-2-=1; 3 —2=-1; 3-2=-1. 3X1XIX1-=3; V3=area of base. 14X/3=141/3 =area of base. 14-/3=-/196XV/3=V/588 =24.2487+cu. in. Ans. (2) 4X4=16;.7854X16=12.5664-area of end. 12.5664X12=150.7968 cu. ft. Ans. (3) 18=18.5: 18.52=342.25: 342.25X.7854=268.80315 cu. in.=area of end. 268.80315X8=2150.4252 cu. in. Ans. (4) 62X.7854=28.2744=area of end. 28.2744X8 =226.1952 cu. ft.=390865.3056 cu. in. 390865.3056 — 2150.4-181.764 bu. An:s. (5) 15X5X4-300 cu. ft.=518400 cu. in.; 518400 —2150.4 =241+bu. Ans. (6) 12X3X5=-180 cu. ft.=311040 cu. in.; 311040 —2150.4 =144.6+bu. Ans. MENSURATION. 165 ARTICLE 323. (1) 5X5=25 sq. ft.=zarea of base. 25X21 —3 =175 cu. ft. Ans. (2) 10'2X.7854=78.54, area of base. 78.54X15- -3 — 392.7 cu. ft. Ans. (3) 37 ft. 8 in.-452 in., 4522X.7854=160460.3616=area of base. 79 ft. 9 in.-957 in. 160460.3616X957-*-3 -51186855.3504 cu. in.-29622.0227+cu. ft. Anis. (4) 720X720=518400=area of base. 518400X477 —3 =82425600 cu. ft.=3052800 cu. yd. Ans. (5) 1 —1.5: 1.52X.7854-1.76715-area of base. 1.76715 X9. -3=5.30145=cu. in. in cup. 231 —-5.30145 -43.57-+times Ans. ARTICLE 324. (1) 3X3=9=area of lower base. 2X2=4=area of upper, base. 9X4=336; 1/36-=6area of mean base. 9+4+6 —19; 19X'2 —76 cu. ft. Ans. (2) 27X27-729=-area of lower base. 6X16X1625G6area of upper base. 729X256 = 186624; 1/186624- 432 -area of mean base. 729+256+432-1417; 1417 X ~4-=105802.6666+cu. in.=61.2283950O+cu. ft. Ans. (3) 2-=2.5: 2.52X.7854=4.90875=-area of mouth. 12X.7854=.7854= area of bottom. 4.90875X.7854= 3.85533225; 1/3.85533225-1.9635-area of imean base. 4.90875+.7854+1.9635=7.65765; 7.65765X, =:12.76275 cu. in. Ans. ARTICLE 325. (1) 133=2197; 2197X.5236-1150.3492 cu. in. 153-3375; 3375X.5236 —1767.15 cu. in. 303=27000; 27000X.5236 -14137.2 cu. in. Ans. 166 KEY TO RAY'S THIRD BOOK. ARTICLE 326. (1) 5 ft. 4 in.X3=16 ft.=circumference of base. 7 ft. 6 in. — 7 ft., 16X7, —2=60 sq. ft. Ans. (2) 8:1X3.1416-26.7036=circumference of base. 26.7036X25. 2=333.795 sq. ft Ats. (3) 2 ft. 11 in. —35 in.; 35X3.1416 —109.956-circ. of base. 109.956X55-2=3023.79 sq. in.-area of conv. surface. 352=-1225: 1225X.7854-962.115 sq. in. —area of base. 3023.79+-962.115=3985.905 sq. in.=27.679S95+ sq. ft. Ans. (4) 79122=62599744; 62599744X3.1416 =196663355.7504 sq. mi. Ans. ARTICLE 327. (1) 10 ft.=120 in., 5 ft..-=60 in., 4 ft.=48 in. 120X60X48 =345600 cu. in., 345600. 231 —1496+gal. Ans. (2) 12 ft.=144 in., 6 ft.=72 in., 10 ft.=120 in., 144X72X120=1244160 sq. in., 1244160 —.2150.4 =578.57 —bu. Ans. ARTICLE 328. (1) 4 ft.=48 in.; 6 ft..=72 in.; 482=2304; 2304X72 =165888; 165888X.7854=130258.4352 cu. in.=564.01.9 +gal.-17.9+bl. Ans. (2) 6 ft. —72 in.; 9 R.=108 in.; 72'2=5l14; 5184X10S559872; 559872X.7854-439723.4688 cu. in. =1903.56-4+gal..=60.43+bl. Ans. ARTICLE 329. (1) 40X30=1200; 40 —30-10; 10X10=100; 100 —. 3=33; 1200H-33'=1233.; 1233 X50=616662; 616663X.7854 =48433 cu. in.=209 4-209.66+gal. Aws. U !ENSUR4ATION. 167 (2) 7 ft. 6 in. -90 in.; 10 ft.=120 in.; 12 ft. 6 in.=150 in.; 90X120-10S00; 120-90=30; 30'2-900; 900. 3 =300; 10800+300=11100; 11100X150-=1665000 1665000>24 — 120 gallons. (2 ) C)60X5=*300: $4X23=892: 60-23-37 bl. $300+151 =,351; $351-$-92= —$29; $259- -37:$7 Ans. (22) 12 cwt. 1 qr. 71 b.-1232 lb. 3. 3 r. 13 lb.=SS lb.: 1232 — 88-14 Ans. (23) 102X3.1416=314.I6 sq. in. 5cts.X314.16 —$15.708 Ans. (2 ) If. ox is worth 8 sheep, 3 oxen wrill be worth 24 sheep, but 2 horses are worth 3 oxen, therefore 2 horses are worth 24 sheep; hence, 1 horse is worth 12 sheep, and at $2.50 a piece, 12 sheep are worth $30 (As. (25) If of $1, buy of a sheep, 1 sheep is worth 5 times S,2=10, andl q of a sheep is wrorth -$ of $.= -; hence, -1' of an ox is worth $1~0, and 1 ox is worth 14 times $ -' =$20; and 10 oxen are worth 10 times $20-=200 Anls. (26) 19 is 2J times 9, therefore the required number must be 2, times 54: 54X2-=114 Ans. (27) of: -. — 1; of1; 1 is 3 times ~X3-1 Ans. (28) 13X5=68-1 hr. required to perform the work. 681- 9 " v273X — 1o"-7- days of 9 hr. each. (29) $40: $100 $15: $37.50 Anzs. (30) From Monday noon till 6 P. M. of Thursday is 78 hr. O9 —-- -------------------- 23 170 KEY TO RAY'S THIRD BOOK. which, is 3l days of 24 hr. each; hence the clock has gained 3 ><7 m in. 71 X3 1 5X -195 = min., therefore the time by the clock will be 6 hrs. 243 lain. (31) If 7 men can mow 35 A. in 4 days, I man can mow v of 35 A., which is 5 A, in 4 days; and in 1 day 1 man can mow 1 of 5 A., which is 1- A.; if 1 man can mow 14 4 4 A. in 1 day, 10 men can mow 10 times 1* A., which is 12a A, in 1 day, and in 3A days they can mow 3., times 121 A. A= 25 < 7. 1 -=43 A. As. (32) 350 —16= y19 d. yd.: $1X6=$20-L5 $ 4X357-=$26:2: $ — X19=8$12': $20 + 5$12- 7 262 9$5$94-4 —$5.67+ Anso I0 a 64q-9 ~1 64 g g —~vo ~:/ (33) 11 hr. X6-66 hr.: 10)X66-660 cu. ft. 660 cu. ft.-.(5X2)-66 ft. Anls. (34) If 4 men can do the work in 20 days, 1 man can do it in 4 times 20 days=80 days; hence 1 man can do Sl, and 3 nen ~-36 of it in one daty. If 6 women can do the work in 20 days, 1 woman can do it in 6 times 20 days —120 days; hence 1 woman can do 1 and 5 women ~ 5- of it in I day. Therefore 3 men and 5 women, in 1 day can do s3 + 5 199 of the work, and therefore they can do it in 1 ~ 1 9 2o 4 o-12 iL days. (35) 40+60 —100 yd.: 5 yd.: 40:: $6: $48: 6 yd.: 60 yd.:: $9: $90: 5 yd.: 100 yd.::$7: $140: 90+$4'S-$138 $140 —$138=82 Ams. (36) After the first drawing;4 or are pure wine, and when 10 gal. more are drawn, 4 of the remaining 40 gal. are pure wine; 4 of 40 gal.=32 gal. Anis. (37) In 14 days of 10 hr. each, there are 140 hours. If 27 men can do the work in 140 hours, 1 man can do it in 27 times 140 hr.=3780 hr.; and since the work of a boy is half that of a man, it will require a boy twice 3780 hr.=7560 hr. to do the work. If it requires 1 boy 7560 es3 ------ -- - - - --- - - ~-~ — ~-~I MISCELLANEOUS EXAMPLES. 171 hr. to perform the work, it will take 24 boys — I- of 7560 hr. —315 hr. to do it. 315 hr. -4- 45=7 hr. Ans. (38) 16' 40"/X9-2~ 30': 2 30'-. 15 gives 10 min. Ans. 10 min. before 9. (39),0 men: 0O men g0.4 ft. 1.: ~ft. 1. $.$ft. w.: 4. ft.w. X.$ ft. d.: IJ ft. d. [9X1_-1=9 da, Ans. 10 hr.: 9 hr.: 0:. da. (40) $1+$.20=$z1.20o: 14.$1.20=$112 A_,s. (41) 9 cows=3 times 3; if 3 cows=2 horses, 9 cows-3 times 2=6 horses: 4+610: 4-2: -=3 2 of $45-=18, A's share 6 of $45=$27, B's share. (42) 35 mi., 2 —14 mi. the sumll of the rates: 14-5-9: 9- -: 2=4 mi. one: 4V+5=9-, mi. the other. (43) $1-$.17= — $.825: $14.85 — $.825=$18, purchase price. $18X. 12-$2.25: $18d-$2.25-$20.25 Ans. qt. bu. qt. (44) 1 bu. —32 qt. 32-4 —28 qt. 28: 2:: 32: 2 bu. 1 pk. 14 qt. Ans. (45) The provisions of 32 men for 5 mon., will last 1 man 160 mon.: the provisions of 1 man for 160 mon., in 8 mon. would be consumed by as many men, as 8 mon. is contained times in 160 mon., 160. 8-20 men, hence 3220-12, the number of men that must leave. (46) Int. for 63 da. at 6 % —$.0105 $1-$.0105=$.989: $1009.29+*~$.9895=$1020, the face of the note. Int. for 60 da. at 12 %-$.02: $1+$.02=$1.02: $1020 —. $1.02 —$1000 Ans. (47) $12 —20=$.60 gain per yard. The difference betwreen 5 % less, and 10 %o more is 15 o: $.60 —.15-$4 Ans. (48) 9X9X,!=40, cu. in. 14X11X5-=961 cu. in.: 401: 96:: $3.24: $7.70 Ans. (49) 120+60-180: 180: 120:: 3 mon.: 2 mon. Ans. (50) 25-20=5 rd. gain in 25 rd.: 5 rd.: 47 rd.:: 25 rd.: 235 rd. Ans. 172 KEY TO RAY'S THIRD BOOK. (51) Int. of $1 for 4 mon. 3 da.=$.0205: $1 —$.02)5 -.979 $300$.97 95-3 - 306.27-+Ans. (52) $L.50X40=$60: 60 —. $1.20=50 gal.: 50 gal.-40 gal.:=-10 gal. Ans. (53) Int. of $1 for 63 da.=$.0105: $1.26- $.0105 —$120 Ans. (54) In running 28 rd. the dog gains 3 rd., but the hare has 45 rd.=15 times 3 rd., the start; therefore the dog must run 28 times 15 rd. —420 rd., to catch the hare. (55) =20 o: $1+$.20=$1.20: $1.50-.$1.20-= 1.25, the cost; $2 —$1.25=$.75 $1.25X.01=$.0125 $.75-. $.0125-60 / Ans. (56) $260+$13=$273 received for those sold: $1.30X.40 =,$.52: $1.30+$.52 —$1.82: $273. 1.82=150 bu. sold: $260. $1.30=200 bu.: 200 bu. —150 bu.=50 bu. Ans. (57) A is ahead 8 mi.X5=40 mi.: B gains 10-8=2 mi. per hour; 40 mi.-. 2 mi.=20 hr. Ans. (58) A in 1 da. mows ~ A.; B - A.; and both can mow -A -=1I A. in 1 da.: 9 A —1 A.-.=6 da. Anzs. (59) Both do 4 times as much as B: hence B could do it in 20X4 —80 da, and A in {- of 80 da.=26' da. Ans. (60) 1 of the first lot cost 21 cts., and 1 of the 2d 22 cts.; hence, 1 of each, that is, 2, cost 21-t+2-=5 cts.: If 5 sold for 14 cts., 2 sold for' of 14=55 cts., 5-5 ct.=gain on 2: - ct.: 52 cts.:: 2: 240 Ans. (61) 2_1_; 31-166; A can do it in $ da. and in' da. he can do', and in 1 da. -: similarly, B can do -og in 1 da.,:.5 +5 -sa=what both do in 1 da.: 1-. — 1 da. Ans. (62) As 2 min.: 3 min.:: 11 rods: 161, the space traveled by A in 3 minutes. Hence B, in traveling 17 rods, gains, rod on A; and the object being to find how many rounds he must make to gain half a round, we have this proportion: As 1 rod: 17 rods:: - round: 17 rounds, the space to be gone over by B: consequently A will make 161- rounds. -- ----------— *Is~~ — MISCELLANEOUS EXArPLES. 173 (63) A does -- in lda., B 1 and C: all can co d3 +, in 1 da.: 1 — — 1 da. s (64) From the question, it is obvious that in I day A and B do -, Aa and C andl and C - of the work. By taking the sums of these fractions, we find tlice -what they all do in 1 day; that is, -a +-,-b+ I -4-+ 5- - 3-o i- 0=-, is twice the part of the work they all do in 1 day; here V of - is the part of the work they all do in 1 day: therefore they all do it in 1 -- 8 days. (65) 30 lb. at 10 cts. $3.00 25 12 - 3.00 4 15 -.60 50 20 = 10.00 109 )16.60(15 5 eta. Ans. (66) If 10 men can dig 1 acre in 11 days, then I man can dig 1 acre in 11X10O110 days, and 1 man can dig T-'6 of an acre in 1 day; tl-len 5 men can dig.T a —X;2 of an acre in 1 day, and 5 men can dig w. X<6=?TL of an acre in 6 days. G - qS=? a:cres to be dug by boys. If 18 boys can dig, acre in 11 days, then 1 boy can dig 1 acre in 11X18=198 days, and 1 boy can dig T 9h of an acre in I day; then 1 boy can dig T,8X6= —-1 of an acre in 6 days. But since 8T acres are to be dug in 6 days by boys, each of whom call dig 3! of an acre, it Will require as many of whoni c i3 ~ 1 boys as -33 acre is contained times in -'- acres: -- 33X -=-63X3-189 Ans. (67) 24 in.=2 ft.: 15+15+12+12=54 ft., the distance around; 54 ft.X10 ft.=540 sq. ft.: 540 sq. ft.-. 2 ft.= 270 ft.-90 yd. Ans. (68) A could do it 3 times in I da., B 5 times and C once: therefore, the three could together do it 9 times in 1 da., and do it once in g da. Ans. (69) 282 —784 7854+50=834 men Ans. 174 KEY TO RAY'S THIRD BOOK. (70) 402=1600: 1600+1600-3200 sq. ft.: V/X3200-56.56 + ft. Ans. (71) Suppose the work to consist of 150 equal parts, then the person does 3 of 150=90 parts in 30 days, or 3 parts in 1 day; 150-90=60 parts done by both in 6 days; hence they both do 60 — 6-10 parts in 1 day; then 10 —3=7, the number of parts done by the assistant in 1 day: therefore 150 +7-213 days. (72) 1-T-4s=B'share; — th — e of the estate=$7420: -~=$ of $7420=$1060, and the estate-$1060X15= $15900 Ans. (73) 242 —576: 18=324: 576+324-900-square of the diagonal on the floor, this is the base in a triangle where the hypotenuse is sought. 122=144: 900+144-1044: V>1044=32.3+ ft. Ans. (74) 56 lb. tea —2~X56=140 lb. coffee-=3.X140-490 lb. sugar: 10 Octs.X490-$49 Ans. (75) From the question it is evident, that all three do T-'~ of the work in 1 day, and that B in 1 day does t1, and C,%; hence A does l f-(- 2 4+3)12 \_ 29__ 34 20 & -8- -0 4 0 86 in 1 day; therefore he can do the 4o5-4o2-4o8-4 084day; do the work alone in 1 - -8 — d'ays. A.ns. (76) A built 4-4 as much as C: 3!B of 4, that is, as much as C: 4+1+-1 O?: A built 4 out of Lo, that is, 4-1=-: B and C each 3 out of 1o0, that is?3: - of 200 ft.=80 ft. A b- Of 200 ft.-60 ft. B and C also 60 ft. Ans. (77) 42.432=1800.3049 square of hypotenuse: since the two sides are equal, -1 of 1800.3049, or 900.15245 is the square of 1 side, which is the area. 900.15245 P.-5.62+A. (78) If he had worked the whole time he would have earned $1.25X(25=$31.25. Now, each day he was idle lie lost $1.75, that is, 50 cents which he spent, and $1.25 which he did not earn. But he lost in all the difference between $31.25 and $19, which is $12.25, and he lost this at the rate of $1.75 per day; therefore lie was idle as many MISCELLANEOUS EXAMPLES. 175 days as 1.75 is contained times in $12.25, which is 7; therefore he was idle 7, and consequently worked 18 da. An1s. (79) B, who works 30 days, will do - the work: hence A must do 2, which will take him 20 days. 20-3-17 days more: 30-17=13 Ans. (80) 1 — 5-=-9- of =' 1-(9-5 +-)=?= C's; 14 — -==$1300. Property $1300X7=$9100 -: of $9100 =$3250=A's; - of $9100-$3900=B'si; - of $9100$1950-C's Ans. (81) By the first it receives - in the hour, by the second 5; by the th o i +; by the third 1 — 75 in 1 hour; 1 -— 7' —2-1 hr. Ans. (82) $l.20X60 —72.00: $72X.10=$7.20: $72+$7.20 — $79.20 the amount to be received: $1.50X20=$30.00: 79.20 — $30 —$49.20: 60-(20+5)=35 gal.: $49.20 — 35 —$1.404 Ans. (83) 1 apple cost 4 cts., and sold for cts.: gain on I apple = - 4-= ct.: 24 cts. — ct.-144 Ans. (84) $4500X.01S=$,1: $500-(-25+$S1+$134)=$500$240=$260, net income; $4500X.01 —$45: $260 — $45 _5' 7 Ans. (85) 60X30=1800 sq. ft.: 12 ft. 4 in.X8 ft. 4 in.=104 sq. ft. 10/: 1800 sq. ft.-104 sq. ft. 10'/1695 sq. ft. 2/: 1695 sq. ft. 2'X4=6780 sq. ft. 8' Ans. (86) $700+$600+$40400=-1700.: -O4-7 —=0-4~:$260= 41 gain: - 7-4 of $260 —$65 and gain= —$65X17-$1105 Ans. (87) 120X.10=12: 12+8-20: 20 -.05=400 Ans. (88) $1200 - 4-30 B's yearly debt: A, spending $350 less, will save 8350-$30 —$320- his annual income; $320 X3-$960 Ans. (89) 53X.5236=65.45: 65.45X2=130.9: 5-125: 125X2 -250: 130.9+250=380.9: V380.9-7.24+in. Ans. |176 I3KEY TO PiAY'S THIRD BOOK. (90) $1.0 —.!-1i6~-i. 83; o2'. 25- 83- 3-$30=cost of 1st. $30 —825 =;5 lost on the first, and therefore gained on the second. 5-5. $.16=$31.25-cost of 2nd. | (9.) I4- f-=; -8-o2i4000; -7=3000 slain':. of 3000-1500 Antes. (92) The 3d=-i of 3 times the 2nd= — the 2nd. 113-K,, of whlch - is -4; 1i-, 19; -4d of 95 yd.=20 yd. 1st.: 1 2 of 95 yd.=60 yd. nd cl: and 3 of 95 yd.-15 yd. the 3d. A42s. (93) $4X60(=$-2-0: Anmount of $1 for 6 mon. —1.03 ~ o -- 401.03$=233.009-+-: $250-$233.009+ —=,16.99+- As. (94) As often as he buys one cow, he buys 6 sheep and f=-2 horses; 1 cow cost $30; 6 sheep $30; 2 horses 9240. While buying 1 cow, he expends $30+$30+$-9240-$300; $1500 $300-5 cows; 5X6=30 sheep; 30 - 3-10 horses. Ans. (95) 4 polls cost 94: $328 —4-$324, the tax on his property; $324..016$ —20250 Ans. (96) C does 1 — 4; does 3- - As. (97) B received of A's —8100; toth received.3 of A's+-100, 3 of A's+$100=z1000; A4 A900: A's=$600, B'shf1000-$600=$400 Ans. (98) 4 of B's= — of A's+$50; - of B's=- of A's-+$25, and B's-= of A's+875, both have Y of A's+-$75=$500:'7' of A's=- 425:.-25: A's=$200: B's=$500$200=$300 An.s. (93) -9 of the time past noon is equal to 4 of the time to midnight, and the whole time past noon-g or 3 the time to midnight; the time from noon to midnight, which is 12 hlr.=-11, 4 the time to midnight: 1 of 12-3 hr. [Aces. (100) In thle 4th store $ was her money; hence, on leaRin.g the 3d sto-re, she had $1; before she spent the $1, she had 82-half what she had on entering the 3d store; hence, on leaving 2nd store, she had $4; before spending $ she had $4 -= of $9, her money on leaving Ist store; $9+$1- -$10-, of $20, her money at first. TO RA YS SECOND BOOK, SOLUTTIO NS TO T.E M'ORE DIFFTICLT OQUESTIONS. ON TEACHING INTELLECTUAL ARITiC'l ET.MIC. IT is to be regretted that many teachers are not acquainted with the great value of mental arithmetic. It is important, whether it be regarided as a mere mental discipline, or as the'oundation of an accurate knowledgle of arithmetici and the various branches of mathematical science. Whenever a pupil is found incapable of anaalyzinlg nlad solving, mentally, all the questions in Ray's 2d Book, the best course he can pursue is, at once, to study mental arithmetic, although in order to do so, it should be necessary to suspend, for a time, the study even of Algebra or Geometry. Persons more than twenty years old have studied it, who afterward declared that the time thus employed was productive of more beneficial results than twice the same amount of time ever employed at any other branch of education. It cannot, therefore, be too strongly recommended to every teacher who desires to render his pupils thorough scholars, to make them well acquainted with mental arithmetic. It should be taught before written arithmetic. It often happens, however, that this subject has been neglected until after the pupil lias commenced the study of written arithmetic. In such cases, the study of both may be attended to at the same time, one part of the recitation being devoted to mental, and the other to written arithmetic. METIIOD or TEACIIINT.-ClcLsses.-Pupils study best in classes; it is almost as easy for a teacher to instruct 15 pupils in a class, as one alone. Keym.-12 (177) 178 KEY TO RAY'S SECOND BOOK. An important point is to keep the attention of the whole class directed to each question; one means of doing this is, for the teacher occasionally to read an example to the class, and then7 having allowed suflicient time for the answer to be prepared, to call upon some one for it. By this means, each one is obliged to solve the example mentally, as he may be called upon to answer it. It is sometimes proper to put the same example to the whole class, giving all permission to answer it; this plan succeeds better in small classes than in large, as it has a tendency to produce noise and confusion. When the class meets to recite, two methods may be followed in recitation. First, each pupil may have his book open, and, when his turn comes, read his question, and then proceed to analyze and answer it. Or, Seconid, the class may close books, and the teacher read each question distinctly, and then call upon a pupil to analyze and answer it, taking the pupils in succession. Both methods are practiced by good teachers; the first, however, is especially applicable to small scholars, while the latter answers very well with those who are older; it is an excellent means of teaching pupils to command their attention-a very important acquisition. ARITHIIOMIETERt.-Some difference of opinion exists among teachers with regard to the extent to which artificial aid should be given to pupils in the study of mental arithmetic. It is certain that visible illustrations may be used, in some cases, to great advantage, especially to give clear and precise ideas in the commencement of a subject; but, it is also true, on the other hand, that a frequent use of artificial aid tends to prevent the pupil from exercising his own intellectual powers, and thus, if carried too far, is productive of positive injury. It should be recollected, that the younger the pupils, the more they require such assistance as can be given them from tangible objects. The superior advantage of the arithmometer consists in this, that it can be seen by the whole class at the same time, and it is so simple in its construction that it can be made by any mechanic. The arithmometer consists of a square frame with 12 strong pieces of wi.;v passing from one side to the other. On each piece of wire there are 12 balls, one-half black, the other half white, so ON TEACHING ARITHMETIC. 179 as to form a contrast, and alternately disposed, so as to be easily counted. As the arithmometer contains 144 balls, it may be used to represent all numbers, from one up to this amount, so as to give pupils an idea of the relative magnitude of numbers. The operations of addition and subtraction are so easily shown by it, that any explanation is unnecessary. To represent multiplication, for instance, the multiplication of 4 by 3; take 3 rods, each having 4 balls on the one end of it, (the other balls being pushed out of the rway toward the other end,) and the pupils will easily see that 3 times 4 make 12; in the same manner it may be shown that 4 times 3 are 12; and hence, that the product of two numbers is the same which ever be made the multiplier. Division is shown by taking the number of balls represented by the dividend, and separating them into parts, each part containing as many balls as the divisor contains units; the number of parts, into which the dividend is tuus divided, will represent the quotient: thus, to divide 15 by 3, take 15 balls and separate them into parts, each containing 3 balls, and it will be seen that there are 5 divisions, so that 15 contains 3 five times. The operations in fractions, so far as they can be explained without dividing a unit, are also easily shown by it. To give pupils the first ideas of fractions, the division of an apple is preferable to any thing else, it being an operation with which pupils are generally familiar. ANALYSIs.-Among the modern improvements in the art of teaching, the analytic and inductive method, as applied to arithmetic, forms one of the most important. To analyze, means to separate into elementary or first principles. Intellectual or mental is used to designate that kind of arithmetic, the operations of which are performed wholly in the mind. This kind of arithmetic is quite as valuable for the habits of close thinking and accurate reasoning, as for the mere amount of knowledge which is acquired. To enable the pupil to realize to the fullest extent the advantages of the study, he should clearly understand every process, and be able to give a reason for every operation. It can not, therefore, be too strongly recommended to the teacher, to require every question to be solved by methods purely analytical, which 0 -0~~~~ —-------—. —" 180 KEY TO RAY'S SECOND BOOK. the pupil can both comprehend and explain. In this respect, the pupil will derive great advantage from a careful attention to the solutions termed ANALYSIS, (Ray's Arithmetic, 2d Book,) which are intended as MIodel Solutions. To explain the difference between an Analytical Solution and one that is not, take the following question. What is three-fourths of 12? Solution. 3 times 12 are 36, and 4 in 36 goes 9 times. Ans. 9. Analytical Solution. 3-fourths are 3 times 1-fourth; 1-fourth of 12 is 3, and 3 times 3 are 9: therefore 9 is 3-fourths of 12. In analyzing we must generally refer to unity, and the pupil should always bear this in mind: thus, if 2 yards of cloth cost 6 dollars, what will 5 yards cost? Here we have given the value of 2 yards, and must find the value of one yard, after which we can readily find the value of 5 yards. Again, suppose we have this question: 2-thirds of a certain number is 8, what is 5-thirds of the same number? ANALYSIS.-1-third is 1 half of 2-thirds; if 2-thirds are 8, 1-third is 1-half of 8, which is 4; if 1-third is 4, then 5-thirds are 5 times 4, which are 20. Ans. 20. If learners pursue the method taught in the book, they will have no difficulty in progressing; but when they depart from it, they will soon be involved in perplexity. The questions are so arranged, that each one depends, in some measure, on that which precedes it; consequently, pupils should understand every thing as they proceed. When a teacher finds his class unable to advance, he will always find, that they are not sufficiently acquainted with the preceding part. In such cases review. The pupils should go through the first six sections at least two or three times before entering upon the seventh and succeeding sections. If they then properly understand multiplication and division, they will find little difficulty in the remainder of the work. ORAL EXERCISES IN MENTAL ARITHMETIC. The following method of exercising pupils mentally, in Addition, Subtraction, Multiplication, and Division, will be found to have CI~~Ba~s x 55~3~."aEL~s;.........BPLBLP.. ON TEACHING ARITHMETIC. 181 the happiest effect in concentrating the attention of the class, and exciting a lively interest in the exercises. It also renders pupils remarkably expert in the performance of the operations. The success of the whole, however, depends upon the instructor; he must propose the exercises orally, and carry on the operations in his own mind, so as not to give examples that are either too difficult or improper, and also to be able to decide on the correctness of the result. The instructor should announce the exercises clearly, distinctly, and deliberately; while this is being done, each pupil should carry on the operations in his own mind, so as to be able to give the result almost as soon as the announcement is closed. To avoid confusion in the class, when a pupil thinks he has obtained the correct result, let him hold up his hand. If his answer is incorrect, let some other pupil who has obtained a different result, announce it; and so on, until a correct answer is obtained. Following will be found specimens of various exercises. These are given to show teachers what kind of exercises may be employed, but not for the purpose of being presented to the class, for let it be distinctly understood, that the success of the whole depends largely on the teachers announcing the exercises orally. In the following specimens the exact language is given which the instructor should use. It should always be brief as possible, no more words being used than are necessary to point out the exercises to be performed. 1. Five, add 6, add 3, add 4, add 2, add 9, add 7. Result? (36). The meaning of this is, that the pupil is to take 5, then add to it 6, then to this sum add 3, and so on. Lastly, for the sake of brevity, the teacher says Result? instead of TIWhat is the result? 2. Four, add 5, subtract 2, add 7, add 5, subtract 6, add 2. Result? (15). 3. Seven, add 9, add 8, add 7, add 3, add 2, add 6, add 3, add 9, add 6. Result? (60). 4. Eight, add 5, subtract 3, subtract 4, subtract 7,-class says can not be done, before the teacher has time to ask for the result, or to propose any thing additional. e~~ a~~"~"~~"~l~~~~""~~~~~"~~ —--------------- — ~~~~5 182 KEY TO RAY'S SECOND BOOK. 5. Twelve, subtract 3, subtract 5, add 6, subtract 7, add 9. Result? 6. Fourteen, subtract 8, add 6, add 7, add 5, subtract 9, add 4, add 7. Result? Exercises involving only addition and subtraction, are less interesting than those which also combine multiplication and division. Such exercises should therefore be used as soon as the pupil is acquainted with these processes. No division should be required that leaves a remainder. 7. Three, multiply by 2, add 4, multiply by 3, divide by 5. Result? (6). 8. Four, multiply by 3, add 3, divide by 5, add 6, multiply by 2, add 4. Result? (22). 9. Seven, multiply by 3, subtract 5, divide by 4, add 1, multiply by 8, divide by 10. Result? (4). 10. Nine, multiply by 6, subtract 4, divide by 10, add 3, multiply by 4, subtract 5, divide by 9. Result? (3). 11. Six, multiply by 5, add 5, divide by 7, subtract 5, add 4, multiply by 10, add 8, divide by 6, subtract 6. Result? (2). 12. Eleven, add 5, divide by 8, add 5, multiply by 4, add 5, divide by 11, add 5, divide by 4, add 5. Result? (7). 13. Twenty-one, divide by 3, add 3, multiply by 3, subtract 3, divide by 3. Result? " (9). 14. Sixty, subtract 4, divide by 7, add 3, multiply by 4, subtract 2, divide by 6, add 5, multiply by 6, divide by 9, add 2, divide by 2, subtract 2. Result? (3). 15. Twelve, multiply by 5, add 3, divide by 9, multiply by 7, add 5, divide by 6, add 1, multiply by 10, add 8, divide by 9, subtract 5, multiply by 6, add 3, divide by 5. Result? (9). 16. Eight, multiply by 8, add 8, divide by 8, subtract 8. Result? (1). 17. Seven, multiply by 7, add 7, divide by 7, subtract 7. Result? (1). .JA ON TEACHING ARITHMETIC. 183 The length and difficulty of the exercises must be graduated by the age, attainments, and ability of the members of the class; if too easy they will not excite sufficient interest, and if too difficult, the class will be discouraged. At first, the examples should be short, afterward they may be longer, like the following: 18. Fifteen, add 5, multiply by 3, add 4, divide by 8, subtract 2, multiply by 7, add 12, divide by 9, add 3, multiply by 10, add 6, divide by 12, multiply by 8, subtract 1, divide by 7, subtract 7. Result? (2). By teaching the pupils the meaning of the termss stuare, squazle 7rot, cube, and cube root, the exercises may be made to embrace the operations of squaring, extracting the square root, etc., as in the following exercises. For the sake of brevity, the terms square root, and cube root, are used to denote that the square root, or cube root, is to be extracted. 19. Three, square, add 7, square root, add 3, square, subtract 1, divide by 6, add 1, square root, multiply by 8, add 1, square root. Result? (5). 20. Seven, multiply by 4, subtract 1, cube root, add 5, cube root, multiply by 12, divide by 3, subtract 4, cube, subtract 1, divide by 9, multiply by 4, subtract 1, cube root. Result? (3). 21. Four, square root, cube, add 1, square root, cube, add 9, square root, subtract 2, cube, square root, divide by 2, add 1, cube, subtract 4, square root, add 5, square root, add 5, multiply by 7, add 1, cube root, square root. Result? (2). To TnAAClfxtRs. — In the following solutions, fbr the sake of brevity, arithmetical signs are sometimes used; instructors, however, in explaining solutions to pupils, should always employ language the most simple and easily comprehended. A ftull analysis is given to some of the more difficult examples, while to others less difficult, the solution is as brief as possible. In all cases the pupil should be required to give a full analysis; the style and manner of doing this being a better test of scholarship thanl the ability to give merely a correct answer. 184 KEY TO RAY'S SECOND BOOK. SECTION 10.-LrESSON IV. (6) 1I each nman received I-sixth, there must have been 6 men; hence each man received 1-sixth of 42 dollars, which is 7 dollars. (IO) One-fifth of 20 cents is 4 cents, what she received from her brother; 1-half of 4 cents is 2 cents, what she received from her sister; hence, she received, in all, 20 cents, and 4 cents, and 2 cents=26 cents. (12) If he gave 4-sevenths, he had 3-sevenths left; 1-seventh of 14 cents is 2 cents, and 3-sevenths are 3 times 2 cents, which are 6 cents. (1'4) One-fifth of 30 yards is 6 yards, and 2-fifths are 2 times 6 yards, which are 12 yards; if 12 yards cost 48 dollars, 1 yard will cost l-twelfth of 48 dollars, which is 4 dollars. (15) One-fifth of 25 cents is 5 cents, and 3-fifthls are 3 tines 5 cents, which are 15 cents: at 2 cents each, 15 cents will buy as many peaches as 2 cents is contained times in 15 cents, which is 7 and 1-half Ans. 7; peaches. (16) One-ninth of 54 is 6, and 5-ninths are 5 times 6, which are 30. To give each girl 1 chestnut would require 3 chestnuts; hence each girl received as many chestnuts as 3 chestnuts is contained times in 30 chestnuts, which is 10. Ans. 10 chestnuts. (18) One yard cost 1-seventh of 7 yards; if 7 yards cost 42 dollars, 1. yard cost 1-seventh of 42 dollars, which is 6 dollars, and 3 yards would cost 3 times 6 dollars, which are 18 dollars. If 9 barrels of cider cost 18 dollars, 1 barrel would cost 1-ninth of 18 dollars, which is 2 dollars. (22 One-fourth of 28 is 7; 1-severth of 28 is 4, and 3-sevenths are 3 times 4, which are 12; 7 cents and 12 cents are 19 cents, which taken from 28 cents, leaves 9 cents. If 3 books cost 9 cents, 1 book will cost 1-third of 9 cents, whilich is 3 cents. PL~~~YI i~~~I*I~~.-~~Fl~~~i-P —~:~~~~"I~~(CV~~~L T'~~M~~5nr-.~~~K ov MENTAL EXERCISES. 185 SECTION I.- LEssoN III. (13) Since 5-sevenths are apple trees, the remaining 2-sevenths are cherry trees. One-seventh is 1-half of 2-sevenths; if 2-sevenths are 12, one-seventh is 1-half of 12, which is 6, and if I-seventh is 6, 5-sevenths are 5 times 6, which are 30, the number of apple trees. (17) Since 25 is 5-eighths, 1-fifth of 25, which is 5, is 1-eighth; hence, S-eighths is 8 times 5 years, or 40 years, the father's age. One-tenth of 40 years is 4 years, the age of the youngest. (18) Since 45 is 5-thirds, 1-fifth of 45, which is 9, is 1-third of the first cost, and 3 times 9, which is 27, is the first cost; hence, he gained 45 less 27, which is 18 dollars. Or, the gain is evidently 2-thirds of the first cost, which is twice 9 dollars, or 18 dollars. (21) Since he gained 2-sevenths, he must have sold it for 9-sevenths of the first cost; therefore, 45 is 9-sevenths, and 1-ninth of 45, which is 5, is 1-seventh of the first cost; hence, 7 times 5, which are 35, is the first cost. SECTION 12. —LESSON I. ('7) One-fifth of 20 is 4, and 2-fifths are 2 times 4, which are 8; 1-third of 15 is 5, and 8 is once 5, and 3-fifths of 5. The following are the answers to the remaining examples in this lesson. Although they are explained in the form of common fractions, the pupil should give them after the manner expressed in the Arithmetic; thus, the answer to question 8 should be read, two tines a acnd 2-fifths of 5. (8) 2. (9) 31. (10) 111. (11) 4. (12) 61. (13) ].: 21. (14) 4.: 33 (15) 4-.! (16) 7 -. LESSON I I. I Answers. (2) A. (3) 18. (4) 15. (5) 9. (6) 40. (7) 35. (8) 44. (9) 4. (10) 2. (11) ]. (12) 6. (34) 24. (15) 71. (16) 3!- (17) 7. l(1) *5. (19) 5 (2 0) 7 (21) - 3 (22) 14. 186 KEY TO RAY'S SECOND BOOK. (23) 1-tenth of 50 is 5, and 7-tenths are 7 times 5, which are 35; if 35 is 5-ninths, I-fifth of 35, which is 7, is 1-ninth, and 9-ninths, or the number, is 9 times 7, which are 63. If 8 is 4-fifths, 1-fourth of 8, which is 2, is 1-fifth, and 5-fifths, or the number, are 5 times 2, which are 10. 63 is as many times 10 as 10 is contained tines in 63, which are 6 times 10 and 3-tenths of 10. Ans. 6 times and 3-tenth times. (24) Ans. 4 times andc 2-sevenths of a time. (25) 24. (26) 12. (27) 2-tlirds of 21. (28) 18. (29) 40. (30) 3 times 7. 8ECTION 13.-LESSoN L (25) In 42 there are 6 sevens, and Joseph husks as many threes as his father husks sevens, that is, 6 threes, which are 18 r>OWS. (28) If the horses eat 16 bushels in 2 weeks, they will eat 1-half of 16 bushels, which is 8 bushels, in 1 week; if they eat 8 bushels in 1 week, it will require them as many weeks to eat 56 bushels as 8 bushels are contained times in 56 bushels, that is 7. Ans. 7 weeks. It is easily seen, that the rnoumber of horses has nothing to do with the solution. The question would be the same if the words, a cerlain nzumber, were substituted for 5. (30) To eat it in 1 day would require 6 times 12=72 horses; and to eat it in 9 days, ~ of 72-8 horses Ans. (31) If 1 lb. of sugar=2 lb. of rice, 1 lb. of sugar and 2 lb. of rice-4- lb. of rice=] lb. of coffee Ans. LESSON II. (22) Ans. 4. (23) To Thomas 1 the most, and he had 6 left. (24) Ans. 5 dollars. (25) Ans. 54 dollars. (26) By subtracting 1-third and 2-fifths of 15 from it, there are left 4 yards. The cost was 30 dollars, and the sales 58 dollars. Difference, 28 dollars. (27) Ans. 1 dollar. (28) 23 less 18 is 5, the number of miles per day that B gains on A. If it requires 1 day to gain 5 miles, it will require 8 days to gpain 40 miles. A'ns. 8 days. - - ----------— ~ - - - 1 -— ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~3~ MENTAL FRACTIONS. 187 (29) 10 less 7=3 yd. gain in I min. The hound will gain 90 yd. in 90-.-3=30 min., during which time the hound goes 30 times 10=300 yd., and the hare 30 times 7 =210 yards. (30) 7 less 4=3 rods. The hound runs 7 rods to gain 3 rods, and will run 35 rods, which is 5 times 7, to gain 5 times 3 rods —15 rods Anes. (31) 3 miles less 1 mile is 2 miles; the first gains 2 miles in going 3 miles, and will gain 4 miles, or twice two miles, in going twice 3-6 miles Ans. (32) Since 1 gains 3 miles on C in traveling 5, and since 15 is 5-threes, it will require him to travel 5-fives, that is, 25 miles, to overtake C. (33) Since 8 gallons arefilled, and 5 gallons emptied in 1 hour, it is evident the cistern is filled at the rate of 3 gallons (8 less 5 equals 3) per hour; hence it will require as inany hours to fill it as 3 gallons is contained times in 24 gallons, that is 8. Ans. 8 hours. (34) Since it is filled by the first pipe in 6 hours, it is filled at the rate of 6 (36 — 6 —6) gallons per hour; and since it is emptied in 9 hours, it is emptied at the rate of 4 (36 —9-4) gallons per hour; hence when both pipes are left open, it is filled at the rate of 2 (6-4-2) gallons per hour; consequently it will be filled in 36-. -218 hours. (35) 8 is 2-fifths of 20 dollars, the price of the coat, and 1-half of 8 is 4 dollars, the price of the vest; hence, the whole cost 8 and 20 and 4 dollars, that is, 32 dollars. S ECTION 19.-Lzsson 11:I. (11) It requires 8X11=88 hours to perform the work; 88 hours - 10 hours —=S8 days of 10 hours each. (12) 8 miles for 37 cents would be 87 cents for 1 mile, and 11 times 37 is 4 —7=507 cents for 11 miles. (13) If it requires 2 pipes to empty it in 17 minutes, it will take I pipe twice as long, that is, 34 minutes, and 3 pipes 1-third as long as 1 pipe, that is, 1-third of 34 minutes] l1- minutes. 188 KEY TO RAY'S SECOND BOOK. (14) 1 horse will require 1-fifth as much as 5 horses, and 7 horses 7 times as much as 1 horse. l-fifth of 18 bushels is'5and 7 times are 1 2 6, which are 251 bushels. 5.5, (18) 4 of 18=36=5q feet in the earth; and of 18-90 126 feet in the air. (19) 4 of 15=30 —31, A's part. l of 15=5, B's part. 5+3 —, and 15_S —S=62, C's part. (20) 48 dollars. (21) 8 cents. (22) 12. (23) 14 dollars. SECTION 20. —LESSON II. (13) It will require 1 man twice as long as 2 men, and 3 men 1-third as long as 1 man. Twice 3-1 days are 64 days, the time in which 1 man can do it; and -- of 64- days is 3of o days, which are 2o days, equal to 29 days, the time in which 3 men can do it. (14).Ans. 5Tr days. (15) Ans. 93 days. SECTION 22.-LESSON IL. (34) Since he gained 1-fourth, he must have sold it for 5-fourths of the first cost. If 18 is 5-fourths, 1-fifth of 18, which is UI, is 1-fourth, and 4-fourths, or the whole, are 4 times -8, which are 72, equal to 144 dollars. (35) Ans. 205 dollars cost; gain 4- dollars. (36) Ans. 44 dollars. (37) Ans. 3- dollars. LESSON II. (6) If 4-fifths of a yard cost 8 cents, 1-fifth will cost 1-fourth of 8 cents, which is 2 cents, and 5-fifths, or a yard, will cost 5 times 2 cents, which are 10 cents. At 10 cents a yard, 25 cents will buy as many yards as 10 cents is contained times in 25 cents, which is 2, times. An.s. 2- yards. (7) Ans. 62 dollars. X- - < —--— ________ MENTAL FRACTIONS. 189 (8) If 6-sevenths of a yard cost 4 dollars, 1-seventh will cost 1-sixth of 4 dollars, which is 4 or of a dollar, and 7-sevenths, or a yard, will cost 7 times 2, which are 14 dollars. 12 dollars will buy as many yards as 14 dollars are contained times in 12 dollars. But 134 is contained in 12 as often as J4 is contained in %6; and 14 is contained in 336 as often as 14 is contained in 36; 14 in 36 is contained 2,-_=-24 times. Ans. 24 yards. (10) Ans. 3} yards. (11) If he gained 1-third, he must have sold it for 4-thirds of what it cost. If 12 is 4-thirds, 1-fourth of 12, which is 3, is 1-third, and 3-thirds, or the first cost, is 3 times 3, which are 9 dollars. If 5 yards cost 9 dollars, 1 yard cost, of 9 dollars, which is 14 dollars. (12) If he gained 2-fifths, he must have sold it for 7-fifths df the cost. If 21 is 7-fifths, 1-seventh of 21, which is 3, is 1-fifth, and 5-fifths, or the first cost, is 5 times 3, which are $15. This at 3-fifths of a dollar a bushel, would buy as many bushels as 3-fifths of a dollar is contained times in $15, which is 25 times. Ans. 25 bu. (13) If I cost 3 of a cent, 5 will cost` of a cent, and one whole apple 1-O of a cent; and 3 apples cost 3a-0=3 cents. (14) 2= 11 5, 22:: If Vy of a yard cost 4/ dollars, 42, which is twice -1, will cost twice 4-3-9} dollars. (15) Then 8 dollars must be - of the cost; of the cost is 2 dollars, and the cost is 5 times 2 dollars, which are 10 dollars; 11 dollars less 10 dollars will be 1 dollar, which is T1~ the cost. SECTION 25.-LESSON I. (3) E of 3-3, 5X =1; X Of - 5 =, and 2X=$5 SAns. (4) 4 of 24=6, 7X6=42; - of 42=3, 5X3=15 cts. Ans. (5) 5 of 15=3, 9X3=27;. of 27-$13-.Ans. (6) ~ of 30=10, 8X10=80; - of 80=5, 7X5=35 trees. Ans. (7) 15-, of 14-2, 5X2=10; 2 —X10=$25 Ans. AW-~; —---------— ~ —— ~ ~ 190 KEY TO RAY'S SECOND BOOK. (8) 1 —,3 5_=i-1; of 241= 2 times 7-7 the cost of 1 ~ ~L, 4 4 3 4 - -2, barrel, and 24 times 7-=345=8 dollars Ans. (9) 331;-~ -Y-, of 20=2, and 3 times 2=6; 25 times 6 =17 cts. Ans. (10) 3=345, t-L of 30=2, 4 times 2=8 miles, the distance traveled in 1 hour; 7~ times 8=58 miles. Ans. (11) -1 of 1I-=, 11 times ~-=13 dollars. Ans. r=12- dollars. Ans. (13) 5, 2 9 45, and 32= 1 1 of 46=,and 4 tinmes 2 4-' 5 3 3 5 5, 8 of ancl 11 times =8=5 o dollars. Ans. T -5 15-15 —I (14) 1 of w hich is of the number, and 7 times. 3-, ancl 4 of 3.S..9-1 0ls. (1L5)- 2=8: 3, =: 21= 5 f 17 7o, and 3 times -' or the number. 21 times 2=1 -21 X -05 -. Ans. (16) -1 of 4=1la, and 4 times 1`=-6 dollars, the price of a barrel; - of 6=- and 3 times - s —53 dollars. Als. (17') of 5-0 and 3 times -o-J-9; - of 9-9 —7, and 5 times'o — dollars. Ans. (18) D of 30-6, and -1 of 6=2, the wages of one man onze day. 4 times 2=8S, and 7 times 8=56 dollars. Ans. (19) - of 36=6, and I of 6=-3 5 ties 3 — and 12 times?1=45 dollars. Anrs. (20) I of 12=4, S of 4=-; 5 times 1, and 3 times 5 =7- rods. Ans. (21) G of 36=6, -- of 6=-; 5 times 3-=3, and 9 times 3= 27 bushels. Ans. (22) If 5 oxen eat 2 acres in 6 days, they will eat 1 acre in 3 days; and 1 ox will eat 1 acre in 5 times 3 days, or 15 days. Then it will require 1 ox 8 times 15 days, which are 120 days, to eat 8 acres; and it will require 12 oxen T1: of 120 days, which are 10 days, to eat 8 acres. Ans. 10 days. (23) 400 dollars for 5 months is 80 dollars for 1 month, and if 8 persons spend 80 dollars in 1 mnonth, one person will MENTAL FRACTIONS. 191 spend 10 dollars in I month. Hence 11 persons will spend $110 in I month, and in 8 months $880. [Ans. (24) 10 oxen on 5 acres for 3 months, is 2 oxen on 1 acre for 3 months; or 6 oxen on 1 acre for 1 month; or 90 oxen on 15 acres for 1 month; or 18 oxen on 15 acres for 5 months; or 126 sheep on 15 acres for 5 months. Ans. 126. LEssoN II. (1) 5 times 18=90; - of 90=10. Ans. 10 men. (2) 8 times 15=120; ~ of 120-10. Ans. 10 men. (3) 9 times 2o-222; 5 of 22-=4.1. Ans. 41, hr. (4) To do the work in 5 of the time will require 5 times as many men; 5 times 5 men are 25 men; twice as large will require twice as many men; twice 25 men are 50. men. _Alzs. 50 men. (5) 8 times 5-40; o of 40 —8. Ans. 8 days. (6) If 6 men can do the whole work in 5 days, they will do one-half of it in 2~ days. The whole work requires 6 times 5-30 days work of one man, and half of it will require 15 days work of one man. Since 9 men (6+3) are employed to finish it, it will require ~ of 15 davs, or 1!- days. Hence the whole time is 2;+1-4' days. (7) It will require 28 days work of one man, and 7 men will do half the work, or 14 days work of one man in 2 days; and since 3 men then leave, there are 7 less 3, or 4 men left, to do 14 days work of one man, which they will do in - of 14 days, or 3i days. Hence the whole time is 2+3>=51 dclys. (8) 5 times 8=40, the number of cent loaves. 1 of 40=10, the number of 4 cent loaves. -j1r of 40-=4, the number of 10 cent loaves. (10) 3 times 20-60, the number of cent loaves, and -1 of 60 -15, the number of 4 cent loaves. of 60=12, the number of 5 cent loaves. 6 h Ace —--------------- 192 KEY TO RAY'S SECOND BOOK. (11) 3 times 8=24, the weight in oz., when flour is $1 a barrel. of 24 of 214=6, the weight respectively, when flour is 2 and 4 dollars a barrel. (13) $3=$-: $21 —$5: 7 times 7=49 oz., when flour is $.! per barrel: 5 of 49=9 4 oz., the weight when flour is per barrel. Ans. 94 oz. (14) 1 man could mow it in 6 times 5-=33 days, and 10 men in' of 33-3- days: 5- -3-2 days. Ans. LESSON III. (2) 3+4+5-12. T3 of 36=9, W's; -, of 36=12, T's; - of 36-15, John's share. (3) 2 of 71-=5 share of Ist; 9 of 7-=2-1, share of 2nd. (4) 40-25-15, what B paid. 2_ and of 56-35 dollars, A's share. z} 3 of 5 lar-s, A'S share. 1 =- and ~ of 56=21 dollarls, B's share. (5) +4-1, and 1- -- C's part. $860-$500=$360:: of 360 —45 dollars, A's loss; 4 of 360=90 dollars, B's loss; 4 of 360=225 dollars, C's loss. (6) 7'='T, ancd'2 —94; r4 of 21=-7 dollars,share of1st, and 9i of 21=13, dollars, share of 2nd. (8) T0C;: } of 25=15 dollars. Ans. (9) -15 3 of 80=12 dollars. Ans. (10) 180 sheep eat as much as 18 oxen. 27-lS-1845. 47 =54. 5 of 25=15 dollars, what A pays. 48, an d of 25=10 dollars, what B pays. (11) 15 cows are 5X3 cows, and 5X2 horses are 10 horses; 10 horses eat as much as 15 cows. 14+10=24. -4 v and _5 1 T and U24; 42 of 60=5, - 7 of 60=35, and 7-a of 60-25. Hence A should pay $35, and B $25. (12) 15 oxen are 5X3 oxen, and 5X2 horses are 10 horses; hence 10 horses eat as much as 15 oxen. Since I horse eat as much as 20 sheep, 6 horses eat as much as 120 sheep; hence 15 oxen and 120 sheep eat as much as 16 horses. 8+16-24,, I 6'-2; of 75=$25, A should pay; and. 2 of 75-$50, B should pay. % w ---------------------— P —-----. MENTAL FRACTIONS. 193 (14) $50 for 4 months is the same as $200 for 1 month, and $60 for 5 -months is the same as $300 for 1 month. 200 4+300=500; A'o_-, A's share, and -~ —, B's share. 2 of $45-$18, A's gain, and 5 of $45-=$27, B's gain. (15) 3 men for 4 days=1 man for 12 days, and 5 men for 3 days-l man for 15 days. 12+15-27;-, — =; 4 of $81=$36, and' of $81=$45. (16) $2 for 5 months is the same as $10 for 1 month, and $3 for 4-months is the same as $12 for 1 month. 10+1222, ~ — T; i~ of $55=$25, A's share, and -6of $55=$30, B's share. (1L7) 4 horses for 5 mon. is the same as 1 horse for 20 mon., and 10 cows for 6 mon., as I cow for 60 mon.; but 1 cow for 60 mon. is the same as 3 cows for 20 mon., and 3 cows eat as much as 2 horses, and 2 horses for 20 mon. is the same as I horse for 40 mon. 20+40-60, 2=o, and 0 - of $27-$9, what E should pay; and of $27-$18, what F should pay. LEssoN IV. (2) Of each 7 cents John gets 3, and James 4. In 28 cents there are 4 times 7 cents, hence John gets 4 times 3 cents which are 12 cents, and James 4 times 4 cents wrhich are 16 cents. (3) 4+3+2=9. 9 in 45 is contained 5 times. 5X4-20, A's share. 5X3-15, B's share, and 5X2=10, C's share. (4) 5+3=8. 8 in 96 is contained 12 times; 12 times 5=60, apple trees; and 12 times 3-36, peach trees. (5) Since there are 2 sheep for each cow, there are 6 sheep for e.ach 3 cows, or each horse. 1+3+6=10, that is, for each 10 animals there are 1 horse, 3 cows, and 6 sheep. 10 in 60 is contained 6 times; 6 times 1=6, the number of horses; 6 times 3-18, the number of cows; and 6 times 6=36, the number of sheep. (6) 2+3=5. 5 in 35 is contained 7 times; 7X2=14, the number of boys, and 7X3=21, the number of girls. e.-~. —! 3 (2 )X,] 194 KEY TO RAY'S SECOND BOOK. (7) The number and 3 times the number, that is 4 times the number, is 48, hence once the number is A of 48, which is 12. (8) 1+2+3=6. 6 in 42 is contained 7 times; 7 times 1= —7, A's plums; 7 times 2=14, B's plums; and 7 times 3= 21, C's plums. (9) 1+4-5; -'1 of 25=5; 4 of 25=20. Ans. 5 yd. and 20 yd. (10) Since Agnes gets twice as many as Emma, and Sarah twice as many as Agnes, Sarah gets 4 times as many as Emma. But 1+2+4 —=7; hence of each 7 cherries Emma gets 1, Agnes 2, and Sarah 4. 7 in 35 is contained 5 times; 5X1=5, 5X2=10, and 5X4=20; hence Emma gets 5, Agnes 10, and Sarah 20 cherries. (11) 3+4=7: of 28 is 4; 3 times 4=12; 4 times 4=16. LESSON V. (12) 2+3=5;, of 25 cts.=10 cts., John's; ~ of 25=15 cts., George's. (13) 5+7=12; i~5 of 48=20, tuhe 1st. g7 of 48=28, the 2d. (14) 3+4-+5= —12; u of 60=15, of 60=20, and A. of 60 =25. (15) 1+2+3+4=-10; I of 70=7, 2 of 70=14, y6 of 70= 21, and -4y of 70=28. (17) 1.+21=i4. 4 in 16 is contained 4 times; 4 times 12 — 6, Thomas' share; and 4 times 2>-10, Moses' share. (18) j4+1+=1 —-. 3 nd in 39 is contained 36 times; 36X-= 18, 36X=12, and 36X= —9. Ans. 18, 12, and 9. (20) 14-1-=4'.:y in 33 is contained 12 times: 12 times 1 — 12, and 12 times 1 —21. Ans. John 12, James 21. (21) i+12 —=. 8 in 32 is contained 12 times; 12X1=12, age of Frank, and 12X12=20, age of William. (a22) 1+21=3 —. 3o in 30 is contained 9 times; 9 times 1=9, the number of unsound, and 9 times 21l21, the number of sound apples. (23) 1+4=g. ~ in 27 is contained 15 times; 15 times 1- 15,..... J MENTAL FRACTIONS. 195 the number of feet the 2d can build, and 15 times -_-12, the number the 1st can build. (24) Since B has, as many as A, and C 1 as many as B, therefore C has 1 of _, or x as many as A. 1++1. —7. 7 in 42 is contained 24 times; 24 times 1=24, A's share; 24 times — 12, B's share; and 24 times 1-6, C's share. (25) Since there are 2 as many sheep as hogs, and - as many cows as sheep, for each hog there is 3 of a sheep, and 4 of 2- = -1 o a cow. l +, in 104=48; 48X1=48, the number of hogs; 48X —=32, the number of sheep; and 48X 2=24, the number of cows. LESSON VI. (2) 19 —5=14; A. of 14=7, the number of apples James has; and 7+5=12, the number Thomas has. (3) 31-7-24; ~ of 24=12, the less number; and 12+7= 19, the greater number. (4) 32-8=24; 3 of 24=12, the number each had at first. (5) 20+4+6 —30; ~ of 30=15; 15-4=11, the number Thomas had left; and 15-6=9, the number William had ]eft. (6) 24+7+5-36. Since Mary bought twice as many as Sarah, they both bought 3 times as many as Sarah, therefore Sarah bought - of 36=12 cherries, and Mary twice 12, or 24 cherries. 24-7=17, the number of cherries Mary had left; and 12-5-7, the number Sarah had left. (7) 50-5=45, and I of 45=15, the required number. (8) 31-10=21, which is 3 of the number; hence 1 of 21, or 7, is - of the number, and 4 is 4 times 7, or 28. (9) 21+7=28, 4 of 28=7 is 1, and.535, the numllber. (10) 4-3=-7. 37-7=30. 4 of 30=10, Henry's age; 10 -t-3=13, Oliver's age; and 10+4=14, James' age. (11) Since Sarah has 3 cents less than Mary, she has only 5 more than Jane. 8+5=13, 43-13=30, - of 30=10, Jane's; 10+8=18, Mary's; 18-3=15, Sarah's. 196 KEY TO RAY'S SECOND BOOK. (12) Twice the age of Frank, plus his age, less 3 years, is 42 years; hence 3 times the age of Frank must be 45 years. - of 45=15 years, the age of Frank, and twice 15, less 3 years, is 27 years, the age of Mary. (13) The ring cost $5, and the watch $12 more than the chain; hence 3 times the cost of the chain, and $17 is equal to $62; therefore 3 times the cost of the chain is equal to $45. 1 of 45=$15, the cost of the chain; 815+5=$20, the cost of the ring; and $15+12-$27, the cost of the watch. (14) 30+6=36, which is of his money. a of 36-18, which is 4 of his money, and 7 times 18=126 cents, his money. (15) The money of John and James=3 times the money of James and $3; hence the money of Frank=3 times the money of James and $10; and the money of all three, $55,=6 times the money of James and $13, hence 6 times the money of James is $42. -L of 42- 7, James' money; 7+7+3-=$17, John's, and 7+17+7-$31, Frank's. LESSON VII. (2) 4+ =0?. 1 of 60=6, which is 4. =4 and = 18; hence Thomas pays $42, and John $18. (3) From noon to midnight is 12 hours. Since the time past noon-' the time to midnight, therefore the time and a the time, or a of the time to midnight=the time from noon to midnight, 12 hours. If 3 is 12 hours, -1 is 3 of 12 hours, or 4 hours; that is, one-half the time to midnight, which is the same as the time past noon, is 4 hours. Hence the time was 4 o'clock, afternoon. (4) 1+-3=-, 1 of 12 hours=1- hours, which is ~ and 4 —.1 hours; hence the hour was half past 4 P. M. (5) 143. of 56=8=4 —; 4=32, and 4-24, the distance traveled each day respectively. (6) Since the first, plus 4 of the first less 8, that is L of the first less 8=100, 12 of the first is 108. 1 of 108S9, is 4 of the first part. =-63, the lst; 4=45, and 45 —8= 37, the 2nd part.. ---- --- - -— L MENTAL FRACTIONS. 197 (7) The first+"+43, that is 9 of the first=45. ~ of 45=5, of the first. -4=20, the first part; ~ of 20=10, the second; and a of 20-15, the third. (8) 2 of 3=twice' of 3= —; hence C has,2 as many as A. 1++2-1t — i of 40-A4, which is. -20, A's; =12, B's; and 5=8, C's. 3,F 5 ac:,,, 1 5 3 11++ = 5 531 4 4 (9) of 0 — 3 times of 5 - 1+j+5-!. 6 hence J4 of the top part-70; 14 of 70=5, which is a; 6-30, top; =25, middle; and 3=15, the lower part. (xo) ~ of s —~ —. 1+2 3 _ 4 _ hence (10) 3 of T= -e% y4. +2+T — -- 0+To+;o-; hence o- of the cost was $34. r- of 34 is 2, which is o; -- $20, the cost of the coat; 4 =$8, the cost of the hat; and =- $6, the cost of the vest. (11) If i the cows=- the sheep, the cows=4 of the sheep. 1.+4-; 1 hence 1' of the sheep is 55. 1l of 55=5, which is -; — 35; the sheep; and 4=20, the cows. (12) If 3 of the less — of the greater, the -3, or the whole of the less=3X2, which are 4 of the greater. 1+-k; hence 5 of the greater-100; 4 of 100=20 is - of the greater. 3-=60, the greater number. 2 40, the less. 3 7 (13) If 4 of Mary's age=- of Sarah's, 4 of Mary's=4 of Sarah's. 1+ -4=. 1 of 14=2, is ~ of Sarah's, 3-= 6, Sarah's age; and 4-8, Mary's age. (15) If of the first-= of the second, of - or of th second, and -, or the whole of the first - - of the second. Hence the second plus 9 of the second, or 7 of the second-51. -T of 51_ —3 is K of the second. 8 -24, the second part; and 9-27, the first part, (16) If 2 of the apple trees-4 of the peach trees, 14=, and or all the apple trees= — of the peach trees. Hence the peach trees and of the peach trees, that is, 1 365. T, of 65=5, is. =35, the peach trees; and — 30, the apple trees. (17) Since - A's distance= 5 B's, -3 A's=- of 4, or - of B's, an of =3 of A's 3X or! of B's; hence B's and 6 of B's, that is l, is 66 miles. IT of 66 is 6 miles, 4 of d- e a - - - ------------- 198 KEY TO RAY'S SECOND BOOK. B's, and since the difference between B's and - of B's is. of B's, B traveled 6 miles further than A. (18) Since once the time past noon+3 hours is 4 the time to midnight, twice the time past noon+6 hours.-the whole time to midnight; but the time past noon+the time to midnight is 12 hr., hence the time past noon, with twice the tinme past noon-+ 6 hr.=12 hr.; hence 3 times the time past noon is 6 hours, and the time past noon is 3 of 6 hr., or 2 hr. (19) See the "Explanation." If 12 hours is 4, of 12, which is 3, is; hence the hour is 3 in the afternoon. (20) Here 12 hours must ble of the whole time; hence 1 must be ~ of 12 hours, which is 4 hours; that is, the hour is 4 P. M. (21) If, the time past noon is ZU of the time past midnight, the time past noon must be o- of the time past midnight; then the time past noon is J-a of the whole time from midnight, and the time from midnight to noon must be T9~ of the whole time. - of 12 hours is 14 hours, that is, the hour is 1~ P. M., or 20 minutes past 1 o'clock. LESSON VIII. (2) 1+-5 l- of 20=4, is 1 of the number; -3=12, the number. (3) I-1=~-4. 4 of 21-3, is 4 of the age; 5-15, her age. (4) Twice the number and 4 of the number equals'3 of it. y3% of 52 is 4, which is;.5=20, the number. (5) Twice the number less its 4, is 14 less 4 or 0; if'O is 40, iT of 40, which is 4, is 4, and o-, or the number is 28. (6) Three times the number less its 4 is 5 less, or; if l is 48,'-1 of 48, 4, is and 4 is 20. 1I o 6.6 4_13 3 -12 (7) 1+A- + -;-+ of 26:2, is 4, and -12, his age. (8) Her age,' and -4 of her age are ~9 +T- or 1'; but twice her agre is 24; hence l4 of her age plus 10 years =4,'d; f.! of her awe is 10 years, and her agre, 24 years. MENTAL FRACTIONS. 199 | (9) 1-4-4.. 4 is 30 cents; hence a is 50 cents. 1 1i 31__ (10) 1+ - Q+ 5 +- +(+G=o 1; treble the number is 4, hence -9- of the number is 27, and the number is 30. (11) 1 —T=,,- 4 of 40= 5, i-s of the father's age, or 55 years; hence the son's-15 years. (12) 1+4 -. 3 times her age is 5. HIence -~ of her age is 18 years, and her age is 15 years. (13) 1 —2=7. of 28=4 yards is B;'=36 yards. Ans. (14) Twice the distance from A to B is equal to the distance from C to D; hence twice the distance from A to B is equal to ~ of the same distance plus 20 miles; therefore 20 miles is 4 of the distance from A to B, which is 15 miles, and from C to D is 30 miles. ( 15) 1+- + - + _9_-+: —4-5-; I of 94 is 1, which is l.; hence the age is 45 years. LEssoN IX. (1) Ans. 4. (2) Ans. 4. (3) Ans. 2. (4) Ans. as many as - of a day is contained times in I day, that is, 2'. (6) He could dig 2 trenches in 7 da., or 4 of 1 trench in 1 da. (7) Hle could walk 5 times the distance in 16 days, or y of the distance in 1 day: and in 2 days a 6~o (8) An~s. -!-+4, which are -3 (9) Ans. 1++ - +, which are ~-. (11) -+ 1- 3 -i=; 1 in 1 is contained 4 times. Ans. 4 da.! I 2 12 n 1 (12) *+q 4-n'; 43 in 1, 2} times. Ans. 2'4 days. (13) -F4 -A- 1 —— 1; hence all three can do it in 1 day. (14) 1+- I +1 =2-o- — 7; O-'in 1, 13 times. Ans. 14 hr. 3 ( ) T' - -T'(l6) I —r i=; in 1 is contained 20 times. Ans. 20 da. I = z in I d a. -! - th e p art (17) A and B can do 4+ -454 in1da. 4 —4fjthe part that C can do in a da.; hence he can do it in 24 da. (19) The first will empty the cistern 3 times in 1 hour, and the second 5 times in 1 hour; hence both will empty it 8 times in 1 hour, or once in -~ hour. 200 KEY TO RAY'S SECOND BOOK. (20) o days in 1 day is contained 1- times. Ans. 1' days. (21) C can reap 4 of the field in 1 day, and since B can reap it in 31 days, he can reap 3 such fields in 10 days, orT 3 of a field in 1 day. 3 =4; hence both can reap V the field in 1 day, or the whole field in 2 days. (22) A can do I of his piece in 1 da. If B can do a piece 3 times as large in 7 da., he can do - of such a piece in 1 da.; hence both can do ~+3 6s of it in 1 da., or the whole in as many da. as 4y is contained times in 1. 4 in1 1, 1 times. To do a piece of work 5 times as large will require 5 times as long. 5X1 -5 da. —6;9 da. Ans. (23) Tj of 100 gallons=5 gallons to be emptied per minute: the 1st and 2d empty 2 gallons in a minute; hence the third must empty 3 gallons per minute, and in 20 minutes 60 gallons. Or, the lst and 2d empty 40 gallons in 20 minutes and the 3dc must empty 100 —40=60 gallons. LEssoN X. (2) Ans. 3,. (3) Ans 51. (4) Ans. 5`. (5) Ans. 14. (6) Ans. 1. (7) Ans. 2 2. (8) Ans. 11. (9) Ans. 20 Thus, if she gave away 4 of what she received, she had 4 left; X of 3=., and - of 60 is 20. (10) Since a remains, 35 is {, and 2 is 14 miles. ~ of 14 is 6, and 14 less 6 is 8 miles, the required distance. (11) Ans. 25. (12) of 27 is 12, -- 8 is 6; 12 and 6 are 18, which is 3 of 30. Ans. $30, (13) After losing 2v she had 4 left, and after giving of to 5 5 __ 3 3 7 Sarah, she had 4 of left. A of 4=4. If 4 is 6, i is 14. Ams. (14) Ans. 20 cents. (15) 4 of 36 is 45, and 45 is 9 of 35. Ans. (16) If Z of B's=2 of A's, then all of B's=4 of A's: 1-4-= of A's-12: -4 7-=28 cts., A's; and 4-16 cts., B's. (17) -41 -4+ = =- 8 ——.; hence 32 is 4-, and the whole is 96, MENTAL FRACTIONS. 201 of which 32 are apple trees, 24 pear trees, and 8 plum trees. (18) The whole less 2 is A, and the difference between - and 2 is -; if 5 are 25, g is 5, and' are 10, the pear trees, and b are 35, the apple trees. (19) Since the plum trees are -1 of the apple trees, i of the plum trees —1 of the apple trees; hence the cherry trees.1 and, that is, 5- of the apple trees; and the whole number of trees=once the apple trees, 3- of the apple trees, and 7 of-the apple trees; that is, 1+ —+ -, or 4~, of the apple trees=69.' of 69 is 3, which I is 2. -14 -36, the apple trees; — =12, the plum trees; and 7T-=21, the cherry trees. (20) 5 of 12 years is 20 years, which is 4 of the ages of Jane and Sarah; if 20, is 9,4 - of 20, or 5, is, and 9 —45 years, the sum of their ages. Then Sarah's age and 7 of her age, or!-5 of her age, are 45 years; 5 of 45 is 3, which is A of her age, and 8, or her age, is 24 years; 4 of 24 years is 21 years, Jane's age. (21) -3 of 55=15; A of 30, or 24, is 4 of 54: twice 54=108; 15 in 108, 71 times. Ans. (22) ] of John's money=3 of - of Charles':=: 9 9 of Charles'+-33=Charles' money; hence $33 is Ad, and Charles has $60, John has of $60$ —36. LESSON XI. (2) Since 4 of John's steps are equal to 5 of Henry's, 1 of John's is equal to 1- of Henry's, and 5 of John's to 64 of Henry's; but while John takes 5 steps, Henry takes 6. Hence John takes 5 steps to gain 4 of one of Henry's steps on him, and therefore must take 20 steps to gain 1 of Henry's steps on him. (3) In taking 7 steps, John gains 2 steps on Henry. Hence, to gain 30 steps, he must take as many times 7 steps as 2 steps is contained times in 30 steps. 2 in 30 is contained 15 times; 15 times 7 steps are 105 steps. Ans. (5) Since 3 leaps of the hound are equal to 10 of the hare, 202 KEY TO RAY'S SECOND BOOK. and the hound takes 3 leaps while the hare takes 5; therefore, in taking 3 leaps, the hound gains 5 of the hare's leaps on her; and to gain 100 leaps, lie must take as many times 3 leaps as 5 leaps are contained times in 100 leaps. 5 in 100 is contained 20 times, and 20 times 3 leaps are 60 leaps. An.s. (6) Since 4 of M's steps=7 of N's, 1 of M's-1 —l of N's, and 5 of M's -=S- of N's; hence while M takes 5 steps he gains 1- of N's steps on him, and to gain 35 steps, he must take as many times 5 steps as 14 steps are contained times in 35 steps. 1, in 35, are contained 20 times, and 20X5 steps=100 steps. Ans. (7) Since 2 of the hound's leaps-3 of the hare's, and the hound takes 2 leaps while the hare takes 3, the hound goes just as fast as the hare, and will not catch it, but will continue at the distance of 8 leaps firom it. (9) Since the tail is as long as the head and 1 of the body, therefore the tail is 4 inches plus X the body, and the body is equal to 4 inches, plus 4 inches. plus 1 of the body; hence -2 of the body is 8 inches, and -1- is 4 inches. body; 3hne 3_-12 inches, the length of the body; 44-4=8 inches, the tail; and 12+8+4=24 inches, the whole length. (10) Since C has as many as A+-' as many as B, C has 10 cents+3 of B's; but B's money —A's and C's; hence B's=10 cents+10 cents+- of B's; or - of B's=20 cents; hence B has 60 cents, and C 50 cents. (11) Since the cow- cost as much as the sheep and 4 of the horse, therefore the cow cost 8 dollars and - of the horse; but the horse cost twice as much as both the sheep and cow; hence the cost of the horse was equal to 16 dollars, plus 16 dollars, plus ~ the cost of the horse; therefore, the cost of the horse was 32 dollars. Hence the horse cost $64, and the cow $24. (12) Since the tail weighs 3 pounds more than the head and V the body, therefore the tail weighs 11 pounds plus, the body; but the body weighs as much as the head and tail; hence the body is equal to 8 pounds plus 1i pounds e —-- a ----- X —- - ---— ~- ----- -- --- ---- - — ~A MENTAL FRACTIONS. 203 plus ~ the body; therefore - the body weighs 19 pounds, and the whole-38 pounds. The tail weighs 11+19-30 pounds, and the whole fish 76 pounds. (13) If 2 of B's steps=3 of A's, then 4 of B's-6 of A's: hence, B gains 1 of A's in moving 6 of A's, that is,, of the distance B travels;. of 30 ft.=5 ft. Ans. LESSON XII. (2) The difference between 4 peaches and 2 peaches is 2 peaches, and the difference between having 9 left, or 3 left, is 6; hence giving 2 peaches more to each would increase the number given by 6. Hence he has as many children as 2 peaches is contained times in 6 peaches. 2 in 6 is contained 3 times. Ans. 3 children. 3 times 2=6, and 6-+9=15, the number of peaches he had. (3) The difference between giving 5 and 8 to each is 3, and between having none left and 21 left is 21; 21 3 — 7; hence she had 7 playmates, and 56 cherries. (4) The difference between paying 1 dollar and 1, dollars a yard is I a dollar, and the difference between having 5 dollars left and nothing left is 5 dollars. Hence she wished to buy as many yards as - dollar is contained times in 5 dollars, which is 10 times; therefore she wished to buy 10 yards of silk, and had 15 dollars. (5) The difference between 29 over and 6 lacking is 35 cts.; 8 cts.-3 cts.=5 cts. 5 cts. in 35 cts., 7 times. 7X8= 56, and 56-6-50 cts. Ans. (6) 12 —10=-2,; 2~ in 40, 16 times. Ans. 16 yards. (8) 3 times the age of James less his age is twice his age, or 10 years; hence his age is a of 10 years, or 5 years; and Thomas' age is 15 years. (9) Since B's age is twice C's, and A's 5 times B's, therefore A's is 10 times C's, and the difference between A's and C's is 9 times C's, which is 45 years. Hence C's age is 5 years; B's 10 years; and A's 50 years. (10) Since A's moneyi- B's, twice A's=B's; and since (10)l Since A' money=~ ----- 204 KEY TO RAY'S SECOND BOOK. B's money is - C's, twice A's money is -1 C's, and 6 times A's money=C's. But the difference between A's and C's money is 815; therefore 6 times A's money less A's money is $15. Hence 5 times A's money is 15, and A's money is $3. Therefore B's money is $6, and C's $18. (171) Since the second contains 4 times as many as the first, and the third 3 times as many as the second, the third contains 12 times as many as the first. Both the first and second contain 5 times the first, and 12 times less 5 times the first, or 7 times the first is the excess of the third above both the first and second, which is 70 sheep; hence 7 times the first are 70 sheep, and the first contains 10 sheep, the second 40, and the third 120. (12) If 3 of C's age=twice A's, C's age must be 6 times A's; B's is twice A's; the difference between B's and C's will be 4 times A's-20 years; hence A's age is 5 years, and B's 10 years, C's 30 years. (13) A daughter and a son together received 3 dollars, hence there were as many of each as 3 dollars is contained times in 18 dollars, which is 6; therefore he had 6 sons and 6 daughters, in all 12 children. (15) If he had worked every day he would have received 9 dollars, but as he received only 5 dollars, he lost 4 dollars by being idle; he lost 50 cents each day he was idle, that is, his wages, 30 cents, and the 20 cents he paid for his board; hence he was as many days idle as 50 cents is contained times in 4 dollars, which is S. Ans. 8 days idle, and therefore he worked 22 days. (17) The difference between 4 yr. ago and 4 yr. hence is 8 yr.; 3 times B's age (which was A's) plus 8 yr. was equal to twice (B's age plus 8 yr.); that is, 3 times B's age plus 8 yr. was equal to twice B's age plus 16 yr., hence B's age was 8 yr., and A's 24 yr. (18) In 10 years, A's age+10 years-3 of (twice A's+10 years)C — of A's+6 years: hence -5 of A's=10 —6=4: 5=20 years, A's age, and 40 years=B's. (20) Since the first horse and $12 are worth 1 the second; 3 MENTAL FRACTIONS. 205 times the first horse and $36 are worth the second; but the second horse and $12 are worth 4 times the first; hence 3 times the first horse and $36, and $12, are worth 4 times the first horse; hence the first is worth $48; and the second $180 (2 1) If: cost E pence, A will cost c, and 1, 3 penny: hence, for 9 pence, you can buy as many as 5 are contained times in 9 or 3, which is 52 Ans. LESSON XIII.-MIS CELLANEOUS EXAMPLES. (1) Ans. 105. (2) Ans. $60. (3) If A walks 10 miles in 1- days, he walks 8 miles in 1 day, and if B walks 8 miles in 1A days, he walks 5 miles a day. It takes A 2A days to travel 20 miles; and in 2, A days B will travel 12A miles. (4) 2 apples for 3 cents and 2 apples for 5 cents are 4 apples for 8 cents, or 1 apple for 2 cents, which is the rate of buying. 3 apples for 7 cents is 1 apple for 2- cents, the rate of selling; hence on each apple sold he gained - of a cent, and on each dozen, 12 times 3 of a cent, which is 4 cents. (5) For each egg at 2 cts. he bought 2 eggs at 3 cts. a piece; hence for each 3 eggs he paid 8 cts.= —22 ts. a piece. By selling them at 3 for 10 cts. he sold them at 31 cts. a piece; hence on each egg he gained 3A-22, or' cts., and on each dozen 8 cts. Had he sold them at 4 for 10 cts., or 21 cts. a piece he would have lost 2 -2-1, or - cts. a piece, or 2 cts. per dozen. (6) 2 for 1 cent is l cent a piece; 4 for 1 cent is I cent a piece. Hence each 2 pears cost +1, or L cent, and each A cent. He sold them at A cent a piece; A less x — 49; hence he gained 19 of a cent on each pear sold; and to gain 18 cents he must sell as many pears as -490 is contained times in 18 cents, which is 80. Ans. 80 pears. (7) Each duck cost $, and each 2 chickens $l; therefore 2 chickens and a duck together cost it _.1. Z$ -f5 20 6 KEY TO RAY'S SECOND BOOK. — $_w what he gained on each lot. Hence he sold as many lots as l is contained times in $21, which is 30; therefore he sold 30 ducks, and 60 chickens. (8) 10X4=40, and 3X40-120, the hours work of 1 man; of 120-15, the hours work of 8 men; and 15 hours-2. days of 6 hours each. Ans. 2. days. (9) Reserving 2 for Mary and 3 for Lucy, there are left 27 to be divided equally. I of 27 is 9, James' share. 9+ 2-11, Mary's share; and 9+3=12, Lucy's share. (10) Ten gal. in 1 hr. is 20 gal. in 2 hr., and 20-17=3, the number of gal. left in each 2 hr.; hence it will require as many times 2 hr. to fill the vessel as 3 gal. is contrained times in 15 gal. Ans. 10 hr. (11) If A can do the work in 4- days, he can do twice the work in 9 days, or 2 of the work in 1 day. If A and B together can do the work in 2' days, they can do 7 times the work in 18 days, or.- of the work in 1 day. T78 2-T-, the part of the work B can do in 1 day. HIence B can do the work in 6 days. (12) A, B, and C can do 1 of the work in 1 day; A and B do of it in 1 day; and B and C I of it in 1 day. ence l5 _. 3 —, the part C does in 1 day, therefore he can do it in 13- days. R-g —, the part A does in 1 day, therefore he can do it in 111 days. 3 the part 4~I~ — aU, the part B does in 1 day, therefore he can do it in 27-3 days. (13) Since 5 men, or 7 women, can do the work in the same time, both together can do it in half the time of either; that is, a of 35 days, or 17- days. (14) Since 1 woman can do 4 of a man's work, 4 women can do 3 men's work; hence the work of 2 men and 4 women =the work of 2 men and 3 men, 5 men; and the work of 1 man and 1 woman-the work of 1' men; hence it will take 1 man and 1 woman as many times 28 days as 1-4 is contained times in 5. J1. in 5-2' times, and 2-X28 days are 80 days. Ans. (15) Both consumed ts in I day, and f -or C. in 6 days. After MENTAL FRACTIONS. 207 were consumed;s were left, and if the woman consumed -- in 30 days, she consumed a in 10 days, or the whole in 50 davs. Bea-a0=l';, the part the man consumed in 1 day; hence it would last him 1 5o_21: days. (16) 36 at 3 for 4 cents cost 48 cents, 24 at 4 for 3 cents cost 18 cents. 36+24=60, and 80 —60-20, which at 10 for 17 cents bring 34 cents. All the egos at these rates bring 48+18+34=100 cents. 80 eggs for 100 cents= — 1ij cent a piece, or 15 cents a dozen. (17) James' money+:- of James' money+S6, or:- James' money+$G6=$34. IHIence 7 is $28, and,- is $4; 4-I$16, James' money; and $18-Thomas' money. (18) HTere A's money plus o of A's money less $5 is equal to $51; or, 7 of A's money is equal to $56. Hence; is - of $56, which is $8, and 5, or A's money, is $40; and B's money is $11. (19) Since A's age is twice B's, therefore 2-) times A's must be equal to 5 times B's age; but -- of B's age plus 44 years is equal to 2 toimes A's age; hence 3 of B's tage plus 44 years is equal to 5 times B's age; therefore 4'~B's age is 44 years; or 2j B's age —44 years; hence -— 2 Hears, and his age is 10 years; and A's age is 20 years. (20) As many, and half as many more, are 1~ times as many dollars as he has; hence 11 times A's money plus $2. is $40, or 1.- times A's money=$37!~, or, of A's money= 87.; hence E is $2;l, or A's money is $25. (21) a of the number=100 less 2,=97'-=1 ~5;, of the number — and the number-39. (22) The whole minus - is aI. I of __nd AS - - J =q —; hence -- of the number at first is G5; therefore ~ is 1 sheep, and -2, or the whole flock, was 72 sheep. (23) Since - of the silver+$10=- of the gold, of the silver +$-2==, 7 of the gold, or the silver+$124 —, or X of the gold, or the silver= — of the gold-$1'. But the gold and silver=$50, and the gold+- o of the gold!- - 9VBL — 6 —9 -. —--— 1 —C-~ 208 KEY TO RAY'S SECOND BOOK. -$12=0 50, or i' of the gold-$62. Hence'io=$2, and 1-, or the gold=-20; therefore the silver is $30. (24) Since - of's — 3 of B's+$5, therefore 1 of A'si ji of B'sA-$8, or the whole of A's= — of B's-'I-16. But since both have $56, therefore B's and - of B's+$16= $5o5; or 3 of B's+$16 is $56; therefore 5 of B's is $40. of 40 is $8, which is 1 of B's, and - tht is, the whole of B's, is $24.,$56 —T24=$32, A's. (25) Since 3 Of A's distance-4 of B's, therefore vof A's distance-3 of 4, or 4 of B's, and 4, or the whole of A's distanCe=7X-A,~ or 01' of B's. I' of B's distance and B's distance, or 44 of B's distance=S6 miles. lHence -1~ of B's distance is?-1 of 86 miles, or 2 miles, and - - is 30 miles. A's distance is 24, or 56 miles. (26) Since - of A's is equal to 4 of B's, 1 of A's is equal to. of B's, and 43, or the whole of A's, is equal to 5 of B's. Hence their difference is I of B's, and 4- of 1 of B's is offthfis5 4 5 $Y15, that is, of B's is $15; hence 1o is I o: $15, which is $5, and 2o, or the whole of B's, is 20 times $5, or $100; and A's is 6 of B's, which is $120. (27) Since the watch and chain are -! of the watch, therefore the chain is a of the watch. 3 times 4- are,, and twice the price of the watch plus 4 of its price are h, which care $100. Hence 1 is -2' of $100, or $5; and 7, or the price of the watch is $35, and the chain is 2, or $10. (28) 7- of 24 is 21; if 21 mi. are 7. of 21, which is 7, is of the distance from B to C; hence'-, or the distance, is 49 mi.; and from A to C is 49+24, or 73 mi. (29) 4 horses for 2 months=8 horses for 1 month. Since 3 cows eat as much as 2 horses, 9 cows eat as much as 6 horses; hence 9 cows for 3 months=6 horses for 3 mlonths-18 horses for 1 month. Again, since 10 sheep eat as much as 3 cows, and 3 cows as much as 2 horses, therefore 10 sheep eat as much as 2 horses, and 20 sheep as much as 4 horses; hence 20 sheep for 5 months=4 horses for 5 months-20 horses for 1 month. Hence,. A has the same as 8 horses 1 month, B the same as 18 tS X,-,, -- MENTAL FRACTIONS. 209 horses 1 month, and C the same as 20 horses 1 month, in all 8+-18+20-46 horses 1 month. 46 for $92 is $2 for 1 horse; hence A ought to pay 8X$2=-$16; B 18X $2 —$36; and C 20X$2=-$40. (30) Since the three eat 8 melons, each eats 22 melons: James eats 2' melons out of the 3 melons he furnished. Of the 2-l melons eaten by Thomas, James furnished 1, and John 21, that is, James furnished'a, and John, both together making s. Therefore, since James furnished i- of the s, and John 9 of the _, James should have 3 of the 8 cents, which is 1 cent; and John - of the 8 cents, which are 7 cents. (31) + — 617 3-; and 1 3- C's pal't. a =-p-; hence is $160, and 2'T is - of $160=-SO; f — 6 times $80=$480, A's share;'7=7 times $80= $560, B's share; and = —8 times $80=$640, C's share. (32) Since A's age is 5 of B's, twice A's age is 1o of B's; hence -- B's age+B's age+- B's age= —0 B's age-2 years; or?1 of B's age=2-0 of B's age-2 years; therefore { of B's age is 2 years, and his age is 12 years; and -9- of B's age, or A's age, is 20 years. (33) A and B can do 1I in 1 day, hence in 4 days they can do -1 or { of it, and 3 remains. If B does A in 36 days, 16 4 4 he does 3 of — 4 in 12 days, or G- of - in 1 day, or the whole in 48 days. Since both can do -f in 1 day, and Bcan do j1 in 1 day, A can do 16 48 in 1 day, and the whole in 24 days. (34) Since v of B's and C's is $12, q is $6, and, or the sum of their shares, is $42. Again, since -9 of C's=:~ of A's and B's, k of C's=T-O of A's and B's; or'~0, that is, 4- of C's=A's and B's; but since A's is $12, therefore 5 of C's=-12 and B's, or 4 of C's less $12=B's. Therefore C's and L of C's less $12 —=$42, or 9 of C's-$54; hence - of $54, which is $6, is 4 of C's, and A, or the whole of C's, is $24; and $42 less $24=$18, B's share. (35) To mowr ~- in 12 days would require 3 men; to do it in, the time, 12 men; in 4 the time, 4 men. Ans. Key -14 210 KEY TO RAY'S SECOND BOOK. (36) During the first 6 months the second has $100 more than the first; to equalize it, then, the first must dsttingt the last 6 months have $100 more than the second, that is $300, and therefore must put in $200 Aims. (37) After A had given B 2, they each had I of 24=12 cts.; hence A had 12+2=14 cts., and B 12-2 or 10 cts. (3$) 3 of 50-30. If 30 exceeds the age as much as 10 is less, 30-10-20 must be twice thle excess, therefore the age is 30-10 —20 years. Ans. (39) T'h difference of 4 cts. in price maknes 30 cts. dif[erence in aniaount; hence there were 3-~-7I dozenl; 10 cts.- —:! cs. loss per dozen; 6-+1l —,7 cts. cost per dozen. (4:0) A difference of 3 cts. in price mnakes -1 50 cents difference in amnount; hence there were $].50- -3=50 lb. Ans. (41) 12 hoys-=4 women-2 men. If 12 boys or 2 mien can do the wolrk in 6 days, it would require 6 tim-es 2 men to do it in I day. Aes. 12 men. (11) 2 lb. iwould be worth 03, -1 —6 —=10 cts., or 5 Ets. a lb.; l-b11). is worth l 2 cts., and foxr 1t we must give 20 lb. (.38) If of the gainu-1- of the selling price, will be -X and the gain - of the sellinOg price hence the cost will be 1-2 —- of i:he selling price. 3 ards at 4 cost $ 15 which is - of $25. 42s. (44) The first package cost 24 3 —21 cts., and contained 21 -:-7-3 lb.; the second contained twice 3=6 lb., and cost 6X8=48 cts., atud must be sold for 43+5 =53 cts. SECTION P2. —PE rCENNTAGE. NoTE.Tile greater part of the questions in this section are of such easy solution, that it is deemed unnecessary to present either their answers, or tle mnethod of solving them. L E SON II. (2) Anis. 25. (3) As. 20. (4) A.s. 50. (5) Ans. 60. (6) Anas. 30. PERCENTAGE. 211 (7) If he cleared ~ he must have sold them for a of the cost. If 6 cents was ~, I is 1 cent, and 5, or the first cost, was 5 cents. Hence, in selling them for 8 cents, he cleared 3 cents on each; 1=60 %. (8) 6 yards for 3 is $, a yard, and 5 yards for $4 is $4 per yardll. 4'1 is3 1 i, is -15. A is -'Io, or 60 %. (9) If he lost 1, 8 cents are 4 of the first cost, and the first cost was 10 cents. 3 for 25 cents is $8 cents a piece. 10 cents-8 — cents is 12 cents. 12 cents is - of 10 cents; therefore by selling them at 3 for 25 cents, he would have lost A of the cost, or 16]- per cent. Again: 2 for 25 cents is 121 cents a piece, a gain of 21 cents. 2- is 4 of 10; hence by selling them at 2 for 25 cents, he would have gained 4 of the first cost, or 25 o. (10) 2 for 3 cents is 1, cents a piece; 3 for 2 cents is - cent a piece. 1ie- is ~ loss on each. 14 is 9, therefore 6 is g of 1- cents, or the first cost; hence he lost 7 of the first cost, or 55 5f. (12) Aces. 20. (13) Ans. 10. (14) A.es. 40. (15) Ans. 50. (16) Ans. 12. (17) Ans. 3. (18) Ans. 25. (19) 1-=2=40 %. At! the loss is l_=121 Yo. At 9 the loss is -1=10 %. At 47 the loss is =15 %. PROFIT AND Loss.-LESSON III. (1) 20 o — 20 Ans. 75 =15 -. Anls. (2) 100 ~%4-00=1), that is, the cost and gain are equal. 150 o-11- o1- or 4; the gain is 1) times the cost. (3) 5 25 Ans. 35 o — UT —7. Ans. (4) The whole, or 1=-I-. Ans. 100 %. = —-- Ans. 300 %. (6) If: of the gain equals ~ of the cost, 4 equals 3, and the whole gain equals 9 — = —13%-036,%. (7) If or, and, or the whole of the gain -= or 3 of the cost. - o=75o A ns. ~~~4~~~~~"~~ Ans.~""I"""~LII"-~~~P r~~ I 212 KEY TO RAY'S SECOND BOOK. (8) 25 %=1To=4 —; hence the watch sold for` of its cost; if $10 is -, 4, or the first cost, is $8. When he sold it for $12, he gained $4, and 4=1_ -u=50 So. Ans. (9) 124 whei, when is lost, the muslin is sold for a- of its cost. Hence 7 cents is s of the cost, and the first cost was 8 cents. By selling it at 6 cents per yard he lost 2 cents. — 4==25 So. Ans. (10) 16B- -—. If 6 was lost the horse was sold for - of his cost; if $35 are 5, - is $7, and a are $42, the cost. $63-42=-$21, the gain; - - o -5 % Ans. (11) 20 2%0i —- —; hence $18 are A of the first cost, or $15. 10 =, and of $15 is $1,. 15-1 —=$13., watch sold for. He lost $18 —13A=$44 Ans. (12) 20 o % —o —4; hence $60 are ~ of the first cost, and y of $60 is $10, which is A, or A's gain. B lost 20 %, or 5 of what it cost him; 5 of $60 is $12, B's loss. Hence B lost $2 more than A gained. (13) Since 25 % is 4, he sold one watch for - of its cost, and the other for 3 of its cost; hence $30 are 5 of the first cost of the first watch, and 4, or the first cost, is $24, and he gained on the first watch $6. If $30 is of the first cost of the second watch, its first cost must have been $40, and he lost $10 by the sale; therefore, he lost $10 —$6=$4 Ans. (14) 4 apples for 3 cents is -3 of a cent each. 50 % is fz; hence 4 is.3 of the first cost, and the first cost is 4 cent. 5 apples for 4 cents is 4 of a cent each; 4-4= 3, the gain on each apple. 1=-%, hence 3 is - of the first cost; -}=60 ~o, the gain Ya. (15) 5 lemons for 4 cents is 4 of a cent each. 20 o is -, and he sold the lemons at 4 of their cost. If 4 of a cent is 4 of the first cost, of a cent is I of the cost, and the cost was 1 cent each. 6 lemons for 5 cents is - of a cent each; 1 —=- -163 %, the % of loss. (16) 2 of 10 % of 60 —- of - t of 60 —4; 4 is Z of 8; 8 is 5. of 40; r=20% A.ns. PERCENTAGE. 213 (17) ~ of 3 of 50 % of 120=-93 of I of 120=18; 18 is 10 less than 28; 28 is 20 %, or ~ of 5 times 28, which is 140. Ans. LESSON IV. (3) Ans. 60 cts. (4) Ans. $2. (5) A2s. $4.80 (6) Ans. $6. (7) Ans. $8. (8) Ans. $9. (9) Ans. $8.40 (10) Ans. $9. LESSON V. (2) Ans. $1. (3) Ans. $2.80 (4) Ans. $2.40 (5) tins. $4.50 (6) Ans. $3.25 (7) Ans. $5.162 (8) Ans. $3.53 (9) Ans. $4.80 (10) Ans. $19.20 (11) Ass. $8.45 (12) Ans. $4.24 (13) Anls. $3.20 (14) Ans. $6.75 (15) Ans. $3.80 (16) Ans. $28. (17) Ans. $44. (18) Ans. $68.20 (19) Ans. $32.80 (20) Ans. $56.80 (21) Ans. $99.12 LESSON VI. (2) Ans. $50. (3) Ans. $60. (4) Ans. $75. (5) Ans. $140. (6) Ans. $240. (7) Ans. $350. (8) Ans. $4000. LESSON VII. (2) Ails. $200. (3) Ans. $500. (4) Ass. $250. (5) Ans. $300. (6) Ans. $25. (7) Ans. $750. LESSON VIII. (2) Ass. 4 years. (3) Ans. 21 years. (4) Aiss. 2-2 years. (5) Ans. 33 years. (6) Asns. 63 years. (8) Ans. 50, 33', 20, 162,, 4, 12, 10, and 8 years respectively. (9) Ans. 40 years. (10) Ans. 25 years, and 20 years. 214 KEY TO RAY'S SECOND BOOK. LESSON IX. (2) Ans. 8. (3) Ans. 5. (4) Ans. 7. (5) Ans. 8. (6) Ans. 7. (7) Ans. 5. (8) Ans. 6. (10) Ans. 8-1. (11) Ans. 10. (12) Ans. 12~, 20, 25, and 50, respectively. LESSON X. (2) Ans. $400. (3) Ans. $25. (4) Ans. $500. (5) Ans. $300. (6) The amount of $1 for 4 years, at 6 %o, is $1.24; and the required principal will be as many dollars as $1.24 is contained times in $496: $496. $1.24 —=400. Ans. $400. (7) $1, at 5 s, for 4 years, will amount to $1.20; hence the present worth of each $1.20 of the given sum, is $1; therefore, the present worth of $24 will be as many times $1, as $1.20 is contained times in $24: $24__ $1.20=20. Ans. $20. (8) The amount of $1 for 5 years, at 6 %, is $1.30, and the present worth of each $1.30 of the amount is $1; therefore, the present worth of $65 will be as many times $1, as $1.30 is contained times in $65: $65. —$1.30=50; hence the present worth is $50. By subtracting the present worth from the debt, we obtain the discount. $65-$50$o=15. (9) The amount of $1. for 5 years, at 5 %, is $1.25; hence the present worth of each 81.25 of the given sum, is $1. And the present worth of $55, will be as many dollars as $1.25 is contained times in $.55: $.55-$1.25 =44; therefore, $44 is the present worth. $55-$44= $11, the discount. (10) 6 years 8 months:-6- yr. If the interest of $1 for 1 year is 6-cents, for 6 years it will be 40 cts. $X; hence the amount of $1 for 62 years, at 6 S%, is $15-7-. Therefore the present worth of $.77 will be as many dollars as 5 is contained times in 77, which is 55. Hence the present worth is $55, and the gain by paying it now $77 —55 —$22. e, -H e, MENTAL FRACTIONS. 215 LESSON XI. MISCELLANEOUS EXAMPLES IN INTEIREST. (1) 6X4(-=25; i —o =5. Ans. (2) 5X5=25; _25. 1+-=4, and l is 1- of 5. A. (3), of — 1F0; hence the interest of 1 year is l1e of the principal; -1 -0 Ans. 10. (4) 2 years 6 months —2=- years. 5- of and 2 times.j ~U, 4-Z, and 2 tim es -1 vgg the part the interest is of the principal for 1 year - T-0 —TO%10 o. Axns. (5) 10 To= ('T j-o; hence for 1 year the3 interest is l-' of the principal, and at the same rate o, the time for which the interest is 3 of the principal, will be as many years as Tl6 is contained times in 3-. -f in 3 is contained 6 times. Ans. 6 years. (6) If 3 times the yearly interest is equal to.9 of the principal, once the yearly interest is equal to s- of the principal. 7-1T-l1o 2 %. A'. (7) - of the interest for 2 years is thle sa.me as 20 Of the interest for 1 yea r. If 2 of the interest for 1 year equals - of the principal, of the interest for 1 year equals j of the principal, and s, or the interest for I yearl equa ls 7 of the princi1' a —% b%-.40%. Ans. (8), for 3 years is the soae as for I yer. f s of tle intcerest for 1 vear equals 9 of the principal,. equals - o 3~ ald g Cclqu'rlS -, fOr T-6o of the principat. 1y' equals 6 /. Ais. (9) 1 year 4 months= 6 mont1hs, a hiclh Is,twice 8 months; heince the inte-rest for I yea- 4 moanthfs wrvill be -, of the principa; of $G'00-m$6.' n". (10) 1 year 4 mon. are I or ir If rhe interest of yr is f of the prrincipal, for year it is I{ of — 1T —O, and for 1 year it is T9% of the principal; that is, the rate YOUB- ____ _ s --- - 216 KEY TO RAY'S SECOND BOOK. is 9 %. Hence for 1 year the interest is $9; for 1 month the interest is $4, and for 8 mon. $6; for 18 days the interest is 1, or 3 of $ or 45 cents. Therefore, the whole interest is $9+-$6. 9=$1529, or $15.45 (11) 4 years at 10 % is the same as 1 year at 40 s, and 1 year at 40 % is the same as 8 years at 5 So. Ans. 8 years. (12) 3~ years are,o years. $40 for 1,o years is $4 for'- of a year, or $12 a year. 5 15 is, hence $12 are I of the principal, which must therefore be 12 times $20= $240. Hence B's money, and twice B's money, or 3 times B's money is $240; therefore B's money is $80, and A's twice $80, or $160. (13) 1: years are -o years. $49 for. years are $7 for 5 year; or $35 a year." 7 %o is TiP, hence $35 are Tlo of the principal, and $5 are T-i, and _1 arec $500, the sum of A's and B's money. Since 2 times A's equals 3 times B's, 1 time A's equals 1 B's; therefore B's money and 1, times B's money, that is, are equal to 500; hence a B's money is - of $500, or $100, therefore B's money is $200, and A's $300. (14) 2 yrs. 3 mos.-=2=9= yrs. $18 for - yrs.=$2 for - year, or $8 for 1 year. 4 o is T4 —; hence $8_=I of the principal, and -Sn, or the whole principal, is $200, which is -I of A's, and V of B's money. Since ~ A's 2 B's money, 4 A's=-, B's, and a A's-B's nmoney; hence B's money and I B's, or. B's=$200; ~ of $200 — 66', which is, B's, and $1331-B's money. Since a A's —-- B's money, A's money==-4 B's, and -a of $133~ -$177', A's money. APPENDIX. TEST EXAMPLES FOR THE SLATE OR BLACKBOARD. SUGGESTIONs. —Almost every teacher, in the course of his instructions in arithmetic, desires to exercise his pupils on the slate or blackboard with examples not contained in the text-book. The following are intended for this purpose. A part of the class, or the whole, if there is room, should be called up together; the examples should be given out slowly and distinctly, all the pupils working the same example at the same time. When there is not room for all the members of the class to work on the blackboard at the same time, a part may work on their slates at the recitation seat. Where the Mental Arithmetic, (Ray's Second Book,) does not contain exercises that may be used in illustrating the rules in the Third Book, there will be found in the following pages, mental exercises preceding the examples intended for the blackboard. AD DI TI ON 1OF STIMPLE lfUMBTElRS. 1. 471 + 724+ 360+117-= how many? Ans. 1672. 2. 258 + 233+ 421 —609= how many? Ans. 1521. 3. 4082+5412+ 340+617- how many? Ans. 10451. 4. 7324+-2465+ 416+- 38= how many? Ans. 10243. 5. 71654. 404+1253+986 —- how many? Ans. 9808. 6. 836 -+1245-+4314+ 980+4207+ 836=12418. 7. 1253+3178+6421-+6893+ 702+1458=19905. 8. 7163+-2461+ 716+3072+ 409+ 516-14337. 9. 1.242+ 563+8000+ 906+1780+- 438=12929. 10. 7362+ 486+3547+1098+ 362+1346=14201. ('217) 218 TEST EXAMPLES. 11. 463 + 246+ 36+ 45+ 37+134+629+89= 1679. 12. 4167+3254+6323+634+705+-248+- 47+89=15467. 13. 78167+8054+11167+9084+ 550+1809+7811+9750 =126392. 14. 2881 +92375+ 648+2437+ 717-+ 2529+2160+360 =104107. 15. 26341+11426+30637+9817+8784+12060+356+1217 =100638. 16. What is the population of the earth, supposing Europe to contain 230000000, Asia 390000000, Africa 60000000, North and South America 40000000, and Oceanica 20000000? Ans. 740000000. S UBTRACTIO I. SUGGESTIONs.-The easiest method of exer- 34682 cising pupils at the blackboard in simple 346820 a cipher subtraction, is to direct them to write down 34682 [added. any number, such as 34682, then to annex a cipher to it, as in the margin, and subtract 312138 from this continually the number first set 34682 down; when, after subtracting nine times, if 277456 the operation has been correctly performed, 34682 the remainder will be the same as the origi- 242774 nal number. The reason of this is evident; 34682 adding the cipher is the same as multiplying - by 10, and the operation really consists in 208092 nothing more than taking 9 times the number 34682 from 10 times. In choosing the first number, 173410 the right hand character should not be a 34682 cipher, for the reason that it gives less variety 138728 to the work. 34682 104046 34682 69364 34682 34682 MULTIPLICATION OF SIMPLE NUMBERS. 219 ULTIPLI CATIONl0. NoTE.-We have given an extensive list of examples in multiplication and division. Exercises in these rules necessarily embrace the operations of addition and subtraction, and are, therefore, well suited to render the pupil familiar with all the elementary rules. Multiplicands. Multipliers. Products. 1. 379 75......... 28425 2. 789 57...... e 44973 3. 2728 71.......... 193688 4. 3570 86.......... o 307020 5. 3259 43.o. o 140137 6. 5630 99...... 557370 7. 426 111........... 47286 8. 375 123........ 46125 9. 572 137 a...... o 78364 10. 729 236.......... 172044 11. 973 297.......... 288981 12. 1334 333......... 444222 13. 1779 347.......... 617313 14. 1986 436.......... 865896 15. 2327 572..... 1331044 16. 3377 635......... 2144395 17. 4323 736....... 3181728 18. 5745 892........ 5124540 19. 7963 997..... 7939111 20. 8381. 819.......... 6864039 21. 2764 785.......... 2169740 22. 6392 527.......... 3368584 23. 3231 492.......... 1589652 24. 6798 987.......... 6709626 25. 324 1476......... 478224 26. 135 1246..., 168210 220 TEST EXAMPLES. Multiplicands. Multipliers. Products. 27. 365 1646 o..... 600790 28. 829 6874..... 5698546 29. 1634 3457..... 5648738 30. 6516 3258..... 21229128 31. 1754 2369..... 4155226 32. 1437 2596..... 3730452 33. 3587 4121..... 14782027 34. 5036 2139..... 10772004 35. 4007 3271.... 13106897 36. 4786 3497.... 16736642 37. 87603 9865.... 864203595 38. 4379 4265.... 18676435 39. 31624 7138...... 225732112 40. 83457 6835..... 570428595 41. 683794 5827...... 3984467638 42. 14278 3236..... 46203608 43. 50361 5267..... 265251387 44. 18014 13184..... 237496576 45. 23930 11491..... 274979630 46. 683794 17481.... 11953402914 47. 54042 39552...2137469184 48. 8735628 35784.. 312595712352 3MULTIPLICATION BY FACTORS. (See Article 33.) 1. 2358 24......... 56592 2. 2596 27..... 70092 3. 4531 28...... 126868 4. 4765 36..... 171540 5. 3589 48..... 172272 6. 5472 64.... 350208 7. 7359 72..... 529848 8. 84622 96.. 8123712 9. 52389 63... 3300507 10. 432234 54........ 23340636 11. 79797 81........6463557 ~~ — ----- ------ ~ -- ----- - --------— ~~~~~~~~~~~~~~~~~~ MULTIPLICATION OF SIMPLE NUMBERS. 221 Multiplicands. Multipliers. Products. 12. 98979 63.......... 6235677 13. 1234 44........ 54296 14. 4729 77.......... 364133 15. 13579 108...... 1466532 16. 24682 120.......... 2961840 17. 65423 132..... 8635836 18. 875643 144......... 126092592 IMULTIPLICATION WHEN THERE ARE CIPHERS IN THE MULTIPLIER. (See Articles 34, 35.) 1. 483 200.......... 96600 2. 4728 40.......... 189120 3. 6823 80.......... 545840 4. 158 350..... 55300 5. 7428 520..... 3862560 6. 9436 870.......... 8209320 7. 369 800. 295200 8. 6230 50..... 311500 9. 8370 240. 2008800 10. 46000 340........ 15640000 11. 5763 3600.2.........0746800 12. 5700 6800........ 38760000 13. 7430 4520..... 33583600 14. 9040 8320.......... 75212800 15. 9651 304........... 2933904 16. 12318 7004........ 86275272 17. 57906 1001......... 57963906 18. 18001 2600.......... 46802600 1 9. 8602 1030..... 8860060 20. 9100 3007.. 27363700 21. 90042 9009......... 811188378 22. 70302 8030..... 564525060 23. 904000 10200..... 9220800000 24. 80307 460001... o. 36941300307 ~ —-------------------------— 0 222 TEST EXAMPLES. D IVI SIO TN. I. SHORT DIVISION. (See Article 45.) Divisors. Dividends. Quotients. Rem. 1. 3 3636......... 1212 2. 2 4264........ 2132 3. 4 5372 e s...... 1343 4. 3 7692...... 2564 5. 4 8364....... 2091 6. 6 12798......2133 7. 7 23694....... o 3384,, 6 8. 8 35826....... 4478... 2 9. 9 39875....... 4430... 5 10. 10 42736...... 4273... 6 11. 11 54387.... 4944.. 3 12. 12 98764....... 8230.. 4 13. 9 52345........ 5816.. 1 14. 8 763279........ 95409... 7 15. 7 53262....... 7608.. 6 16. 9 30205...... 3356... 1 17. 7 1234567....... 176366... 5 18. 9 4326721..... 480746... 7 19. 12 7295432...... 607952... 8 II, LONG DIVISION. (See Article 46.) 1. 21 4524.....,.. 215...9 2. 42 9870..... 235 3. 58 87654..... 1511.. 16 4. 61 28609....... 469 5. 29 153598..... 5296.. 14 6. 36 4104..... 114 7. 23 3496...... 152 S. 88 15362..174.. 50 9. 57 44973.... 789 10. 86 307020..... 3570 11. 23 461.25..... 2005.. 10 12. 75 8425..... 379 13. 71 661419........ 9315. 54 @-Gwrausrns~~asa-~l~av.-ru,~~~~,~u~a SHORT DIVISION. 223 Divisors. Dividends. Quotients. Rem. 14. 53 982700... 18541.. 27 15. 58 4327865...... 74618.. 21 16. 91 764231..... 8398.. 13 17. 92 333222..... 3621.. 90 18. 93 426891..... 4590.. 21 19. 94 5432687.... 57794.. 51 20. 85 444333... 5227. 38 21. 87 71778543..... 825040 63 22. 124 46215...... 372.. 87 23. 136 24374...... 179.. 30 24. 137 78364....... 572 25. 297 288981..... 973 26. 347 617353...... 1779 40 27. 436 865896......1986 28. 472 251104.. 532 29. 223 52135.. 233.. 176 30. 572 1331044...... 2327 31. 635 2144395..... 3377 32. 746 549056...... 736 33. 7'76 297208...... 383 34. 725 5828921.... 8039.. 646 35. 806 328848...... 408 36. 842 70173290..... 83341.. 168 37. 846 353628...... 418 38. 8 76 379308...... 433 39. 855 38923458... 45524.. 438 40. 906 405888... 448 41. 967 48325859..... 49975.. 34 42. 975 8971021304.. 9201047.. 479 43. 1111 142372.. o. 128.. 164 44. 1212 2346521... 1936.. 89 45. 1432 347854... 242.. 1310 46. 2312 453276... 196.. 124 47. 3587 5497630.. o 1532.. 2346 49. 1563 4917361... 3136. 113 j50. 1774 5967854. 3364 118 224 TEST EXAMPLES. Divisors. Dividends. Quotients. Rem. 51. 2140 4560700........ 2131.. 360 52. 4625 3456745....... 747. 1870 53. 2507 9736205....... 3883. 1524 54.. 6425 23470525....... 3653 55. 7898 3875642....... 490. 5622 56. 7569 57289761....... 7569 57. 8832 45785088....... 5184 58. 5476 29987000...... 5476.. 424 59. 7396 54701000....... 7396.. 184 60. 6787 86167752....... 12696 61. 6315 103415000....... 16376.. 560 62. 7171 142308495....... 19845 63. 23785 237854...... 10.. 4 64. 43587 41258724....... 946. 25422 65. 442211 66335522....... 150. 3872 66. 1.23456 87654321....... 710.. 561 67. 538723 1.00002002....... 185 338247 68. 400203 96700054....... 241 251131 69. 639406 12707070....... 19 558356 70. 4670012 513287321....... 109 4256013 71. 32378 4012301...... 123. 29807 72. 56000 5600001....... 100... 1 73. 111111 2222222...... 20... 2 74. 333771 43723521...... 130 333291 75. 21432 23786543....... 1109 18455 76. 333 999999999. 3003003 77. 414141 462462462..... 11.16 281106 78. 47217 37854321....... 801 33504 79. 785433 785432179..... 99 784612 80. 153269 175432798....... 1144 93062 81. 182766 18276543....... 99 182709 82. 297208 88332595264...... 297208 III. DIVISION BY COGMPOSITE NUMBERS. (See Article 47.) 1. 24 56592....... 2358 2. 1 ]5 735379..... 49025... 4 M 5G -_f. < DIVISION OF COMPOUND NUMBERS. 225 Divisors. Dividends. Quotients. Rem. 3. 36 171540.... 4765 4. 48 172272..... 3589 5. 72 529848... 7359 6. 63 3300507... 52389 7. 56 249738....... 4459....34 8. 54 23340636.... 432234 9. 81 6463557... 79797 10. 72 84630986..... 1175430.. 26 11. 81 34210390..... 422350... 40 12. 16 3254632..... 203414... 8 13. 49 78787878... 1607915.. 43 14. 108 871992.... 8074 15. 121 5200769... 42981... 68 16. 108 991254......9178... 30 17. 132 491832.... 3726 18, 120 425717.... 3547... 77 19. 144 729814...... 5068... 22 20. 180 29070..... 161... 90 IV, WHEN THERE ARE ONE OR MORE CIPHERS ON THE RIGHT OF THE DIVISOR. (See Articles 48, 49.) 1, 20 32785..... 1639.. 5 2. 30 437856..... 14595... 6 3. 40 812569... 20314... 9 4. 50 437694... 8753.. 44 5 60 328320....5472 6. 70 830547..... 11864...67 7. 80 545840....6823 8. 90 765932.....8510... 32 9. 100 457834.... 4578... 34 10. 110 363587.... 3305... 3 11. 120 7395173..... 61626... 53 12. 70 67216540..... 960236... 20 13. 450 3613500.... e. 8030 14. 2600 413798..... 159.. 398 15. 3500 4448623.... 1271.. 123 Key.-15 226 TEST EXAMPLES. COQIPOUND NUMiVBERS. I. REDUCTION FRO A HIGIIHER TO A LOWER ORDER. 1. Reduce 3 bu. 5 qt. to pints. Ans. 202 pt 2. Reduce 17 bu. 3 pk. to pints. Ans. 1136 pt. 3. In 7 lb. 11 oz. of gold, how many gr.? Ans. 45600 gr. 4. In 5 lb. 17 pwt. of silver, how many grains? Ans. 29208 gr. 5. At 3 cents a grain, how much will 1ib 5 ~ of musk cost? Ans. $244.80 6. Reduce 17 lb 7 3 1 gr. to grains. Ans. 98341 gr. 7. Reduce 5 cwt. 3 qr. 12 lb. to ounces. Ans. 9392 oz. 8. Reduce 4 T. 1 cwt. 3 qr. to pounds. Ans. 8175 lb. 9. At 5 cents a pound, what will 5 cwt. 1 qr. 15 lb. of cheese cost? Ans. $27.00 10. At I cent 5 mills a pound, what will a load of hay cost, weighing 15 cwt. 2 qr. 20 lb.? Ans. $23.55 11. In 3 mi. 8 rd. how many rods? Ans. 968 rd. 12. In 17 yd. 2 ft. 9 in. how many inches? Ans. 645 in. 13. In 1 mile there are 5280 feet; how many times will a wheel 17 ft. 6 in. in circumference turn round in running 14 miles? A2ns. 4224 times. 14. H-low many times will a steamboat, 136 feet 3 inches long, run her length, in going from Pittsburgh to the mouth of the Ohio, the distance being 981 miles? Ans. 38016 times. 15. In 3 sq. mi. how many perches? Anzs. 307200. 16. In 7 sq. yd. how many sq. inches? Ans. 9072. 17. In 5 cubic yards, how many cubic inches? Ans 233280. 18. In 17 cords of wood, how many solid feet? Ans. 2176. 19. Hlow many nails in 7 pieces of cloth, each containing 5 yd. 2 qr. 1 na.? Ans. 623. 20. Reduce 15 E. Fr. 5 qr. 3 na. to nails. Ans. 383. 21. In 2 hhd. of wine, how many gills? Ans. 4032. 22. How many times can 3 gills be filled out of 31 gal. 2 qt. of cider? Ans. 336. 23. How many times can a bucket of 3 gal. 2 quarts, be filled out of a cistern containing 55 hhd.? Ans. 990 times. 24. From 5 vessels, each containing 3 hhd. 12 gal. of cider, how many kegs can be filled, each containing 2 gal. 3 qt. I pt.? Ans. 349 and 13 pints left. 25. From a barrel of beer, how many bottles can be filled, each capable of holding 1 pt. 2 gi.? Ans. 192. COMPOUND NUMBERS, 227 26. From 14 hhd. of beer, how many kegs can be filled, each capable of holding 5 gal. 1 qt.? Ans. 144. 27. WVhen a person is 36 years old, how many minutes has he lived, reckoning a year 365 days 6 hours? Ans. 18934560. 28. How many seconds in a lunar synodical month, which contains 29 da. 12 hr. 44 min. 3 sec.? Ans. 2551443. 29. How many seconds in a lunar siderial month, which contains 27 da. 7 hr. 43 min. 11 sec.? Ans. 2360591. 30. How many seconds in 18 centuries, the solar year being 365 da. 5 hr. 48 min. 51 sec.? Ans. 56802475800. II. REDUCTION FROM A LOWER TO A HIGHER ORDER. 1. In 240 pints, how many bushels? Ans. 3 bu. 3 pk. 2. How many bushels are there in 10 bags, each containing 195 pints? Ans. 30 bu. 1 pk. 7 qt. 3. In 639 pt., how many bu.? Ans. 9 bu. 3 pk. 7 qt. 1 pt. 4. In 12960 grains of gold, how many ounces? Ans. 27. 5. If a piece of gold weighing 341304 gr. be divided into 8 equal ingots, what will be the weight of each? Ans. 7 lb. 4 oz. 17 pwt. 15 gr. 6. ltow many ounces in 916 pills, each weighing 5 grains? Ans. 9 3 4 3 1 D. 7. In 14769 ounces, how many cwt. Ans. 9 cwt. 23 lb. 1 oz. 8. What is the weight of 373 bags of salt, each weighing 85 pounds? Ans. 317 cwt. 5 lb. 9. In 647 in., how many yards? lAns. 17 yd. 2 ft. 11 in. 10. In 1745 rods, how many miles? Ans. 5 mi. 3 fur. 25 rd. 11. In 4459 perches, how many acres? Ans. 27 A. 3 R. 19 P. 12. In 12959 square inches, how many square yards? Ans. 9 sq. yd. 8 sq. ft. 143 sq. in. 13. In 1664 cubic feet, how many cords? Ans. 13. 14. Reduce 5966781 cubic inches to tuns of hewn timber. Ans. 69 tuns 2 cubic feet 1725 cubic inches. 15. In 432 nails, how many yards? Ans. 27. 16. In 75 English ells, how many yards? Ans. 93 yd. 3 qr. 17. In 17 pieces, each containing 27 ells Flemish, how many yards? Ans. 344 yd. 1 qr. 18. In 9072 pints of wine, how many pipes? Ans. 9. 19. In 10080 gills of wine, how many hhds.? Ans. 5. 20. In 13248 pints of beer, how many barrels? Ans. 46. 21. In 72015 hr., how many weeks? Ans. 428 wk. 4 da. 15 hr. 228 TEST EXAMPLES. 22. Reduce 900610 seconds to days. Ans. 10 da. 10 hr. 10 min. 10 sec. 23. How many days from the Sth of May till the 22d of July? Solution. 31-8=23, the number of days left in May; then 23+30+22-75 days Ans. 24. From August 10th, 1844, till April 26th, 1845, how many days? Ans. 259. 25. How many days from January 10th, to July 30th, 1847? Ans. 201. 26. How many days from January 20th, to April 25th, 1848? Ans. 96. 27. If the 8th of August be on Saturday, on what day of the week will the first of November be? EXPLANATION.-The number of days between these dates is found to be 85, which, being divided by 7, becomes 12 weeks 1 day; counting 1 day, therefore, after Saturday, we find that the first of November must be on Sunday. 28. If a leap year begins on Friday, on what day will the 17th of June, the anniversary of the battle of Bunker Hill, happen? Ans. Friday. 29. If a common year begins on Wde'nesday, on what day will the 4th of July happen? Ans. Friday. 30. In the year 1830, the 4th of July happened on Sunday; on what day did the year commence? Ans. Friday. NoTE. —The eight preceding questions do not properly belong to reduction; they are inserted on account of their utility. ADDITION. 1. A man bought 3 bu. 3 pk. of wheat at one time, 5 bu. 1 qt. at another, 2 bu. 1 pt. at another, land at another 9 bu. 6 qt. 1 pt.; how many bushels did he buy in all? Ans. 20. 2. A jeweler has 4 pieces of gold; the 1st weighs 7 oz. 11 pwe. 3 gr., the 2d 5 oz. 9 pwt., the 3d 10 oz. 15 gr., the 4th 5 oz. 19 pwt. 6 gr.; how much do all weigh? Ans. 2 lb. 5 oz. 3. An apothecary made a composition of 3 ingredients; the first weighed 73 23 19 10 gr., the second 1lb 23 2D, and the third 2 lb 4. 2 3 2 D) 18 gr.; what did the whole mixture weigh? Ans. 4 tb 8 gr. d" - XI —-~I...-~`. ---------- --------- ~W ADDITION OF COMPOUND NUMBERS. 229 4. A bre*er bought 5 bags of hops: the first weighed 2 cwt. 1 qr. 10 lb., the second 3 cwt. 20 lb., the third 2 cwt. 2 qr., the fourth 3 cwt. 1 qr. 12 lb., the fifth 2 cwt. 3 qr. 11 lb.;' what was the weight of all? Ans. 14 cwt. 1 qr. 3 lb. 5. There are 3 pieces of road; the first is 3 mi. 1 fur. 18 rd. long, the second 2 mi. 3 fur. 9 rd., the third 5 mi. 3 fur. 20 rd.; required the length of all. iAns. 11 mi. 7 rd. 6. Add together 5 A. 2 R. 15 P., 4 A. 3 R. 25 P., 6 A. 1 R. 19 P., 8A. 1 R., and 3 A. 15 P. Ans. 28A. 34 P. 7. Add together 15 C. 40 cu. ft., 19 C. 85 cu. ft., 20 C. 34 cu. ft. 12 C. 94 cu. fi. Ans. 67 C. 125 cu. ft. 8. Add together 4 gal. 2 qt. 1 pt. 3 gi., 3 gal. 1 qt. 1 gi., 5 gal. 1 qt. 1 pt. 1 gi., and 3 gal. 3 qt. 3 gi. An's. 17 gal. 1 qt. 9. Add together 1 da. 5 hr. 10 min. 15 sec., 3 da. 7 hr. 40 min., 5 da. 12 hr. 35 sec., and 9 da. 9 min. 10 sec. Ans. 19 da. 1 hr. 10. Add together 6~ 3' 12//, 40 7' 15", 8~ 4' 40//, and 1~ 45' 13". Ans. 20~ 20". SUB3 TRACTION. 1. From 12 bu. of wheat, there was taken 2 bu. 7 qt. and 1 pt.: how much remained? Ans. 9 bu. 3 pk. 1 qt. 2. From 2 lb. 5 oz. of silver, there was taken 2 lb. 10 pwt. 5 gr.; how much remained? iAns. 4 oz. 9 pwt. 19 gr. 3. From 3 lb 5 3, take 2 lb 7 2 15 gr. Ans. 5 X 4 3 2 D 5 gr. 4. From 9 lb. 13 oz. 14 dr., take 3 lb. 13 oz. 15 dr. Ans. 5 lb. 15 oz. 15 dr. 5. From 50 miles, take 5 miles 1 rod. Ans. 44 mi. 7 fur. 39 rd. 6. A lot of land contains 11 A. 3 R.; how much will remain after selling 3 A. 2 R. 20 P.? Ans. 8 A. 20 P. 7. Bought 8 cords of wood; after using 5 cords 87 cubic feet, how much will remain? Ans. 2 C. 41 cu. ft. 8. If from 11 yd. 2 qr. of cloth, there be cut 5 yd. 3 qr. 2 na., how much will be left? Anls. 5 yd. 2 qr. 2 na. 9. If from 31 gal. 2 qt. of cider, you draw 17 gal. 2 qt. 1 pt., how much will remain? Ans. 13 gal. 3 qt. 1 pt. 10. There are two brothers, whose ages are 11 yr. 2 mon. 20 da., and 9 yr. 5 mon. and 15 da.; what is the difference between them? Ans. 1 yr. 9 mon. 5 da. 230 TEST EXAMPLES. 11. The city of Washington is 5hr. 8 min. 6sec. of time west of Greenwich, and Cincinnati is 5 hr. 37 rmin. 36 sec. west of the same meridian; what is the difference of time between Washington and Cincinnati? Ans. 29 min. 30 sec. 12. The latitude of Philadelphia is 39~ 56' 58"M, and that of Greenwich 51~ 28' 39//; how much nearer to the equator is the former than the latter? Ans. 110 31' 41//. MIVULTI PLIC AT I O E. 1. In 5 bags of wheat, each containing 2 bu. 3 pk. 4 qt., how many bushels? Ans. 14 bu. 1 pk. 4 qt. 2. What is the weight of 9 silver spoons, each weighing 1 oz. 9 pwt. 13 gr.? Ans. 1 lb. 1 oz. 5 pwt. 21 gr. 3. What is the weight of 3 packages of drugs, each weighing 2 t 11,~ 23 2D 7 gr.? Ans. 8it- 10~ 1D I gr. 4. What is the aggregate weight of 8 bars of lead, each weighing 6 lb. 11 oz. 2 dr.? Ans. 53 lb. 9 oz. 5. What is the weight of 5 tierces of rice, each weighing 4 cwt. 3 qr. 20 lb.? Ans. 24 cwt. 3 qr. 6. If a man walk 20 mi. 3 fur. 15 rd. in one day, how far can he walk in 12 days? Ans. 245 mi. 20 rd. ~7. How many acres in 5 lots of ground, each containing 7 A. 2R. 8P.? Ans. 37 A. 3 R. 8. How many cords of wood, in 8 piles, each containing 2 C. 38 cu. ft. 432 cu. in.? Ans. 18 C. 50 cu. ft. 9. Ilow many yards of cloth, in 7 pieces, each measuring 5 yd. 3 qr. 2 na.? Ans. 41 yd. 2 na. 10. If a man drink 3 gal. 2 qt. 3 gi. of wine in a month, how much would he drink in 12 months? Ans. 43 gal. 1 pt. 11. htow much beer in 11 casks, each containing 6 gal. 3 qt. I pt. Ans. 2 bl. 3 gal. 2 qt. 1 pt. 12. If it require a man 1 hr. 15 min. 25 sec. to walk 1 league how long to walk 12 leagues? Ans. 15 hr. 5 rain. 13. If a ship sail through 2~ 10'15// of latitude in one day, how far will she sail in 8 days? Ans. 17~ 22/. 14. The moon moves 13~ 10' 35/' in a solar day, and the sun 59' 8//; now, supposing the moon at any given time, to be 10~ in advance of the sun, how far will they be separated at the end of 5 days from that period? Ans. 70~ 57/ 15// GREATEST COMMON DIVISOR. 231 DIVISION. 1. Divide 12 bu. 3 pk. 4 ct. by 8. Ans. 1 bu. 2 pk. 3 ct. 1 pt. 2. Divide 31lb. 8 oz. of gold into 6 equal ingots; what is the weight of each? Ans. 5 lb. 3 oz. 6 pwt. 16 gr. 3. Eight equal packages of medicine weigh 16lb 7 3 2 3, what is the weight of each? Anzs. 2 Db 7 3 15 gr. 4. Divide 92 lb. 10 oz. 4 dr. by 7. Ans. 13 lb. 3 oz. 12 dr. 5. Divide 53 cwt. 10 lb. by 9. Ans. 5 cwt. 3 qr. 15 lb. 6. Divide 40 A. 3 R. 39 P. by 7. Ans. 5 A. 3 R. 17 P. 7. Divide 97 cu. ft. 1032 cu. in. by S. Ans. 12 cu. ft. 345 cu. in. 8. Cut a piece of cloth, containing 41 yd. 1 qr., into 12 equal parts, what the length of each? Ans. 3 yd. 1 qr. 3 na. 9. Divide 21 gal. 3 qt. by 6. Ans. 3 gal. 2 qt. 1 pt. 10. Divide 14 da. 5 hr. 6 min. by 12. _Ans, 1 da. 4 hr. 25 min. 30 sec. 11. Divide 810 30' by 8. Ans. 100 11' 15" GREATEST COMlMION DIVISOR. MENTAL EXERCISES. 1. Name the divisors of 6; of 8; of 10; of 12; of 14; of 16; of 18; of 20; of 22; of 24, etc. 2. What is the greatest common divisor of 2 and 4? 6 and 9? 10 and 15? 15 and IS? 2 and 6? 6 and 10? 12 and 15? 15 and 21? 3 and 4? 8 and 10? 7 and 14? 15 and 25? 3 and 6? 8 and 12? 14 and 21? 18 and 24? 4 and 6? 9 and 12? 21 and 28? 20 and 35? FOR THE SLATE OR BLACKBOARD. i. 45, 12.. Ans. 3. 9. 583, 1961. Anls. 53. 2. 120, 70.. Ans. 10. 10. 3285, 9125. Ans. 365. 3. 752, 156.. Ans. 4. 11. 1001, 5291. Ans. 143. 4. 280, 142.. Ans. 2. 12. 693, 763. Ans. 7. 5. 4590, 4410.. Ans. 90. 13. 799, 1071. Ans. 17. 6. 377, 551.. Ans. 29. 14. 966, 1495. Ans. 23. 7. 909,1717.. Ans. 101. 15. 3763, 4757. Ails. 71. 8. 481, 703.. Ans. 37. 16. 33522,41223. Ans. 453. 232 TEST EXAMPLES. LEAST COMIWMON MULTIPLE. IENTAL EXERCISES. 1. Name a multiple of 3; of 5; of 6; of 7; of 8; of 9, etc. 2. Name a comulon multiple of 2 and 3; of 3 and 4; of 4 and6; of 5and7; of 4and8; of 6and9; of 8and12. 3. What is the least common multiple of 2 and 4? of 3 and 4? of 6 and 8? of 6 and 9? of 6 and 10? of 8 and 12? of 9 and 12? of 10 and 12? of 8 and 10? of 10 and 15? FOR THE SLATE OR BLACKBOARD. 1. 4, 14, 20............ ns. 140. 2. 3, 4, 5, 6, 7, 8....... ns. 840. 3. 2, 3, 4, 5, 6, 7, 10, 14...... Ans. 420. 4. 3, 6, 9, 12, 15, 18..... Ans. 180. 5. 2, 4, 8, 16, 32, 64, 128, 256.... As. 256. 6. 2, 3, 4, 5, 6, 10, 15, 20..... A. Ins. 60. 7. 2, 3, 4, 7, 8, 16, 20, 25, 9... Ans. 25200. 8. 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 Ans. 27720. COMMON F 1lAC T IONS. I, REDUCING FRACTIONS TO THEIR LOWEST TERMS. 1. 75.... 25. 7. 3109. 7 2. 208_ 5* ~ 5 4 ~ 33 9 _ 192 I o 5 S170 235 3 7 6-e o o 3. 8 4 3 4 =. c o 97 q7 4. s2 55 10. 5l28 6=5 547. 18_ 3 -o -4' 1 1T4 4 z*'. - *- 1 1 4 e 10 3' II. IMIPROPER FRACTIONS TO FWHOLE OR MIXED NUMBERS. 1 2. IS 1. 6.'1'... 711 3. 45=-. 0. 8-. 23 - 27"g..... 23 4. T2.. 7.5... 37-4. 5. -48 M 8 F. T 166 2. 10. 47...2734R.....~~~~~~~ T ~ ~ ~.~ COMMON FRACTIONS. 233 ITI. COMPOUND FRACTIONS TO SIMPLE ONES. 1. of of =.. ~ ~~ *... 4I 2. 5 of 4 of -- 55 2. 9..... 3.. 3. 9% of - of t of........ - 4. 4 of of of of=.... 7. 4 of of of of...... 6. j of,j of -- 11..4... 7. 4 of of of 4 of ~ of of.. 4. 8..-f of 4 of 4 of -of - =.... 3-4. 4 4 IV. FRACTIONS OF DIFFERENT DENOMINATORS, TO EQUIYALENT FRACTIONS HAVING THE LEAST COMMON DENOMINATOR. 1. 1 1 1 1 A1s. 35 7q4 110 70 2' -8- 1' T T.. ^ An. TOO' 4 D, ol) T7O42. 5 7 1o 1 3. Ans. 30 2.8 33 2 -1':2' TO'4.. 42 427 4'1 48 3., ~ -~ —~4 1. Ans 84'105 Sn 86 3. -2 of 9 o * o Ans.........3. 6 9of7 -. T2 12f f 42_' T 4, o 7 -... S.. 10 200 140 2 5 7 38 60 5 0 6a CN O' DI F FR ET DNOHATOR S TO' ] OIA6, 7 ~ 1 7 Ans. 40 42 33 34 *' I1 T 4- 42 *4 *', 42', 48 37 8 13 36 42 32.27 7 T -., Ans. S I 019 1+ 3 1 6 900 6.4 3 30 0 6 V. ADDITION. a 7- +2 T 4 +q = -o e 3 -;l 3o 3, T-+,'~ 4... 4-, 4,..' 3. +4 +4 +5 7.1,!5 4+ +.n.2 6. 4 +_I 4+ +_4 _..4...... 3.. 4 Q5.; 3 + 7 1 4 8, 4+4,++ 7-4..... 9. i.... -,........2O 3 51 I 1 1_ 3 91 4 a 4, 4' ~%,, I+......... 3. -+7+1'+21-. ~ iUU - 9 -6, zJ~U,39 4 ~- --...4. I3...... —-------------- 234 TEST EXAMPLES. 10. 1 -+13+17+ l+2-.. S 11. 3+ I + I + +1 1 +=..... 13 3a6 +Tl -1- 7 +~-1-.-+ 3=.....,3-9-,~7. 14. ~+ + + ++ - + -6 -- *+ VI. SUBTRACTION. 4 3 T = — 8. 1~ — 5=.. 1 11 of 9 1 6 4 1 7B * 91 1 __ *991i G 1.., 2 14-l-o-. 5.2 -., *4 VII. MULTIPLICATION. (See Article 151.) 1 _ 1 X9 5 1 1 * 7=s * 22 2. 21-1.... 1 2 of. 2=. 6. 5. T7- * 92. 10. 15X12= 7 2 o 1 29 (See Article 152.) 1... 2 1X 9. 5. lX-5 1. 140. 4. 39X * e 14q.. 43X5=. *.. 253. 1 _ 4 2. -- X 12 7-W= 7..2 68. 1 3 1 1 3 2' 1' 5 20 515. 6. o X16-. y.. 134X1 -- 190 2L 48X.... 42. 6. 25x97..4.. ~g a...... f 4. 39X16:...6. 9. 134X14....190. COMMON FRACTIONS. 235 (See Article 153.) 1... 6. 5AX 5' =. 30o. 2.... ). 7. 33X 44... 18. 3. 1. x4.= 1 3. 8. 2 X 14-=.. 3 7 4. 4>X 3=,...3. 9. 12'X184=.. 234g. 8 4 3 ~ 4 5. 5 X -=X.... 3. 10. 52X 63. 3812 IL X'7X 4XI143...X5 =....... 4->. 12. 3-X21.X1 X3 54. 13. 3-X4VX 1 X25=.69. 14. 21X24X9X45X3X=....... 242. VIII. DIVISION. (See Article 154.) i. 25 ~ ~... T. 2. -- 10. 6. 3 — 5 -=.. 3 4 3. -7... j. 9 7 53.7 * ~ —J7 - - 7. 5ai ~7 4. 5.-4-22=. P' 5 68 5.. (See Article 155.) 1. 36__..2... 54. 5. 6 2B.... 2. 2. 1 —= 11. 6. 1. 3i=.. 3, 12+i.. 18. 7. 4 1 1.. 3. 4. 1012=. 8. 9 3.... 2. (See Article 156.) 1. 9. s 3 9. 2. 2.. o. _. 9_ 4 4 2. 3. 1= 9.. 14. 1 74 — 3=. * 5. 4 * 1 * 3. Us * = 11. 12.4 —. 2 24... - 1*. 12. 15-3 —. 4 *. 5. ~ 1.4 4- 4 -- 7. - 13. 12-22 -. 5. 4' — = 31. = 5 5.. ~,~. 1-~. 4~'~ 236 TEST EXAMPLES. REDUCTION OF COMPLEX TO SIMPLE FRACTIONS. (See Article 158a.) 36- 133 28' 31.3 =A. 5 of -- of' of 2- 43 3. -= 5. o f.o. 123.48 144 38] 38 47 2. 4 of 2 32 0 1 9. l of 0 of 2 of.. 273 44(See Article 14 60.) 911. 72-. 4. 44to te faction of an oz. AnS 5= 5. educe of an b. to the fraction of a pwt. Ans. 6. 1 of a. to the fraction of an inch. Ans. 7. g of an acre to the fraction of a perch. Ans. 83, T f acwt.to sthe fraction of an oz. o~. o. 1. Rdc1, of an. to the fraction of a D. Ans. -. 2. 1 of a bu. to the fraction of an inch. Ans. sT. 7. ~i of an acre to the fraction of a perch. Ans.:~.33 8. o of an hour to the fraction of a second. Ans.. NOTE.-The same examples may be applied conversely, in reducing fractions of a lower, to equivalent fractions of a higher denomination. 8~ -- — ~~~~~~~~~~~~~~~~~~~~~~~~~~l COMMON FRACTIONS. 237 II. FINDING THE VALUE OF A FRACTION IN INTEGERS OF A LOWER DENOMINATION. (See Article 160.) 1f. What is the value of 44 of a bushel? Ans. 1 pk. 7 qt. 2. Of 4 of a pound troy? Ans. 7 oz. 4 pwt. 3. Of 2ss of a mile? Ans. 6 fur. 105 yd. 4. Of of anacre? Ans. IA. 2R. 3~P. 5. Of ] of an acre? Ans. 3 R. 32 P. 6. Of ~o of a day? Ans. 21 hr. 36 min. 7. Of 4 of a pound apothecaries? Ans. 5 9 1 3 84 gr. 8. Of 3 of a hogshead? Ans. 37 gal. 3 qt. 1 gi. II. IN REDUCING A QUANTITY, COMPOSED OF ONE OR MORE DENOMINATIONS, TO THE FRACTION OR PART OF ANOTHER QUANTITY, COMPOSED OF ONE OR MORE DENOMINATIONS. (See Article 163.) 1. What part of 35 bushels is 15 bushels? Ans. 4. 2. What part of 3 bushels is 6 quarts? Ans. y1c. 3. What part of 4 bu. 3 pk. is 3 pk. 4 qt.? Ans. -.77 4. What part of 1 yard is 2 feet 8 inches? Ans. A. 5. What part of 3 A. 2 R. 20 P., is 1 A. 1 R. 20 P.? Ans. s.1 6. What part of 3 pounds 2 ounces 15 pennyweights, is 1 pound 9 ounces 5 pennyweights? 4ns. 31. 7. What part of 5 pounds 3 ounces 3 drams, is 4 pounds 4 ounces I dram? Ans. -1~. S. What part of 3~ l yards is 24 feet? Ans. 9. 9. What part of 14 hours is 154 minutes? Ans. -W. 10. What part of 1 hr. 63 min. is 4 min. 32-T sec.? Ans. 44 11. What part of 12 cents is ~ of 2 cents? Ans.'. 12. WVhat part of $1.69 is 91 cents? Ans. 73. IV. ADDITION AND SUBTRACTION. 1.; Add together ~ oz., U oz., and lbs. Troy.: Ans. 7 oz. 19 pwt. 20 gr. 238 TEST EXAMPLES. 2. Add together - of 3 A. 1 R. 20 P., - of I A., and - of 3 R. 15 P. Ans. 3 A. 2 R. 334 P. 3. From 2 of 5 pounds Troy, subtract g of 31 ounces. Ans 3 lb. 2 oz, I pwt. 2- gr. 4. From 15T miles, take 7444 furlongs. Ans. 4 fur. 9 yd. MIISCELLANEOUS EXERCISES. 1. 5723 — 5 — = what number? Ans..7 2. What fraction is that, from which if you subtract sq, the remainder will be? Ans. 3 3. What fraction is that, to which if you add -31, the sum will be s? Ans. m301 4. What number is that, which, being divided by 3 -, the quotient will be 5? Ans. 14 5. What number is that, which, being multiplied by 33, thp product will be 1? Ans. 1-4 6. Divide T; of 4 by oft. Ans. 7. What number, multiplied by ~, will produce 11 9? Ans. 264j S. What number, added to 11-7, will produce 17-r? Ans. 6': 9. A expends $4 in cloth, at $7 per yard, and B, $5 at $g per yard. What quantity does each purchase, and how mucI4 does one purchase more than the other? Ans. A 4 yd., B yd., and B 3 yd. the most. 10. A merchant bought 35 yards of cloth for $25, and sold it for $30; what was the purchase price, and the selling price per yard? Ans. $q and $4. 11. If 5 of a yard of cloth cost $2, what will be the entire cost of two pieces, each containing 5- yards, and a third one containing 5&- yards? Ans. $574. 12. Sold to A 25 barrels of apples for 564 dollars; to B 12-. barrels for 40 dollars; and to C 10 barrels for 260 dollars; what did the whole amount to, and what was the average price per barrel? Ans. $1231 and $24 8a. 13. Sold to A 54 yards of cloth at $3-1 per yard, and to B 34 yards at $54 per yard; what did the whole amount to, and COMMON FRACTIONS. 239 what was the average price per yard? Ans. $36.~ and $458 14. At 183 dollars per tun, what will be the cost of 4 of a tun of iron? Ans. $11-.l 15. How far will a man travel in 5~ days, at the rate of 22 miles per day for 21 days, and the remainder of the time at the rate of 18 miles per day? Ans. 111 miles. 16. How far will a man travel in 54 days, allowing him to travel 1 of the time at the rate of 24 miles per day, 4 of the time at the rate of 30 miles per day, and the remainder of the time at the rate of 36 miles per day? Ans. 1673 miles. 17. Bought 40 yd. of cloth at $32 per yard. Sold 204 yd. at 3; per yd.; 93 yd. at $34 per yd.; and the remainder at $4 per yd. What gain by the sale? Ans. $6 9. 18. Divide 1-9A by v-3 of 10 Ans. 4-. 19. What part of 20 is? Ans..1 20. If 34 yards of cloth cost $144, what is the price per yard? Ans. $4.07- 5 21. Divide 4 by g of 4. Ans. 54. 22. Find the difference between 54 and of 4 1. Ans. 4- 3'I 23. Divide 4 of 44 by of 2 Ans. 1.. 24. Divide 14 by 2 —, and to the quotient-add the quotient of 5, divided by 34. Ans. 2-0. 25. A man having 4 of a boat, sells ~ of his share for $96384; at that rate, what is the whole vessel worth? Ans.. $20080. 26. What part of 31 is 4 of 4? Ans. 4-. 27. Two persons bought a piece of cloth, for which the first paid 24, and the second 2-1 dollars; what part of it should each have? Ans. and 2 s 28. Two men bought a lot of wood for $28.35, of which one got 54 cords, and the other the remainder, which was 8 cords; what ought each to pay? Ans. $11.55 and $16.80 29. Of an estate valued at $6000, the widow receives 1;, each of two sons 4 of the remainder; and the residue is divided equally between three daughters; what is the share of each daughter? Ans. $11424. 240 TEST EXAMPLES. 30. A, B, and C, bought a haystack, weighing 103 tuns, for $45'. A got 3. tuns, B 31, and C the remainder; what ought each to pay? Ans. A $145; B $14-1.4; C $164 9 DECI1MAL FRACTIONS. I. ADDITION. 1. 3.2746+0.38 +2.763 +1.89743 +2.7=11.01503 2. 6.274 +2.34689+-3.004786+2.3478 +3.7+ 2.648329 -20.321805 3. 2.4789+3.26 +4.37897 +2.178643+0.004+1.3478 =13.648313 4. 0.7+0.37+0.2370.1237+0.91237+0.491237 =2.834307 5. 123k4 + 1 #45234 o 5 1240o5+2 +12345 1 —=13717.319616 6. 14 T -+3T'1 o+2 - T +-3 a — +3 ]j-+72TgF % 2 a%$ +T oUo-24.7573 7. 3.7468+2.0374+3.04789-+2.6+324+1.73+2.672898 +3.478-11.00746+2.897304=347.217752 II. SUBTRACTION.. 8.342-2.518........... 5.824 2. 14.273-8.786...... 5.487 3. 9 2......... 6.3778 4. 3 1..... 1.7378 5. 15.27468-10.4879 —.. 4.78678 6. 4.67432-1.894746=.... 2.779574 7. 3.04-1.68=............ 1.36 8. 2.4063-1.8794.......... 0.5269 9. 3T2 o —1 T......... 1.564 10. o ~28-8 23s=.......... 19.77 11. 16.4-8.326=........... 8.074 12. 4.31 -2.3872-..... 1.9228 13. 17-3.472.......... 13.528 ^9 __4J —-— ~ —— ~ —~ — ~ —~ DECIMAL FRACTIONS. 241 14. 21.3-2.4796 -....... 18.8204 15. 3-2.2641= 4........ 0.7359 16. 26.37-8.'17341-......... 18.19659 17. 3.4-1.763214-......... 1.636786 18. 3000 —0.003-...... 2999.997 III. MULTIPLICATION. SoUGESTroNs.-By changing te7e place of the decimal point, each operation, with the same figures, may be made, both in Multiplication and Division, %,o give Several examples. Thus, the product of 32.7 by 12.5 is 408.75; hence, 3.27X1.25=4.0875, and.327X.125=.040875, and so on. In like manner, 552.5. 1'7= 32.5; hence, 55.25~ ^ 17= 3.25; 5.525 — 17 =.325;.5525 — 17-.0325; 552.5 — 1.7 325; 552.5.17-= 3250; 552.5 —..017= 32500, and so on. 1. 8X324-2592. 8X3.24-25.92 2. 23X57.98=..... 1333.54 3. 36X5.769=-..... 207.684 4. 543X7.8906... 4284.5958 5. 12 X,5789...... 6.9468 6. 11OX.23456....... 25.8016 7. 48 X9876 —.. 474048. 4.8X9876. 47404.8 8. 2.34 X123456=..... 288887.04 9. 3.456X28767 =.. e. 99418.752 10. 0.325X50807 =........ 16512.275 11. 3.9875X9804=........ 39093.45 12. 0.0025X375=.. 9.6875 13. 11X3457=38027. 11X34.57=380.27 1.1X34.57 =38.027 14. 5.43X7.8906=........42.845958 15. 1.804X45.607= e.... ~ 82.275028 16. 0.4X57.98....... 23.192 17. 1.7X0.579=......... 0.9843 18. 0.31 X0.579-......... 0.17949 242 TEST EXATIPLE:S. 19. 0.07X0.08....... 0.0056 20. 3s,ts - 20. (-..... 0.0000432 21. - 3 X 2. 0.000000416 22..45X0.3456 —... 0.15552 23. 0.92X0.9>0.5.5-...... 0.414 24. 0.15X0.16XO.05..... 0.0012 25. 1.23X34.5XO.567XO.0789- 1.8983848905 26. 0.0025X0.015XO.16..... 0.000006 IV. DIVISTON. 1. 404.341-47........603 2. 5.76 —- 12-..... 0.48 3. 5.44- -17........ 0.32 4. 0.474048- -48-.. o.. 0.009876 5. 46.5- 6......... 7.75 6. 27.3-:8........ 3.4125 7. 56.7 —:4......... 2.3625 8. 73.091- 16...... 4,619375 9. 1120 —14=.... 80. 112- 1.4. 80. 10. 192.1.6.....120. 11. 924:' 1.32 o.....700. 12. 2769 0.213=..... 13000. 13. 43... o — 13.4375 14. 19 0.128=......... 1.48.4375 15. 32 —1.5 —.......... 21.333-+ 16. 42 —0.014 o...... 3000. 17. 53- 0.00008 —...... 662500. 18. 9 —0.562 5........ 16. 19. 7- -0.21875...... 32. 20. 11 0.04296875-..... 256. 21. 4,8 1.2=........ 4. 22. 32.2_1.4....... 23. 23. 5.76_0.4 8....... 12. 24. 449.503_ 6.709=..... 67. 25. 0.343 — 0.049 o. o.. 7. 26. 3.24-: 1.44......... 2.25 DECIMAL FRACTIONS. 243 27. 12.18. 2.9=... 4.2 28. 44.928 —1.3-..34.56 29. 4.94795 — 2.11=......... 2.345 30. 3.8788620.669 —...... 5.798 31. 0.208728. 0.5798-........ 0.36 32. 1.236-.0.48=.......... 2.575 33. 51.2 —1.28 —.......... 40. 34. 85.12 1.216........... 70. 35. 2.4 —.0048. —..... 500. 36. 3386.88 —0.3456-..... 9800. 37. 44.8 2.56. —. 17.5 38. 3.27-.0096.-...... 340.625 V. REDUCTION OF COMMON FRACTIONS TO DECIMALS. 1. =. 0.]36 2. U7...... 0.00875 3............ 0.006875 4... 0.01171875 5.........0.0135546875 6. 1 UDa -a -a. 0.0001 7......0.222464 8.:if~ evv —....... 0.00000048828125 9. 3. 0.085714285+, (Period 857142.) 10. }-. 0.88235+, (Period 8823529411764705.) 11...0.000566+ 12...... 0.0000001238+ VI. REDUCTION OF DECI-MALS TO COMMON FRACTIONS. 1. 0.2976_....' 2 a * 2, 0.06125=........... g49. 3 0.19921875=.... - 4 0.0406640625-.......... 04 VII. REDUCING DECIMALS OF ONE DENOMIINATION TO THEIR EQUIVALENTS IN ANOTHER DENOBINATION. 1. Reduce.03375 of a yd. to the fraction of a qr. Ans..135 2..0225 of a lb. T. to the fraction of an oz. Ans..27 244 TEST EXAMPLES. 3. Reduce.00390625 of a bu. to the fraction of a pt. Ans..25 4..0025 of a minute to the fraction of a second. Ans..15 5..0006875 of a day to the fraction of a minute. Ans.99 6..7 of a quart to the fraction of a bushel. Ans..021875 7..21 of an in. to the fraction of a yd. Ans..005833+ 8..9 of a dram to the fraction of an ounce. Ans. 0.05625 9..3 of a pint to the fraction of a gallon. Ans. 0.0375 10..84 of a 3 to the fraction of a lb..Ans. 0.00875 VIII. FINDING THE VALUE OF A DECIMAL IN INTEGERS OF A LOWER DENOMINATION. 1. What is the value of 0.865 of a bushel? Ans. 3 pk. 3 qt. 1.36 pt. 2. Of.325 of a yard? Ans. 1 qr. 1.2 na. 3. Of.00625 of a day? Ans. 9 min. 4. Of 0.78125 of an acre? Ans. 3 R. 5 P. 5. Of.54296875 of a pound?.Ans. 8 oz. 11 dr. 6. Of.34619140625 of a cwt.? Ans. 1 qr. 9 lb. 9 oz. 14.5 dr. 7. Reduce 12 gr. to the decimal of a lb. T. A2ns. 0.0020583+ 8. 15 perches to the decimal of an acre. Ans. 0.09375 9. 3 R. 21.52 P. to the decimal of an A. Ans..8845 10. 3 min. 4.032 sec. to the decimal of a day. Ans. 0.00213 11. 1 foot 1.5 in. to the decimal of a yd. Ans..375 12. 1 pk. 6 qt. to the decimal of a bu. AAns. 0.4375 DiIISCELLANEOUS EXERCISES. 1. What part of a year is a day, reckoning the year at 365 days 6 hours? Ans. 0.0027378507+ 2. VWhat part of a year is a day, reckoning the year, as is right, at 365 days 5 hours 48 minutes 48 seconds, or 365.24222+ days? Ans. 0.002737909+ 3. Reduce the following fractions to decimals, and find their sumn: )-4-, 5 T,' Gus T 63, 630, and I. Ans. 6.0025 4. How many times is 0.0002 contained in 0.01? Ans. 50. 5. What is the product of 0.0107 by 0.00206?.Ans. 0.000022042 6. What will 6 pounds 13.504:ounces avoirdupois cost at $.3125 per pound? Ans..$2.13875 7. What is the difference between.11 of a pound and.11 of an ounce Troy? Ans. 1 oz. 4 pwt. 4.8 gr. _b, _> ~_~ -- _0 v ~ -- PROPORTION. 245 8. How many times is 0.0015 of an ounce Troy, contained in 47 pounds?.Ans. 376000. 9. How many times is 0.03125 of a nail contained in 25 yards? Ans. 12800. PRO P 0 R T ION. SIMPLE PROPORTION. MENTAL EXERCISES. 1. If the product of the means is 24, and one of the extremes is 3, what is the other? If one of the extremes is 4, what is the other? If one of the extremes is 12? 2. What is the fourth term of the proportion 3 2:::12. 3. What is the fourth term of the proportion 6 9:: 8. 4. What is the fourth term of the proportion 10: 2:: 20. 5. Complete the proportion 7: 4:: -: 8. 6. Complete the proportion 3:-:: 6: 16. 7. Complete the proportion-: 7:: 8: 28. FOn THE SLATE OR BLACKBOARD. 1. If 6 lb. of tea cost $5, what will 8 lb. cost? Ans. $6]. 2. If 5 men can do a certain piece of work in 12 days, in what time ought 8 men to do it? Ans. 7- da. 3. If a man travels 55 miles in 2 days, how many miles ought he to travel in 5 days? Anls. 137, miles. 4. If 5 yards of silk cost $4.40, what sum, at the same rate, ought to be paid for 11 yd. 3 qr. 2 na.? Ans. $10.45 5. If a certain quantity of provisions last 30 men 25 days, how long ought it to last 20 men? Abns. 37- days. 6. If 12 oxen require 20 A. 2 IR. 30 P. of pasture Sfor a summer, what quantity ought 30 oxen to have for the same length of time? Ans. 51 A. 2 R. 35 P. 7. A cistern is filled by 3 pipes, of a certain size, in 32 min. 5 sec.; in what time would it be filled by 7 pipes of the same size? Ans. 13 min. 45 sec. 8. If a barrel of flour last 7 persons 8 weeks, how long will it last 15 persons? Ans. 31-l- weeks.,.PQ — - --- - - -- -- --— ~ 246 TEST EXAMPLES. 9. If a note at 5 % yield $185 interest, at what % would it yield $259? Ans. 7. 10. If 3- of flour cost 252 cents, what will 143 lb. cost? Ans. $1.08' 11. If 6 2 yards of cloth cost $134, what part of a yard will.$. buy? Ans. 3 5 12. If when the days are 121 hours long, a man can perform a journey in 84 days, in how many days can he perform it when the days ale 74 hours long? Ans. 134 days. 13. Find a number which shall have the same ratio to 7 that 27 has to 3; also, a number to which 39 has the same ratio that 31 has to 25. Ans. 63, and 31O. 14. Complete the proportion of which the first, second, and fourth termus are, 35, and 3-; also, that whose first, third, anfd fourth terms are.35, 1.25, and.0145 Ans. Third term is.a: second term,.0000406 15. How much in breadth that is 15 rods in length, will be equivalent to an acre of land, which is 40 rods in length and 4 rods in breadth? Ans. 10 rd. 3 yd. 2 ft. COMPOUND PROPORTION. 1. If 4 men in 7 days, earn $17, how much will 14 men earn in 10 days? Ans. $85. 2. If 4 men, in 6 days, can build 23 feet of wall, in how many days can 12 men build 460 feet? Ans. 40 days. 3. If 400 men consume 5 bl. of flour in 12 days, how many men will consume 15 bl. in 20 days? Ans. 720 men. 4. If 12 men can mow 105 acres of grass in 10 days, howv many men can mow 7 acres in 4 days? Ans. 2 men. 5. If 3 men, working 10 hr. a da., can reap a field measuring 150 yd. by 240 yd., in 5 da., how many men, working 12 hr. a da., can reap a field 192 yd., by 300 yd. in 4 da.? Ans. 5 men. 6. If 7 men can build a wall 245 yd. long, 8 ft. high, and 18 in. thick, in 35 days of 12 hours each, what length of wall, 10 ft. high and 27 in. thick, could 12 men build in 43 days of 10 hours each? Ans. 229 yd. 1 ft., ~ —---------------- ALIQUOTS 01 PRACTICE. 247 ALIQUOTS OR PRACTICE. MENTAL EXERCISES. 1. At 25 cents a yard, what will 8 yards of cloth cost 10 yd.? 12 yd.? 15 yd.? 20 yd.? 25 yd.? 2. At 50 cents a yard, what will 8 yards of cloth cost? 10 yd.? 11 yd.? 15 yd.? 18 yd.? 21 d.? 3. At 20 cents a bushel, what will 5 bushels of oats cost? 10 bu.? 12 bu.? 18 bu.? 33 bu.? 64 bu.? 4. At 12. cents a yard, what will 8 yards of cnusli;n cost? 10yd.? 12 yd.? 15 yd.? 18 yd.? 20 yd.? 24 yd.? 5. At 16-2 cents a yard, what will 6 yards of calico cost? 9yd.? 12 d.? 1 yd.? 2yd.? 20 y? 6. At 12 cents a yard, what will,- of a yard of muslin cost? 21 yd.? y-,d.? 44 yd.? 7. At 20 cents a bushel, what will 3 pecks of potatoes cost? 2 bushels 1 peck? 2 bushels 2 pecks? F'OR THE SLATE OR BLACKBOARD. 1. At $2.31~ per yd., what will 27 yd. cost? iAns. $62.433 2. At $3.93- per yd., what will 64 yd. cost? BAs. $252. 3. At $3.50 per yd., what will 5 yd. 2 qr. cost.? Ans. $19.25 4. At $1.28 per yd., what will 7 yd. 1 qr. 3 na. cost? Ans. $9.52 5. At $2.24 per bu., what will 9 bu. 3 pk. 4 qt. cost? Ans. $22.12 6. At $3.84 per bu., what will 11 bu. 3 pk. 7 qt. 1 pt. cost? Ans. $46.02 7. What will 11 lb. 7 oz. 12 pwt. of silver ore cost, at $122 per pound? Ans. $139.60 8. What will 3 pounds 11 ounces 19 pennyweights 23 grains of metal cost, at $5.76 per pound? Ans. $'23.039 9. What will 5 weeks 5 days' work amount to, at $5 per week, of 6 days each? Ans. $29.16 10. What will 5 pounds 11 ounces of butter cost, at 12 cents per pound? Ans. 68{ cents. _ 248 TEST EXAMPLES. 11. What will 5 A. 3 R. 12 P. of land cost, at $11.20 per acre? Ans. $65.24 12. What will 13 pounds 13 ounces 13 drams of spice cost, at $2.56 per pound? Ans. $35.49 13. At the rate of 216 mi. in 24 hr. how far does a ship sail in 5 hr. 24 min. 22 sec.? Ans. 48.655 miles. 14. July 1st, 1845, the sun's longitude changed 57' 12".8 in passing from meridian of Greenwich to that meridian again; supposing the meridian of Cincinnati to be 5 hours 37 minutes 22 seconds of time west of Greenwich, how much did the longitude change while the sun passed from meridian of Greenwich to that of Cincinnati? Ans. 13' 24".244+ PE3RCENTAGE. 1. A merchant, with a capital of $2150, increased it 10 %o the first yr., and the second yr. lost 5 %O of the amount had at the close of the first yr. What sum had he at the close of the second yr.? Ans. $2246.75 2. A merchant, with a capital of $2235, gained 20 So the first year, and the second year lost 16 — S% of the capital at the close of the first. What sum had he at the close of the second year? Ans. $2235. 3. If a man gains 25 %o of his investment, and then loses 20 % of both investment and gain, what will be the gain or loss on the first investment? Ants. Nothing. 4. A and B have each $5000; A increases his 12- %, and B diminishes his 8, %'b. How much more, then, has A than B? Ans. $1050. 5. An agent receives $5.65 for collecting $226; what % is this? Arls. 21,. 6. An agent, after deducting his commission of $15.47, returns to the owner $348.53: what % did he receive for collecting? Ans 4,'. 7. What %o of any sum is } of it? Ans. 12. S. A vessel containing 39 gallons, lost by leakage 3 gal. 1 qt.; what:O is this? Arns. 8. PERCENTAGE. 249 9. What is the commission on $150 worth of groceries at 21 %, and $250 worth, at 11 0o? Ans. $6.50 10. An agent receives $231.84 to purchase goods, at 3. %o on the amount of his purchases; how much must he expend? Ans. $224. 11. A factor receives $554.85 to purchase dry goods, at 2'- So on the amount expended; what will be his commission? Ans. $14.85 12. What is the annual premium for insuring a brick house worth $3840, at A %? Ans. $33.60 13. A house worth $5280, is insured on " of its value, at a';i what is the premium? Ans. $21.12 14. A man pays $15.50 for insurance on his house, at.; at what sum is the house valued? Ans. $3100. 15. A man pays $29.40 for insurance, on a house valued at $1680; what is the rate %? Ans. 1. 16. At 1~ o, what sum must be insured on $1580, to include the premium in case of loss? Ans. $1600. 17. At 2 -%, what sum must be insured on $457.08, to include the premium in case of loss? Ans. $468.80 18. Bought 25 shares of stock, (par value, $50 a share), at 3 So advance; what did it cost me? Ants. $1287.50 19. The owner of the stock, (Ex. 18), sold it at 2 % discount; what did he receive for it? Awls. $1225. INTEREST. 1. What is the interest of $240, for 3 years 9 months, at 1 go per annum? Arns. $9. 2. What is the interest of $120, for 2 years 6 months, at 21 % per annum? Anrs. $7.50 3. What is the interest of $240, for 6 years 4 months, at 10- % per annum? Ans. $163.40 4. W hat is the interest of $432, for 2 years 7 months 21 days, at 7' %o per annum Ans. $85.59 5. VWhat is the amount of $234.45, for 1 year 4 months 10 days, at 8j3 % per annum? Ans. $262.372+ 6. What is the amount of $841.50, for 3 years 7 mouths 20 days, at 9' S per annum? An7s. $1132.401+ 2 _... _....,.. hi —---- -- 250 TEST EXAMPLES. 7. What is the interest of $319.89, for 2 years 9 months 10 days, at 36 % per annum? Ans. $319.89 8. What is the interest of $243.75, for 3 years 5 months 10 days, at 11 %o per annum? Ans. $96.552+ 9. What is the amount of $450.25, for 5 years 2 months 20 days, at 15 % per annum? Ans. $802.945+ 10. What is the interest of $625, for 4 years 9 months 18 days, at 6 % per annum? Ans. $180. 11. What is the interest of $781.25, for 4 years 9 months 18 days, at 7 lo per annum? Ans. $262.50 12. What is the amount of $720, for 12 years 11 months 29 days, at 7 Yo per annum? sAns.,1375.06 13. At 6 o, what is the interest of $250, from Jan. 10th, 1840, to Ilarch 10th, 1841? Ans. $17.50 14. At 7 %, what is the interest of $60, from Anay 10th, 1841, to Jan. 25th, 1842? Ans. $2.975 15. At 5 %7, what is the interest of $241, from Feb. 12th, 1840, to April 24th, 1842? Ans. 826.51 16. At 4' 0o, what is the interest of $124.60, friom Nov. 25th, 1843, to Jan. 5th, 1845? Ans. $6.23 17. At 10 %, what is the interest of $344, from June 11th, 1845, to May 1st, 1846? Ans. $30.577+ 18. At 6 %, what is the amount of a note for $367.40, dated Nov. 8th, 1829, paid March 4th, 1834? Anzs. $462.679+19. At 6 %, what the amount of a note for $108, dated Feb. 12th, 1836, paid 3March 1st, 1838? Ans. $121.302 20. At 8 ~o, what the amount of a note for $325, dated Dec. 12th, 1843, paid Jan. 7th, 1846? Ans. $378.805+ PROBLE 1S IN INTERlEST. 1. A't a5 interest, in what time will $600 gain $125? Ans. 4 yr. 2 mon. 2. At 3 %7 per annum, in what time will anly principal double itself? Ans. 33- yr. 3. A man pays $15 for the use of $360, for 8 months; what is the rate %? Ans. 64 PERCENTAGE. 251 4. If $96.25 is paid for the use of $440 for 2 yr. 6 mon.; what is the rate %? Ans. 8.3 5. What principal, at 6' %, will gain $23.075 in 1 yr. 3 mon.? Ans. $284. 6. What fund at interest, at 7 t, will yield an annual income of $1299.20? Ans. $18560. DISCOUNT. 1. What the present worth of $500, due 10 mon. hence, when money is worth 5' per annum? Ans. $480. 2. What the present worth of $415, due 1 yr., 4 mon. hence, money worth 8 %O per annum? Ans. $375. 3. What is the present worth of $281.26, due 1 yr. 5 mon. 6 da. hence, money worth 10 % per annuma? Ans. $246. 4. What is the discounit of $355.25, clue 4 months hence, at 4. %? ils. $5.25 5. What is the discount of $690.183, due 9 months hence, at 3 0? Ans. $15.183 6. A note not bearing interest, dated Jan. 6th, 1849, for $943.68, is payable March Ist, 1850; what was its value on the dclay it was dated, reckoning money worth 8 7o per annum? Ans. $864. 7. A note not bearing interest, dated Jan. 1st, 1845, for $176.30, is payable July 1st, 1847; what is its value, April 1st, 1846, reckoning moneyworth 6%o per annum? Ans. $164. 8. How much stock, the selling price per share $92], and par value $100, must be sold to pay $715.85, due 9 mon. hence, reckoning money worth 4 %o per annum? Ans. $750, or 71 shares. BANK DISCoUNT. 1. At 6 %, what is the bank discount of a note of $75, p)ayable 30 days after date? Ans. $0.412. At 6 %, what is the bank discount of a note of' $230, payable 60 days after date? Ans. $2.413. At 6 1o discount, what is the proceeds of a note of $420, payable 30 days after date? Ans. $417.69 4. At 6 %o discount, what is the proceeds of a note of $375, payable 60 days after date? Ans. $371.064 art —--- 252 TEST EXAMPLES. 5. If $393.80 be received at a bank, for a note payable in 90 days, discounted at 6 %, what is the face of the note? (See Art. 241.) Ans. $400. 6. If $592.65 be received for a note payable in 60 da., discounted at 7 %, what is the face of the note? Ans. $600. 7. WVishing to borrow at 6 %, $1000 for 30 days, for what sum must I give my note? Ans. $1005.53+ MISCELLANEOUS EXERCISES IN PROFIT AND LOSS. 1. A merchant bought cloth for $225, sold it at 20 go advance; how much profit did he make? Ans. $45. 2. A cask of wine, containing 70 gallons, lost 15 So by leakage; what quantity remained? Ans. 59 gal. 2 qt. 3. A man bought a buggy for $75, and sold it at a profit of 8 s; how much did he get for it? Ans. $81. 4. When hats that cost $3.75 a piece, are sold for $4, what is the rate oc of profit? Ans. 62. 5. When hats that cost $4, are sold for $3.75, what is the rate o/ of loss? Ans. 6.. 6. Bought sugar at 4 cents a pound, and sold it at the rate of 18 pounds foi' a dollar: what %/c profit did I make? Alns. 38S. 7. Supposing the same sugar sold at the rate of 7 lb. for 25 ct., what is the rate 5/ of loss? Ans. 10'. 8. A grocer bought a barrel of fish for $11.75, and sold it for $13.65: what %o profit did he make? Ans. 164s~. 9. Bought 12 bl. of apples, each containing 23- bu., at $2.50 a bl., sold them at $1.20 a bu.: required the whole gain, and gain So? Ans. Gain, $9.60; rate 32 %. 10. Bought eggs at the rate of 5 for 4 cents, for how much per dozen must they be sold, to gain 25 So? Ans. 12 cts. 11. Bought 8 barrels of flour, each containing 196 pounds, at $5.25 a barrel, and sold it at 25 cents for 7 pounds: required the whole gain, and the gain per cent.? Ans. Gain, $14; rate, 331. 12. A merchant, by selling cloth at $1.82 per yard, loses at the rate of 9 5o: at what rate per yard ought it to be sold, to realize a profit of 7 %? Ans. $2.14 PARTNERSHIP. 253 13. A merchant, by selling cloth at $2.43 per yard, gained - of the first cost; what 6 would he gain by selling it at $2.64 per yard? Ans. 22g. 14. Sold quills at 11 cents a bunch, of which - was clear profit; afterward raised them to 13A cents a bunch: what 1 did I clear at the latter price? Ans. 96 4T. 15. A merchant, by selling 4 yards of cloth for $13, gained 25,o: what % would he lose by selling 4 yards for $10? Ans. 31l. 16. By selling bacon at 12' cents per pound, a retailer makes 25 So profit: what % profit will he make by selling it at 13 cents a pound? Ans. 30. 17. A merchant, selling flour at $3.22 a barrel, loses 8 %o; when he bought it, he expected to sell it at $4 per barrel: what %o profit did he expect to clear? Ans. 147. 18. A trader purchased $856 worth of goods, which he sold at a loss of 12,1 SO; he invested the proceeds in goods, which he sold at a profit of 20 %: by the two transactions, what % profit did he make on his first investment? Ans. 5. ASSESSMIENT OP TAXES. 1. A tax of $2441.25 is to be levied on a corporation, of which the taxable property is valued at $420000, and which has a list of 525 polls, each taxed 25 cents: what ro is the tax on property? Ans..0055, or 5' mills on the dollar. 2. What is A's tax, whose property is valued at $1200, and who pays for 1 poll? AAns. $6.85 3. What is B's tax, whose real estate is valued at $8500, his personal property at $260, and who pays for 3 polls? Amzs. $48.93 PARTNERSHIIP. MENTAL EXERCISES. 1. James has 2 cents, and John 1 cent, with which they buy 9 apples: what is the share of each? 2. A has 2 cents, and B I cent, with which they buy 20 chestnuts: what is the share of each? IC, --— ~~ -------------- 254 TEST EXAMPLES. 3. A has 2 cents, and B 3 cents, with which they buy 10 peaches:'what is the share of each? 4. A has 3 cents, and B 4 cents, with which they buy 14 pears: what part of them must each have? What is the share of each? 5. C has 5 dollars, and B 3 dollars, with which they buy 12 yards of cloth: what part of it should each have? What is the share of each? 6. A has 3 cents, and B 2 cents, with which they buy 12 peaches: what part of them should each have? WVhat is the share of each? FOR THE SLATE OR BLACKBOARD. 1. Two men hired a pasture for $24.09. A pastured in it 5 horses, and B, 6 horses: what ought each to pay? Ans. A, $10.95; B, $13.14 2. Three men together perform a job of work for $57.50. A worked upon it 20 days, B, 17 days, and C, 13 days: what should each receive? Ans. A, $23; B, $19.55; C, $14.95 3. Three partners put into business the sums of $300, $400, and $500; they gained $375: find the share of each. Ans. $93.75, $125, and $156.25. 4. Divide $1000 among three persons, so that their shares shall be as the numbers 2, 5, and 9. Ans. $125, $312.50, and $562.50 5. A person bequeathed by will the following legacies; to A $1500, to B $875, to C $525, and to D $350: but when his property was sold, it produced only $2437.50: how much did he really leave to each? Ans. to A, $1125, to B, $656.25, to C, $393.75, and to D, $262.50 6. Divide $60 between A and B, so that A shall have $ — as often as B has $a. Ans. A $24, B $36. 7. Divide $115 between three persons, so that the parts shall be to each other as l, 4, and 3. Ans. 4$40, $30, and $45. 8. Gunpowder is composed of 76 parts of nitre, 14 of charcoal, and 10 of sulphur: how much of each is there in 112 lb. of powder? Acns. 85;% lb., 15 7 lb., and 111 lb. ~ ~ 1~ lb. —-— e EQUATION OF PAYMENTS. 255 PARTNERSHIP WITHI TIBIE. 1. Two men finished a job of work for $45. A had 5 men upon it 3 days, and B 3 men 7 days: what is each man's share of the money? Ans. A's, $18.75; B's, $26.25 2 A, B, and C traded in' company; A put in $100 for 3 months, B, $90 for 5 months, and C, $35 for 10 months; they gained $198: what was the gain of each? Ans. A $54, B $81, C $63. 3. Two men hired a pasture for $14; in it A pastured 4 horses for 5 months, and B, 5 horses for 6 months: how much should each pay? Ans. A $5.60, B $8.40 4. A advances $150 for 9 morn., and B, $120 for 6 mon.; they gain $115: what the share of each? Ans. A $75, B $40. 5. Four men hire a pasture for $50.25, in which A pastures 5 cows 9 weeks, B, 4 cows 10 weeks, C, 6 cows 10 weeks, and D, 7 cows 8 weeks: how much should each pay? Ans. A $11.25, B $10, C 15, D $14. 6. Three persons with a joint stock, gain $183.60: the first furnishes: of the capital for 4 of the time; the second, r of the capital for 4 of the time, and the third, the remainder of the capital for the whole of the time: find their shares. Ans. $24.48, t36.72, and $122.40 EQUATION OF PAYMIENTS. 1. What will be the equated term of payment of $200, due in 3 months, $300, in 8 months, and $500, in 12 months? Ans. 9 months. 2. If $75 be due in 4 months, $125 in 5 months, and $150 in 7 months; what is the equated time? Ans. 5 9T months. 3. A owed B $750, to be paid in 15 months, but at 12 months he paid himn 8250: at what time was the remainder due? Ans. 164 months. 4. Find the equated time of payment, when.) of a sum of money is due at 3 months, 5 at 8 months, and the remainder at 15 months. Ans. 7 — months. 256 TEST EXAMPLES. 5. If a person owing $1000, payable in 7 months, pay $300 down, how long after the 7 months may he delay the payment of the remainder? Ans. 3 months. ANALYSIS. 1. If 9 barrels of flour cost $29.25, what will 11 barrels cost? Ans. $35.75 2. If a railroad car can run 609 miles in 29 hours, how far will it run in 13 hours? Ans. 273 miles. 3. If $79.25 will pay for 25 barrels of flour, how many barrels can be purchased for $60.23? Ans. 19 barrels. 4. If a man walk 3 miles in X of an hour, how far can he walk in 63 hours? Ans. 30,4 miles? 5. If 4 of a pound of silver ore is worth $4.05, what is a pound worth? Ans. $6.75 6. If 42 of a bushel of pecans is worth 714 cents, what is C) of a bushel worth? Ans. 1.389 7. If 4 of a steamboat is worth $3750, what is -'-& worth? Ans. $6875. 8. If 24 bushels of wheat cost $2;04, what will 54 bushels cost? Ans. $5.46 9. If 3~ bushels of wheat cost $2.16, how many bushels can be purchased for $7.50? Ans. 11.9. 10. If 11 - bl. of flour serve a family (of 15 persons) 45. wk., how many bl. will serve them 17- wk. Ans. 4,. Observe that the answer is entirely independent of the numzber of persons in the family. 11. If a barrel of flour serve 5 men 17 days, how long will it serve 3 men? Ans. 284 days. 12. HIow much wheat, at $4 a bu., must be given in exchange for 5 yd. of cloth, at $1~ per yd.? Ans. 8-3 bu. 13. If 44 bu. of corn cost $3, what will 8$ bu. cost? Ans. $7'3. 14. If 34 lb. of coffee cost 3564~ cents, what will 251 lb. cost? Ans. $2.55 - --- ------- -----------— e ANALYSIS. 257 15. If 4-g acres of land cost $27-, how much will $50 buy? Ans. 7-A A. 16. If 12 yards of muslin cost $3$, what will 11 yards cost? 17. The distance from A to B is 28 miles, or 1 of the distanuce from B to C; what the distance from B to C? Ai, 49 miles. 18. A has 88%-; anLd of A's mloney is equal to $ of B's:' lwhat is B's money? Ans. $7.35 19. A has $36, and 3 of A's mon ey is equal to - of B's, and g of B's is equal to -~s of C's: required C's money. A ns. $55 i 20. If 5 men can do a certain piece of work in. of a day, in what time ought 3 men to do the same work? Ans. 1~ days. 21. If 4 men eat 32 pounds of bread in 2 days, how many pounds will 9 men ea-t in 5 days? AlZs. 180 pounds. 22. If a man can travel 195 miles in 5 days, when the dclays are 12 hours long, how far can he travel in 6 days, when the days are 1 hours long? A's. 2141 miles. 23. If 9 men can build 54 rods of wall in 8 days, how much would 12 men build in 5 days? AIs. 45 rods. 2-14. If 5 men mow 20 acres of gross in 6 days, how mnany men nwould mow 12 acres in 9 da.ys? Als. 2 men. 25. If 10 bushels of oats supply 18 horses 20 days, how many bushels will supply 60 horses 36 days? Ais. 60 bushels. 26. How many men will it reqcuire to dig a trench i35 yards long in 8 days, when 16 men can dig 54 yards in 6 days? LAs. 30 men. 27. If 20 bushels of whealt are sufficient for a family of 8 persons 3 months, how much will be required for 7 persons 2 months? -A4s. 11:- bushels. 2o8. If' 3 barlrels of flour sserve'14 persons 5,. weeks, how long will 5-/ barrels serve 7 persons? Ails. 19; weeks. 29. If 5 len earn $26.2c5 in 7 davys, llo how much xwill 7 men earn in 11 days? Lt[s. $57.75 30. If 9 horses eatr 28.8 bushels of ota-s inr 8 days, how many bushels will 13 horses eat in 15 days? AZns. 78 bushels. 31. If $24 will buy d of an acre of land, what part of an acre will $15 buy? A us. -,. 258 TEST EXAMPLES. 32. A man can do a piece of work in 32 days; what part of it can he do in 1 day? Ans. ]3, 33. A man can mow a certain meadow in 3- days; what part of it can he mow in 27 days? Ans.:26. 34. If 24 yards of cloth cost $34, what part of it, and how much, will 3- quarters cost? Ans. 2-7, and $1711 35. A cistern, containing 1500 gallons, can be filled by one pipe in 4 hours, and emptied by another in 5 hours: in what time can it be filled, if they are both left open? Ans. 20 hr. 36. A cistern, containing 800 gallons, is filled by one pipe at the rate of 100 gallons in 3 hours, and emptied by another at the rate of 144 gallons in 5 hours; in what time will it be filled, if both these pipes are left open? Ans. 176-8 hr. 37. A can do a piece of work in 2 days, and B in 4 days; in what time can they both do it? Ans. 1i- days. 38. A and B can dig a trench in 20 days; A can dig it in 34 days: in what time can B dig it? Ans. 484 days. 39. A can dig 32 yards of a trench in 6 days; B, 29 yards in 5 days, and C, 54 vards in 10 days; in what time can they finish 100 yards, if' all work at the same time? Arns. 6~3 days.'40. If A can finish a certain piece of work in 6 days, of 7 hours each, and B can do 4 times as much in 15 days, of 9 hours each, what is their comparative strength? Ats, The strength of A is to that of B, as 45 to 56, or as 1 to 141. 41. A, B, and C, dig a trench in 4 days; A can do it by himself in 7 days, and B in 14 days; in what time can C perform the work alone? - Ans. 28 days. 42. A and B can perform a piece of work in 2 days; A and C in 3 days; and B and C in 5 days: in what time can each do it by himself? lAns. A in 31-, B in 5 T, and C in 60 days. 43. A, B, and C, can do a piece of work in 10 days; B, C, and D, in 12 days; C, D, and A, in 14 days; and D, A, and B, in 16 days; in what time can each do it by himself? As.A, in44 B 293, C 23-l3o, D 173;- days. 44. A leaves Cincinnati for Dayton at 7 in the morning; and B leaves Dayton for Cincinnati, at 9 the same morning; ANALYSIS. 259 A travels 3V, and B 44 miles per hour; how far will each have traveled when they meet, reckoning the distance from Cincinnati to Dayton 59 mi.? Ans. A 30T~, B 28 —l miles. 45. A cistern has 3 pipes; by the first, it can be filled in 1- hours, by the second, in 34 hours, and by the third, in 5 hours: in what time can it be filled by all these pipes running together? Ans. 48 minutes. 46. After paying away 4 and 5 of my money, I had $33 left; how many dollars had I at first? Ans. $60. 47. After paying away 1 of my money, and 4 of the remainder, I had $57; how much had I at first? Ans. $95. 48. A father's age is 4 times that of his son, and the sum of their ages is 40 years: what is the son's age? Ans. 8 yr. 49. The greater of two numbers is 24- times the less, and their sumn is 14: what are the numbers? Ans. 4 and 10. 50, The smaller of two numbers is - of the greater, and their sum is 25: required the numbers. Ans. 74 and 17k. 51. The smaller of two numbers is 2 of the greater, and their difference is 21: required the number. Ans. 6 and 27. 52. In a certain s.chool there are 32 pupils, and there are 12 more boys than girls: required the number of each. Ans. 10 girls, 22 boys. 53. Three-fourths of a certain number exceeds 4 of it by 25; what is the number? Ans. 140. 54. The 4,,, i, and 4 of a certain number, added together, make 127; what is the number? Ans. 84. 55. The 4, -, and 4 of a certain number, added to the number itself, make 175; what is the number? Ans. 56. 56. There are 195 sheep in 3 flocks; the second contains 10 more, and the third 20 more than the first: required the number in each flock. Ans. 55., 65, and 75. 57. At what time between 7 and 8 o'clock, are the hour and minute hands of a watch opposite? Ans. 5 T min. past 7. At what time are they together? Ans. 38y min. past 7. 58. A number of men, each at 75 cents a day, and half as many boys, at 25 cents a day, together earn daily $10.50: required the number of each. Ans. 12 men, and 6 boys. 59. A, B, and' agreed to dine together. A furnished 2 260 TEST EXAMPLES. loaves, B, 3 loaves, and C, for his share, contributed 20 cents, to be equitably divided between A and B3: required the share of each. A's. A's, 4 cents; B's, 16 cents. 60. If 30 mnen, in 40 hours, can dig 80 cubic yards; how many men, who are stronger in the ratio of 4 to 5, would it require to dig 120 yards in 90 hours; supposing the ground, in the latter case, is harder than that in the former, in the ratio of 8 to 9? Ans. 18. EXCHIIAITGE OF CiURRENTIES. 1. Reduce ~130 10s. sterling, to U. S. money, exchange being 9 %o advance. AAns. $632.20 2. Reduce ~318 7s. 6d. sterling, to dollars, exchange being 7 % advance. Ans. $1514.05 3. Reduce $1106.35 to pounds sterling, when exchange is 9 %0 advance. Anzs.: 228 7 s.: 6 d. 4. Reduce $2119.67 to pounds sterling; exchange being 7 7 advance. Arls. ~445 14 s. 6 d. 5. At $1 for 5.'30 francs, what will be the cost of a bill of exchange on Paris for 1457.50 francs?; Ans. $275. 6. At $1 for 5.45 francs, what will be the cost of a bill of exchange on HIavre for 2633.44 francs?:Ais. $483.20 7. Reduce $385.25 to French money, reckoning 1 worth 5.30 francs.. Ans. 2041.825 francs. 8. Reduce $217.25 to French nmoney, reckoning $1 Worth 5.44 francs. iAns. 1181:.84 francs. DUO D E C I Mv A L Find the product of the following lengths and breadths. Lengths. Breadths. ft. /'.'ft. /' ft. // 1. 7 9 by 3 6... Ans. 27 1 6. 2. 8 5 by 4 7... Ans. 38 6 11. 3. 8 1 by 3 5.. Ans. 27 7 5. 4. 7 5 9 by 3: 5 3.. Ans. 25 8 6 2 3. 5. 10 4 5 by 7 8 6... Ans. 79 11 0 6 6. 6. 7 6 by 5 9. Ans. 43 1 6. TABLE. 261 A Table of Squares, Cubes, and Roots, TO EXERCISE PUPILS AT THE BLACKBOARD. 1 1 1 1.009000 1.000000 43 1840 79507 6.5 7438 3.5033.08S ) 4 8 1.414213 1.259921 44 1936 85184.63 349 3.53:0348 3 9 1 7 1.732050 1.442250 45 2025 91.125 6.708203 3.556893 4 16 04 2.000000 1.587401 46'2116 97336 6.782323,0 3 58018 5 25 125 2.236068 1.709976 47 2209 103823 6.855654 3.608826 G | 6, 216 2.449489 1.817121 48 2304 110592 6.928203 3.634241 7 49 343 2.645751 1.912933 49 2401 1117649 7.000000 3.659306 8 G64 512 2 828427 2.000000 50 2500 125000 7.071067 3.6840:31 9 81 729 3.000000 2.080084 51 2601 1:32651 7.141428 3.70S430 10 100 1000 3.162277 2.154435 52 2704 140608 7.2[1102 3 732511 11 121 1331 3.316624 2.223980 53 2809 148X77 7.2801.09 3.756286 12 144 1 7,2 3.4641.01 2.289.12 5-1 2916 157464 7.348469 3.779763 1:3 169 2197 3.605551 2.;351335 55 3025 166375 7.416108 3.8029-53 14 196 2744 3.741657 2.410142 56 31:36 175616 7.483314 3.825Q62 15 225 3375 3.8729S3 2.466212 57 3249 185193 7.541 834 3.849501 16 256 4096 4.000()00 2.519S42 58 3364 195112 7.615773 3.870877 17 289 4913 4.123105 2.571282 59 3481 2015379 7.681 145 3.892996 1S 1321 58312 4.242640 2620741 60 3600 3 16000 7.745966 3.914867 19 3611 659 4.358S98 2.066102 61 3721 2o6981 7.810249 3.936497 20 400 8003 4.472136 2.714418 62 3844 238328 7.874007 3.957892 21 44t11 1261 4.582575 2.758924 63 3969 250047 7.917253 3.979057 22 4854 1068l 4.690415 2.802039 64 4096 2621.44 8.000000 4.000000 23 529 2167 4.795831 28438'67 65 4225 274625 8.062257 4.020726 24 576 13524 4.S98979 2.884499 66 4356 287496 8.124038 4.041240 25 625 15625 5.000000 2924018 67 4489 300763 8.185352 4.061541S 26 (i76 17576 5.099019 2.962496 68 4624 314432 8.246211 4.081655 27 7-29 19683 5.196152 3.00(0000 69 4761 328509 8.306623 4.101566 28 784 21952 5.291502 3.036589 70 4900 343000 8.366600 4 121255 29 841 24389 5.38516,1 3.07231.7 71 5041 357911. 8.426149 4.140818 30 900 27000 5.477225 3.107232 72 5184 373248 8.485281 4.160168 31 961 29791 5.5677641 3.141381 1 3 5:329 [389017 8.544003 4.179339 32 1024 32768 5.656854 3.174Q802 74 5476 405224 8.602325 4.198336 33 1089| 35937 5.744562 3.207534 75 i 5625 421875 8.660254 4.217163 34 1156 39104 5.830951 3.2396L2 76 5776 438976 8.717797 4.23582-1 35 1225 42875 5.91.6079 3.271066 77 5929 456533 8774964 4.254321 36 1296 46656 6.000000 3.301927 78 6084 474552 S.831760 4.272659 37 1369 50653 6.082762 3.332222 79 6241 493039 8.888194 4.290841 38 1444 54872 6.164414 3.361975 80 6400 512000 8.944271 4.308870 39 1521 59319 6.244998 3.391211 81 6561 531441 9.000000 4.326749 40 1600 64000 6.324555 3.419952 82 6724 551368 9.055385 4-'344481 41 1681 6S921 6.403124 3.448217 83 68S9 571787 9.1104133 4.362071 42 1764 74088 6.48(0740 3.476027 84 7056 592704 9.16s5151 4.379519 262 TEST EXAMPLES. INTVOLUTION AND EVOLUTIOiN. NOT-E.-The preceding table may be used to extract the square root of any number that can be separated into two factors, one of which is a perfect square, and the other not greater than 84. Thus, the square root of 700=-/100X7_10/ 7; hence, we find the square root of 7 in the table, and multiply it by 10, and the result is the square root of 700. In a similar manner, we may find the cube root of any number composed of two factors, of which one is a perfect cube, and the other a number whose cube root is given in the table. MISCELLANEOUS EXERCISES IN EVOLUTION. 1. The base and perpendicular of a right-angled triangle are 42 and 5G feet respectively: what is the hypotenuse? Ans. 70 feet. 2. The base and hypotenuse of a right-angled triangle are 21 and 35 feet respectively: what is the perpendicular? Ans. 28 feet. 3. Two men start from the same point; one travels 60 miles west, and the other 50 miles south: how far are they then apart? Ans. 78.1+ miles. 4. A ladder, 32 feet long, stands against a house, the foot resting on the ground, on a level with the foundation of the house, and 18 feet distant from it: how high on the wall will it reach? Ans. 26.45-+ feet. 5. There is a room 32 feet long, 16 wide, and 10 high; what is the longest straight line that can be drawn on the floor; and what is the length of the line joining the opposite upper and lower corners of the room? Ans. 35.77+ ft., and 37.14+ ft. 6. How far apart are the diagonal corners of a cubical block, each side of which is 6 inches? Ans. 10.392+ in. 7. HIow long is the diagonal of a square, whose side is 10 feet? Ans. 14.14+ feet. 8. HI-ow long is the diagonal of a rectangular block of marble, whose length, breadth, and thickness, respectively, are 10, 4, and 2 feet? Ans. 10.954+ ft. MAISCELLANEOUS. 263 MISCELLANEOUS EXABIPLES. 1. If each of the side walls of a house contains 80325 bricks, and each of the end walls 47975, and 1 partition, 46560; how many bricks are there in the house? Ans. 303160. 2. A has $5231 in cash, and property worth $8427; B owns a farm worth $20265, and owes $6987; how much is A worth more than B? Ans. ~380. 3. How many pounds can be carried in 729 wagons, each capable of holding 2793 pounds? Ans. 2036097 lb. 4. If 327 horses cost $28449, how many can be bought for $69861? As. 803. 5. What is the difference between $97.03 and $3.97? Ans. $93.06 6. A man bought 640 acres of land at $217.28 per acre, and a mill for $47000; how much did the land cost more than the mill? Ans. $92059.20 7. Dividing $2323 equally among 25 men, what is the share of each? Ans. $92.92 8. What will be the cost of 11 bu. 3 qt. I pt. of salt, at 2 cts. a pint?.Ans. 214.22 9. What cost 11 lb. 11 oz. 11 pwt. 11 gr. of gold, at 3 cents a grain? Ans. $2067.45 10. How long will a star be in moving 830 15', if it moves 9' in 1 mlin.? Ans. 23 da. 3 hr. 11. What cost 3 lb 2 5 2 3 2 D of Iodine at 2 cents a scruple? Ans. $18.40 12. If a man breathes 20 times in 1 min., how often will he breathe in 3 wk. 3 da. 3 hr.? Ans. 694800. -13. How many cannon balls, weighing 2 qr. 5 lb. each, can be made from 16 T. 10 cwt.? Ans. 600. 14. If a man takes 6 steps to each rod, how many steps will he make in 18 mi. 7 fur.? Ans. 36240. 15. What cost 3 hhd. of ale at 3 cents a pint.? Ans. $38.88 16. How many acres in a field 4 fur. 16 rd. long, and 3 fur. 10 rd. wide? Ans. 143 A. 17. If a wine glass holds 3 gills, how many times can it be filled from 2 p. 27 gal.? Ans. 2976. 18. What will be the cost of a pile of wood, 288 ft. long, 8 ft. high, and 6 ft. wide, at $4.57 a cord? Ans. $493.56 19. If 39 lb. 7 oz. 15 pwt. of silver be divided among 5 men, what will each lman's share be worth at 6 cts. a pwt.? Ans. 114.18 264 TEST EXAMPLES. 20. The longitude of a certain place is 84 VW.; what is the longitude of a second place, where it is 26 min. past 1 P. M. when it is 10 A. M. at the former place? Ans. 320 30/ W. 21. Two pieces of cloth, one containing 35, and the other 55 yd. are to be cut into pieces, the longest that will admit of the same number of yards in each; what will be the length of the pieces? Ans. 5 yd., the greatest common divisor of 35 and 55 yd. 22. Three piles of wheat, corn and rye, containing respectively 42, 54 and 78 bu. are to be put into sacks, the largest that will contain the same number of bushels in each: what number of bushels must each sack contain? Ans. 6 blu. 23. What is the least sum, for which I can buy, either cows at $12 each, oxen at 620 each, or horses at t.40 each? Ans. $]20. 24. A can build 4 rd. in 1 day, B, 6 rd., and C, 15 rd.; what is the least length of fence that will employ either an exact number of days? Ans. 60 rd. 25. What are the denominators of all the fractions that can be reduced to twenty-fourths? Ans. 2, 3, 4, 6, 8 and 12. 26. What -wil ~Tv of a mi. of railroad cost, at, 312.50 a ft.? -Ans. $42000. 27. H-low nmany apples, at the rate of 21 for 31 Ct8s., can be bought for $? Ans. 60. 28. What fraction of a mi. will i38 rd. 4 yd. i ft. be? Ans. 4-3. 29. At $-15.08 per yd., what will eight-thousandths of a yd. of cloth cost? Ans. 6.36064 30. What decimal of 2 bu. I qt. is 1 pk. 5 qt.? Ans..2 31. If 11 men in 12 da. plow 300 A. of land, how much can S men plow in 22 days? Ans. 400 A. 32. If 3 bu. 5 qt. of corn cost 62,02, what will be the cost of 17 bu. 3 pk.? Ans $11.36 33. If there are 2000 bricks in a pavement, 20 ft. by 30 ft., how many will there be in another, 57 ft. by 119 ft.? Ans. 22610. 34. A man bought a house for,4800, and sold it for 85700: what %o did he gain? Ains. 18 -5 35. A ii.an bought stock in a railroad at 4- Jv advance, and MISCELLANEOUS. 265 sold at 9 ~% discount; what } on his investment did he lose? An s. 12.1 %o 36. WThait cost 327 T. S cwt. 3 qr. 17 lb. of sugar, at $150 a tun? Ans. $49116.90 37. When railroad stock is at a discount of 18 %, how many shares can I buy for,$3690, par value of each share being $50? Ans. 90. 38. AWhat must be paid to insure 2 of the value of a house worthl $3000, at -9 o, t'he policy costing $1.25? Auts. $31.25 3o). What is the interest on $873.27 for 4 yr. 11 mon. 22 da., at 8 %? Ans. $347.75540. By selling goods for $400, I lost -- 7, what had they cost? Ans. $402.01+ 41. Wfhat would be thle proceeds of a note for 8575, due in 77 da., discounted in bank at 9 %? Ans. $563.50 42a Sold flour at $5.85 per bl. gaining 17 %; wha;t % would have been lost by selling 12 bl. for $57? Ans. 5 % 43. A put in $600 for 2 yr., B $900 for 8 mon., C $1000 for 4 mon.; they gtin $320; how should it be divided? Ans. A $18(, B $90, C $50. 44. If 17 lb. at 12 cts., 14 lb. at 14 cts., 23 lb. at 11 cts., and 211 lb. at 10 cts., be mixed, what will 1 lb. of the mixture be worth? lAns. 11+ ects. 45. If 2 of a pt. cost 7- cts., what will " of a bu. cost? AnRs. $4. 46. At what time between 4 and 5 o'clock, is the min. hand 9 min. behind the hr. hand? Ans. 12 min. past 4. 47. Divide $37 between A and B, giving A 1-I cts. as often as B receives 1~ cts. Ans. A $16, B $21. 48. How many gal. of beer at 15 cts. per gal., together with an equal quantity of water, must be mixed with 43 gal. of beer at 23 cts. per gal., and 35 gal. at 21 cts. per gal. to make a mixture worth 16 cts. per gal.? Ans. 28 gal. 49., of a numnber exceeds its 5 by 14;,what is the number? Ans. 114 50. Reduce ~47 16 s. 9 d. to U. S. money. ins. $231.5335 51. How many cu. ft. in a room 26 ft. 3 in. long, 21 ft. 4 in. wide, and 10ft. 5 in. high? so-?. 5833 — cu. ft. C~~ —------------- ~~X~s "'~"~~ —--------- 266 TEST EXAMPLES. 52. A ladder 46 ft. long reached as far up a wall as its foot was distant from the bottom of the wall: how high did it reach? Ans. 32 ft. 6+ in. 5.2 4-d 53. T of of =what decimal? Ans..675 54. A ball is 6 in. in diameter; what is the diameter of a ball twice as large? Ans. 7.559+ in. 55. A man whose property increased $578 annually, was, in the year 1856, worth $22471; what was he worth in the year 1842? Ans. $14379. 56. A man commenced business with $100000, lost $327 each mon., and closed with $88228; how long was he in business? Ans. 3 yr. 57. A piece of cloth 1 E. Fl. 2 qr. 2 na. wide, contains 55 sq. yd.; how long is it? Ans. 40 yd. 58. The sides of a triangular field are 5 fur., 12 fur., and 13 fur.; what will be the cost of fencing and plowing it; the fencing at 23 cents a rd., and the plowing at 10 cts. a R.? Ans. $396. 59. In a field in the form of a trapezoid, the parallel sides are 80 rd. and 76 rd. long, and are 36 rd. apar.t; what is the value of the field at $50 per A.? Ans. 8877.50 60. The circumference of a circle is 20 ft.; what is the radius? Ans. 3 ft. 2.197+ in. 61. A tree exactly round, is 7 ft. in circumference; what is the width of the widest board that can be sawed from it? Ans. 2 ft. 2.73+ in. 62. What is the diameter of a circle, whose area is 42 %o of the area of a rectangle 17 rd. long, and 11 rd. wide? Ans. 10 rd. 63. What is the solidity of a pyramid 8 ft. high, whose base is 2 ft. 3 in. square? Ans. 13. cu. ft. 64. If a cube 4 in. in length weighs 6 lb., what will be the weight of a globe of' the same material, 5 in. in diameter? Ans. 6.13+ lb. 65. What is the area of a circle whose circumference is 10 ft.? Ans. 7.9577+ sq. ft. ~~~~~~~~~~ ------ - - - - - --- --.- - — ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~4~~~~~~~~ MISCELLANEOUS. 267 66. A circular measure 8 in. high, contains 1 bu. or 2150.4 cu. in.; what is its diameter? Ans. 18.495+ in. 67. If 47 yd. of cloth 4 ft. 2 in. wide, cost $29.375, what will 37 yd. cost, 5 ft. 4 in. wide? Ans. $29.60 68. A crib 15 ft. long, 8 ft. wide, and 7 ft. high, was filled with grain as follows; 120 bu. of corn at 31~ cts. a bu., 150 bu. of wheat at 87' cts. a bu., and the remainder oats at 20 cts. a bu.; what was the value of the entire contents? Ans. $249.75 69. HIow many apples must be bought for $5, so that 20 S may be gained by selling them at the rate of 2 for 3 cents? Ans. 400. 70. How many must be bought for the $5, so that 20 % may be gained by selling them 3 for 2 cents? Ans. 900. 71. How many sq. ft. of boards, 9 in. thick, can be sawed from a log 15 ft. long, and 3 ft. 4 in. in diameter, allowing j-3 for waste? Ans. 1523.2 sq. ft. 72. How high must a triangle be, whose base is 24 ft., to be equal to a rectangle 18 ft. long, and 14 ft. wide? Ans. 21 ft. 73. 20 % of -=what? Ans..1S 41 74. How many lots, each 33 ft. by 112 ft., can be laid out from 10 A. of land, allowing 16 %o for streets? Ans. 99. 75. How many bars 8 ft. long, 5 in. wide, and I in. thick, are equal to 85 bars, 4 yd. long, 4 in. wide, and -1 in. thick? Ans. 68. 76. A borrowed $300 July 21st, 1850; 8400 Sep. 7th, 1851; $500 May 5th, 1852, all at 9 o: how much does he owe Jan. 1st, 1857? Ans. $1774.90 77. What sum will amount to $665.50 in 2 yr. at 10 rO compound interest? Ai71s. $550. 78. The time till noon to-day, is ~ of the time till noon tomorrow; what is the time? Ans. 9 o'clock A. AM. 79. An American borrowed 800 francs in Paris, Jan. 20th, 1852: what will it cost him to pay the debt in Cincinnati, July 20th, 1853, interest being 5 15, exchange 2 % above par, and $1=5.375 francs? Ans. $163.20 268 TEST EXAMIPLES. 80. A book contains 400 octavo leaves; if the same paper had been folded in duodecimo, how many leaves would it have made? Ans. 600. SUGGESTrOn.-See table, page 100, 3d Book. 81. HIow many leaves can be made fromn a quire of paper, folded in octavo?;Ans. 384. 82. Wha.t is the bank discount on $120, payable in 40 da., at 9 %C? Ans. $1.29 83. Wlhat %o of any number is 433 Yo of 3 of it? 84. A. person who spent $200 more than 4 his yearly income, had left $50 more than 4 of it; what was his income? A)zs. $1000. 85. A owns - of a house worth $3600, B,:, C tihe remainder; it is insured for $2160; what will each lose by its destruction? Ans. A $,q180, B $360, C $900. 86. A note for $300, dated Jan. 15th, 1856, at 60 cIa., was discounted in bank at 6 %, March 6th; what -was the proceeds? Ants.,299.40 87. A man having lost - of his property, and afterward gained ~ as mluch as the remainder, found that he then had $2520; how much had he at first? Anzs. $3000. 88. A ship can move 14 mi. per hr. by' steam alone, or 17 by steam and sails; how long will it be in going 3410 mi., using the sails half the time? Ans. 220 hr. or 9 da. 4 hr. 89. If 3 apples are worth 2 peaches, and 3 peaches are worth 5 pears, and 2 pears worth 3 plums, how many plums should be given for 48 apples? Ans. 80. 90. Divide $60 among A, B and C, giving A half as much as B, and B $5 more than C. Ans. A $13, B $26, C 821. 91. What is the least floor, that can be exactly covered either with marble slabs 1 ft. 6 in. square, or iron castings 2 ft. 8 in. square? Ans. one 24 ft. square. 92. How many cans 6 in. high, 4 in. long, and 1 in. wide, can be made from 26 sq. ft. of tin, making no allowance for joints? Ans. 48. 93. If 4?1 lb. of flour cost 15:} cts., what will 5 bl. cost? Atzs. 33.60 MISCELLANEOUS. 269 94. What will 180 windows cost, each having 2 sashes, and each saslh, 1.2 panes of glass, 8 in. by 10 in.; the sashes costing 37 - cts. each, the glass at $4 per box of 100 sq. ft., and the glazing 1' -cts. a pane? Ans. $303. 95. How many bu. of coal will a boat 100 ft. long, 12 ft. wide, loaded 3 ft. 6 in. deep, contain; counting the coal bu. at 5 pk. or 2688 cu. in.? Ans. 2700 bu. 96. C lent D $500 Jan. 1st, 1852; when must he lend himn I8500 more, so that he may receive $1200 on settlement, July 19tfi,:1856, money;:worth 6 % per annum? AnOs. June 7th, 1854. 97. Wfhat is the least number of lb. that would make an exact number of bu., either of wheat, corn, barley or oats? (See note 2, page 83, 3d Book.) Ans. 3360 lb. 98. How many yd. of carpeting - yd. wide, will cover 2 floors, each 22 ft.' 6 in. square? An&S. 150 y d. 99. Cincinnati, is 7~ 18/' 30" west of Washington, at what time could an event; be announced in C., that happened at noon in W., allowing 4 min. 14 sec. delay in telegraphing? Ans. 25:min. before noon. 100. If I lose 10. by selling cloth at $3.33 per E. En., what w will I gain by selling the same cloth at $2.96 per E. Fl.? Ans. 33~ o%. 101. If it:require 600 square tiles to- cover floor 10 ft. by 15:ft.; how large are the tiles? Ans. 6 in. square. 1:02. A man bought as follows, at'6 rmen. credit: Jan. 1.5th, $234; Mar. 13th, $2M05; June 1sti, $128;- What does he owe Dee. 31st? (6 %o) Ains. 8577.804 103. HIow many yd. of carpet, 32 in.a wide, will be required for a 1roln 22 ft. square, leavinfg 4 in.: around each side uncovered? A ns. 565 yd. 104. A- wheel is 6 ft. in diameter; how many revolutions will it make in going 10 mi. Anls. 2801.1]2-+ 105. Hiow many cans 4 in. long, 4 in. wide, and 6 in. high, can be filled from- 1 bu. 1 pk? Ains. 28. 106. What is the difference between Tv and.15? Ans. lQ, 107. If 317 mi. 3 fur. 2 rd. 2yd. 2 ft. 3 in. of railroad is built by 1 000 men, how much does each build? iAns. 2 fur. 21 rd. ~ yt. 3. in..o os in.s TEST EXAMPLES. 108. I pay $2100 for a note having 2 yr. 8 mon. to run; what is the face of the note? Ans. $2436. 109. A street 60 ft. wide crosses another 40 ft. wide, at right angles; what is the distance between the opposite corners? Ans. 72.11+ ft. 110. A note dated Jan. 5th, 1855, and running 90 days, was discounted in bank, Mlar. 15th, 1855, and the holder received $398.40; find the face of the note. Ans. $400. 111. At $28125 a mi., how much railroad can be built for $100000? Ans. 3 mi. 4 fur. 17 rd. 4 yd. 10 in. 112. In how many ways could a class of 8 boys be arranged, if one particular boy keeps the first place the entire time? Ans. 5040. 113. Divide 2.24 ft. by 6~ in. Ans. 4.32 114. Having mixed 7 gal. at 43 cts., 9 gal. at 55 cts., and 14 gal. at 46 cts., how many gal. at 32 cts. must I add to make a mixture worth 47 cts. per gal? Ans. 2 gal. 115. A man bought some apples 2 for 1 ct.; and twice as many pears 5 for 1 ct.; he sold them 7 for 2 cts.; did he gain or lose, and what So? Ans. He lost 4 1). 116. What is the cube root of 7 units of the 7th order? Ans. 191.293+ 117. A, B and C took in succession from 30 marbles; B took; of what A left, and C who took the rest, had 12 less than A, how many did each take? Ans. A 16, B 10, C 4. 118. Howmany wk. in the 19th century? Ans. 5217 wk. 5 da. 119. How would 3 lb 3 ~ 3 3 3 gr. be expressed in Troy weight? Ans. 3 lb. 3 oz. 7 pwt. 15 gr. 120. Divide 1 T. of sugar among 5 men in the ratio of their ages, which are 57, 53, 51, 49, and 46 yr. respectively. Ans. 1st. 4 cwt. 1 qr. 20 lb. 5 oz., 2nd. 4 cwt. 141 b 1 oz., 3d. 3 cwt. 3 cqr. 23 lb. 7oz., 4th. 3 cwt. 3 qr. 71b. 13 oz., 5th. 3 cwt. 2 qr. 9 lb. 6 oz. 121. Divide 5 T. 3 hhd. 51 gal. by the square root of the ratio of 1 bu. 5 qt, 1 pt. to 42 bu. 6 qt. Ans. 3 hhd. 61 gal. 122. A bankrupt whose debts are $30000, can pay 20 o; if he pays one of his creditors, in full, $2000, what % can he pay the others? Ans. 142 % MISCELLANEOUS. 271 123. A field containing 102 A. is 6 fur. in length; how wide is it? Ans. 1 fur. 28 rd. 124. What is the area of a triangle, whose sides are 12 ft., 35 ft., and 37 ft., respectively? Ans. 210 sq. ft. 125. How much would the fences around 4 separate square fields, each containing 10 A., exceed in aggregate length, a fence around a square field containing 40 A.? Ans. 1 mi. 126. A had $10 more than B; he then wvon $20 of B, when he had twice as much as B had left: how much had each at first? Ans. A $80, B $70. 127. For how much must I give my note at 60 da., so that when discounted in bank, the holder may receive $237.48? Ans. $240. 128. What do quills cost per thousand, on which I gain 20 % by selling at 9 cts. per dozen? Ans. $6.25 129. Which is worth the more; $600 due 5 yr. ago, or $1200 due 5 yr. hence, reckoning 10 So? and how much more? Ans. the first, worth $100 more. 130. The time since Tuesday noon is y57 of the time since Sunday noon; what is the time? Ans. 8 o'clock Wednesday morning. 131.' of A's money —-7 of B's, and together, they have $215; how much has each? Ans. A $105, B $110. 132. When must a note for $300 have been given, so that it may be worth $501.20 on Oct. 1st, 1857? Ans. July 27th, 1846. 133. The time to Wednesday noon is i-4 of the time to Saturday noon; what is the time? Ans. 4 o'clock Tuesday morning. 134. What sum of money, when divided between A, B and C, in the proportion of a,, and 4, will give C $22.50? Ans. $66.25 135. The solidity of a sphere is 65.45 cu. in.; what is the surface? Ans. 78.54 sq. in. 136. A town of 10000 inhabitants increased annually 10 % for 4: years; what was its population at the end of the 4th year? Ans. 14641. 137. Two cubes have their sides in the ratio of 4 to 5; the solidity of the first is what o of the solidity of the second? Ans. 51 0. 272 TEST EXAMPLES. 138. If 833 bl. of flour cost $402, what will 93-1 bl. cost? IAns. 844S. 139. William spent 20 o of his money: he then spent 43i' Yo of the remainder, after which he found that -- of all he had spent was $24 less than the amount he had left; how much had he at first? Ans. ~$200. 140., of A's money is $3 less than B's, alnd together they have $28; how much has each? znls. A $15, 1 $13. 141.'What is the area of a floor, whose length is 18 ft. 2' 3//, and breadth 16 ft. 5' 4//? Ans. 299 sq. ft. 1/. 142. A lost 5 % of his money; he then gained 6 % on what he had left, when he had $14 more than at first; what sum had he at first?.Azis. $2000. 143. A dog ru-ns 6 rd., while a rabbit is making 21 leaps, and the rabbit runs 4 rd. while the dog is making 9 leaps; if the rabbit is 48 rd. ahead, and makes 7 leaps while the dog makes 3; how many rd. must the dog run to overtake the rabbit? Ans. 144 rd. 1.44. The driving wheels of a locomotive being 15 ft. in circumference, the tender wheels 6 ft. 8 in., and the car wheels 10 ft. 5 in.; how far must the train run, in order that each may mlake an exact number of revolutions? Ans. 500 yd. 145. A man owed $600, payable in 4 yr. 2 mon., he pays $300 now; what sum should he pay at the expiration of the time? Ans. $225. 146. What number is that, whose half is as much less than 30, as the number itself is greater than 30? Ans. 40. 147. x of the distance from A to B — 5 of the distance from B to C; and 2 of the distance from B to C ( — of the distance from C to D; from A to D is 88 miles; how far is it from B to D? Ans. 52 mi. 148. John and William built a wall; John built 10 rd. more than - of it, and William 5 rd. less than I of it; how long was the wall? Ans. 75 rd. THE END.