ECLECTIC EDUCATIONAL SERIES. PiA Y'S ARITHMETICAL KEY: CONTAINING SOLUTIONS TO THE QUESTIONS IN IRAY' S T H IlRD B 0 OK, AND TO SOME OF THE MOST DIFFICULT QUESTIONS IN THE SECOND BOOK: BY JOSEPH llRAY, M. D., IROFESSOR OF MATHEMATICS IN WOODWARD COLLEGE. N:EW AND IMPROVED EDITION. CINCINNATI: WVIL0S N51, I INKL E & Co. PHIL'A: CLAXTON, RtEMSEN & IHAFFELFINGER. NEW YORK: CLAR K & MAYNARD. f~~~ —--- ~............ RAY'S SERIES, EEIBRACING A Thoroubgh and Progressive Course in Aritfhmetic, Algebra, and the Higher Mathematics, Primary Arithimetic. Higher Arithmetic, Intellectual Arithmletic. Test Examples in Aritllnretic tldhlimaents of Arithmetic. New Elenientary Algebra. Practical Arithmetic. New Higher Algebra. Plane avnd $olid Geometry. BY ELI T. TAPPANi, A.M[., Pres't KIeln yoe College. 12azo, cloth, 276 pp. Geomnetry anad T i'igoanometry. By ELI T. TAPPAN,, A.M., Pres't 7Kenyo7 College. 8vo, sheeip, 420 pp. Analylie GeomnI etry. By Gco. I-I. HIowisoN, A.Ai., Proqf. i 1 lI(ltss. Institute of Techllology. Treatise on Analytic Geormetry, especially as applied to the Properties of Conics: including the;Modern MIethods of Abridlged Notation. Elements of Astronomy. By S. H. PEABODY, A.SM., Prof. of Physics and Civil Lnfginseerissg, Amherst College. Handsomnely and profusely illustrated. 8vo, sheep, 336 _pp. Ray's Arithmetical ]Key (To Intellectualt and Practicatl) ]iey to Ray's Higher Arithmetic; Iey to Ray's New Elementary and Higher Algebras. _ -Descriptive Cir.culars anrl PT.rce _,ist rpon A ppli~cation to the I'tblisshers. Entered according to Act of Congress, in the year Eighteen ilundrcd and Forty-five, by WINTHno P B. SMrITH, in the Clerk's Office of the District Court of the United States for the District of Ohio. PRE F ACE. The leading objects in the preparation of the following work, have been:First, 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 exami)le 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. 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 owvn 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 plpils both rogues and liars. - -- -- - NOW PUBLISHIED. AY'S HIGHER ARITHMETIC, embracing the principles of the science, analyzed and applied; designed for advanced students and business men. An admirable and highly commended 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. OnsEwVE.-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 TEACIIING WRVITTEN 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 correctizess, 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 tc the class. In this way, they make a more decided impression on the pupils, than when presented in the text-book. M1ental exercises, suitable for the illustration of those subjects, when they can be used with advantage, will be found in a succeeding part of this volume, under their appropriate heads, and preceding the examples for exercise at the black-board. D3- _ (G 3t~IKEY TO RAY'S THIRD ]BOOK. When practicable, the pupils should be arranged in classes, due 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 arranged 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 memorizingr 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 he 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 answers, the pupils in succession may read them. When there is a black-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 the method of performing the operation. The time required to examine the questions is generally short, while the habit 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,', worthy of particular attention. its order to teach either intellectual or practical arithmetic well, the in2structor must be familiar with the subject, so as to be able tc explain, promptly and clearly, the solution of every question, as wvell as the principles on which each rule is based. lie should have, also, a strong desire to render his pupils familiar with the subject: for, it may be stated as anz acknzocwledled truth in education, thlat, a teachec' who is indiferent to the advancencent of his pupils, Zlill render them indifferent like-wise; aned that, on the contrary, one that -is deeply inter'ested isn the studelnt's progreess, will rarely fail to enkindle the same spirit. OF SYMBOLS. 7 OF SYMxBOLS. 1. The sign = is called the sign of equality; it denotes that quantities between which it is placed are equal. Thus, $1100 cents; read, one dollar equals one hundred cents. 2.'The sign + is called plus; it denotes that numbers between which it is placed are to be added together. Thus, 4+3= 7; read, four plus three equals seven. 3. The sign - is called minus; it denotes that the number placed at the right of it is to be taken fromn the number on the left. Thus, 5 —3=2; read, five minu.s three equals two. 4. The sitgn X is called the sign of mulliplication; 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, 12.3-3=4; read, twelve divided by three is i e(lll to four. Division is also denoted thus, U1, 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 number. 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 sum is seventeen. 7. A number placed over another number, a little to the right, is called an exponent. Thus, 2 and 3 are called the exponents, in the e.xpressions, 42, 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, 4 L-:4X4, 23=2X2X2, etc.'ithe exponent 2 denotes the second power, or square of the number over which it is placed. The exponent 3 denotes the te! _ a —-- 8 I(KEY TO RAY'S THIRD BOOK. third power, or cube. Thus, 3-= 3 X 3 = 9 - the square of three; and 33 —2X2X2=8=the cube of two. v. The sign 1/ denotes that the square or second root of the number before which it is placed, is to be extracted. Thus, /106-4; read, the square root of sixteen is equal to four. In a similar manner the sign / denotes that the cube or third root of the quantity before which it is placed is to be extracted. Thus, V27=3; this is read, the cube root of twenty-seven is equal to three. NUMB R ATI ON. SUGGESTIONS TO TEAciInRS.-lle careful to impress the mind of the pupil with a clear idea of a uznit. 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. Make the pupils well acquainted with reading numbers iunder 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 " ART." in "Ray's Third Book." The examples in the Key are numbered, and refer to the corresponding number of example, under the same article, in the Arithmetic. ARTICLE 13. 1, Five. 2, Sixty-three. 6, One hundred and forty3, Ninety. 4, One hundred. seven. 5, One hundred and four. 7, Two hundred and eight. NUMERATION. 9 8, Two lltndred and eighty. 37, Seven hundred and four 9, Four hundred and three. thousand two hundred and 10, Seven hundred and twen- 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 cigh- 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, Thllree 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, Tw-elve 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 thouancl 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 thousand 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, Thiree millions, thirty- 69,'Tw'\enty-three bIillions, six thousand. four millions, ninety thousand 63, Nine millions, one thou- nine hundr'ed and one. sand and three. 70, Nine hundred and fortv64, Thirty-three millions, two billions, twenty-nine milseven thousand eight hundred lions, three hundred and seven and twenty. thousand and twenty-nine. 65, Sixty-one millions, one ARTICLE 14. 1. 100. One hundred. 9. 5 in the first order is five, 2. 203. Two hundred and or five units. 4 in the seecthree. ond order is forty. 3 in the 3. 3020. Three thousand and sixth order is three lun~twenty. dr~ded thousand. 8 in the 4. 5004.y Fifty tbousandio third order is eight hunhundred 4nd four. de5004. 7 in the fourth order is seven thousand. 9 5. 9000500. Nine million, five in tis seventh or is nine hunred. in the seventh order is nine 6.70hun050Sred venty.thou sa millions. 4 in the fifth or6. 70050. Seventy thousand der is forty thousand. 2 in and fifty. the eighth order is twenty 7. 600004000. Six hundred millions. S in the 6th order million, and four thousand. is eight hundlred thoustrd. 8. 200501200. Twenty rnil- 4 in the 9thl order is funir lions, fifty thousand two huoodred millions. 5 in the hundred. 10th order is five billions. NOTATION-ADDITION. 11 NO T T I O N. ARTICLE 15. 1. 4 13. 65 25. 222 37. 503 49. 45083026 2. 16 14. 97 26. 999 38. 827 50. 870743579 3. 18 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. 4t 22. 435 34. 101 46. 30000 58. 41)0205 11. 46 23. 502 35. 21 47. 163000 59. 800669 12. 59 24. G25 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. ADDITION. SUGGESTIONS TO TEACHERS. —Simple Addition is the method of adding numbers of the same name or kind into one sum. WVe can not add numbers of different 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 sumn would be neither 5 apples nor 5 pears. If it is asked why we write units 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 understand +he reason for carrying in Simple Addition. Clerks, who add long columns of figures with great ra'pidity, never call the names of the figures in adding; they 12 KEY TO RAY'S THIRD BOOK. louk at each figure as they proceed, and either mentally or orally name the sumn which it makes in connection with those that pre.. cedle it. In this manner the addition is perfornmed in about half the time that it could have been, if each figure had been named aloud. 1 ARTICL LE 20. (Ex. 3) sheep. (Ex. 4) dollars. (Ex. 5) 4321 (Ex. 6) 50230 43 210 1254 3105 21 142 3120 423 14 35.- - 78Ans. 8695 Ans. 53758 Ans. 78 Ans. 387 ARTICLE 22. (Ex. ll) Ans. 21023. (Ex. 12) Ans. 27910. (Ex. 13) Ans. 89569. (Ex. 14) Ans. 2499593. (Ex. 15) Ains. 24194086. (Ex. 16) 23 (Ex. 17) 45 (Ex. 18) 51 (Ex. 19) 264 41 1 9 48 1 04 74 32 76 3 21. 83 74 85 155 16 55 4 Ans. 843 Anis. 237 Ans. 225 Ans. 264 (Ex. 20) 94753 (Ex. 21) Jan. 31 (Ex. 22) June 30 2847 Feb. 28 July 31 93688 March 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, 151 50 245 Next five, 153 65 325 November, 30 42 187 December, 31 89 96 Ans. 365 Ans. 246 Ans. 853 SUBTRACTION OF SIMPLE NUMIBERS. 13 (Ex. 26) 1732 (27) 1656G 67.164 1SO 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. 1.570 (32) 3005 (33) 275432 (34) 880000989 42627 402030 2002)02 105 300005 77436}(00 307004 872026 206005207 800791 4002347 490);3 320600 990019919 Ans. 5851840 Ans. 1474132 Ans. 2155513020 SUBTRAOTION. SUGGESTIONS TO TEACIERS.-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 eachl 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 KIEY TO RAY'S THIRD BOOK. ARTICLE 25. (2) sheep. (3) dollss (4) 734 245 875 531 123 751 - -.Ans. 2003 Ans. 122 Ans. 124 (5) 8752 (6) 529 (7) 79484 3421 8 251 63 An,s. 5331 Ans. 521 Ans. 54321 ARTICLE 27. (12) 4444444 (13) 91516171 (14) 63046571 1234567 15lG61718 34992884 Ans. 320987 7 Ans. 763544533 Ans. 28053687 (15) 153425178 (16) 100000000 5384524t 8 10001001 A1s. 99579930 Ans. ns. 89998999 (17) cents. (18) dollars. (19) feet. (20) years. 63 137 75 1837I 17 65 37 1492 Anzs 46 Ans. 72 Anis. 38 Ans. 345 (21) dollars. (22) dollars. 1840 10104 475 7426 Ans. 1365 An1. 2678 (23) dollars. (24) 20010 1 (0t 0) 1 901) 1 9 11 Anis. 991 Ais. 99989 (25) 912010 (26) 4000000 (27) 2020930 50C,) 4004 1 0)t)0(t, Ans. 861928 Ans. 3995996 Ans. 10:119'21 MULTIPLICATION OF SIMPLE NUMBERS. 15 MULTIPLICATION. SUGGESTTONS TO TEACHERS.-It is important that the Fupil should understand the nature of multiplication, namely, tlhat it is a short method of performing addition, when the numbers to be added together are all equal to each other. The multiplier shows how 1117my tin2es the multiplicand is to be repeated, hence it is always an abstract umnber. The multiplicalnd, however, may be either an abstract or a concrete number; thus, the abstract number 5, repeated three times, makes 15; and the concrete number, G apples, repeated three times, nmakes 15 apples. It is obvious from this, that the product is always of thie same name as the multiplicand. Some, who hlave only examined the subject superficially, suppose that two concrete numbers may be multiplied togethler, as in tile 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 3 yardls, and that tile product is 15 dollars. This view arises from an ignoroance of analysis and the elementary principles of numbers. The correct solution is as follows: T'he price of 1 yard is 5 dollals, and 3 yards being three timzes as mnlch as 1 yarc, the price wZill be three tisnes as mnuch, 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 Ans. 147 Ans. 228 Ans. 640 Ans. 2202 (16) 1427 (17) 19645 (18) 44386 7 8 9 Ans. 9989 Anzs. 157.160 Ans. 399474 (19) 70832)4 (20) 964578 (21) 9643~2 7 9 10 Ans. 4958268 Ans. 8681202 Ins. 964320 ;1 FKEY TO RAY'S THIRD BOOK. (22) 46782 (23) 86458 (24) 680323 11 12 11 Ans. 514602 Ans. 1037496 Ans. 7143553 ARTICLE 32. (25) 1236839 (29) 235 (30) 346 12 13 19 Ans. 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 (3S) 4''9 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 Ans. 94672 (40) 342 (41) 376 (42) 476 364 526 53(; 1368 2256 2856 2052 752 1428 1026 1880 2380 Ans. 124488 Ans. 197776 Ans. 255136 IULTIPLICATION OF SIMPLE NUMBERS. 17 (43) 2187 (44) 3489 (45) 1646 215 276 365 10)935 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 98'65 6835 33502 438015 417285 43074 525618 250371 191,44 700824 667656 14358 788427 500742 Ans. 16736642 Ans. 864203595 Ans. 570428595 (52-) 31624 (53) 126 (54) 823 71.38 6 12 252992 Ans. 756 dollars. Ans. 9876 dollars. 94872 31624 22.1368 Ans.'225732112 (55) 675 (56) 496 (57) 152 13 24 28 2025 ] 984 12 16 675 992 304 As. 875 cents. s. 1 cents. lns. 11904 ce Ans. 4256 miles. Key. — 2 18 IKEY TO RAY'S TIIIRD:BOOK. (58) 1760 (59) 365 (60) l130 209 24 109 15840 1460 10170 3520 730 1130 Ans. 367840 yards. 8760 hours. Ans. 1;23170 feet. 8 Ans. 70080 miles. (61) 2029 (62) 80401 (63) 101032 ]007 60007 20001 14203 562807 101032 2029 482406 202064 Ans. 2043203 Ans. 4824622 807 Ans. 2020741032 ARTICLE 33. (2) (3) 2395 (4) 2873 124 124 8 9 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. (1) 245 (2) 138 (3) 428 (4) 872 100 1000 10000 100000 Ans. 94500 Ans. 138000 Ans. 4280000 Ans. 87200000 -- _ —--- -- -- - ------- DIVISION OF SIMPLE NUMBERS. 19 A ITICL IE 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 9:.00 108006 25806 3007 36002 8602 27063 Ans. 468026000 Ans. 8860060 Ans. 27363700 (8) 80600 (9) 70302 (10) 904000 7002 80300 10200 1619 210906 1808 5642 562416 904 Ans. 564361200 Ans. 5645250600 Ans. 9220800000 DIVISITOn. 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 can 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 (lividend 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, IHow 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 - A_ - -- fi 29 KEY TO RAY'S TIIRD 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 A.ns. 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 Ans. 12312 Ans. 44234 ARTICLE 44. (4~) 3)462 (5) 5)1170 (6) 4)948 Ans. 154 Ans. 234 Ans. 237 ARTICLE 45. (12) 2)8652 (13) 3)406235 (14) 4)675043 Ans. 4326 Ans. 135411~ Ae2s. 1687604 (15) 5)984275 (16) 6)258703 (17) 6)523408 Ans. 196855 Ans. 43117- Ans. 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. 183002-4i Ans. 54841 Ans. 3472834. —.- ---- ------- DIVISION OF SIMPLE NUMBERS. 21 (24) 3)8934 (25) 4)140 (26) 4)812 Ans. 298 oranges. Ans. 35 cents. Ans. 203 barl. (27) 9)423 (28) 12)540 (29) 4)321276 Ans. $47 Ans. 45 years. Ans. 80319 gals. (30) 8)1736 (31) 7)734566 (32) 11)495.Ans. 217 barl. Ans. 104938 weeks. A~ns. 45 (33) 9)3552 (34) 2)56 (35) 2)3725 Anzs. 398 Ans. 28 Ans. 1862' (36) 3)147 (37) 4)500 (38) 5)1945 Ans. 49 Ans. 1205 Ans. 389 (39) 6)44-76 (40) 7)2513 (41) 8)5992 Ans. 746 Ans. 359 Ans. 749 (42) 9)8793 (43) 10)1090 (44) 11)4125 Ans. 977 Ans. 109 Ans. 375 (45) 12)5556 (46) 4)144 (47) 5)195 Ans. 463 3)36 each son's share. 3)39 Ans. 12 Ans. 13 ARTICLE 46. (3) (5) (6) 13)4056(312 Ans. 14)11.577(826}- Ans. 15)48690(346 Ans. 39 112 45 15 37 36 13 28 30 26 97 69 26 84 60 13 90 90 ------:- - ----— 9I 221, KEY TO RAY'S THIRD BOOK. (7) (8) 23,)1110960(48302-4 Quotient. 67)1228TS(1834 Quotient. 92 67 190. 558 184 536 69 227 69 201 60 268 46 268 14 (9) (10) 53)12412(034-~ Quotient. 72)146304(2032 Quotient. 106 144 181 230 159 216 2'22 144 212 144 10 (11) (12) 51)47100(87254 Quotient. 88) 71104(808 Quotient. 432 704 390 704 378 704 120 108 12 (13) (14) 66)43956(666 Quotient. 99)121.900(123135 Quotient. 396 99 435 229 396 1'98 396 310 396 297 130 99 31 rub __)~P.~.P —" —-' —— ~- C.-"- - - s I -~~~-PI;! DIVISION OF SIMPLE NUMBERS. 23 (15) (16) ll1).;2312(2'6 Quotient. 123)381600(3102-343 Quotient. 224 369 291 126 2'4 123 672 300 672 246 54 (17) (1s) 204)105672)518 Quotient 1234)600000(486l2-.z Quotient. 1020 4936 367 10640 204 9872 i632 7680 1632 7404 276 (19) (20) 4674)47263488(10112 Quot. 2034)26497935(13027a-ol Qouot. 4674 2034 5234 6157 4674 6102 5608 5593 4674 4068 9348 15255 9348 14238 1017 (21) (22) 9876)48905952(4952 Quotient. 12345)4049160(328 Quotient. 39504 37035 94019 34566 88884 24690 513.55 98760 49380 98760 19752 19752 24 KEY TO RAY'S THIRD BOOK. (23) (24) 973)552160000(567489$s43 Quotient. 15)3465(231 Ans. 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)95233(89 bushels. 95 856 133 963 133 963 (29) (28) 63)14868(236 hhd. 365)25U00(68 dollars per day. 126 2190 2'6 3100 189 2920 378 $180 rem. 378 (31) (30) 365)37960($104 1235)6!571435(5321 Ans. 365 6175 1460 3964 1460 3705 25943 2470 1235 1235 DI-VISION OF SIMPLE NUIMBERS. 25 (32) (33) 405)1'247400(3080 A4ns. 1006)10401000(103308- %f Quotient 121 5 1006 324.0 3410 324() 3018 0 3920 3018 9020 8048 972 A RTICLuE 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 Ails. 98 Ans. 1.46-2 6X2+1 —— 13 renm. (7) 11)19008 (8) 8)7840 12) 1 728 8)980 Ans. 144 Ans. 122-4 8X4-32 1ren. (9) 8)14771 (10) 9)10206 9)1846-3 9)1134 Ans. 205-1 Ans. 126 8X1+3=11 rem. ARTICLE 48. (2) 110)268512 (3) 1100)47100 (4) 1100)372101 Anis. 268 -2 Ans. 47 Ans. 3721 —oI (5) (6) (7) (8) 1100)4621,50 1100)620134 11000)181003 11000)3751000 An.s. 462 0% Ans. 620~3% Ans. 18y-O0 An.s. 375 6-6 T -au _6 -a -aI-~~~~~~ KEY Tt RAY'S TI-IRD BOOK. AATICLE 49. (3) 4:000)73,005 (4) 9j000)63o001 Aes. 188- Ans. 4- oD (5) 111000)107810OU (6) 1810)401617(2023 2,7 Quotient. 36 Ains. 98 41 36 56 54 2 (7) (8) 21100)9072(137(432 i i Quot. 64j00)364(006(506- 8 a Quot. 84 320 67 440 63 384 42 56 42 (9) 2510000)765416037(30o6~o~30%7 Quotient. 75 154 150 4 (10) 634100)435637154(687944 Quotient. 3804 5523 5072 4517 4438 79 A;!- - ------ - i —------- 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 1]612 181.(056 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. A- ns. 493 27! 79 772 225 3d. (3) (4) 157 3) 4928 9503 57068 264 186 586 16967 305 -479497 614 1234 Ans. 40101 123 850 946 16967 614 Ans. 332 28 KEY TO RAY'S TH-IIRD BOOK. (5) $123078 $237 spent in Jan. 538 8706 301 spent in Feb. 10 months left. 131784 538 53S0 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)31 173(362 Ans. 49 258 156 3285 273 537 3]40 273 5i6 17885 30186 pages. 213 99 2 172 160965 60372 columns. 41 160965 67 Ans. 1770615 422604 36 2'232 4044924 lines. (12) 63 gallons. 28 (11) 10 5 3 -- $ 40449240 words. 15)315(21 Ans. 25)1400(56 Ans. 30 125 4044924 lines. 47 15 150 25 150 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=g-llons 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 $30 70 6=gallons left in cistern ---- per hour. 2100 $6000 2100 6)360 25)3900(156 acres. 60 hours. Ans. 25 70 acres. 140 226 acres Ans. (18) 222 125 125 150 Ans. 347 150 30 KEY TO RAY'S THIRD BOOK. (19) 500 (20) 45 (21) 340 (22) 27 135 15 11 98 Ans, 365 225 9)351 1'25 diff. 45 246 39 675 15 Ans. 371 65 - Ans. 24 Ans. 740 ARTICLE 61.a-CANCELLATION. 4X3$ 4J1X 11 (2) — =4 Ans. (3) =11 Ans. 3 15xS2 f4XX4 (6) =30 Ans. (7) -- 12 Ans. 31xX1 3 36X (8) 37X =111 Ans. (10) -— =12 Ans. 14X9 7X$X$X3 (1i) - - =21 Ans. 6 $X$ ARTICLE 61.b (1) 42X25X18 _. 2X2XX5X;X6 -- -- 2X 5 X 6=60 Ans. 21X15 ZXIX X0 2 X0>23 (2). -=2X 23=46 Ans. 14X15 2XlX3X$ (3) - - 6 Ans. 35' x; COMPOUND NUMBERS.-U. S. MONEY. 31 (4) 21X11X 6X26 7X $X 11XgX3XT$XX g =1X3 —33 13X3X14X2,X X X X X Ans. 21X15X33XSX14X17 (5) 20 X 34 X 22 X 27 7X $X OX$XXX $X 4X X X 7X,X I — =7X 7=49 A4x x1 xx A Xnux tx X;x $ XX Ans. (6) 21X95X6 7X$3X0X19X2X$ — _ — -- 19 X 2=38 Ans. 35s x9 IxOx:x FEDERAL, OR U. S. MONEY. ARTICLrE 67. -NuIE RATION. (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 68.-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. > ------ -------- — ~ —- ---- 32' KEY TO RAY'S THIIRD BOOK. ARTICLE 71.-REDUCTION. 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, wouhl1 consist in nothing more than writing down the question to be solved, and then placing the answer under it. ARTICLE 72.-ADDITION. (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 ] 7.63 8.37 87.33 8 2;68.95 12.31 718.0o)7 Ans. $222.19 9 Ans. $68.45 Ans. $1077.58 (5) $ cts. m. (6) $ cts. m. (7) $ cts. m. 200.t)0 0 504.06 3 5.07 0 43.87 5 420.19 7 30.20 3 56.93 7 105.50 0 10(.00 5 8.50 0 304.00 5 60.02 0 2.31 3 888.45 5 700.01 1 1000.10 0 Ans. $31.1.62 5 Ans. $2222'.22 0 40.00 4 64.58 7 Ans. $2000.00 0 ARTICLE 73. —SUBT RACTION. (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) $ cts. m. 10.00 0 50.00 0 1000.00 0 1000.43 5 1 50 5 1.01 1 900.68 7 As. $9.99 9 Ans. $49.49 5 Ans.$998.98 9 Ans. $99.74 8 t~~ —----------— ~ —-----------------— =-~~~~~~~~~~~~~~L CO)MPOUN!) NUMBERS.-U. S. MONES. O33 AItTICLE 7,.-M- LT IPL ICATION. (2) $ cts. (3) $ cts. im. (4) $ cts. In. (5) $ cts. 7.83 1 2.09) 3 23.01 8 35.14 8 9 16 53 iAns. $62.64 Ans. $108.83 7 138108 1054~ 23018 17570 Ans.$368.28 8 Azis. $1860.42 (6), cts. (7). tt8. (8) cts. n. (9) $ cts. 125.02 40.04.12 5 3.-28 6 2 1)2 17 38 25004 8008 875 2624 7501.2 4004 125 984 Ans. $7751.24 Ans. $4084.08 Ans. $2.12 5 Ans. $1.24.64 (10) $ cts. (%1) $ cts. (12) 465 Here, instead of 1.06 5.75 7 multiplying 7 338 38 cents by 465, tAns. 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. An. $358.28 (13 ) cts. (14) $ cts. (15) $ cts. 34 5.67 2.69 89 24 169 306 2268 2421 272 1134 1614 269 Ans.;30.26 Ans. $136.08 An-s. $454.61 34iL.IKEY To RAY'S THIRD BOOI. (16) $ cts. (17) 73 (s3) -St ].25 63 4 219 124 11)b. 125 438 65 1125 75() 4599 gal. 5620 5;5 67 44 Ans. $S6375 - 22 995 Ans. $73.060 __.........._. _ Ans. $2529.45(19) 35 (20) 312 (21) 10 11 18 $ cts. 3 1. 25 350 yd. 33432_ hours. 54 1 1 3 Bushels 54 ~~~- ~- ~~~509 Ans. $3.5() 1 0296 625 3432 Ans. $67.50 Ans. $446.16 (22) icts.s. (23) 17 ets. m. 10.00 1 51 14 7 1]50 - 867 17 500050 85 1] 029 O 0001 882 867 lb. 11 76 Ans. 1500.15 0 Ans. $127.44 9 ARTICILE 75.-DIVISION. (4) (5) cts. yd. (6) M1. 8)280 25)300('12 Ans. 805)16;100(!(20 Ans. 25 ] g61) Ans. 35 yd. - 50 0 50 COMPOUND NUMABERS-U. S. ATONEY. 35 (7) 75)1200(16 Ans. (8) 1125)'2'4000(208 bu. Aiis 75 2235 450 9000 450 9000 (9) 4)92.250 (10) 8)57.500 (11) S cis. --- la16)25.76(1.61 Ans. Ans. $23.062qj- 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-t Ans. 3(020 626 2642 1740 2965 1565 3775 1750 3775 1565 1850 1565 (14) $ cts. m. $ cts.m. (15) $ cts. m. ets. m. 133)10000.00 0(75.18 7q-Ans. 154)'2705.01 0(17.56 5 Ans. 931 154 690 1165 665 1078 25)0 870 133 0, i) 170 1W001 1064 924 1060 770 931 770,~~~ ___ —------- — E |3 6 iKE~Y TO PtAY'S TIIIRD BOOK. (16) 25 (17) 235 15 8 ----- m. n1. cts.nm. 125 lb. 1880)122900(65=6 5 Ans. 25 11280 - cts. cts. -- lb. 375)6000(16 Ans. 9400 375 9400 2250 2250 ARTICLE 76. (1) $ ots. (2) $ cts. (3) $ cts. (4) 47.50 35.25 18,38 $ cts. $ cts. 38.45 23.75 81.62.75 5.00 15.47.- 35 3.1.0 19.43 59.00 100.00.50 59.00 100.00 1.50 Ans. $1.90 Ans. $120.85 1.00 100.00 $3.19 Ans. $119.00 Ans $300.00 (5) (6) $ cts. $ cts. cts. $ cts. 50.00 8.10.75 20.00 30.50 1.65 8 19.75 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 19.25 200.85 --- 2.50 400.00 429 2175 1.50 236.75 143 1740 124.34 9.10O.5.5o $19.75 999.00 12,60 $ ets. 8S9.25 - $400.00 21.7 5 Ans. $5.99 195. 75 9.10 Ans. $109.7 5 -.Ans. $204.25 Ans $122.65 ra —--- _ _ _, —-- -.I COiMPOUND NUIBERS.-U. S. MONEY. 37 (11) 365 (12) 21 (13) 19 (14) $ cts..65 4 2000,00 --- - 163.75 ].825 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 An s.$162.75 (15) $ cts. (16) $ cts. (17) $ cts. 4)516.00 410)910.00 22.2X11 1000.00 _1 - ~500.00 4)1_29.00 10)2.25 cts. 2) 1500.00 43)32.25(75 Ans. Ans. cts. 22 5 m. 301 11)750.00 215 Ans. $68.18+215 (18) $ ots. (19) $cts. 9 lbs. at 16 cts.. $1.44 8 yards at $1.10........80 4 lbs. at $1.25............5,0O 18 yards at 25 cts.......4.50 45 lbs. at 9 cts...........4.05 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 " " 31 cts........ 23.25 Ans. $152.10 Ans.,158.88 38 KEY TO RAY'S THIIRD BOOK. COM POUND NUIiMBERo$ In the remaining part of this work, instead of writing out the solution in fall, 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. —DRY MIEASURE. bu. pt. pk. pt. (11) 2X4XSX2-12S Ans. (12) 1LXSX>2=192 Ans. bu. qt. (13) SX4X'=I-'56 Ans. bu. plk. pk. pk. pt. (14) 1X4+1=5: 5XSX2=80 Ans. bn. ph1. pk. qt. qt. qt. Pt. (15) 2X4=8: 8XS+-2=66: 66X2=132 Ans. bu. pk. pk. pk. qt. qt. qt. Pt. (16) 4X4 —2=18: 18X8+1-=145: 145X2-=90 Ans. bu. pk. pk. pk. qt. qt. qt. pt. Pt. (17) 7X4+3-31: 31X8+7=55 5: 255X(2+1=511 Ains bu. pk. pk. qt. qt. pt. pt. (18) 3X4=12: 12X8'-96: 96X2+1-193 Ans. pt. qt. qt. pk. pk. bu. (19) 384 —. 2=19: 192. —. =24: 24 -. 4-6 Ans. pt. qt. pt. qt. pk. qt. pk. qt. pt. (20) 472 —=23 1:23- 8 —2 7. Ans. 2 7 1' tpt. cqt. 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. Cf~~~~~~~~~~ REIDUCTION. 3) pt. qt. qt. pk. qt. pk. bu. pk. bu.pk.qt. (22) 508 * —2 — 541: 24 *. 8 31 6: 31 -.t-=-7 3. As. 7 3 6 AIRTICLE 82.- T:RoY WEIG 1IT. lb. gr. lb. oz. oz. (2) 4X1X><20X24=-.23U40 Ans. (3) 5X1-+4I —64 Ans. lb. oz. oz. oz. pwt. pwt. (4) 9X12+3=111: 111X00+5='..5 Ans. lb. oz. oz. oz. pwt. pwt. (5) 14X12+11=179: 179X20+19-=3599: pwt. gr. gr. 3599X24+23-=86399 Ans. lb. oz. oz. oz. pwt. pwt. (6) 8X12+9=105: 105X20+13 —2113: pwt. gr. gr. 2113X24-4+17-50729 Ans. 0gzr. pwt. gr. (7) 171 —-244=7 3 Ans gr. pwt. gr. pwt. oz. pwt. oz. pwt. gr. (8) 505-. 24=2-' 1: 21 - 20=1 1. Ans. 1 1 1. gr. pwt. gr. pwt. oz. pwt. oz. lb. oz. (9) 130. 24=522 2 522 -. 20= 6 2: 26. 1.=2 2. lb. oz. pwt. gr..Ans. 2 2 2 2. pwt. oz. pwt. oz. lb. oz. lb. oz. pwt. (10 8U5 —. 20=40 5:'40~12=3 4. Ans. 3 4 5. gr. pwt. gr. pwt. oz. pwt. oz. lb.oz. (11) 2591. 24=1066 7: 10(6 —. 20=53 6: 53 12-=4 5. lb. oz. pwt. gr. Ans. 4 5 6 7. ARTICLE 83.- APoTHECARIEs' WEIGoT. lb gr (1) 3X12X8X3X0 —=172'S0 Ans. - -- -------— Cie 40 KEY TO RIAY'S THIRD BOOK. b 9 2 3 3 3 9 (2) 4X12=43: 48X8+5=389: 389X3=1167: 9 gt'r. gr. 1. 1i67X20-V-2z)3342! Anls. lb 3 3 3 D 9 (3) 7X12 —$4: 84XS —-672: 672X3+2=?201) Ans. lb 3 2 3 3 3 D 9 (4) 7X12+2-= 86: 86X8 —688: 688X3+1=2065: 9 gr. 20U65X20-=41300 Ans. (5) 67-12-=5 7 Ans. 3 3 3 2 lbl 3 (6) 431 —=53 7' 53- 12-14 5. Ans. 4 5 7. D 3 3 g 3 2 3 9b 3b3 (7) 975- -3 25: 325-.-S-40 5: 40-.12 —=3 4. A5 ns. 3 4 5. gr. 9 gr. 9 3 B 3 2 3 (13) 6321. 20=-316 1:316 -3=105 1: 105 —8=13 1.: 13-12-1 1. Ass. 1 1 1 1. gr. D gr. 3 3 3 3 3 (9) 30941 —. 20=1547 1: 1547_ 3=515 2: 51.5 8 —=64 3: lb ~ lb iin93 5Dgo. 64. 12=5 4. Ans. 5 4 3'2 1. gr. 3 gr. 3 3 g 3 (10) 29239 20=1461 19: 1461+. 3-487: 487.8 — 60 7: b Tlb 3gr. 60_- 12 —5. Ans. 5 7 19. ARTICLE 84.-Avo IRDUPOIS WEIGHIT. cwt. lb. cwt. qrlb. ct. r. lb. (1) 2X4X25=200 Ans. (2) 3X4+3=15: 15X25=-375 [Ans. i REDUCTION. 41 T. cwt. cwt. cwt. lb. (3) 1X'2o0+ —222: 22X4X5-=2200 -Ans. T. cw t. c wt. qlr qr. qr. lb. (4) 3X20-60: 60X4+3-243: S243X25=G 075 Ans. cwt. qr. qr. qr. lb. lb. (5) 4X4+1-117: 17X25+19=444 As. T. cwt. cwt. 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+2=-277: 277X16+-12=4444 cwt. qr. qr. lb. lb 11. oz. (8) 2X4=-8: 8X25+17=217: 217X16=3472: oz. dr. dr. 3472X16H-3-55555 ArLs T. cwt.cwt. cwt. qr. qct. lb. lb. (9) 1X20+-6926: 26X4=104: 104X25+4=2604: lb. OZ. OZ. drz. dr. 2604X16+2=41666: 41666X1+-10=-66G6666 Ais lb. qr. lb. qr. cwt. cwt. lb. (10) 4803 25=19_2 3: 1092.4=48. Ans. 48 3. lb. qr. qr. cwvt. cwt. T. cwrt. (11) 22400~-25=896: 8964=-294 224-. 20=11 4 Ans. dr. oz. oz. lb. (12) 2048000 -.16=128000: 128000+. 16=8000: lb. qr. cqr. cwt. cwt. T. 8000. — 5 —320 320 —4 —80: 80 20=4- As. dr. oz. dr. oz. lb. oz. (13) 61546 16=4034 2: 4034 - 16=-52 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. (1i) 97203 -- 16-6075 3: 6075-. 25=243: 243 -4=60 3: cwt. T. T. qr. oz. 60. 20-3. Ans. 3 3 3. _ ~~~~~L~-L~I~P - -~ 42 KEY TO RAY'S THItRD BOOK. clio. oz. oO. b1. oz. (15) 544272-. 16=3417: 34017 —. 16=216 1: lb. qr. lb. qr. cwt. qr. cwt. T. cwt. 2126 2o5-=85 1: 853. 4=21 1: 21-1 20=1 1. T. cwt. qr. lb. oz. Ans. I I 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. ARTICLE 85.-LONG M3EASURE. yd. ft. ft. ft. in. in. (1) 2X3q+2 —8: SX12+7=1.03 Ans. yd. ft. ft. in. in. (2) 7X3:2-: 21X2+11=9263 Ans mi. fur. fur. rd. (3) -12X8=9G: 96X40=3840 Ans. nmi. fur. fur. flir. rd. (4) 7X8+6=62): 62'X40-2480 Ans. mi. fur. fuir. rd. rd. (5) 9X8-72: 72X40+31-9911 Ans. in. ft. in. ft. yd. ft. yd. ft. in. (6),133 —11 —11 1 11-+-3-3 2 Ans. 3 2 1 in. ft. in. ft. yd. yd. in. (7) 1S1 —12= —15 1: 153 — 5. Ains. 5 1. rd. -fir,. fur. mi. (8) 22'40-40=56: 56-~-8=7 Ans. rd. fur. fur. mi. fur. (9)'2200- 40=55: 55-.8-=6 7 Ans. 3 -- -- --- -- ----------- 2 REDUCTION. 43 ARTICLE 88. —LAND JM EASUltE. sq. yd. sq. in. A. P. (1) 8X9lX144=13O68 Ans. (2) 4X4X40=640 A;bs. sq. mi. P. (3) 1X64OX4X40=-102400 Ans. sq.yd. sq ft. sq.ft. sq.ft. sq. in. (4) 2X9+3-21: 21X144(..3024 Ans. A R.P. R. P. P. (5) 5X4+2-=2: 22X40+20=900 An3. P. R. R. A. (6) 960-40=-94: 24 —-4=6 Ais. sq. in. sq. ft. sq. It. sq. yd. (7) 3888-144=27: 27. 9=3 Anss. P. R.P.. AR. A. R. P. (8) 943-.40=-6 3: 6-4=1 2. Ans. 1 2 3. P. R. P. R. A. R. A. i. P. (9) 603 —40=15 3: 15-.4=3 3. Ans. 3 3 3. sq. in. sq. ft. sq. ft. sq. yd. sq. ft. (10) 4176-.-144 —2'9: 29 —9=3 2 Ans. ARTICLE 89. ft. ft. sq. ft. yd. yd. sq. yd. (2) 16X123=192 Ans. (3) 5X4=20 Ans. ft. yd. ft. yd. ft. yd. ft. yd. (4) 18. =6: 12 3=4: 21+-3=7-: 15-+-3=5: sq. yd. sq. yd. sq. yd. 6X4=24: 7X5=35: 24+-35=59 Anss. ft. ft. sq. ft. sq. ft. sq. yd. (5) 18X14=2 52: 252. 9=28 Ans. rd. rd. P. P. R. R. A. (6) 35X32=1120: 11i0 +.40=28: 28. 4-47 As,. 44 KEY TO RAY'S THIRD BOOK. ARTICLE 90. sq. ft. ft. ft, sq.yd. sq. -ft. sq. ft. ft. ft. (1) 13. 11=-12 Ans. (2) 30X9 —270 270 — 18=15 Als., A. P. P. rd. (3) 9X4X40-1440: 1440 — 45=32 Ans. A. P. P. rd. (4) 21X4X40=3360: 3360-*-35=96 AtAs. ARTICLE. 93.-CUBIC MiEASURE. cu. yd. cu. in. C. cu. ft. (1) 2X27X1728=93312 Ans. (2) 2SX128-=3584 Ans. C. C. ft. C. cu. in. (3) 34X8=-272 Ans. (4) 1X128X1728=221184 Ans. cu. in. cu. ft. cu.ft. 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 Anws. ft. ft. ft. cu.ft. yd. yd.yd. cu.yId. (7) SX5X4=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. f. C. (1O) 40X12X8=3840: 3840 128=30 Anis. ARTICLE 94.-CLOTI MEASURE. yd. na. E. F. na. (1) 19X4X4=-304 Ans. (2) 14X3X4=168.Ans. yd. qr. qr. qr. n. nna. (3) 5X4+2=92: 22X4+3=91 Ans. E. En. qr. qr. qr. an. (4) 13X5+1=66: 66X4=264 Ans. 0 REDUCTION. i45 E. Fl. qr. cr. cr. na. na. ha. (5) 23X+-3=141: 141X4+-2=566 Ans. na, qr. na. qr. yd. qr. yd. qr. na. (6) 159 —4=39 3: 3.9-4=9 3. Ans. 9 3 3. na. qr. na. qr. E.Fr. na. E.Fr. qr. na. (7) 287 -4=z71 3: 71 6=11 5. Ans. 11 5 3. yd. qr. qr. E. Fl. (8) 6X4=24: 24-.-3-8 A8 s. yd. qr. qr. qr. E. F1. (9) 94+-3=39: 39. — 3=13 Anso yd. qr. qr. qr. E. Fl. q. (10) 1-X4+-1=49: 49 -.3=16 1,Ans. E. F1. qr. qr. yd. qr. (ii) 37X3=111: ll —. 4=27 3 ~As. E.En. qr. qr. E.Fr. (12) 36X5=180: 180-6 —6=30 Ans. E.En. qr. qr. qr. E.Fro (13) 22X5+4-114: 114 —46=19 Arns, E.En. qr. qr. yd. qr. (14) 47X5=235:,235_. 4=58 3 Ans. ARTICLE 95.-LIQUID MIEASURE. ga l. pt. gal. gi. (1) 17X4X2=136 Ans. (2) 13X4X2X4=416 Ans. lhh d. pt. (3) 2X63X4X2=1008 Ans. T. gi. (4) 5X4X63X4X2X4=403'20 Ans. T. hhd. hhl hhd. hd. gal. (5) 3X4-F3=15: 15X63=945 Ans.. o3 _... —-— L~~Bmsmg. 46 3KEY TO RAY'S THIRD BOOK. lhhd. gal. gal. gal. qt qt. pt. pt. (G) 1XG3-4-6O-l?23 1'23X4-,'192: 492>X2+1 —'385 Ais. hhl~d. gnl. gal. gal. q qt. qt. qt. gi. (7) 2X63-L17-14: 143X4+3-575: 55XX4 —==4600 T. Ilhhd. hhd. gal. gl. gal. pt. pt. (8)'X4-S: 8XG3-1-62566: 566X4X2+12-4 5;9: pt. gi. 4529X4=-18116 Ans. gi. pt. t. qt. qt. gal. (9) 96+ —*. 24 — 4. 2 — 12: 1- -4-3 Ans. gi. pt. pt. t. qt. gal. (10) 6048 —4-4=151i2: 1512- -75: 756 —4=189: gal. hhd. 189.-63-3 Ant. 9g. pt. pt. q.t. t. gal. (311) 32256-i-4 -8061: 8064 —9-4032: 4()3 4 —4-1008: gal. Mhd. hhd. T. 1008 —63= 16: 16 — 4-4 Ans. gi. pt. gi. pt;. qt. qt. gal. qt. (12) 4050-.-41012 2: 1012 —-2=506: 5064 —4=1.26 2: gal. hhd. hhd. qt. gi. 126+-63=2. Ans. 2 2 2. gi t. gi. qt. qt. gal. (13) 3039,'93-4-7584 3: 7584 —-2-3792: 332 —4 —-948: gal. 1t1id. gal. bhcd. T. h h d. T. hhd. gal. gi. 918 63 —'15 3: 15 —43 3. Ans. 3 3 3 3. g i. pt. g;i. pt. qt. pt. qt. gal. qt. (14) 101i25I42531 1: 2531+ —2=1265 1: 1G265-4 =316 1: 316-+ 63=5 1 5-4=1. Ans. 1 1 1 1 1. gal.. _ _al. l__d. T. hbd. T. hhd. gal. qt. X REDUCTION. 47 pnl. gi. (15) 3X84X4X2X4=8064 Ans. p. qt. (16) 5X126X4 —2520 Ans. pt. qt. qt. gal. gal. tr. (17) 5,712" —2-2856: 2S56 —:=714: 714.42 —17 Ans. ARTICLE 96.-BEEaR I MEASURE. hhd. pt. (1) 4X54X4X2-172S Ans. hhd. gal. gal. qt. qt. qt. pt. (2) 7X54:378: 378X4+3=1515 1515X2=3030 Ans. pt. qt. qt. gal. gal. bl. gal. (3) 1000- -2500: 500 —4-125: 125- 36-3 17. Ans. Pt. qt. pt. qt. gal. qt. gal. Ihhld.,gal. (4) 443:=.221 1: -)1 —4=-55 1: 55-.-54=1 1. hhd. gal. qt. pt. Anrs. 1 1 1 1. ARTICLE 97.- TIME. hr. sec. da. min. (1) 2X60X60-=7200 Ans. (2) 7X24X60-1000 A;ns. da. hr. hr. hr. min. min. min. see. sec. (3) 1X24+3-27: 27X60+44 —1664: 1664X60+3 —99243 [.AlS. rk. da.. da. da. hr. hr. (4) 9X7-+6=69: 69X24+-10- 16IGG: hr. min. min. 1666X60+40:100000 Arns. mon. wk. wk. da. da. da. hr. (5) 1X4=4: 4X7+3=31: 31X24=744: hr. min. min. 744X60-+4=44644 Ans. oa. —-----------. [n ~ 48 KEY TO RAY'S THIRD BOOK. yr. da. da. d. hrd. hr.. h min. (6) 1X365+20=3 85: 3S5X24+19=92;59: 0259XG0+15 min. mil. sec. sec. =555555: 555555X60+33=33333333 Ans. da. hr. hr. hr. min. min. (7) 365X94+5=8765: 8765X60+48=z-525948: min. sec. sec. 525948X60+48=3155699S8 Ans. sec. min. min. hr. (8) 10800 —. 6018: 180 — 60=3 Ans. sec. min. min. hr. hr..da. i (9) 432000 60-7200: 7200_ _60-120: 120 *'24-5 Ans. sec. min. sec. min. hr. min. hr. min. sec. (10j) 7322-. 60=122 2: 122 -. 602. Ans. 2 2 2. Uin. hr. mr.in. hr. dci. di. rlin. (11) 4323 —60 —72 3: 72-. ~24-3. Ans. 3 3. muin. hr. hr. da. hr. Cla. wk. wk.hr. (12) 20280__60=338: 338 -. 24=14 2: 14-. 7=2. Ans. 2 2. min. hr. mi hr. da. da. wk. da. (13) 41761. 0 —— 696 1: 696+~i24=29 29 7-7-4 1: wk. mon. monr da. min. 4 41. Ans. 1 1 1. ARTICLE 98.- CIRCULAR MEASURE. (1) 50X60+3'-=303' Ans. (2) S~X60+41/-591': 521/X60+45"/=31305"'/ Anls. (3) 3SX30 —90~900~ 0X60+-25/=5425' Ans. (4) 1cX360X6OX60=-12960002, Ans. (5) 244"//60-4/ 4"/ Ains. (6) 915/- -60=15o 15' Ans. (7) 1861 0l/0=31~ 1/:31 —30=1s 1. Ans. lP 1 1/ REDUCTION. 49 AltrICLE 0O0, bl. pt. cts. cts. $ (1) 2/4X8X24=I28: 5X128-640=6.40 Ans. bu. pk. pk. cts. cts. $ (2) 3X44-2-14: 50X14 —700=7.00 Ans. -pk. qt. qt. pt. P ts. Ct,$ (3) 3X8+3=27: 27X'=54: 3X54=1.62 Ans. $ ctS. cts. cts. pk. pk. bu. (4) 3300: 300 —15=20: 20 —4=5 Ans. $, cts. cts. cts. qt. (5) 1.66z-166: 166 —4=41 and 2 cts. over, which will bu5 qt. pk. qt. 1p. pk. 1k.,. I pt. at 4 cts. a qt. 41-8=5 1 5a4=1 1. Ibu. pk. qt. pt. AxIs. 1 1 1 1. Or thus: 4 cts. a quart is 2 cts cts. cts. pt. lbu. pl. qt. pt. a pt.; and 166 —2=83=-1 1 1 1 Ans bu. pk. pk. bul. pk. pk. pk. bags. lb3. pwt. gr. cts. ets. $ (7) 1 3-5833: 3X583217496=174.96 Azs. lb. oz. pwt. pwt. oz. pwt.prt. pwlt. pwt. (8) 2' 5 5=585: 2 5=45: 585. 45=13 Ans. lb. oz. pwt. gr. gr. pwt. gr. gr. gr. gr. (9) 1 8 18 1=10033: 5 7=127: 10033. 127T=79 Ans lb 3 gr. gr. gr. doses. cts. mn. m. (10) 1 4=6000: 6000-+-15=400: 12 5=125 ml. m. $ 125X400 —. 50000=50 An.s lb 5 3 B cts. ets. $ (11) 1 1 1=315: 4X315=12960=12.60 Ans. cwt. qr. lb. cts. cts. $ (12) 6 1=625: 625X3 — 1875=18.T75 Ans. _.~~~~~~~~~~~~~~~~~~~~~a 50 (KEY TO RIAY'S THIRD BOOK. T cwt.cwt. qr. $cts. $ (13) 1 1 —21=84: 2.25X84=189.00 Ans. lb. oz oz. cts. ts. $ (14) 7 8=-120: 5X190=600=6 Ans. gr. gr. lb 3 D (15) 2OX316=-6320=1 I 1 I Ans. lb. oz. dr. dr. dr. dr. (16) 15 9 12-3996: 3996 —.4=999 Ans. cwt, qr. lb. oz. lb. oz. oz. oz. oz. (17) 44 2 2=71232: 9 15-159: 7132-9-1.59=- 48.Ans cwt. qr. lb. lb. lb. lb. (18) 14 1 3-14-28 1428- -84. 17 Ans. cwt. qr. lb. lb. lb. lb. (19) 7 2 6=756: 756 — 12=63 Ans. cwt. lb. oz. lb. oz. oz. oz. oz. (20) 6 10-9760: 3 13=61: 9760 — 61=160 Ans. A. R. P. P. ets. cts. $ (21) 2 3 5=445: 20X445=8900-89 Ans. sq.yd. sq.ft. sq.ft. sq.in. cts. ets. $ (22) 2 2 = 20=2880: 2880X5=14400-144 Ans. sq.yd. sq.in. in. in. sq.in. sq.in. sq.in. (23) 1 = 1296: 3 X 2 = 6: 1296 - 6=216 Ans. A. R. P. P. A. R. P. P. P. P. (24) 16 1 13 —2613: 1 1 1 O=-'01: 2613-.201=13 Ans. cu. ft. cu.ft. cu.in. (25) 2X2X2=8: 8X1728=13S24 Ans. oz. oz. lb. oz. (26) 1000X5=5000=312 8 Ans. oz. oz. T. (27) 1.000X128-1'28000-4 Ans. C. cu. ft. oz. oz. T. cwt. (28) 9X128=256: 950X256=2d43200-7 1.2 Ants. ~~ — - ---— ~ - I - -- - - - ----— p — g~~~~~~~~~~~~~~~~/ REDUCTION. 51 yd. qc. na. na. yd na. na. na. pieces. (29) 13 O 2 —2'2: 666-10656: 10656. 222-=48 Ans. yd. qr. qr. yd. qr. qr. qr. suits. (30) 5 1-21: 147=588: 588 -. 1=28 Ans. yd. qc. na. na. yd. na. na. na. (31) 2 1 1-37: 37=592: 592. 37=16 Ans. yd. q. qr. E.FI. qr. qr. qr. suits. (32) 3 2=14: 70=210: 210 — 14=15 Ans. hhd. gi. cts. cts. $ (33) 1=2016: 5X2016=1-0080=100.80 Azns. gal. qt. qt. qt. qt. cts. cts. (34) 31 2=126: 126X5-630: 10X630=6300-63 Alts. $ cts. cts. cts. pt. pt. gal. (35) 2=200: 200 -40: 40=5 Ans. $ m. 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) 165-1 I20 1 1=11: 1320. 11=120 Ans. hhd. qt. cts. cts. $ (40) I=-216: 3X216=648=6.48 Ans. da. min. beats. beats. (41) 1=1440: 70X1440=100800 Ans. 52 LKEY TO RAY'S TItIRD BOOK. (42) ]840 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 wvill be found that 18238 and 1839 are not leap years, and that 1.840 is one; by adding 4 to this, we find that 1844 and 1848 are both leap years. da. hr. min. see. seec. sec. sec. (45) 365 5 48 48=31.556I928: 1609403328~- 3135536928 yr. — 51 Ans. yr. dca. da. mininm.in. da.' hr. (46) 60X365=21900: 30X21900=657000=456 6 Ans. wk. da. hr. hr. mi. ]i. (47) 3' 3 —555: SX.555=4440 Ans. da. (48) 2s 40=640 64~.2=32 Ans. (49) 900-324000": 43' 12"-'= 592'": 324000"'/ 592//=125 hr. min. =2 5 Ans. cts. cts. $ (50) 2X20=40 quires: 20X40=800:8 Ans. cts. cts. $ (51) 3X24=-72 sheets: 2X72144 —1.44 Ans. dcloz. cts. cts. (52) 3=36 apples: X36=-72 Ans. (53) 1 dozen dozen —12X1 —=.44, and 144X4-576 oranges: cts. cts. $ 3X576=1728=:17.28 Ans, (54) 1 gross =-1 dozen, and 5 gross- 60 dozen: cts. cts. $ 5X60=300 —3 Ans. REDUCTION. 53 c cs. $ts. (55) 4X196=784=-7.84 Anzs. m. m. $ (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. cts. $ 1X792=792 — 7.92.As. NoTE.-In the solutions to several of the preceding examples, 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 wllere 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 occupyfng 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 othler examnples. The quotient in all cases of division is an abstrct r nInbr, al-d the analysis of the question under consideration a11lways determlines whsat name or denomination is to be given to it. 54 KEY TO RAY'S THIIRD BOOK. ADDITION OF COMPOUND NUMBERS. SuGGESTIONS TO TEACHERs. —The attention of the pupil should lbe 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 tlhis, to explain more clearly the reason for carrying in simple addition. ANSWERS TO THE (UESTIONS IN COMPlOUND ADDITION, TO WHICH THERE ARARE NO ANSWERS IN THIE BOOK. ARTICLE 101. (4) 871b. 9oz. 17pwt. 22 gr. (5) 190b 11 33 2D lgr. (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. 090 cu. in. (14) 25 yd. 3 qr. 2 na. (15) 37 E. Fl. 2 cr. 3 na. (16) 4 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) 3209 5' 5"// ~~~~ ---------------------- - - -- — ~~~~~~~~~~~~~~~~~~~~~~~4. COTMPOUND NUMBERS. 55 (21) bu. pk. qt. pt (22) bu. pk. qt. pt. (23) lb. oz. pwt. gr. 2t 3 2:1 200() 3 0 1 2 10 15 21 14 0 5 "0 143 0 2 1 1 1 16'14 23 2 -0 1 400() 3 0 0 2 8 5 1I 18 1 0 1 255 1 5 0 22 7 _1-Ans.1000 0 0 - Ans. G 8 17 23 Aris. l,000 0 0 0 Ans. 100 0 0 0 (24) It 3 3 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 0 18 5 0 19 14 14 0 6 5 2 18 7 3 0 16 0 10 7 3 9 19 7 12 Ans. 1 3 2 016 3 19 9 Anis. 33 3 3 17 9 Anis. 99 9 9 (27) mi. fur. rd. (28) A. R. P. (29) sq. yd. sq. ft sq. in. 104 1 10 186 3 14 17 3 119 95 6 30 286 0 17 18 0 141 00 0 0 113 2 9 23 7 00 Ans. 586 2 0 29 5 116 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 0 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 0March 31 March 31 5 2 0 0 April 30 April 30 1 2 1 MaIy 1 May 31 7 2 1 - JuLe 30 18 3 0 Ans. 121 July 4 33 0 0 Ans. 1SG Ans. 10 0 0 Q (36) Cape Hlorn, 55~ 58' 30// S New York, 40 42 40 N. Ans. 96 41 10 h, m__ ___ s-~ —---------- 1 56 KEY TO RAY'S THIRD BOOK. SUBTRA CTO N. SUGGESTIONS TO TEACnERaS.-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 QUESTIONS IN COMPOUND SUBTRACTION, TO WHICH THERE ARE NO ANSWERS IN THE BOOK ARTICLE 102. (4) 8:,b. 1 oz. 18 pw\t. 7 gr. (5) 4 T. 2 cwt. 3 qr. 5 lb. 15 oz. 3 dr. (6) 7 mi. 38 rd. (7) 1 yd. 2 ft. 11 in. (8) 250 A. 38 P. (9) 2 sq. yd. S sq. ft. 104 sq. in. (10) 55 da. 5 hr. 55 min. 55 sec. (11) 90 59' 56" bu. pk. qt. bu. pk. qt. pt. lb. oz. pwt. gr. (12) 4 0 0 (13) 100 0 0 0 (14) 19 0 0 0 2 14 0 0 1 49 9 10 10 Ans. 1 2 7 Ans. 75 3 7 1 A,,s. 9 1] 9 14 lb. oz. pwt. gr. tub 3 3 gr... 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 AIs. 5 10 3 8 Ants. 1 8 0 1 14 A42s. 27 8 12 cwt.. qr. 11). mi. fur. rd. A. R. P. (18) 32 2 16 (19) 25000 0 0 (20) 146 2 ) 8 2 17 110 1 1 6 2 1 4 I ins. 23 3 24 A,,s. 24890 6 39 Ans. 59 3 26 N- --- -- -- - -- - ---- -- --- -----------— r;~I COMPOUND NUMBEIS. b 7 C. cul.ft. El. r.1 i a. da. q lr. n'a. (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. clda. hr. miin. see. yr. mon.da. (24) G63 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 As. 3 6 13 46 Ans. 80 8 1 yr. mon. da yry. mon. da. (28) 1 848 8 1 (29) 1832 6 28 (30) 55~ 58' 30" 1.822 6 24 1807 11 25 33 55 I5 Anas. 26 1 7 Ans. 24 7 3 Ans. 22 3 15 cquires. sheets. (31.) 71~ 10' 0" (32) 500 0' "Y' (33) 20 0 36 6 30 10 25 48 5 11 Ans. 35 3 30 Ans. 39 34 12 Ans. 14 13 (34) 1 dozen dozen = 12X12 —144 G dozen dozen =144X 6=864 g a dozen dozen =144_ 2 — 72 Ans. 792 ARTICLE 104.-MULTIPLICATION. bul. pk. qt. pt. lb.oz.pwt.gr. lb. oz. pwt. gr. (4) 4 3 3 1 (5) 2 11 6 (6) 10 10 1( A1m 58 l 2 0 l 3 6 10 ~Ans. 58 1 2 0 1 3 S 9 4 4 ~~~~~3 1 8044 —-~ —-------- — C i8 KEY TO RAY'S TIIIRD BOOK. lb h 3 9 gr. T.cwt. qr. lb. T. cwt. q. lb. (7) 4 0 2 15 (8) 8 2 12 (9) 10 3 14 s 9, 7 Ans.'2 8 7 1 0 A-4s. 3 17 2 8 Ans. 3 16 0 23 mi. fur. rd. mi. fur. rd. A. PR. P. (10) 90O 4 16 (11) 30 4 10 (12) 130 3 30 15 12 4 7Aits. 3128 2 0 Azns. 366 3 0 Anzs. 523 3 0 cu.yd.cu.ft.c u.in. yd. qr. na. gal. qt.pt. (13) 23 9 2os (14) 26 2 2 (15) 62 1 1 12 6 8 A 8s. 280 1 100S Ans. 159 3 0 A~ns. 499 0 0 gal. qt. pt. da. hli.min.sec. (,16) 123 2 1 (17) 365 5 48 48 (18) 4 11I 15" 5 12 8 Ans. 618 0 1 Ans. 4382 21 45 36 AIs. 1 3 30 0 doz. bu.pk.qt. mi.fur. rd. A. I. P. (19) 1.44 (21)2 3 5 (22) 3 5 16 (23) 6 3 30 12 4 6 8 1 728 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 Ans. 9 10 1 0 9 14 58 2 0 0 Ans. 6 31 2 0 0 COMPOUND NUMBERS. 59 T. cwt. lb. yd. qr. na. hhd.gal. ct.pt. (26) 16 2 2-1 (27) 37 3 2 (28) 47 3 1 l'19 - 89 5 A2s. 99 12 0 6 Ans. 3370 3 2 Ans. 44 52 2 1 (29) mi. fur. rd. (30) C. CU. ft. 27 3 35 7 938 31 17 Anzs. 852 0 5 Ans. 132'2 ARTICLE 106.-DIv ISION. (4) (5) (6) bu. pk. qt. pt. lb. oz, pwt. gr. oz. pwt. gr. 5)67 3 4 1 8)14 8 1] 16C 10)8 1i2 12 A,s. 13 2 2 1 Ans. 1 10 2 2 Ansi. 17 6 (7) (s) cwt. qr. lb. oz. dr. mi. fur. rLd. 11)35 1 17 3 7 7)39 7 8 Anls. 3 0 22 0 5 Ans. 5 5 24 (9) (11) (12) yd. qr. na. bu. pk. qt. lb. oz. dr. 5)6.2 0 3 10)490 2 4 5)3 66 9 10 Ans. 12 1 3 10)49 0 2 10);53 5 2 A2s. 4 3 5 As'. 5 5 5 (13) (i4)i llb. oz. pwt. gr. da hr. i nin. sec. 6)339 7 9 18 C)11.4 22 45 3) 7) 5 7 4 23 9)19 3 47 3 5 a're1n. Aits. 8 1 0 17 Ats. 2 3 5 172 rCm. 2X6+ —a=17 sec. relm. ---- -- - - -- - - ~~~~~~~~~~4 160 K:EY TO RAY'S THIRD BOOK. (16) (17) lb). oz. dr. lb. oz. dr. lb. oz. pvw-t. gr. lb.oz.pwt.gr. 23)'1027 1 8(44 10 8 Anzs. 245)17 7 6 6(0 0 17 6 9' 19 LAns. ] 07 245)2~1(0 oz. 16 245 91 1776 15 1715 23)241(1.0 oz. 61 23 24 11 250 16 122 74 245)1470(6 gr. 1.1'1470 23)184(8 dr. 184 (18) (19) bu. pk. qt. bu. pk. qt. mi. fur. rd. mi. fur. rd. 78)309 2 2(3 3 7 Ans. 319)788 4 9(2 3 31 Ans. 234 638 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. 5 46 9)57 319 310 REnMARK.-For cases involving the division of a compound nulmber by a fraction; also, one compound number by another, see " lRay's Iqilher Arithmetic." COMPOUND NUMBERS. 61 ARTICL,E 107. (l) (2) (3) b 3 3 B gr. oz. pwt. gr. A P]1. P. 1 0 0 0 0 8 10 12 104 2 37 4 2 0 13 9 3 5 87 I 38 8 9 7 7 5 2 7 8 1 0 35 3 1 2 1.4 26 3 0 40 1 0 5 12 -Ans. 4 3 2 1 5 3)1.- 3 35 3)25 17 12 As. 50 2 2 Or, add the two qu anti- ) - ]2 ties together, and take 5)8 their sum from 1 l. Ans. 1 14 12 (4) (5) (0) mi. fur. rd. yr. mon. da. 35~0 30' 0// 3 A25 1845 2 3 Sub. 20 35 0 4 1843 6 27 1 4 ____ --- _ _ _ _ _ _ _ - 14 55 0 1 6'20 1 7 6 Add 14 20 0 30 5 230 15 0 Ans. 54 3 0 Ans. 8 0 0 Sub. 25 4 30 4 10 30 Add 6 1 9 20 Ans. 10 29 50 ARTICLE 108. (1) 15)180 25' 30// (2) 30. 15=-2. Ans. 2hr. Aiis. 1 hr. 13 min. 42 see. (3) 1.5)710 4' 0"// (4) 15)100 35' 0"// Ans. 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 41 5 15 15 Ans 90 20' 0/" Ains. 16~ 14' 0// Ans. 770 1/ 0// 62 I( 3KEY TO RAY'S THIIRD BOOK. ARTI CI, 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. An&s. 12 37 20 Ans. 1 0 0 P. M. (10) hr.min.see. (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 lAns. 11 22 40 A. MI. Ans. 9 55 4 A. M. (12) 1940~-S0 42"-43~ 18': 430 1S'. 15= 2 hr. 53 min. 12 sec. h1r.min.sec. N o T E. -In performing the subTime at XW. 1 0 0 traction wve can not take 3 hr. fi'om Subtr. diff. 2 53 12 1 hr., but 1 P. hl. is the 13th hour Anls. 10 6 48 A. DJ. from midnight, from which, after taking 3 hr., the remainder is the 10th hour from midnight, or 10 A. AI. ARTICLE 113.-FACTORING. N o T E.-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. Ans. 3, 5, 7. (23) 168-. 92=84: 84 2=42: 4 2 — 21: 213=7. Ans. 2, 2, 2, 3, 7. (24) 216 - -2=108: 108 2=54: 54-. -2-27: 27- -3=9: 9*3 —3. Alss. 2, 2, 2, 3, 3, 3. (25) 330-. 2=165: 165 — 3=55: 55 —. 5=11. Ans. 2, 3, 5, 11. (28) 110=2X5Xl11 21.0=2X3X5X7. Ans. 2, 5. (27) 105=3X5X7: 231=3X7Xll. Ans. 3, 7. (28) 330 —~X3X5X Ill 390=29X3X5X13. Alns. 2, 3, 5. (29) 231-3X7Xl 1: 330=2X3X5Xll. Ans. 3, 11. R._'# GREATEST COMMON DIVISOR. 63 ARTICLE 114. (1) 42=2X3X7: hence the divisors are, 2, 3, 7; anid 2X3=6; X7-=14; and 3X7=-21. Ans. (2) 105=3X5X7: hence the divisors are, 3, 5, 7; and 3X5-=15; 3X7=21; and 5X7=35. Ans. (3) 20=2X2X5: hence the divisors are, 2, 5; and 2X2-4; and 2X5 —10. Ans. (4) 24:-2X2X2X3: hence the divisors are, 2, 3; and 2X2 =4; 2X3 —6; 2X2X2=8; and 2X2X3=12. Alns. AIRTICLE 116.-RULE I. (4) 42=)3X7: 54=iX3X3X3: hence 2X3=6 gr. c. d. (5) 70=2X5X7 11'0=2X5X 1 hence 2X5=10 gr. c. d. (6) 105 —=3X5X7: 165=3X5X1: hence 3X5=-15 g. c. d. (7) 60-2X2X3X5: 90-2X3X3X5: hence 2X3X5 =30 g. c. d. (8) 140=2X2X5X7: 210=2X3X5X7: hence 2X5X7 =70 g, c. d. (9) 66:-X3Xll 1.54=2X7Xll:hence 2Xll=22 g. c. d. (10) 154=2X7Xll: 280-=2X2X2X5X7: hence 2YX7 =14 g. c d. (11) 231-3X7Xll:'73-3X7X13: hence 3X7=-91 g. c. d. (12) 30=2X3X5: 422=2X3X7: 66=2X3X11: hence 2X3=6 g. c. d. (13) 60-2X2X3X5: 90-=-X3X3X5: 150=2X3X5X5: hence 2X3X5=30 g. c. d. ARTICLE 117.-R-ULE II. (4) 495 —42 —11+33 r.: 42 —33=1+9 r.: 33- -9-3+6 r.: 9 —6 —1+3 r.: 6 —3=-2. A1s. 3. (5) 323 —247=1+76 r.: 247-.-76-3+19 r. 76 —19=4. Ans. 19. (6) 465 285=1+180 r.: 85 —. 180=1+105 r.: 180 — 105 -1+75 r.: 105_75=1+30 r.: 75 - 30=2+15 r 30 — 5=-2. 1 A5 s. 15. 6(4 KIEY TO RAY'S TIIHIRD BOOK. (7) 127'4 -- 5 32=2+i210 r.: 53;. 2102 -I -]2I 1 r.: 210 —. 12 -1+98 r. 112_ 98 — 1+14 r.:.98 -14-7. Als. 14. (8) 759_ 337=2-+85 r.: 337 —. 85=3+82 r.: 85.382 =1-3 r.: 82 -3=27+1 r.: 3 -1=3. AJIs. 1. (9) 69087 -- 9873=6-949 r.: 973- -9849 —1+24 r.: 9849.'24=410+9 r. 24 - -2i+6 r. 9 -6 —1+3 r. 6 _3 —. A= 2 s. 3, (10) 1814- -1776=-1-38 r.: 1776 — 38-=46+28 r.: 38 -08 — 1+10r.: -.- 1.0=2+8.: 10 —. 8=1+-2 r.: 8 — =4.,Alls. 2. (11) 1815 —693:2+429 r. 693 —-429-=1+264r.: 4294-264 -— 1i6 15.: 2i64-*-165-14-99 r. ~ 1 — )99=1- 66r..: 99 - 66=1+-33 r.: 6G- 33 —'. AAns. 33. (12) 34'1-'14 —1 +1 36 r.: 214). 132'6=1+819 r.: 1326.. S 19 —] +507 r..: 819 5. 07i=1+31 1r.: 507 312 =1-+195 r.: 3.12) — 5.11. 7. 1954-1I7 1 17=1 — 8 r.: 117: 78= 1+- 39 r. 78 3. 39 Ans. 39. (13) 1],01.7.840=20+217 r.: 840 —217=3+1-89 r.: 17 —lS89 — 1+-2 r.: 189-.'28=G+- 1 r. 2 5 — 21=1-t7 r.: 21 —7 -3. Ans. 7. (14):153452 — 6284=2+0S084 r. 66284 —.20884 3-3 -3632 r.: 20884-363': 5-''2I4 r.: 3632 -2724 =1+908 r.: 2724 -- 08=3. Anis. 908. (15) 55 40=1+15 r.: 40 15-+=210 r.: 15.+10=-1+5 r.: 10.-.-5=2. Again, 10 5 5-2 n1. A1s. 5. Or, b)yresolving the ntimlbers into their pr-ime factors, and taking the protluct of those factors thit are common. 40= 2XX2X2 5. 55:=5X L 105)-3X7X5. Since 5 is the only factor common t- each of the numlbers, it is their only coinmon divisor. (16) 154 — 70=2-+14 r.: 70 —1-i-45. Augain, 819. 14 — 55+7 r.: 147. -' zs. 7. (17) 168-S-120:1 —48 r.: ]204 —48-2+24 r. 48 — 24- 2. Again, 1768 —24=-734-16 r.: 24-1.16=1+8 r.: 16 —8=2. Alrs. O. Or, 120-2X2X2X3X5. 168=:2 X0X2X3X7. 1768=2X2X2X)13X1 7, hence, 22X':X-8, is the greatest common divisor.:V - LEAST COMMON MULTIPLE. 6.5 AIRTICLE 119. -RULE I. (3) 12= x2x () C6=2X3 30o=_ x$x5 10o=,x5 2X2X3X5=60o Ans. 1S-8=4XX3 2X3X5X3=90 Ans. (5) 6 = X3 () 6 X 3 8=gX2X2 15=$X5 9=4 x3 35=,X 7 2X3X2X2X3=72 Ans. 2X3X5X7=210 Ans. (7) 10=2X5 (8) 9=3X3 12-=-2X2X3 1.5 —X5 15=:X X S = 18=2X$X$ 2X5X 2X3=60 Ans. 24-2X 2 X 2X $ 3 3XX5X2X2X9K=360 Ans. (9) 8= _X2X2 (10) 14=2X7 15= —3X5 2133> 02=X~ x' 30=2X X5 30=x >, X 35=-.6X 2X2X2X3Xs5=120 Aiis. 2X7X3X5=210 AnIs. REMAARIC. —When the numbers are small, or when they are few, BRule 1st is preferable; but in other cases Rule 2d is the best..It w ill be advantageous to require the pupil to work several of the same examples by both rules. ARTICLE 120.-RUILE II. (2) 2)6 10 (3) 3)15 21 35 (4) 3)9 12 5 5)5 7 35 3 4 2,3X=30 Ans. 7)- 7 7 3X3X4(:396 Ans. 1 I 1 3X5X7=105 Ans. (5) 7)14 21 (6) 3)6 9 15 (7) 2)4 14 35 2 3 2 3 5 7)2 7 35 7,<2X3=42 Ans. 3X2X3X5-90 Ans. 2 1 5 2X7X2X5=140 Ans. A/ey.-5 66 KEY TO RAY'S THIRD BOOK. (8) 2)6 10 15 18 (9) These numbers contain no common factor, there3)3 5 ]5 9 fore their least commorn 5)1 5 5 3 multiple is their product. 1 1 1 3 7XllX13X3=3003 Ans. 2X3X5X3=90 Ans. (10) 2)63 12 84. 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 2 3 7 3 1 1 31 X3X3X2X3X7=1134 Ans. 2X2X3X7X3=252 Ans. (12) 2)8 12 20 24 25 (13) 2)9 10 24 25 32 45 2)4 6 10 12 25 2)9 5 12 25 16 45 2)2 3 5 6 25 2)9 5 6 25 8 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 1 1 1 5 5)1 5 1 25 4 5 2X2X2X3X5X5-600 Ans. 1 1 5 4 1 2X2X2X3X3X5X5X4 =7200 Ans. (14) 2)98 72 64 21 18 (15) 2)2 3 4 5 6 7 8 9 2)49 36 32 21 9 2)1 3 2 5 3 7 4 9 2)49 18 16 21 9 3)1 3 1 5 3 7 2 9 3)49 9 821 9 1 1 5 1 7 2 3 3)49 3 8 7 3 2X2X3X5X7X2X3=;25'0_ Ans 7)49 1 8 7 1 7 1 8 1 1 2X2X2X3X3X7X7X8= — 8224 Ans COMMON FRACTIONS. 67 COMMON FRiACTIONS. SUG_ ESTIONS To TE.CcIERS.-A thorough knowlodge of fractions is indispensable to a proper understanding of all the most impQrtant and useful parts of Arithmetic, and is also of great utility in almost every branch of science; therefore, no pains should be spared to render the pupil perfectly acquainted with the subject. Previous to commencing fractions in written arithmetic, it is desirable that the pupil should have studied Mental Fractions. (See Ray's Arithmetic, 2d Book, page 86, following.) When this has not been attended to, the instructor will find it advantageous, at the commencement of each case, to prepare questions to be solved mentally, similar to those which follow. As far as it may be practice;ble, let the pupil be made to understand clearly the nature of every operation he is required to perform. In doing this, the examples and illustrations derived from mental arithmetic will be found of great value. ARTICLE 138.-CAsE I. NoTE.-For mental exercises, and the method of illustrating the subject, see Pay's Arithmetic, 2d Book, page 95. The only difficulty in solving the examples in this section, consists in finding the greatest common divisor of both terms of the fraction. The following are the greatest common divisors. (2) 6. (3) 30. (4) 6. (5) 15. (6) 30. (7) 14. (8) 16. (9) 5. (10) 18. (11) 14. (12) 15. (13) 97. (14) 111. (15) 101. (16) 23. (17) 117. (18) 199. ARTICLE 139. —CAsE II. From the 1st and 2d questions in this Article, it will be readily seen that the reduction of an improper fraction to a whole or mixed number, involves the same principle as reduction from a lower to a higher denomination. Thus, to reduce 17 pecks to bushels, and to reduce 17 fourths to units, or U17 to a mixed nunm4 ---- ---- - - - ---- ------ -- 6t8 KEY TO RAY'S TEHIRD BOOK. ber, all involve the same principle, and are performed in the same manner. NOTE.-FOr mental exercises embracing the principles of this case, see Ray's Arithmetic, 2d Book, page 90. The examples in this Article being all performed by division, the exhibition of the manner of doing the work would consist simply in writing down the question and then the answer; thlls, Q. (6 ), 5 =s. ARTICLE 140.-C-ASE III. From the first six examples it will be seen that the reduction of a whole or mixed number to an improper fraction, involves the same principle as the reduction from a higher denomination to a lower. NOTE.-For mental exercises embracing the principles of this case, see Tlay's Arithmetic, 2d book, page 89. (7) 8X4+3-35. Anzs. 3s. (8) 12X5-3=63. Ans.?(9) 5X10+3=53. Ans. ~ (10) 15X6+3=93. Anls.. (11) 26X24+13=637. Ans. 347. (12) 8X2+1=1-7. Ans.'7. (13) 5X4+1=21. Ans.'4. k(4) 3X55+17=182. Ans. a8sa (15) 46X8+5=373. Ans. 33. (16) 21X583+117=12360. Ans. 1286o (17) 1X1000+999=1999. Ans. 1-9. (18) 14X71+6=1000. Ans. 0O0. (19) O1Xlll-+=llll. Ans1. -1 l. A R T I C L E 141. 4X7 28 8X9 72 (2) 4. -Ans. (3) 8s- =-Ans. 1X7 7 1 9 19X19 361 37X23 851 (4) 19- = —- s. (5) 37 = —- ns. 1 X19 1 1 X23'23, COMMON FRACTIONS. 69 ARTICLE 142.-C-ASE IV. (5 ) i-?': I X-.;3 S. (6) 3>8 -— 3 ",-L i ] /",3 —-33 ARTICLE 143, (12) -X —X — Ains. (13) -X -X-=- Ans. ~ 9 9 f 9'3 9 I 17 35 (14) 1-= —. -X-X 2 Ans. 3 7 $ S 7 8 (X X -=-=2 -Ans. 6 13 O6' $ 1 (16) 1-= —. X.X - Ans. 2 1~1 ~ P (17) X-X-X-X-' — A. Ans. 2 1 $ 4 8 $ 0 1 I s) X-X-X-X-X = Ans. ARTICLE 144.-CASE V. 1X4X5 20 1X3X4 12 1X5X7 35 2X4X5 40 2)X3X4 24 2X5X7 70 3X2X5 30 1X2X4 8 3X2X7 42 4X2X5 40 3X2X4 24 5X2X7 70 4X2)X4 32 1X2X3 6 6X2X5 60 5X2X4 40 4X2X3 24 7 X2X5 70 70 KEY TO RAY'S THIRD BOOK. 1X3X4X5 60 2/57/,560 (8) -X3x4x5 -o (9) -X5x7x8 s80 2'X3X4X5 120 aX5X7XS 840 2/2X4X5 80 2X3X7X8 336 3X2X4X5 120 5X3X7X8 840 3X2X3X5 90 3X3x5x8 360 4X2X3X5 120 7X3X5X8 840 4X2X3X4 96 5X3X5X7 525 5X2X3X4 120 8X3X5X7 840 6 X 7 X13 546 (11) 1 of -=3 (12) 4 of -=511X 7 X13 1001 2. 3 4 X11X13 572 3X3X1 9 5X7 35 7 X/11/X3 1001 8X3X1 24 8X7 56 8 X11X 7 616 7XSX1 56 1X8 8 1.3X1.1X 7 1001 3XSX1 24 7X8 56 3X8X3 72 1XSX3 24 ARTICLE:E 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 1X2 1X5 (2) 4 (3)- = (4) - = (5) X -T 2/2 2/4 3/2 2X5 4 —4 X2, 1X3 2X2 4X2 — 2X3 5X2 3 - 3 3 G -G a ~T C;S —-----—' —---— 9- 8 -f -- q- T -- COMMON FRACTIONS. 71 (6 X4 —-- (7) x7 (8) x4- (9) I 3X4,)X2 4X2 5X2 3X3 ".0 8 1 0 9 6X2 7X24 8X2 4X3 14-1 4 - 6 1 ARTICLE 146.- CASE VI. (4) Least common multiple of 2, 3, and 4=12. (Art. 119, 120.) g' 6,;=_4, 3; 1X6:6 —, 1X4=4, 3X1=3. Anis. T6F,, 4- -13 (5) Least common multiple of 3, 6, and 9=18. (Art. 119, 120.) 3 -,, -' -. 2;' 1X6=6-, 1X3=-=3, 1<4+3=7: 7X40+16=296: 296X301+-25=8979 Ans. sq.yd. P. sq. yd. P. P. (26) 7506. 30-=248 4: 248-40=6 8: R. A. R. P.sq.yd 6-4=1 2. Ans. 1 2 8 4 ch. 1. 1. in. in. in. in. in (27) 5 15=515 7T92L0=T' 9: 7 z95X515=- 7o8 — 4078 lAnls. aI 0 I O 84 KEY TO RAY'S TIIIRD BOOK. P. P. (28) 401-=8 8T~ X32=81X16=1296: 1296.:40 Rg. P. R. A. A.P. — =32 16: 32-~-4-8. Ans. 8 16. da. d. da. hr. (29) 3654X4=1461: 1461X24=-35064 Ans. hr. da. hr. da. yr. da. (30) 914092_. 24=38087 4: 3807 —. 365 —104 101: yr. cen. yr. cen. yr. da. hr. 104 -100=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 dividing by 4, makes 243 days. Ans. 21 yr. 24U da. ARTICLE 160. —CAsE I. lb. oz. lb. oz. (33 w8X16i 8=4 Ans. (4)? X12== -- 4 fAns. yd. qr. qr. na. (5) X1-X4=-: X4=- Ans. A. R. R. P. (6) 7~ X4=~`u: 97 X40=- Anls. $ ct. (7) 3aX10=-30 o — Ans. da. hr. hr. min.'(8) 1-5X24: 61e x60.e=v. Ans bu. pk. pk. qt. qt. pt. (9) X4=-: X8=-o:' 3DX2=- Ans. ARTICLE 161.-CASE II. na. qr. qr. yd. (3) 4 —. -: 4 4=-;' Ans. gr. pwt. pwt. oz. oz. lb. (4) - -24l 4= O *2-: 2 g- *. 12=1 1U A~nls. 8- - - - - -7` --- -- U'go g COAMMION FRACTIONS. 85 -—.) 1 " 4 4 n i D r7 S lb pt. qt. qt. pk. pk. biu. (6) 3. =- S: -l3 3 ~- S 3 3ns. oz. lb. lb. qr. cwt. (7) -4 16-1 1 *. +25=-q: 1-. +4= AIns. in. in. na. na. qr. qr. E.E n. 3 - 21_33X4_-4 4 4 Rlin. hr. hr. da. (9) 8 *O60=-: 2 25 -; 24= 1 - Ans. dr. oz. oz.. lb. qr. (10) -16 -'5 ARTICLE 162.-CASE III. cts. cts. (3) 3 of 100=3o0=60 Ans. fur. fur. fur. rd. rd. fur. rd. (4) ~ of s= 16.=3. ~'-I of 40=8. Ans. 3 8. oz. oz. oz. pwt. pwt. oz. pwt. (5) a of 12 —a8=9~-9 3=3 of 20=z5~o-12. AnzS. 9 12. oz. OZ. oz. oz. dr. oz. dr. (6) 4 Of 16 —4-=9' — of 16=1 7P-2. AnS. 9 27. R. R. R. P.. R P. (7) - of 4-=-=2:' -- of 40=20. Ans. 2 20. hhd. hhd. hhd. gal. gal. gal. qt. qt. (8) S of 4 7=3 =1 of 63-311 -I of 4 - | Ans. 3 hhd. 31 gal. 2 qt, ARTICLE 163. —CAS E IV. hr. min. muin. da. min. (5) 13 30=810: 1-1440: ~8sL4~O —T9 Ans. fur. rd. rd. m. rd. (6) 3 25-145: 1=320: 1 -i Ans. S0 -: —--------------- 8 KEY TO RAY'S THIRD BOOK. ft. in. in. yd. in. (7) 2 8=32: I —36 8 As. ($): —5 —= Aits. that is, 96 pagoes arle of 432 pages. mi. fi'r. rd. rdcl. mi. fur. rd. rd. (9) 15 3 3=4923:35 7 7=114S7 4 23T3 A s. A. P. P. A. Pt. P. P. (10) 168 28=26908 37 2 14-6014:'- A) oz. dr. dr. lb. dr. (11) 7 1 9-_ 1 04:.i4 0 - A:ns. qt. pt. pt. bu. qt. pt. pt. (12) 2 I —1)6: 1 1 1j —23- A-ns. I6 yd. ft. in. in. yd. ft. in. in. (13) 1 1 1 3830 3 2 8 7 17= 47 _3 8 3 r.__9 0 8 Ans. ARTICLE 164. (2) (3) (4) hr. min. da. hr. in. da. hr. min. see. da.=1G6 0 1 wk. —1 18 0 wk. =4 16 0 0 h11 lir. - 40 0 Ans. 16 45 ~ hr. Ans. 2 0 15 i min.= 40 Ans. 5 6 00 40 (5) gal. qt. pt. gi. (6) A. P. l, ga1. 3 1 1; 2 R. 2 34 -. hhd. =5 1 0 0' 9 2 6 Ans. 6 0 1 1! 3 28 Ans. 2 1 (7) (0) (9) hr. iin. sec. cts. pwt. gr. das.=:18 40 405 $-l' 5 3A o. 11 -~ hr.= 3 20 - d.= 7 7- p'wt.= 21 Ans. 18 36 40 IAs. 55 Ans. 11 3 A..- _~L-~ _ __. -------— ~ COTMMION FRACTIONS. 87 (:o) () (2) fur.rd. ft. in. hr. nin.sec, qr. ia. i.1 31 da.=-20 34 17 E.E.=4 ] 1-tfur.= 25 7 6 hr.= 40 0 - -yd. =1 ()4 2An.s. 1,5 1() 4 An!s. 19 54 17' Ans. 3 01-' Or, 1 5 10 10 ARPTICLE' 165. —PROmiscuous EXAIMPLES. (1) 3'29 __9- _X_ _ 1__ 2'999Xl Ains. (1) +S1 1,' 3, 5: 2 o1-3-5 7 " 195 j1 6+51 __n + + 7'8 _1 a +2 5 9 = -2_, 1-, -19= 13 52 5 S. (2 3 ) 4 o t AT I, = I 3_.- T -U =-j = _.) 9 _ - 573 8 59 oil Als I=7'22- 1o4-, +(4)5 o ~5 8 o I 6 30__0 9 + 29 21, -g;,1 s- 9 +1!Q='9Ans. (3) 14._ 2-=4Xa=-17; 518 31 8-4 -1 8 3 34_14 l(3) _U A9s. ( 5'1 2 7Xf- 3=o -2'27 35 9 1 of I 1 ofA (6) lOX =33 6'S. (7) 10 @ OX5 3162 As.0 and if 4 is 1-seet, 7 42. is the n -u el. (9) to ny ting tere e addd of i, the sum ll be (5); i 0 is 1-seveths, -tet of, ic is 2,_ 44 TGe-1-5-5 0...... B0 - -t5s. (6) Anxs:a )=6 Ans. (7) 10 a 15=~=! (8) If - of any thing- are taken from it, there will be 4~ left; if 16 is 4-sevenths, 1:/otuth of 16, wilich is 4, is I-sevenzth, and if 4 is I-seventh, 7X4=28, is the numnber. (9) If to any thing there be added of it, the sum will be -~;i -T- 7 73 10 r if 20 is 10-seven-ths, 1-tenlh of 20, whiet is 2, is 1-seve,nth; and if 2 is i-seventh, 7 times 2, Awhicll a.re 14, is the numlber. (10), of -- and: of $000 P' —ileft ad s —~: _~-~ P., f-: part left. Or, the part left ilay be found tlius. If hel sell' of his sha1lre, 1he has 2 of L it left, d of = -. $goo=f 4$ 9 00:~v=:375 Ans. |y of 2)0=4 - 2. 88 E3 LKEY TO RAY'S THIRD BOOK. (11) _] of y-17, 1944 75O0 IIaving 7-thirty-sixths, if we divide by 7, we get 1-thirty-! sixth; then, multiplying this by 36, Are get the value of the whole. HIence the answer -will be found by nmultiplying the - of the value by 36 17%6 ~X3i -$10000 Aus. I(12) -2 of 2=4; 3: Ans. 3) 7- 93 8 8 6 ~7' =.1,, 07 Ans. 16X 11TXX. 1 | $ ~T —-iT gi- -'7 6 —; a (15) 4uiX:,:; ~?o off, of; 2 T ~'' o 5 o 7! 41 3 O1 vl 4 4Ifii 59 A 1K 16. 1 4oT4 4 U24 0 t4l(17) If a man walk 2044 rods in 7-fifteenths of an hour, he will Nwalk 1-seventh of 2044 rods in 1-fifteenth of an hour, andi 15 times this distance in an hour; this, multiplied by 1 4, will give the distance walked in 1 — hours. Thuil 7 of 2044-292, and 292X15:4380-rods walked in an hour; and 4380X1'4 -4-'-~ X?9-28408 rods Ans. 2 0 4 4x U 0 2 9 — 2044X29 292X29 Or, 2T4x 757. 8468 Ans. 7 1 (18) 14 feet-15 inches= —: 31 —'1~ 1 0 10- s. (1 9]) 3 +3.= U6 + I I_ 4 8 +15 15 -1- 3 A 4 n 0 3' nd $$ $ S $ (20),. of 2400 —5X(300=1 500 1 5 200-500:2000 If 5 of B's money is 000 i of $2000, i of 000, which is $1600 A)ns. (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'-1-0-`=440, and 44OX12-=520-=elder son's sharle; for a similar reason, divilhing this last sum by 16, and ultiplying, by 35, wrill give the whole estate: -l of 5280-330, and 330X35=11550, whole estate: Youngest son's share 2200 11550 Eldest son's share.. 5280 7480 7480 3)4070 Each daughter's share:-$1356'~ D E C II: AAL FRAC TI ON S. 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 familiar 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. ARTICLE 175.-NUME RATION. To E READn.-lst Column. (1) Five tenths. (2) Six hundredths. (3) Three thousandths. (4) Twenty-eight thousandlhs. (5) Three hundred and forty-one th]ousanzdths. 2d Column. (1) Three ten-thousandths. (2) Six hundred and twenty-five ten-thousandths. (3) Two thousand three hundred and seventy-four' ten-thousandths. (4) Two thousand and six ten-thoulsandths. (5) One hundred and four ten-thousandcths. 3d Column. (1) Four hundred-thousandths. (2) One hundred andI thirty-seven 7hlndred-thotsandths. (3) Two thousand, three hundred and seventy-six hlldred-thousanldths. (4) Thirty-one thousand four hundred ans1 fifty-six h,?2dred-thousandths. (5) One thousand and seven hundredz -thousandths. 4th Column. (1) Seven mzillionths. (2) One hundred and thirty-three 1nillilonths. (3) One thousand seven hundrled and sixty-eight millionths. (4) Forty thousand and thirty-five a:dillionths. (5) Three hundred and sixty thousand and four rillionths. ~~ ~ — — ~ ------ ---- --- ~ - -MYII,~-~ —~ I~~J~OI~LIIIIU~Y ~.e KEL FY TO IY' SIC THIRD' BOOK. 5th Columnn. (]) Eight ter-nillionleh s. (2) Nine h.undred-evilioanths. (3) One million ten thousand one hundred and one ten-millionths. (4) One hundired thlousand thiree hundred and four hzlZdred-mlilldonths. (5) Forty thousand and five hndrecldntillionths. 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 thousanzdths. 7th Column. (1) Six tenths, or Six million ten-nillionths. (2) Eighllt thousancdths, or Eighty thousand tez-millionths. (3) One lhundred ancl two thousanzdths, or One million and twenty thousand te.n-illionths. 8th Column. (1) Forty thousand five hundred and four units, and one thousand andl thirty-seven hundred-thouslandth/s. (2) Fifty-four million units, and fifty-four seillionths. (3) Thirty million seven hundred and one thousand units, and one million thiipty-seven thousand and twenty-five teC-nvillioniths. ARTICLE 176. (1) T-0-.5; T-6.o. 0 6. 0 0 (l s; ~a=oo35,0020o3. T o O o=.000 400 (2) 13- =13.7; ~4-r 24.0S; 301 )30.025; — i0o-o=6.0004; 8~y go-e-S.000010 5 (3).4 (9).4125 (15).300004 (21).00000007 (4).26 (10).020; (16).000203 (22).20020003 (5).35 (11).0008 (17).000007 (23).01010001 (.F).08 (12).20304 (18).0000024: (24) 90.25 (7).005 (73).006005 (19).0080006 (25) ]06.037 (s).304 (A,:).0)0009 (20).o00000002 (2s) 1000.001 (27) 200.0 25 (30) 200.0000000000 (28) 29.()00020 (31) 65.006005 (29).001000005 AIRTICLE 177. (22) 14.3 (33) 32.04 (34) 1.0502 DECIMAL FRACTIONS. 91 ARTICLE 178. —-ADDITION. 6.4 6 88.3 * 466.3:1 |.35 3.4.167 6516.14 74.(19 Als. 10.00 Ans. 6913.3477 2 3 -Ans. 133.333 Alns. 7231.0967 (6) 45.019 (7) 432.432 (8) 16.0,il 7.00071 6 1.0793 9. 000094 93.4'327 100.07 1794 33.27 6.o40l 6.0o09 s.969 I ns. 11.49231 000.1001 32.7199OO06 Ans.!51'(92' Ans. 1599.693834 -Ans. 100.00I0000 (9) 204.0009 (10).0035 103.00000009.00035 42.009(09.000035 430.9900.)000035 Al(0Os..0038885 Alns. 999.99999999 Alt'TILE 179.-SUBTT ACTION. (2) 5.03' (3) 24.0042 (4) 170.0035 (5).0142'2.1 i]5 * 13.7013 6.0(1 81.()()5 4Ins. 2.915 Ans. 10.3029 Ans. 102.00169 Ans..0092 (6).05 (7) 13.5 (8) 3.00000 (9) 29.0029.0094 8.037.00003 1).003 A ns..0476 Ans. 5.463 Ans. 2.99997 Ands. 9.9999 (10) 5.00ooo0 (11) 10000.0000 (12) 1.00000 (13).025. 125.00() 1.()0001.t I ()0()2 i nJ s. 4.875 Ans. 9999.99'399 As..9 A9ns..0. -i5 R TI C LE 8!. — I U LTIPLICATTION. (2).35 (5) 15X 7- Xo7O=' lo 0 0=.10 A's. 7 Aos. 2.4"5 (8) 1.035Xi7 d17.595 Ans. |92. fEY TO RAY'S THIRD BOOIK. (9) 19 (10) 4.5 (11).6'5 (12) G1.76 __2__4 64.0071, 95 A2s. 18.0 2500 617l 35 3750 43232 1- Amts. 40.000 Ans..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 Aains. 6241.000 Ans..0948700 (17).043 (18) 40000 (19).00375.0021.000001.1000064 43 Ans..040000 375 00 86 56250 Ansi..0000903 9 7 5 Ans..09375600000 DIVISION. SUGGESTIONS TO TEACHIERs.-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 unmerstand 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 conutins more decimal places than the dividend, it is best, before conmmencing the division, to reduce thenm both to the same denomination, that is, to make the number of decimal places the same in bothll; the quotient will then be a whole number. ARTIOLE 183. 1.125 2 2.000 (5).37.5 Ans. (6) -— =. - 250 0 Ans..03.008.008S.^,. —- - --- - ---.. Y~ f DECIMAL FRACTIONS. 93 37.2 37.20 86.075 (7) - 7.44 Ans. (8) — 3.13 Ans. 5. 5. 27.5 24.73704 206.166492 (9) d- -=7.191 Ans. (10) 23 — 50.004 Ans. 3.44 4.123 100.8788.000343 (11) - =.2222 Ans. (12) -.0001 Ans. 454 3.43 9811.0047 9811.004700 (1.3) -90300 Ans..108649.108649.21318 (14) -131.81122 Ans..1 9 102048 102048.0000 (15) - 320000 Ans..3189.3189.102048 9.9 9.9000 (16)' -.000032 Ans. (17).-. =440 Ans. 3189.0 225.0225 (18) 10).o0 (19).1)1.0 (20).01)10.00 Ans.01 Ans. 10. Ans. 1000 1.7 1.7000000 (21) -- 065625 Ans. 64 64 (22) 80).080 (23) 1000).001000 (24).001).001 Ans..001 Ans..000001 Ans. 1. (25).0000001).0001000 (26).000004).010000 Anws. 1000. Ans. 2500. 1.5 1.5000000 (27) 5- - 2.142857+.7.7 ARTICLE 185.-CASE I. 1.000 _4.0 (2) 1= -—.1925 Ans. (3). 8 Ans. 8 5 _ ---- - ---------- ~ —— e (4),.0. -- - -- A - (4) 16 Ans. (5) 3. 07. An. ~~~25 ~40 15.0000 1.oooo (69) I> -. 93 75 Ans. (7) f...oo 000 A,, 16 1250 9. 00000 (81) 1-'3-3=022~5 Ans. 400 1.00000000 I(9) 1 250 003905 Ans. 2,56 5.0000 (1o)..=-..8333+ A ns. 6 (i1) 4.4000000_.12124- Ans. 33 (12) 1.- 000000.090909 — Ans. ARTICLE 186.-CASE I1. (2) 6- = 5 _3 b Ans. (3) ~;1 — P n. (4 ).37,5= Pk.-7 4=.1 XJZS (4).- 3.7-5=Tj== Ans. (5). 4.02=4 l.<27=44o Ans.. (6) 8.415=8f —ks' Ans. (7).033=Ta, Ans. 52(8) -9- o__ Ans. (9).34375=..). 1- As. (10).14s43750 19' a0-j =1 Ails. A.x~o~o T J. ARTICLE 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.,, —- -- --- ---- DE CIIMAL FRACTIONS. 9| (a).0004 375 mli.X8X40=-.14 rd. Ans. (9).25 pt. — 2 =.li5 qt.:.125 qt. 4=.03125 gal. Ais. (10).6 pt.-.-2=.3 qt.:.3 qt. — 8=.0375 pk.:.0375 pk. —4-.009375 bu. Ans. (11).3 min. —GO=.OO hr6.: 0 05 h r.:.4=.00050S3 da.+ Aizs. ('12).7 rd. —40-=.0175 fur:.0175 5fur. — 8=.0021875 mi. Ans. ARTICLE 188.-CASE IV. (2).875 yd.X4-3.5 qC.:.5 qr.X4=2. na. As. 3 qr. 2 nm. (3).7 ib. X1-=8.4 oz.:.4 oz.X920=8. pwt. Ans. 8 oz. 8 pwt. (4).8125 bu.X4-3.2'5 pk.:.25 pk.><8=2. qt. Azs. 3 pk. 2 qt. (5).3375 A.X4=1.35 R.:.35 R.X40=14. P. Ans. 1 IR. 14 1). (6).04318 mi.X8=.34544 fur.:.34544 fur. 40 =13.8176 rd.:.8176 rd. X51=4.4968 yd.:.4968 yd.X3 -=1.4904: ft.:.4904 ft.X1'2=5.884S in. Ans. 13 rd. 4 yd. 1 ft. 5.8848 in. (7).33625 cwt.X4-1.345 qr.:.345 qr.X25=-8.625 lb.:.625 lb.XlG —10 oz. Ans. I qr. S lb. 10 oz. ARTICLE 189.-CASE V. (2) 2 pk. 4 qt.-20 qt.: 1 bu.=-32 qt.:.~-8-. 6'25 Ais. (3) 13 hr. 30 min.= S10 min. 1 da.5=1440 min. 1I4 4 ~=- I=.i6'S. (4) I ib.=256 aldr.,: 9 =. 03515625 Ass. (5) 1 A.=160 P'* _.000175 Ans. 160 (6) 1 da.=1440 mnin. T-47z —-.0043611+ Ans. (7) 4 gal. 1 qt. 1.28 pt.-35.98 pt.: I hhd.-504 pt. 35.28 00 7 Ans. 504 96 KEY TO RAY'S THIRD BOOK. (8) 3 pk. 7 qt.. 1 pt.=63 pt.: 2 bu. 2 pk. 4 qt.=168 pt. fGLf-=-.375 Ans. (9) 99 pages= —',- of 512 pages: 9 —-.193359375 Ans. (10) 55 A. 2. 17 P.SS97 P. 7 P.-2421875 Ans. 229 A. 2 R. 16 P.=36736 P. ARTIC LE 190, (1) $0.4X9=$3.6: 80.1875X12:=$2.25: $3.6+$2.25=$5.85 Ans. (2) $0.45X2.3=$1.035: $0.375X1.5=$.5625. $1.035+$.56"25=$1.5975 Ans. (3),2.6875X16=$43: $2.6875X16l=$43.671875 Ans. (4) 835.25 — $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=$36d.8 8884- 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) 1.3 A. 2 R. 35 P.=13.71875 A. $17.28X13.71875=z$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 qr.=1.685 qr.:.685 qr.X4=2.74 na.: 2.74 na.+.4 na.=3.14 na. Ans. 1 qr. 3.14 na, yd. qr. na. (15) 1.53 yd.=l 2 0.48 1.32 qr.=- 1 1.28 Ans. I 0 3.20 DECIMAL FRACTIONS. 97 (1l) 365.25X.05=-18.2625 d'.2625 da.X24 —6.3 hr. 6.'3 hr.-.5 hr.-5.8 hr..8 hlr.XGO=48. min. Anis. 18 da. 5 hr. I8 nuin. (17).41 da.=9.84 hr.: 9.84 hr.-.16 hr.=9.6e hr.:.6S hr.X60=40.8 min.:.8 min.X60-48. sec. Ans. 9 hr. 40 min. 48 sec. (lS).4 T.X4=1.6 hhd.:1.6 hhd.+.3 hhd.-1.9 hhd.: 1.9X63=119.7 gal.: 119.7 gal.-..8 gatl.=120.5 gal.: 120,5 gal. X4-482 qt.-964 pt Ains. (19) 365,25 da.X.3-109.575 da.:.575 da.X24=13.8 hr.: o.8 hr. X60=48 min. A!s,. 109 da, 13 hr. 48 min. (520) 365.25 da.X.005X24XOOXG0X l157788 sec. Anls. (21) 1 cord=128X1728-221i184 cu. in. I,-1g=.000004+ Aims. (22) 17 ami. 3 fur. 15 rd. —7.421875 mi. $690.35XI7.421875$ —12027. 19140625 Ans. REMARr.-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 iunto 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 fiactions. Thus, they may lbe 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. 1By decimals. I + 3 +3=2. (1) 4+= 4 + = Ans. -=.375 =.625 AR,s. Ke. -— 7 938 IKE:Y TO RAY'S THIRD BOOK. (2) g + -. IAts -. 3125. (3) Ans. =.373 7 7.-. (6) a 4 A1-.,s. 9 The great advantage of decimals is, that the operations are perf'ormed with the same facility as in whole numbers; and, that the most complicated arithmetical expressions may be redCed to equivalent decimals that are comparatively simple. PROPORTION. SUcESTION'S TO TEAcInERs. — Simple proportion is justly regarded as one of the most important subjects in Arithmetic; p articular care should be taken, therefore, to render the pupil well acquainted with its principles. First, let him thoroughmly understand the meaning of ratio; if the questions and exercises in the Arithmetic are not sufficient, let the instructor add such as lie may deem necessary. It should be kept before the pupil that there is no such thing as ratio between quantities of different. kindls, and 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, 1an d, 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 oto the second. Thus, the question, if 2 yards of cloth cost 5 dollars, how mtany 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 metlod of reasoning will apl ly to every case, both in simple and comlpound proportion. ARTICLE 203. (8) As 4 hats: 10 ats: $14: $35 Anis. e> - s. —~- -- ~ —-------- ----- ---- ---------------— eS;L~A PROPORTION. 99 (9) As 3 caps: 11 caps:: 9 cts.: $2.53 Ans. (10) As 4 yd.: 9 yd.: $7 $15.75 Ans. (I1) As 8 yd.: 12 yd.:: 32: $48 Ans. (12) As 12 yd.: 8 yd. $4-8: 832 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. 1 gal.: 21 gal.:: 65 cts.: $13.65 Ans. (20) As 5 cwt. 3 qr. 10 lb.: 35 cwt. 1 qr.:: $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.::'$22.6: $113.00 Anls. (26) As 4 ft. 6 in. in shadow: 180 ft. shadow:: 3 f't. 120 ft. Anls. (27) As 12 hr.: 9 hr.:: 60 da.: 45 da. Ans. (28) As $1: $2200:: $.60: $1320 A's, Ans. As $1: $1800: $.60: 81080 B's, Ans. (29) $800.30+$2;500+375.1 0+$500+$115-$0420 40 As $2040.40: $612.12:: $1: $.30 Ans. 30 ets. (30) As $6: $8:: 9 oz.: 12 oz. Anls. (31) As $300: $250:: 6; mon.: 5 mon. Ans. (32) 27X7-1S9, 36-27 —9. As 9 mi.: 189 mi.:: 1 da.: 21 da. Ans. (33) 09 hr.: 1 hr.::,$15: 4$0.S88, one month's services when lhe wolrks 12 hr. a cday. $20.88sX4`$91.91g A us. when lie works I') Z 100 IiEY TO RAY'S THIRD BOOK. (34) As 5 lb.. lb.: $. $ Ans. (35) As G yd. 7yd.::$53: $~6 A'ls. (36) As bu. - bu.:: $. A12. 3NX1-=9 (37) As 1- yd.: 2 yd.:, $. Ans. 4-X X27X.:z.. (38) As $294: 31::: 59-' yd. Anzs. T-TX125X'-19= 125=62-1yd. 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 7 yd.: - E. En.": $: Ans. 20 -45 7 x4 ix 3=$ -I qr.: 401 qr.:: $ -5: Is. 7X4As$. inS. (42) As 3 yd.: 174 yd.: $4: Ans. X' 1- 9.X =5$178.38- Ans. cogs. cogs. rev. rev. (43) As 26: 35':: 1: 19. Hence, the smaller wheel gains 96 of a revolution in each revolution of the larger wheel. rev. rev. rev. of larger. Then,'2: 10 1: 2S= —revolution of larger, Ans. (44) 1 gal. -32 gills, 32-1 —31. As 32: 31:: 100 gal.: 96' gal. Ans. (45) As 70 p.: 20 p.:: 60 sec.: 17 sec. iAns. 1142X17-1=95771 ft.=3 mll. 5 fur. 14.5 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.: $1G.53+ As. (49) As 56 lb.: 1000 lb.:: 63 cts.: $11.25 Ans. (50) As 32 lb.: 3000 lb.:: 24 cts.: $22.50 Ans. AR TI C LE 205. ( 2 )da. 10 da. } 4 mi.' 240 mi. Ans. 4 hr.: 8 hr. 240 SUGGESTION TO TAcs.CERs.-In statring questions in Colnpornd Proportion, it, will be found both interesting and instructive forl PROPORTION. 101 the ptpil 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. Ains. 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 meon. The more months, the more dollars.:: $150. Ans. $937.50. (6) As 7 da.: 9 da. The more days, the mnore miles. 6 hr. 11 hr. The more hours, the more miles.:: 217 mi. Ains. 5114 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 ts.: $60.60 The more money, the more mi. 2: 20 mli, 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. I.: 87 rd. 1. The more length, the more days. 5 ft. h.: 8 ft. h. The more bight, the more days. 4 ft. t.: 5 ft. t. The more thickness, the more da.:15 days.'Anes. 58 4 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 vd.w.: I ycl.w. The more wridth, 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. 102 KEY TO RAY'S THIRD BOOK. ALIQUOTS OR PRACTICE. ARTICLE 207.-CASE I. (3) $24=cost of 24 vd. at $1 (5) $173 50.1 12=cost of 24 yd. at 50 cts. 25 —|!$ 43.25 Ans. 12, -- 3 =''.4.. at 124 cts. $15 Ans. (6) $45 (7) $54 25 -- 111.25 25 =- 13.5(0 2 g 25. 8 161 0 288 41 -- 124I~l 4 25 = 20 50 =-.' 18 -. ~_a- j-i - 4.50 AIns. 3(4) 0 Ans. $190 A2S. $3 19.50 (12) 77 (13) 175 (14) 224 1 14 3 50 =- 38.50 15 25 =, 56 124., 2 -- 126=1 28 1 4 9.62 2450 50V _E IIAns. $125.121 25 43.75 64= 14 1,)',= 21.8371 Ans. $770 Ans. $251 5.62(15) 462 (16) 185 1 1 I>! o 1 (; I~s4 6~T'O 28.87) ~ 331l 28 Ans. $490.87 Ans. $2456.62 3 46 185- - -— ~ —-— e PRACTICE. 103 (17) 150 (18) 24 3 1 450 24= 331a=_ 50 L 4 33 A-s 5ns. $28 Ans. $550 AtTICLE r 208.-CAStE II, (2) $1io (3) $12.50 (4) $12 14, 28 5 140 10000 60 2 R.=,' 5 2500o R.=4- 3.00 Ans. $14 1350.00 10 P.Z 4_.75 2 R. —- 6.5 Ans. $03.75 1 R.=. 3.12" Ans. $359.3374 (5) $IS (6) $12.50 (7) $1.75 12 14 3 216 5000 5.25! R.=l 4.50 1250 2 qr. -.87 5 P1.= j.391 l ~ ft~~=,.lz. 175.00 Ans. $IS6.362 l (8) $1.50 (9) $17.25 (10) $6 4 56 83 6.00 035a0 498 2 q. -— J.75 8625 2 pk. —-. 3.00 1 qr.=~.37~ 966.00 1 pk.- 1.50 Ans. $7.1'21 2 ctr. S.6 2 qt.4.3. 1 qr.. 4.31a Ans. $502'.87 Ans. $978.934 10 )4 KEY TO RAY'S THIRD BOCiK. (11>) $4 (12) $3.20 (13) 2. 24 40 17 6 128.00 1792 2 pk. I 9.00.2 pi —-- 1.60 256 1 pk.:,} 1.00 1 pk..80sO 43.52 4 qt.=.:.50 4 qt.-'.40 1 pk.-1.64 2 qt.=-.5 2 qt.-.20 1 qt.=-.08 3 qt.=.{ 12. I qt. —-.10 1 pt.-.,.C04 S 1 pt.=}.05AnilS. $99.874 1 p Ans. $44.28 Ans. $131.15 (14) $.24 (15) S.56 (16) $3.20 5 3 17 1.90'[.6 240 8 oz.=-.1 oz.-t. 3 2 oz. —4.03 4 oz.:.6Z 4 4.40 oz.= 01O 1 Oz. —.16 2 R..? 1.60 141 d 0420 P.=.40 2 ( -.:.02 10 P=.40 I di [.02.20 A.Is3. 09.91 4 P.=. Ans. $57.5S ARTICLE 209.-PERCENTAGE. 7 %.O 7 4=' %.04 51 %=.0o525 10 %o=.105: 37, 7=.375: 10 1"0 o=1.20 ARTICLE 210.- CASE I. (2) 7 %o=.07 * $175X.07 =$12.25- AIns. (4) 6 7 -.06. $250X.06 =$15.00 Ans. (5) 7 Y =.07: $162X.07 =$11.34 Ans. (6) 5 o.05: $118X.05 =$ 5.90 Ans. (7) 8 =08 $ 11 X.08 -$ 0.88 AIns. (8) 1 %o=.01: $,278X.01 $ 2.78S Aris (9) 21 /f =.02o:: $ GSX.02o=$ 1.53 Anls. et^l- -------- -------- PER;TCENTAGE. 105 (o)) 4- %L.045: $220.50X.0 5=$9.922+ Ats. (11) 7 - %.0,:75 $115.42X.075=-$8.656- + iAns. (12) 5= %o 5. 05.: 243.1i6X.05,-=$13.0S14- Ars. (13) 3: 0X=.03-o: $1250X.03.$40.625 Ans. (14) 25 %-.25: $25X.25=86.25 Ans. (15) 101-A -— 1.015 $82002X1.015=$2032.03 _Ans. (16) 208 %=-2.08: $650X2.08=$1352. Ans. (17) 1.000 % —10: $24.75X/10-$247.50 Ars. (18) 11' %.1=.001 $400X.001 —0.40 Anls. (19) ~ o=.00-: $464X.00-=-1.74 Aris (20) - 4=.00-. -:,$150X.00-1l$1.6'2). Arts, (21) 8s %=.08.: $72x.o0S-%-.G.1 ~ ~7-$6.12 =865.88 Ars. (22) 36 =.-36: S800X.36: $2SS: $800 —$288s$5112 Als. (23) $56X<.13-=$7.35: $51X.14, =$7.48: $7.48-7.35 — 80.13 Ans. (24) $1764X.23=-$405.72: $1764X.31X546.S4: $1764 —-.09-$1158.76 $1764X.03=:857.33. By adding together the amounts expended, we find their sunm is $11G.65, wh-lich, being deducted friom $1764, the remainder is $5905.35, 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% to be expended in nmuslin: 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) 3290X05=16: 320 —16=304: 304X.25-76 304-76 -2218 Ans. (28) 18-+15-f23+12=68: 100 —68-32 % left: $800X.32 —$256 Ans. Q, - X 106 KEY TO RAY'S TIIIRD BOOK. ARTICLE 211. —CASE II. (2) $15X,01=$.15: $3.00 —. $.1520 Alns. (3) $50X.01=$.5: 6-.-$.5-=12 Ans. (4) $75X.01-$.75 $4.50 —$.75-6 Ans. (5) ==.331 —331 Yo Ans. (6) $25X.01=$.25: $0.25 —.25=1 Ans. (7) $142.60X.01=1.426: $7.13 — 1.496=5 Ans. (8) $9X.01-$.09: $9 —$.090100 Atns. (9) $9X.01=$.09: $13.50 —.09=150 Ans. (10) $243X.01=$2.43: $8.50.5- $2.43=3.5=3- Ans. (11) $2X.01=.02: 8.002 -.029=.1-= — j' Ans. (12) $3532X.01=$35.32 $13.245- $35.32.375=.- Ans. (13) $300X.01 —3 $95-. -$3=-8' Ans. (14) $500X.01=-5: $75-+$5=15 Ans. (15) 3=.G6O=GO0 Ans. (16) 5 bu.-160 qt.'- -- -.034-=3- % As. ARTICLE 213. -COIMIISSION. (1) $240X.05=$12.00 Ais. (2) $460X.0a2=$11.50 Ans. (3) $180X.04=$7.20~ $119X.03=$3.57: $7.20+$3.57=$10.77 Ans. (4) $240X.03-=$7.20 $225X.05=811.25: $7.20+$11.25=$18.45 Ans. (5) $4S0X.00-=$10.80~ $275X.03-=$8.80: 8216X.022=$ 5.76. $10.80+$8.S0+$5.76 -— 5.36 Ans. (6) $275)X.03==;8.25 341X.15=$51.15: $964X. 25=$241.00: Si217X.02-=4.8825 $8.25 +$5 i.15+$24 41.00+$4.SS25-$305.'93.S5 ARns. PERCENTAGE. 107 (7) $13X2-$325: $325X.0Q2=$8. 121 Agent receies: $325-$8.12>'$31.6.871 Owner receives. (8) 1114 1b.X15=167101 lb.: 8 cts.X16710=$133G.80: $1336.80X.034=$43.446 commission: $1336.80-$43.446-$12993.354 Anis. (9) $ZEX2,50-.$3 750: $7X175=$1225: $0.25X1456 =$364.00: $3750+$1225 +$364=$5339: $5339X.03 =.-$160.17 com.: $5339-$160.17-=$5178.83 Anas. ARTICLE 214. (11) $1.00+.04:=$1.04 $312- -$1.04=$300 Ans. (12) $1.0-t.0S=$1.08: $1323.54-. -$1.08=81225.50: $13293.54-$1225.50=$iS. 04 Anis. ARTICLE 215.-INSURANCE. (1) $"'_,50X.01=$33.75: $33.75+81.00-$34.75 Ans. (2) $12600+$14400=-$27000: 3 of $27000=$18000: $18000X.04,=8z810.00: $810+-$1=$811 Ans. (3) 4 of $5000-$3750: $3750X.00=,$81S.75, $18.75+$1.50-=20.25 Als. (4) 4 of 21000=$12o000: l12000X.10-'=$1500 $1200X.01-=$18.00: $15005-$18+851+$1-81590 Ans. ARTICLE 216. (6) 1-.01-=.99 $2475,-..99-$2500 Ans. (7) 1-.125=.875: $131255 —.875=$15000 Ans. (S) 1-.01=4 1-.0175-=.9825:2358-.9825-2400 Ans. (9) 1-.00=-1 -.003 5=.99G 2O: $2287.39-.99625 =$2296 ~ $296-9G-$22S7.39 —8.6 1 Ans. Or, 1 —,0~0=i.99 — 7 2 2 87.39 - 7'9 7=$2296, etc 108 KEY TO RAY'S THIRD BOOK. A RTI'CLE 217.-STo cK S. (1):$1400X.04-=,56 $] 400+-$56 —= 1456( Ans. Or, 104 % —1.04 $1400X1.04 —$1456 Alms. (2) 81400X.96=$1344.00 Ans. (3) $50Xll- $550 $550X.05-=$27.50': $5501-%27.50 =-$577.50 Anis. (4) $75X15=$1125: $11i25X.10=$112.50 - $1125-$112.50=-$1012'.50 Ans. (5) 108 — 104-1=3-: $1500X.03=$5750 A2s. Or,' -1500X1.04j-$1567.50: $1500X1.085-$1 —625.00: $16'25.00-$-1567.50,$57.50 A4ns. (6) 3+5 5 —8.i- $1300X.08:=-$110.50 Ans. Or, $1300X.97-$1961.00 cost ~ $1300X1.050-.$1371.50 sold for $: 1371.50 —1261.00 =5110.50 Ans. (7) 4+2:=6: $860>X.06.=$55.90 A.ns. Or, $860X1.04=$S94.40 cost: 860X.97 —=$83S.50 sold for: $894.40-8S38.50=$855.90 Ans. A. T I CLE 218.-Bno KERAGE. (1) $600X.01l —$9.00 Ans. (2) $289X.01 —$3.611 Ans. (3) $360X.003=$1.35 $360 —1.35:=$358.65 Ans. (4) $134X.02o 1$3.35: $134 —$3.35=$130.65 Anzs. (5) $200X.00=-$1.25: $200+81.25 —$201.25 Alns. (6) $+!=3. 8500X.003=$3.75 Ans. Or, $500X.001=$2.50 gain in buying. $500X.00' =$1.25 gain in selling. $2.50+$1.25-.-.3.75 A41s. ARTICLE 220.-CASE I. (3) $65 X.05-$3.25': $3.25X4=$13.00 Ans. (4) $200X.08=$16.00 Aits. (5) $150X.05=$7.50 Ans. -~~~~~~~~~~~~~~~~~~~~~U PERCENTAGE. 109 (6) $300X.o061S.o.00: $1s8.ox -=$36.00 Alls. (7) $2375X.06$16.50:,$16.50X3-=$49.50 Ans1. (8) $1S7.50X.05=$9.u375: $9.375:X4-$37.50 Ais. (9) $233.SOX.06-$14.02: $14.028X10 =-$140.28 Ans. (11) $215.00X.06-$12290. $215++1'.90==$27.90 Ans. (12) $45..0X.0=$3.60:,$3.60X2 —7._: -45++$"7.20$52.2 0 Ats. (13) $80.00X.07= $5.60: $5.6oX4-$22.4 0: $0+$'0-22.40=$102.40 Ans. (14) $420.(00)X. 05 3 —o.o 4 420.00-+$422.40$4 0 A ns. (15) $,237. 6X.03-1=$8.8935:.$S.935X'2-$17.7s87: $237.16+ —$17.787=$.254.947 Ans. (16) $74.75X.04-= 2.99::2.99X5=z14.95: $74.75+$14.95=$z89.70 Ans. (17) Q85.45X.06-z-$5.127: $5.127X -4= —0.508: $S5.45 -$20.50-, 8=$105.958 lAns. (18E) $325.00Xoo5-=z17.55: $17.55X3.4)5-$2.56: $325q$52.65-$377.65 Ans. (19) $129.36X.04}:$G5.6595: $5.6595X4=$22.638: $129.36+$22.638 —$151.998 Ans. ARTICLE 221. —CASE II. (2) $240X.08=$19.20: 2 mon. —-: yr.: G of.$19.20=$3.20 As. (4) $360X.06=,$21. 60: -11 of $21.60-_1.80: 2 d.=: -:x mon.:' - of $1.80-=.12 Als. (6) $86X.06S$5.16: 3 mon.- yr.: I of $5.16=$1.29 AIs. (7) $50X.0S=$4.00: 4 mon. — yr.:. of 4.00-$1.33 1.Ais. (8) 150.05X.08=$12.02: 6 mon.=- yr.: of,1'2,02 —$6.01 Asis. (9) $36oX.05-13lS.00: 7 mon.==-7 yr.:,f $818.00-=$10.50 Anls. ~~~~~u~~~~~~~~~~~~~___ _ ~~~~~~~~~~~~~~~~~~~~~~~~~~~_~~~~~~~~,~~~~~~~~~~~~~0~~~~~~~~~241~52-) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~L~~~~~~~~~ICIII~~~~~~~~~~~~~~~~~~~~~~~Y~~~~~~~~i~~~~~~~~ 1 10 KEY TO RAY'S THIRD BOOK. (10) W$204X.07-$14.23 ~ 11 mon.= r.: 1', or, 14.2s=$13.09 Ans. (11) $,26X.06=$43.56: ~.- of,$43.56-=, 3.63. 10 da.= mon1110 of' $,.G3=$1..21.2 As. (192) $1200X.06 $72 of.00o$f.:) 15 -.=.l" mllon. of 86,00-$3,00 Ans. (13) $180X.08=-$14.40: 72L of 814.40=$1.20: 19 dt.::i8 o mon.: 3 8 of $1.20=$0.76 Ans. (14) $240X.07=$16.80: - of $16.80=-$1.40. 27 do. 7-T= Il ion.: -9 of $1.40-$1.96 Ans. (15) $100.SX.05 0 $5.04: 0 -1 of $5.04-=.42: 2.S I = s _I 4 14 of' $0.42=$0.39.0 Ans. (16) $2SX.o06=1 3.i8 ~ 9 Ilion.= I=- 4 y. - of $13.68S$10.26: $228+$10.26=238.26 Ans. (17) $137.,50X.06$8G-$S. 8 8 yr. 3 of 88.25=I5.50: $137.50+$5.50 — $143.00 Ands. (18) $1-50X.05 —`7,50: ~- of $7.508.G625: 18 da.=-~ -= Ion of $: 2?._ 35 1500+$8.375 — $150.35 A is. (19) 8360X.06=$21. 60~ of $01.60=$1.80: 11 da.='I mon.: 11 of $1.80=$.66 $3060+$g.66=$360.66 Ais. (20) $264X.06=$15.84: -l of $15.84=$1.32: 9 da.=-9% —3 on.: - o- of. 3= 2.396: $264+$.396 —$264.396 iAns. ARTICLE 222. (2) $350 (3) $150 (4) 37:5.40.04.()06.06 14.0O 9.00 i22. 4(0 7 4 6 mon.=:l 1.262 |98.0)0 7 o 2 mlon.1 3.754 3 imon. —=4 3.50 2 mon.=6 1.50 Aims. $37.540 Ans. $ l0i.50 Aits. $37.50 PERCENTAGE. 111 (5) $92.75 (6) $500 (7) $560.06.06.)S 5.5650 30).00 44.80 3 1 Imon. 2.T 2.50 _ 16.6(j95 15 d. 2_ l' 8. mon.~= 1.8;55 3 da. =- -5 4 mon.=1 14. 93. I mon.==.463-+ Ans. $34.00 15 da.-= 1.86Ans. $19.013+ Ans. $1.06.40 (8) 75 0 (9) $456 (10)._21.06.n 5. I () $4t5.00 $22.80 21.30 4 3 5 1 ~80.00 68 t.40 08.)0 3 mon.-= 11.25 4 mon.- 7.t60 6 mon. —-- ]().80 t6 da. =-a 0.75 1 mon. 1.90 I mon.1 —[ L.80 Ans, $192.00 15 da. -.9 15 d.90 3 da.. =.19 10 di..6 Aits. $79.04 Ans. $122.'22) (11) $380 (12) $300 (13) $;50.15.06.06 190o0 $18.00 1j.o00 380 3 6 mon.= l 7.50 $57.00 54.00 1 mon.= I' 1.25 3 6 mon.=l 9.00 Irit., $23.75 171.00 2 mon.- 3.00 25().0)) 6 mon.=- 28.50 Int., 66.00 Ans. $273.75 3 mon.= 14.25 3(0.1)0 9 da. o 1.425 Ais $66.00 Ans. $'215.175 112 KEY TO PRAY'S THI-IRD BOOK. (14) (15) (16) $205.25 8i50.62 $210.25 -.06.05.07 $12.31I50 *7.53J 0 $14.717 15 2 3 2 4.(30 22.593 29.4350 6mont 0. 1,575 4 mon.=- 2.510+6 mon.-= 7.3587+'2mo.-.0 o.=.627 2 1 mon.=.627 1 on.=- 1.2264 15 da. 5 I53l 12 da.-j.251 20 da. - 8.817G Int., $33.3531+ Int., $25.981+ Int., $38.8377+ 205.25 150.62 210.25 AbIs. $238.6031+ Ans. $176.601+ Ans. $249.0877+ (17) (18) (19) $57.85 yr. morn. da. yr. mon. da..05 1849 4 19 1849 4 27 1847 1 9 1848 2 15 $2.8925 _ 2 2 3 10 1 2 1'2 35.8 7850 $150 $240 3 mon.== 4,2-31-+.06.08 2|0% 2 d_ 0160 2 2 mo0n.:l 3.20 2 dt~.:-=~t.0160 18.0o 0 2 2cl.:n..64.t $ 92s8+ 3 mon.- 2.25 As. 253.04 5 7.8 15 0 4 ibs. $23.04A Io da. =5 [.-5_ Aas. $64i.5428q-t A.s. 220.50 PERCENTAGE. 113 / (20) (21 (22) yr. mon. da. monr. da. mon. da. 1845 8 28 11 27 S 28 1843 5 14 7 3 3 1 2 3 14 4 24 5 27 $180 $137.50 $125.40.07.09.os0 12.60 $12.3750 100320 2 6270) 4 mon.- 4.125 25.20 20 da. -.6875 $10.6590 10IdOa 1 3.15 4 dn. =-'.1375 4 mon =- 3.553 da.2 dam'.07 A3s. 4. 1.9500 1 mon.=0l.8S8+ 2 de =1 15 d.-..444 2 da. -.07 l d'44 10 da. =~J.296 Ans. $28.84 2 da. =.059 Int., $5.240+ 1 25o 40 ~('?') Ans. 8130.644yr. moa1n (1t (24) 1848 3 9 yr. mon. da. 1 P4 7 8 2 1847 7 24 1846 10 25 7 7 $234.60 8 29.05 15s3.s80.05 11.7300 05 5865 7.6900 12.3165 6 mon.-' 3.845 6'A,')r:',:I 6.15825 6py. r~r; 6.X 1 58i25:2 mon =. 1 8216+ I mon..-::I 1.02637+ 20 da -' -4172 6 da. ='.20527 5 da. =.1068 t da. -6..034251 4 da. =.0854 V d. _ _ _.03421 Int., $7.42410 1n, 87460 234.60!n s. 22,4-2.0?4+ Ans. $159.546+.;D.-t 1114 KEY TO RAY'S THIRD BOOK. AR3TICLE 223. (3) 16 mon. 13 mon. 4 mon. 18 da. 7 mon. 12 da. Ans. $.08 Ans. $.065 Ans. $.023 $.035 Ans. $.037 10 mon. 13 dh. 5 mon. 17 da. 11 mon. Ajis. $.0521 $.025 Ans. $.055 25 Ans. $.027i 2 mon. I da. 9 mon. 3 da. 14 mon. 4 da. Ans. $.01o0i Ans. $.0454 Anls. $.0o70-3 17 mon. 27 da. 33 mon. 20 da. $.085 $.165 41 31 Ans. $.0891 Ass. $.168 - ARTICLE 224. (2) $.04X65=-$2.60 Ans. (3) $.055X36==$1.98 Ans. (4) $.08X28=$2.24 Ans. (5) $.125X500=$62.50 Ans. (6) $.022X75-=$1.65 Ans. (7) $.048.X186=$9.083 Ans. (8) $.072X129-5=$9.00 Ans. (9) $.2()4X210.25=$42.891 Ans. (10) $.087A1X134.45=$1 1.764+ Ans. (11) $.138~X144.24=$19.929+ lAns. ARTICLE 226. (2) Principal.. $00.00 Interest to 1st pa/yment, (12 mon.,)... 1.00 Amount due, Jan. 1st, 1816.212.00 1st payment, (greater than int.,) to be deducted 7.o00 Balance due Jan. 1st, 1846..... 142.00 Interest for 1 yealr.........52 Ans. Amount due Jan. 1st, 1847. $15(.52 9-, _ 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,.. 200.0() Interest to 2d payment, July 1st, 1844 (6 mon.,) 6.00 $206.00 2d payment, (greater than int.,) to be deductel 1)00.00 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 payment (1 yr. 4 mon.,).. 2.00() 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 fronm Sept. 10th, lS51, to Nov. 10th, 1852, (1 yr. 2 mon.,)........... 9.10 139.10 Deduct 2d payment......... G.8( 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 thie interest,).... 50.50 Interest from Mar. 5th, 1841, to Dec. 5th, 1842, (1 yr. 9 mon.,)..... 35.00 $235.00 Deduct the sum of the 1st and 2d payments. 70.50 tllalnce due Dec. 5th, 1842,...164.5 nlterest on balance to settlehent (1 yr. 6 mon.,) a24.75 Ans. Amount due June 5th, 1844. $18'.175 116 KEY TO RAY'S THIRD BOOK. (6) Principal.. 50.00 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 du'e)... 21.50 Interest to 2d payment, (12 mon.,).... 7.50 267.50 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.,). 8.40 AAns. Amount due July 1st, 1846. 248.40 (7) Principal... $180.00 Interest to 1st 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. ist, 1846, (11 mon.,).. 8.80 168.( 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 Jns. Amount due July 1st, 1846. $138.535 S4 PERCENTAGE. 117 (8) Principal. $400. 00 Sept. 1st, 1584. 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 st 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 3d and 4th payments.... 91.00 304.20 Interest to March 1st, 1847, (6 mon.,)... 9.126.Ans. Amount due MIarch 1st, 1847 $313.326 (9) Principal...... $450.00 1st payment, Jan. Ist, 1847, less than interest then due.$..20.00 2d payment, April 1st, 1847, less thani interest then due 14.00 3d payment, July 16th, 1847, in all, mnore 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 thtn interest then due. 18.00 Interest to June 1st, 13~49, (22 mon. 15o da.,). 64.50 494.50 Deduct sum of 4th and 5th payments... 28.00 Ans. Balance due June 1st, 1849. $466.50 ~;~ —.nn~~sn~rLIde~;a~~ 118 1KEY TO RAY'S THIIRD 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 dae,... 20.00 3d payment, Dec. 16th, 1840; less:in all than interest then due...... 15.00 4th palyment, Ap. 10th, 1841, less in all than interes' t*hen 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...... 1 92.00 Balance due July 13th, 1841,. 900.00 Interest to Dec. 23, 1841, (5 mon. 10 da.,). 24.(0) Amount due Dec. 23, 1841......... 924.00 6th payment, (gTreater than int.,)... 324.00 Balance due Dec. 23, 1841,......... 0.)( Interest to Oct. 1st, 1843, (1 yr. 9 mon. 8 da.,) 603.8) Anis. Amount due Oct. 1st, 184-3. $663.80 REM.-RKl.-In accordance with sugg-estion preceding Example 21, in Arith., in all the preceding solutions the coimputation of interest has been deferred, until the period when a-payment, or the sum of two or more payments, exceeds the interest due. ARTICLE 227. (1) Principal.........$.0.00 Interest to Jan. 1st, 1847, (1 yr.)...... 19.20 330.20 ]st payment, May 1st, 1q4... $50.00 Iliterest to JanL. 1st, 1847, (8 mon.,). 2.00 2d pavmlnent, Nov. 1t'th, 146... 100.t 0 lut. to Jan. 1st, 1847, (1 mlon. 15 da.,).75 $I152.75 152.7i Ans. Balance due Jan. 1st, 1847. $186.45 I"""""~"U"~"1~~f7 PERCENTAGE. 119 (2) Principal................. $540.00 Interest to Jan. 1st, 1848, (10 mon.,)..... 36i.00 $876.00 1st payment, May 1st, 1847..... $90.00 Int. to Jan. Ist, 1848, (8 rmon.,).. 4.80 2d payment, July Ist, 1847... 100.00 Int. to Jan. 1st, 1848, (6 mon.,).. 4.00 3( paymernt. Aug. 1st, 1847..... 150.00 Int. to Jan. 1st, 1,848. (5 mon.,),... 5.00 4th payment, Oct. 11th, 1847,... 180.00 lat. to Jan. 1st, 1848, (2 mlon. 20 da.,) 3.20 $537.00 537.00 Ans. Balance due Jan. 1st, 1848. $39.00 ARTICLE 228. (1) Principal of the note.......... $875.000 Interest to Aug. 10, 1834, (43 mon.)..... 188.125 Amount...............$103.125 1st payment deduct..........260.000 Due Aug. 10, 1834, forminge a new principal, $803. 1 25 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 $602.268 Amount of $50 from March 1, 1836, to Dec. 16, 1836, (9 imon. 15 da.,) to be deducted.. 52.375 Due Dec. 16, 1836, forminc a new principal, $549.893 Interest from Dec. 16, 1836, to Dec. 16, 1837 32.9ui3 $582.886 Amount of $150 f'rom July 1, 1837, to Dec. 16, 1837, (5 loon. 15 da.,)........ 154.125 Due Dec. 16, 1837, formine a new principal, 428.76 L Interest to Sept. 1, 183S, (- mon. 15 da.,). 18.222 Ans. Due on settlement. $446.9 83 — ~, ---- - I-., - -- 120 KEY TO RAY'S THIRD BOOK. ARTICLE 230. (2) $ GOX.05=$ 3.00 $12 —$ 3=4 yr. An7s. (3) $375X.0S-$30.00 $90 — $30=3 yr. Ans. (4) $225X.04-=$ 9.00: $66 —$ 9=7~ yr. 7- yr.=7 yr. 4 mon. Anis. (5) $'250X.06=$15.00: 834.50 —$15 -=2.3 yr.=2 yr. 3 mon. 18 da. Ans. (6) $60X.06=$3.60: $13.77 — $3.6 =3.825 yr.=3 yr. 9 mon. 27 da. Arns. (7) $800-$500=$300: $500X.10=$50: 8300 —$50=6 yr. Ans. (8) $7098-$600=$198: $600X.09=$54: $198 —$54-3r. y rr. yr. 8 mon. Av s. (9) $200X.06= — 12: $200 —$i2=16- yr.-16 yr. 8 mon. Ans. (10) $100: 84$ 25 yr. Atns. (ll) $100-$5 —520 yr. Ans. (12) $100 —.$7=14- yr. Anzs. (13) $100o$-+s3=121 yr. AnS. (14) $100.$10=10 yr. Ans. (15),$100 $12z=Sl yr. Ants. (16) $100 -. G16-61 yr. As. (17) $100X3=$300: $300-$100o=4200: 200 — $15 — 13- yr.,Ars. $10OX4=$400: $400-$100=$300: $300 —$15 =-20 yr. Ans. ARTICLE 231. (2) $S0X.01=$.80: $.80X5=$4.00: $28. —4 — 7 Yo Ans. (3) $300X.01=$3: $3X1-$5: $30-+-$5=6 %o Arns. (4) $1000X.01=10: $I10X2-=$95: $200:$25=8 %c Ans. (5) $260X.01=$2.60: $2.60X2 —5.20: $93.40 — $5.2= 4.5= 4- o Ans. (6) $640X.01=$6.40: $6.40X6=$38.40: $110.40-.-$38.40:2.875 —=27 % An2s. _ PERCENTAGE. 121 ARTICLE 232. (2).05X3 $.15: $8.25-S.15-$55 Ans. (3) $.05X3=$.15: $341.25 - $.15=$2975 Ans. (4) Int. of $1.00 for 16 mon., at 6 %o (Art. 223)= —$.0: $'2.26-.-$.08-$28.25 Ans. (5) 8]023.75-tP. 05-$20475 Ans. (6) Int. of $1 for 12 yr. 3 mon. 20 da. at 9 % (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: $t216+$17.28-$233.28, 2d yr. $233..28 —$200-$33.28 Ans. (3) $50OX.06=$30: $500+$30 —$530, ist yr. $530X.06=$31.80: $530+g31.80=-$561. 80, 2d yr. $561.80X.06z=,33.708: $561.80-+$33.708=$595.508 Ans. (4) $0jX.J 06-$48.00: g48+$800-S848, 1st yr. $84SX.06-='$50.88: $50.88+$848-$898.88, 2d yr. $898.88X.06-$53.9398: 853.9328+$898.88 =$952.8128, 3d yr. $952.812SX.06=$57.168+: $57.168+952.81928 =1009.98+- Ans. (5) $250X.06=$15: $15+$950-=$65, 1st yr. $65X.06 —$15.90: $15.90+$265-=$280.90, 2d yr. $280.90X.06-$16.854: $16.854+$280.90 =$297.754, 3d yr. $297.754-$950=$47.754 Ans. (6) $300X.0o5=15 $15+i$300=$315 ]1st yr. $315X.05=$15.75: $15.75+$315-$330.75, 2d yr. $330.75X.05 —- 1.6.5375: $16.5375+8330.75 -=347.2875, 3d yr. $347.2875X.05=$17.364+: $17.364+$347.287 —.$364.65+, 4th yr. $364.65-$300-64.65+ Ans. 122 KEY TO RAY'S THIRD BOOK. (7) LInt. of $1.00 for 6 mon. at 6 % (Art. 2:33) =$.03.0O3X'20)0-$6: i$6+-,;$200S-z-I$206, 1st 6 mon. $.03X V06 —8=6.1: t6.18+Y$206-$ 212.18, 2(6 mon. $.03X212.18=$6.3654: $6.3654q-2+ 12.18=$218.5 l51, 3d 6 mon. $.03X218.5454-$6.5563+:$6.5563-+$218.54,54 -=$)225.101+, 4th 6 mon. $225.101-4200=$25.101 Ans. (8) 20 %o per annum is equal to 5 %o quarterly, the question then is the samle as to find the amount of $500 for 8 years, at 5 %o per annum. $51)0X.05=_$25: $25+ —p500=$595, 1st quarter. $525X.05 —$26.23:,~25.25 +$525 $.551..25, 2d quar. $551.2o5X.05 —27.5625: 27.562-5+$551.25 -$578.8125, 3d qr. $ 57.8 12 5X.05: $28.9406+ $28.9406+$578.8125 =,607.7531, 4th qr. $6o07.75310)X.05$30.3876+: 30.3876+,$607.7531 =o4638.1407, 5th qt. 363s.1407X.05-$ 31.007 +- $31.907+G638.1l407 =-$670.0477, 6ti qr. $670.0477X.05=$33.5023+:$33.5023+S670.0477 — $703.55, 7th qr. $703.55X.05=-35.177+: 35.177+$703.55 =$738.727+ zAns. 9).3nOX.06=$18.00: $18.00+$300=$318, 1st yr. 031SX.056:19.03: i19.0S+$318 — $337.05, 2d yr. $33-7.0SX.06-$200224: $20.22484-2- -100. 112: $t0.1124+$337.08=-$347.1924 $34.14: $347.194-$300 =$47.1924I Als. (10) s1000o X.06=$60: $60+.1000=$1060, 1st yr. 81060.06= —$63.60 *: 6.G0)+$t10(0-= 113.60, 62 -yr. Irt. off' $1123.(60 for $ mon. 15 d;a. (ArLt. 2:-32) =$47.753: $-11.753+$1123.60=$117 1.353: $1171._353 —. 1000 -$171.353 Ans., --------- -- ----------— r PERCENTAGE. 123 AR.TI CLE, 234. (12) Tab. num. 1.1790848X3a45=$61 7.84+ Ans. (14) 30 yr.=-20 yr.+10 yr. Tab. num. for 20 yr.=3.207135: Tab. numn. for 10 yr.=1.7O0848. 3.207135X890 =-2854.35015: 2854.35015X1.790545 $ 111.70+ $5111.70 —890=04221. 70+ Ans. (15) 70 yr. —25 yr.+-25 yr.-+-0 yr. Tab. num. for o2 yr. — 3.38 6355 Tab. num. for 20 yr.=2.653298. 3.386355X200=677.271: 3.386355XG77.271 -2293.4 8: 2.65329)(2293.48 -$6085.28+: $6085.28- 2000. — aSj85.28+ Ans. ARTICLE 236.-DiscoUNT. (2) q.04X9=$.08 $1+$.03=$:1.08 $81 —$1.08 -=$s-5 Ans. (3) $.06X5-$.30 $1A-$.30-$1.300: $1300-4-1.30 -$l1000 Asis. (4) $.06X3-=18: $1 +$.18 —$1.15: $4720 —$1.1 — $4000 Ans. (5) 2l+-2.OG6$1.06 $.257.50-{ 1__.o=S2)4_904+ An'zs. (6) Int. of $1 for 1 yr. 10 mon. (Art. 223) —-$.1 +$.11=$1.1ll: $199.80 — $1.1 l-$180.00 Ans. (7) Int. of $1 for 5 yr. 10 mon. (Art. 023) =$.35: $1+$.35o=l.35: $6.75 s $1.35$500 Ans. (8) Int. of $1 for 5 mon. (Art. 223) -$.025: $1+$.025-$1.02; $37.50. $1.025= — 300.00 00Ans. (9) Tnt, of $1 for 7 yr. 9 mon. 00 da. (Art. o23) =$.468:3 it+$.4681 $1.468.: 0$493.290 i. 468, -—,335.89 - Anis. (10) $.06X4S.24. l.:, $+..24=901.' 49G 01.24 — S400, present worth. $496-' $400(-$96 As. (11) 0X-: 1 +2.1+.2$1.12: ~ 276.64 -$1.1 2-$247, present worth. $27;6.64 —$247.00:-$29.64 Ans. S _ _ Rs 12 4 KEY TO R-AY'S THIIRD BOOK. (12) Int. of $1 for 3 yr. 4 mon. (Artt 223)=$.o20: $1+-$.920:1.20: $330 —- 1.g0= —-$2O75 present worth. h330 —275=$55 Anis. (13) lnt. of $1 for S mon. (Alt. 293)=$.04: $1+$.04=;$1.04: $260_ $1.04$2-50 Aln (14) $1+$.12=$1.12: $235.20-. $1.12-=$210 Ans. (15) $1+$.07=-$1.07 $401.25 -- 1. 07-$375 present worth. $401.2:5-S375=$26.25 Aes. (16) $10S0X.06=$64.80: $64.80X 10-$648-interest. $.06X10-$.60: $1+$.60=$1.60: $1080 —$1.60 =$0675. $1080-$675-$405 discount. $648 —$405 — $'43 Ais. (17) IQ of $.12=$.10: $1+$.10=$1.10: $1221. $1.10 -$1110. $1122 —$1110=$12 Als. ARTICLE 237. (18) $956.34,-3 —$318.78, each payment.. $.0o5 +81- 1.0o: $31S.78-.-$i.05-$303.60, present worth Ist payment. $.05X92=$10:,$+$.1 0= 10$1.10: $318.78 —$1.10 =$289.80, present worth 2d payment. $.05X3-$.15: $1+$.15=$1.15: $318.78_. $1.15 -$277.20, present worth 3d payment. $303.60+$9289.80+.$277.20-$870.60 Ans. (19) $1440- -.-=-720, 1st payment: $1440 —. 3=$480, 2d payment: $1440-($720A-.$4S0)-$240, last payment. Int. of $1 for 3, 6, and 9 mon. (Art. 223) is $.015, $.03, $.045 $1+$.015=$1.015:,$720-. $1.015-8709.3596+, present north 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 paynments together, their sum is $1405.044+ Anls. _-.............................................................................. - PERCENTAGE. 125 (20C) 700-(G60+$180A-$.$60) —$200, last payment. Int. of $1 for 6, 12, 18, and 24 mon. (Art. 223) is $.03, $.06, $.09, $.1? $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: 1.09 —$238. 5321+, present worth 3d payment. $1q-$.12-$1.1: $8200-. $1.12-=$178.5714+, present worth 4th payment. Sum of payments 8645.167+- Ans. ARTICLE 238. (1) $.06X5= $.30: $1+$.30=$1.30.: $650-. $1.30 =$8500 An"s. (2) $.05X9=$.45: $1+$.45=$1.45: $725 —$1.45 =$500 Ans. (3).05X-4=$.20 $1+2 —0=$1.20: $571.20 —-$1.20 =$476, principal. $571.20-$476=$95.20, interest. Ans. (4) S.06X2=$.15: $1+$.15=$1.15: $690.+$81.15 -$600, principal. $690-$600=$90, interest. Ans. ARTICLIE 240.-B&AN DISCOUNT. (2) $137X 93- -60-212+. Int.=-$2.12+Ans. (3) $1780X93- 60-=89X31=2759. Int.-=$27.59 Ans. (4) $375X 33 -60=75X1ll 1-4=2061. Int.=$2.06{ Ans. (5) $165X 63 —. 6033X21-. 4=173~. Int.=$1.73{ Ans. (6) $140X123-. 60= 7X41=-287. Int. —$82.87 Ans. (7) $ 8OX183_.60- 4X61 —244. Int.-$2.44 Ans. (8) $180X 33 —60-= 3X33=99. Int.=99 cts. $180 —$.99 —$179.01 Ans. 126 KIEY TO RAT'S THIRD BOOK. (9) $960X33_ 60-16X33=528. Int.-$5.28 ~$960 — 5.28-$954.72 An? s. (10) $875X93 —. 60=175X31. 4=13564. Int —$13.561 $875-$13.564z-$S61.4o3 Ans. (11) $3900X63_. 60-65X63 —4095. Int.=$40.95 $3900-$40.95 —$3859.05 Ans. (12) $2580X103-.-60-439X03=44429. Int. at 6 %. =$44.29 Int. at 5 5o. (Art. 225, Rem. 3.,) -- of $44.29=-$36.90S+. $2580-$36.908 =$2543.09+ A,8s. (13) $3.50Xo25= 7878.50, purchase price. $4X225=-$900: $900X 183 — 60=15X 1S3-2745. Int.- $27.45 $900-$2'7.45-=879.55, proceeds of note. t872.55- 787.50=$85.05 gain Aus. (14) $814$.07=$1.07: $535_.$1.07=$500: 8535 —$500 =$35, true discount. $535X.07:$37.45, baInk discount. $37.4 —$35 —$2.45 Aas. (15) $.06X4=$.24. $:1+$.24-$1.24.: $1209-.-$1.24 -9,75: $1209 — 975-=$234, true discount. $1209X.06=- 2.54_: $72.54X4-$'-90.16, bank discount. $290.16-$234=-G56.16 Ans. ARTICLE 241. (2) Int. of $1 for 3 mon. 3 da. (Art. 223) =$.0155 $1 —.0155=$.9845: $393.80+-$.9845-=$400 Als. (3) Int,. of $1 for 63 da. (Art. 223) ='$.0105: $i-$.0105 —=$.9895: $500 —-$.9895-$505.305+ Ans. REMARK.-For a full discussion of the difference between Bank and true discount, see Ray's iligher Arithmetic. ARTICLE 243.-CASE I. PROFIT AND LOSS. (3) $15OX.10-$15 Ans. (4) $8 74X.25 5-$21S.50 Ans. (5) $500X.12= $60: $500+$60o=$560 Ans. (6) $382.50X.04 —$15.30 $3S2.50 —$S15.30=$367.2,0 Al.s. _~ —------ ---- ------ PERCENTAGE. 127 ARTICLE 244.-CASE II. (2) $.90X..5.22': 90 cts.+22-'=$1.12. Ans. $.90X.10-.09: 90 cts.-9 cts.-81 cts. $.S1 Ans. (3) $4.37. 33 $1458 75.458=$5.83+Ans,,$4.3LX.20-$.871: $4.37-! —'$.87 =.$3.50 Ans. (4) $8.50X.20=$1.70: $3.50+$-1.70-$10I.20: and $8.50-41.70=$6.80 An3. (5) $300 —40-$7.50: $7.50X.20-$1.50: $7.50+1.50 — $9.00 Ams.: $7.50-$1.50=-$6.0 A17ns ARTICLE 245. (6) lOts.-+-o of 1Octs.=1cts. 15cts.+~-,1 of 15cts.=1G.)cts. 20cts.+-i' of'20cts. —22cts. 30cts+y'O of 30cts.=33 cts. 40cts.+T'I of 40cts.z-44cts. 50cts. +j0 of 50cts. —-55 cts. 60cts.+-i'- of 60cts.=66cts. (7) 8cts.+ 8 of 8cts.= 9cts. 12cts.~ of 12cts.=13.cts. 16cts.+ 4- of 6cts.-=18cts. 20cts.+ 4 of 20cts.-22 cts. (8) lOcts +. of l0cts.-]1cts. 15ets.+ I of 15cts.=18cts. 25cts.+ 4 of 25cts.=30cts. (9) $1+} of 1 =$l1.25 $1.20+4 of $1.20-$1.50 $1.50+4 of $1.50=$1.87 2 of $2!$2.50 $3+4 of $3-$3.75 $4+- of $4=$5.00 $6+4 of $6=$7.50 (10) 25cts.~+~ of 25cts.=331 ts. 50cts.+-' of 50cts.=G66cts. (11) $2-q- of $2-=$3.00. $3+' of $3=$4.50 $4+~ of $4=$6. $5+= of $5=$7.50 $7+4 of $7. -=10.50 ARTICLE 246.-CASE III. (2) $105 —$75=$30 3: 7o_=.40-40 7b Ans. (3) $ 40-$-30=$10 i~O —=.33-33a' % Ans. (,\ 30 cts. —25 cts.=5 cts.,: ==, 2 —-20 )0 % Ans. ~~Os~~~~lli~~~~~l~ Arr*C~LI - L —CIs.~- - ------— ~ 128 KEY TO tRAY'S THIRD BOOK. (5) 21 cts.-20 cts.=l ct.: 0 —=-.05=5 % Ans. (6) $-$SI: — =.12 =12 1o Ans. (7) 30 cts.-24 cts.=6 cts.: — =.20 —20 % Ans. (8) $40X40=$1600: o -=-. 44_44.o. Ans. ARTICLE 247 —CASE IV. (3) $1+$S.20 =$1.20: $6-$1.20=-.5 Ans. (4) $1 —.25 =$.75: $3..75-=$4 Ans. (5) $1+ —$.25 =$1.25: $75s $1.25=$60 Ans. (6) $1 —$.25 — $.75: $75 —..75=$1-00.As. (7) $1{-$.10 =$1.10: $.22- $1.10=$.20 Ans. (8) $1+$.12,= —$1.125 $8.10- -$1.125=$7.20 Ans. (9) $1-$.15 =$.85: $1.19 —$. 5=$1.40 Ans. ARTICLE 248.-PRoBIIscUOUS EXAMPLES. (1) 6 cts.X30=$1.80: $1.80X.25=$.45 Ans. (2) $250-. 40=$6.25, purchase price per yard. $6.25X.20 =$1.25: $6.25+$1.25=$7.50 Ans. (3) 1 hhd.=252 qt.: $.40X2524=$100.80: $100.80-$75 — $25.80 2.80 =.342=342 % Ans. (4) 2428921-1918604=510317: 1'. 0317_2 6 5 + =26.5+ % -Ans. (5) 1519467 —937903=581564: I ss4- =62+ =62+ %o Ans. (6) 81+$.10=$1.10: $.42 —$1.10=-$.38 -, purchase price. $.51_ $.389_~= $1_12 T-1.4 T-11__47_'3394 3 — T -- TT ~ ~ A ns. (7) 1-.-$.25=81.25: $1 — 1.25=$.80, purchase price of rye. $1NX.5-=.2S5: $1 —$.25=$.75, selling price of wheat. $.80-,.7'5=$.05 Anls. PERCENTAGE. 129 (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. $14.40X5-$72.00: $12X4-$48.00: 14-(5+4) =5 pieces left. $72.00+$48.00=$120.00: $161.28-$120-$41.28, what the remaining 5 pieces must sell for. =$41.28-*-5=$8.2256 Ans. (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: $1.36- -$.8=-$1.60; $1.856-$1.60 -$.256 "s'=.16=16 % Ans. (11) $1+$.12=$1.12: $.96 — $1.12 —$1.75, purchase price. $1.75X.16 —$. 28 $ 1.75 —$.28-$1.47 Ans. (12) $1+$.16=$.16 $1.682_ $1.16 —$1.45, purchase price. $1.45-$1.247-$.203: L-2 ~t-_.14=14 % Ans. (13) $5yX —=$1.2 5: $100 $1.25=80 yd. Ans. (14) $2.50X.20=$.50:.$2.50+$.50-33.00, selling price of each cask. $3.00 —$.05==60 lb. Ans. The number of casks has nothing to do with the solution. (15) $1+$.25-$25- 25: $1-.-$1.25=$.80=-purchase price per bu. $.SOX.35=$.28 $.S0+$.28=$1.08=selling price per bu. $59.40 —$1.08=55 bu. An4s. ARTICLE 250.-AssEss IENT OF TAXES. (1) $500 —. $125000-.004, or 4 mills on the dollar. $1650X.004=$6.60 Alns. t2) $1.25X1258=$1572.50, amount raised on polls. $9057.60-$15792.50=$74S5.10, amount to be raised on property. $7485.10 —$534650=.014 =1 ct. 4 m. on $1 Ans. 4.-. 0 _... _ 1.30 KEY TO RAY'S THIRD BOOK. (4) $1000 is taxed $14.00 (5) $3000 is taxed $42.00 200() " " 2.80 500a'.' 7.00 40 " 6.56 8s 0. " 1.12 3 "6.042 9. ".126 3 pollk,..3.75ii 4 polls 5.00 Ans. $21.152 74ns. $55.246 ARTICLE 251.-AM1ERICAN DUTIES. (1) 100+182+264=546: Draft=1+2+3=6 lb. 546-6-540 lb.: 540X.15=81 lb.: 540 —81459 lb. 2 cts.X459-$9.18 Ans. (2) 400 lb. —4 lb. draft=396 lb.: 396X.05-19.8 lb. tare: 396-19.8-3T6.2, net weight of each bale. 376.2X40-15048 lb. net. $.25X1504S=$3762. $3762X.20=$752.40 Ans. (3) 196 lb.-2 lb. draft=-194 lb. 194X.15=29.1 lb. tare: 194 —29.1-164.9 net: 164.9X75=12367.5 lb. $.05X 12367.5=$618.375 Ans. ARTICLE 252.-PARTNERSIIITP. (2) -3o= —: ~ of $239 —$S7, A's share. oQo= —' A of $232=$145, B's share. (3) $70+$150+$80=$300, whole stock. 7so0 o-U=-ao U74 of $10=$28, A's share. 51 ~-1.: of $120=$60, B's share. -3%-4a -. 4 4 of $120=$32, C's share. (4) $200+$400+$600 —$1200, whole stock. o%-0=, - ~t o= Qo, ~o%=,. —I T ~- _4 0 -Q — ) 00 o1 00 —. of $427.26=i$71.21, A's share: - of $427.26 — $14.49, B's share: and ~ of $427.26 — $213.63, C's share. $427.26 — $1200 —.35605-35.605 7 gain. Ans. (5) 1-t3+5=9. x of $90=$10: -A-1 of $90=$30.: | of $90=:$50 Ans. PARTNERSHIP. 131 (6) 2 —33-5+7 —17. -7~ o(if $735.93-$86.58:;37 of $735.93=$120.87: 57 of $735.93-$216.45: -f of $735.93-$303.03 Ans. (7) 3+6+9+11+13+1 7-59. 3 of $22361 —$1137: 1) of $22361-$2274:,9 of $22361=$3411 ~- of $22361=$4169: 13 of $22361=$4927: 7 of $22361 —6443 Ans. (8).', 4 o 72 1 o0 Since the denomlinators are the 3 i ~ 7:2 — Y, Y;2, 1-f:2 same, the fractions are to each other as their numerators. 40+72+-105=217. ~40jv of $692.23z —;27.60: -;7 of $692.23=$229.68: of $692.23 — $334.95 Ans. ARTI C L E 25. —B A N K I RU PTCY. (9) $175+$500+$600+$210+-$42.50+$20+$10-$1.557.50. As $1557.50: $175:: $934.50: $105.00, A's share. As $1557.50: $500:: $934.50: $300.00, B's share. As $1557.50: $600:: $934.50: $360.00, C's sharn. As $1557.50: $210:: $934.50: $126.00, D's share. As $1557.50: $42.50:: $934.50: $25.50, E's share. As $1557.50: $20:: $934.50 $ 12.00, F's share, As $1557.50: $10:: $934.50: $6.00, G's share. (10) $234+$175+$326-$735: $492.45. $735=$.67 = sum paid on each dollar of indebtedness. $234X.67$15)6.78: $175X.67-$117.25: $326X.67==$218.42 Avns. ARTICLE 254. —GENERAL AVERAGE. (11) 48 _4 3 6= 24__ ( o) — i;8 -; T —O. O 3'-f 45=20, A's loss. - of 45=15, B's loss: - of 45=10, C's loss. (12) $1.0000-$15000 —$25000. $1 25- -$25000=.041-~ 4,V %, general average. $2150X.04 = -$96.75, A's loss. og f $10000-=,$3750 83750X.04 1=$168.75 Ans. 132 KEY TO RAY'S THIRD.BOOK. ARTICLE 255. —PARTNERSHIP WITH TIMIE. (3) 23X27-621 21X39-819: 621+819-1440. 1 69.819 91 69 14 4 94 WT 4: - 149 6=9 6 9Uof $54 —$23.283, A pays. %610 of $54=$30.71~, B pays. (4) $300X5=$1500: $400XS=$3200: $500X3=$1500. $1500+$3900+ $1500=$6200. o-.- ~oo01. of200- - - $2.1 I U5QOi 0=5. 6.3~0 —2 6-; of 1 100=5 -24.19"T, A's and C's loss: ~ of $100=$51.61-T. B's los. (5) 6X30=-180: 5X40=200: SX28-=224. 180+200+224-604: o- s 0',41 1s; -4.5 of $18.12-=$5.40, A: of $18.12=$6, B 1-51 of $18.12=$6.72, C. (6) A $300X 8-$2400, $300+100=$400, $400X8=$3200, $2400+3200=$5600. B $600X10=$6000, $600-300='$300, $300X6==-$1800, $6000+1800=$7800. C $500X12=$6000, $500+400=$900, $900X 4=$3600, $6000+3600=$9600. $5600+7800+9600=$23000. As 23000 5600:: $759: $184.80, A's. 23000: 7800:: $759 $257.40, B's. 23000: 9600:: $759: $316.80, C's. (7) $800X12=$9600: $500X12=$6000: 12 mon.-7 mon. =5 mon.: $9600-$6000=3600: $3600 —-5=$720 Ans. EQUATION OF PAY2MENTS. ARTICLE 257. (2) $2X4==$ (3) $8X5=$40 6X8=48 4X8= 32 $8) 56(7 mon. Ans. $12) 72(6 mon. Ans. ALLIGATION MEDIAL. 133 (4) $250X2=$500 (5) $looX 6=$600 500X5=2500 75X 8= 600 750X8-G6000 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 $200) 1200(6 mon. Ans. ARTICLE 258. (2) Counting from April 2d, it is 90 days to the first payment, and 150 days to the second. $200X 90=$18000 $300X150= 45000 $500) 63000(126 da. from April 2d, which will give Aug. 6th Ans. ARTICLE 259. (8) $103 —$41.03=$100 $lOG —t1.06=$100 $103~+ 10=$209 $209 — $200=$ 9 $200X.06=$ 12 $9 —$12=.75 yr.=9 mon. Ans. ALLIGATION MEDIAL. ARTICLE 260. (2) 6 lb. at 3 ets.=18 cts. (3) 25 lb. at 12 cts.=$3.00 4 lb. at 8 cts.=32 cts. 25 lb. at 18 cts.= 4.50 10 lb. cost 50 cts. 40 lb. at 25 cts.=10.00 10 lb. cost; 50 cts. 50 cts. — 10=5 cts. Ans 90 lb. cost $17.50 $17.50-.-90-$.19j A1s. 134 KiEY TO RAY'S TIIRD BOOK. (4) 3 al.. cost $0.00 (5) 10 at 3-3$30.00 12 gal. at 50= 6.00 12 at 4= 48.00 8 at 9 — 72.(0O 15 gal. cost $6.00 = $G.00 —15=$.40 Azs. 30 worth $150.00 $150 — 30=$5 Ans. (6) 6 to 10= 4 hr.: 63~X 4=2520 10 to 1- 3: 70)OX 3=2100 1 to 3- 2: 27~X 2 —150~ 3 to 7 4: 73~X 4=992~ 7 to 6=11 hr.: 55~X1l=605~ 24)1509~(62o Ans. A N A tL Y S I S. REmaAnRS.-The subject of analysis is one of the most important in Arithmetic. With a thorotugh knowledge of the elementary rules, fractions, and analysis, nearly all the operations that are purely arithmetical may be solved withdut the tid of special rules; because these rules are mostly derived fromn anal.tical solutions of particular questions. Tile pupil, however, Mwho desires to understand analysis fully, and apply it readily, must first make himself wrell acquainted with Mental Arithmetic. See Ray's Arithmetic, 2d Book. ARTICLIE 261. (2) $3(6 (3) $24 (5) 7bl. (6) 10 lb. (7) 10 yd. (9) 7, yd. (10) 4. (11) 4i. (13) 12: 20. (14) 28: 27. (16) 10. (17) 8S. (19) I18. (20) 28. (23) 16 ets. (24) If 1-fifth of a bushel cost 14 cents, 1 bushel will cost 5 times 14 cents, which are 70 cents; an(ld of a bushel will cost y9 of 70 cents, which are 63 cents. Ans. (25) 1-eighth will cost 1-third of 3-eighths; if 3 cost 21 cents, } will cost ~ of 21 cents, wrhich is 7 cents, and if 8 cost 7 cents, 1 yard, or S-eighths, will cost 8 times 7 cents, Nwhich are 56 cents; then, of a yard will cost g of 56 cents, which are 48 cents. A ns..].IP -------------- ~~B — U ANALYSIS. 135 ARTICLE 263. (2) $2.16 — 4 —9 cts.: 9 cts.X23=$2.07 Ans. (3) $32.50_ 13-$2.50: $2.50X14-$35 Ans. (4) $8 — 4=$: $24 — 2 —-l1yd. Ans. (5) $20.67 —7.59 =$.13: $34.71 —$.13-267 yd. Ans. (6) $147. 34=$41-: $4-4-X9 —=$38 Ans. (8) 25 da.X17-=45 da.: 425 da.-: 104 —, da. Ans. (9) If 6 men can build a wall in 10 days, it will require 10 times 6-=-60 men to build it in I day, and 1-fifteenth of 60 men —4 men, to build it in 15 days. AIs. (10) If 5 men eat a barrel of flour in 12 days, I man will eat a barrel in 5 times 12 days, which are 60 days, and 4 men will eat a barrel in 1-fourth of the time of 1 man, that is, in, of 60 days=15 da. Ans. (11) 10 hr.X7-70 hr. to do the work: 70 hr. —. 1 hr. —5s' da. of 12 hr. each. Als. (12) If IS men reap 72 acres in 7 days, 1 man would reap it in 7X18-126 days, and 8 men in.- of 126=154 da. Ans. (13) Each 9 sheep are worth 5 hogs: in 54 sheep there are 6 times 9 sheep; then 54 sheep are worth 6 times 5 hogs, which are 30 hogs. Ails. (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. Ans. (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 32 oz.-6' oz. Ans. (16) If 3 stacks of hay keep 12 horses 5 mon., they would keep 1 horse 5 mon.X12=60 mon., and 20 horses 1-twentieth of 60 nion.-3 nion. Ans. (17) If g of a yard cost 40 cts., - of a yard will cost. of 40 ets.=20 cts., and 3 will cost 3 times 20 cts.=60 cts. AIzs. 1436 KEY TO RAY'S THIIRD BOOK. (18) If ~ yd. cost $2, one yard iwill cost 3 times $2 —6, and yd. will cost -3 of $G=6-$4.50. Anms. (19) If -$ of a tun cost $4.25 one-seventh will cost w1 of $4.25-$.85, and a tun will cost 7 times.$.85 —$5.95: then 7 of a tun will cost of $5.95:,3.85 Ans. (20) If -T of at cord cost $2.52, one-eleventh will cost - of $2.52-$.28, and a cord will cost 11 times $.28 —$3.08: then 143 of a cord will cost:-! of $3.08=$2.86 Ans. cost 4 1, 4 and (21) If 3-a lb. cost $, ne-serenth will cost' of t —,1, and a lb. will cost 7 times $-T,:~ —T; then lb. will cost of $T.-7 4 Ans. ARTICL, 264. (23) If 2 men can earn.~16 in 4 days, 1 man will earn - of $16-$8 in 4 days, anct il 1 day he will earn 4 of $S8 $-2: then if 1 man earn $2 in 1 day, 7 mlen will earn 7 times $2=$14, in 1 day; and in 3 days they will earn 3 timeo $14-$42 Als. (24) If 2 men oan build 12'd. in 9 days, 1 man can build' of 12 rd.=6 rd. in 9 days, and in 1 day he can build - of 6 rd, —= lrd.: then if 1 man can build rd. in 1 day, 7 men can build 7 times - rd.=4: rd. in 1 day, and in 6 days, they can build 6 times 42 rd.:28 rd. Ans. (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 oxenX3 — 25 oxen to plow 33 A. in 1 day; anrd to plow 33 A. in 9 days, it would take ~ of 225 oxen-=25 oxen Ans. ARTICLE 265. (26) Since the first pipe fills 41 gal. in 1 hr., and the second empties 6 gal. in the same time, therefore there are left in, 41 —6=_35 gal. per hour; it will therefore require 2 5 O= o5 0 hr.-7 hr. s mi. 342 sec. to fill the cistern. (27) 60GO0-S —75 gal. filled by A in 1 hr. G: 00-. 12-50 gal. filled by B. in 1 hr.: then 75+50:125 gal. filled by both in 1 h.; d - 4 hr. Aizs. ANALYSIS. 137 (28) 900 —. 10-90 gal. filled by A in 1 hr. 900 — 1-7=5 gal. ernp-ied by B in l hr.: then 90-75=-15 gal. left in per hr., and 900-. 15_-60 hr. to fill the cistern. (30) A does -B1, B y, and C ~l. in 1 day, hence in 1 day they __1 1 ~ _ 1+4 all do o-Lr.+T+ =y1-':-o- in one day, and therefore all do it in 1 d-aa5 days. Ans. (31) Reasoning as in the preceding example, A does -5, B Ti' C T~, and D.4 in 1 day, and in 1 day they all do:l;,14=;$6(0-2X -t~o 4 si 1 5o3_8; hence it will require 1.~ j9 -a-4 4g days to do the whole. (32) 1+.__ + s _8,7 and 1 —71-4 4 da. An s. _-r- z ---, 7 — T Or, suppose the meadow to consist of 72 parts, then A mows I of 72=9 parts in one day, and B - of 72=8 parts in one day: 9+8=17 parts both do in one day; 72 — 17 —4 4v da. Ans. (33) Since A and B together, mow it in 12 days, they mow -l, in one day, and since A can nmow it in 20 days, he can mow IO in one day, hence B can mow l - - =6~ — so d in one day; hence he can mow it in t1-.]-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 T1 in 1 day; hence the man eats X-T-5 U — 3- - in 1 day, therefore he will eat in 1 — 1o=10 da. Ans. (35) The first will empty 4, the second a, Land t'he third 1T in 1 minute, hence they will all empty I4 - _ 320+ 1I 5 + - 8 5 a in 1 minute; and 1 — 520 -= o-_5 2 mmin. 154 sec. Ans. (36) Since A can mow D in 6 days, he can mow in - of 6 days, which is 2 days, and since he can mow J in 2 days, he can mow the whole in 5 times 2 days=10 days; therefore he can mow _-T- in 1 day. Again, since B can mowV in 10 days, he can mow 7 in I of 10 days, wDhich is 2 days, and since he can mow ] in 2 days, he can 138 KEY TO IIAY'S THIRD BOOK. mow the n-hoie in 7 times 2 days= —14 days; therefore ble can monw if in 1 daly. T1 i —75O+-7 70 —-3-%;, the part both mojw in one clay, hence 1-. 3%:=5. dfays, the time in which both can do it. (38) $3000-$500-=$2500: $2500 2-2 $12-0, less share, and $1250+-500=$1750, greater share. (39) The wife received $SOO more than the son, and the son $300 more than the daugllter; hence, the wife received $800+-$300=$1100 more than. the daughter. $300+$1100-=$1400, what the iwife and son received more than twice the dauglhter's. $3500-$l1400=-$9100 $2100 — 3-$700= daughter's share: $700+-$300=-1000, son's share: $1000+-t800 =$1800, wife's share. (41) In this case the minute hand must gailn 25 min. on the hour hand. Now, according to the solution of Example 40, in the Arithmetic, r - of this 25 min. is the distance over which the minute hand must move to overtake the hour hand. I; of 25=3y~-~=27-3 minr. Ans. (42) When the hour hand is at YIII, the minute hand is at XII. If the hour hand did not move, the minute hand.-ould have to plas over 10 minutes to be opposite to the hour hand, but since the hour hand does move, the minute hand lmust gail 10 minutes before it is opposite to the 1,our 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 Exanmple 40, we find j" of 10=10 L min. Ans. (43) -5 of $3276-.$1365: Since.-2 of whal, A put in, was equal to s of what B put in,, of $1365=$273, is - of what B put in; hence $273X9-$2457, what B put in. ):, of $24578$1001; $3276 —lU.01=$1.275, w-hich is of' what C put in; hence - of $275 — t- $55,= } of whalt C put in; hence C put in $455X4 — $1820. Since A received a sumn thle of,ihich wats equal th tie amount A ANALYSIS. 139 he put in, 8 of the amount he put in, multiplied by 5, must equual his sha.re; - of $3276=$546, and $546X5 =$2730, A's share. 5 of 1820=$2184, C's gain. $7000 — ($2730+$9] 84) -$2062, B's gain. ARTICLE 266. (45) By taking 1 11). at 4 cts. I gain I ct.; and by taking 1 lb. at 8 ets., I lose 3 cts.; hence, in order to equalize the gains arId losses, as often as 1 lb. 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 cts.; hence, to equalize the gains and losses, there must be taken 2 lb. at 3 cts., which will give a gain of 6 cts., 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 1 et., hence on 10 lb. there will be a loss of 1 ct.X10-10 ets. 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. must be taken to gain 10 cts., which is evidently as many as 2 cts. is contained times in 10 cts., that is, 5 lb..Ais. ARTICLE 267. (51) 3 —-T; hence. of the number is 8, and the number is 12 times 8 —906 Ans. (52) TO-~ S-T; hence -l of the number is 7, and the number is 10 times 7 —0 Ans. (53) 1-: — T~, and l —' 7T, the part of his money left. Then -1 of his money is $35, and x' is I of 35, which is $5, and 1.5-fifteenths, or the whole, is 15 times $5, which is $75 Ans. (54) 1 + +1 0 + 8- hence 773, and -., is 73; of' 73, which is 1, and,o, or the whole, is 30 times 1 30' Ans. (55) 1 4 5, part remaining; he then had left 20-15=35 cts.; if 5 is 35 cts., I of 35 cts., which is 7 ets., is, and the whole is 9 times 7 cts.=63 cts. An1s. 140 KEY TO RAY'S THIRD BOOK. (56) 1- A; 3 Of 1 i-t3 part left; then — 10 cts.; andl - of 10 cts., which is 5 cts, is b; hence ], or tlhe whole, is 15 times 5 cts.-75 cts. Ans. (57) Of each 4 days he worked 3, earning 86, and idled 1 day, forfeiting $1i; hence, for each 3 days he worked he received $5, and since lie received $25, which are 5 times $5, he worked 5 times 3 da. —15 da. Ans. (58) 1, +4 + 1= 1 0 + 4+ =1 9o; 1 9 o — 10 — part bearing pears; hence,' is 4, and ~, or the whole, are 20 times 4=$ 0 Ans. (59) As many, (1,) as many more, (1,) half as many more, (i,) and one-fourthl as many more, (-,) are equal to Id1+I+- =2-3 =-j; then since 1L of the number are 110, one-fourth will be 1'1 of 11.0, which is 10, and the whole, or 4-fourths, will be 4 times 10=40 Ans. (60) 1-71 -=-,-part left a.t close of 1st day.' of 3 3 - 319 -— part left at close of 2d day. 4 of =; - 9 9 6 94_2- part left at close of 3d day. of 7-' 7' 7 2 7_ 8 1 part left at close of 4th day.'['hen since s-;-=$1.62,1-eighty-first part of $1.62, which is 2 cts., is T5 part of the whole; hence 5 or the whole, are 256 times 2 cts.-$5.l12 Als. (61) I + 1 —-=:part of capital at close of lot yr. 3 of 3 3.+ =12=partoft' capital at close of 2d yr. I of a2 3;'-9- p-art of capital at close of 3d yr. Then since 9=-3375, one-ninth of $3375, which is $375, will be one-fifth of the capital, and -, or the whole, will be 5 times $375=$1875 Ans. EXCHiANGE OF CURiRENCIES. A-RTICLE 269. (1) ~5X20+-3 s.=103 s.: 103 s.X12+2 d.- 1238 d.: 1238 d.X4+1 far.-4953 far. A.is. (2) 8675 far. —4-2168} d.: 2168{ d. — 12=180 s. S{ d. 180 s. —20 —=~9. Ans. ~9 j I lei......... q EXCHANGE OF CURRENCIES. 141 (3) ~ s. d. (4)~ s. d. 3' 0 6:- 17 6 5 5 10 41 8 5 11 2 15 8-l Ans. 9 0 5 Ans. 11 6 0 (5) ~ s. d. (6) ~ s. d. 3 12 2j 6)25 0 101 8 -ans. 4 3 54 Ans. 28 17 8 (7) X~.605X20=12.5 s.:.5 s.X12=6 d. Ans. 1] s. 6 d. (8) ~.796875X20-15.9375 s.:.9375 s.X12911.25 d.:.25 d. X4 — far. — d. Ans. 15 s. 114 d. (9) 7 s. 6d. =90 d.: ~1=240d.: T-4='=~.375 Anls. (10) 8s. 9 d. -105 d.: 1=240 d.: o`-__7i=~.4375 Ans. ARTICLE 270. (11) ~75 10 s.=~75.5: ~75.5X.04-3.02: ~3.02X2~ =~7.55: ~.55X20=11 s. Ans. ~7 11 s. (12) 12 s. 6d.-~.625: ~85.625X.06-=-~5.1375: ~5.1375X 1- (1 yr. 9 mon.)= —.99000625: ~.990625X20 =19.8125 s.:.8125 s.X12:9.75 d.:.75 d.X4-3 far., or.- d. Ans. ~8 19 s. 9-c d. ARTICLE 271. (4) ~25X4.84=$121 Ans. (5) 8 s.=z~.4. ~15.4X4.84-=74.536 Ans. (6) 15 s. 9 d.=-~.7875. ~36.7875X4.84=$178.054- Ans. (7) $179.08-.- 4.84=~37 Ans. (8) $124.388 — $4.84=-~25.7=X~25 14s. Ans. (9) $1000-. $4.84=-~206.61157-+=-~206 12 s. 23- d.-+ Ans. 142 KEY TO RAY'S THIRD BOOK. AR}TICLE 272. (a) ~150)X4A, $666': ~GG6-X.OS0 —$33.~ prem. GG6 l$653 3 —7'() Anls. (3) 10s.=~.5 ~80.5X4 —$=357.777+: ~3357.777X.09 — $33.988+ prem.: $357.777+$33.988 —$391.76- iAns. ARTICLE 273. (13) $120.50X -/3 -~36.15-~36 3s. Ans. (2) $ 75..i>< — ~30.1 =~30 2s. Ans. (3) X98X< -~36.75=-~36 15s. Ans. (4) ~30 15 s.=~30. 75: ~30.75 -i -=$102.50 Ans. (5) ~25 17 s.=e25.5: ~25.85-.- a -$64.6s5 Ans. (6) ~,29 8Ss.X~20.4 ~29.4 -70=$12.00 A s. ARTICLE 274. ELx. 1166 fr.-:-5.30 fr.-=$22'0 Ans. ARTICLE 276.-DUODECI31ALS. (2) ft. in. (3) ft. in. (4) ft. in. / 5 3 5 7 8 6 9 1 5 - 1 10 7 3 5 3 5 7 59 11 3 /// 2 2 3 4 7 10 2 1 8 3 Anls. 7 5 3 Anzs. 10 2 10 Ans. 62 0 11 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 6 1 1 4 6 /// 9 9 6 1 10 3 6 Ans. 21 10 5 0 Ans. 10 2 2 9 6 INVOLUTION. 143 ARTICLE 279,-I NVOLUTION. (2.) 1 —4 — 9 16 —25-d-36-49-64-81-100 —121. (2) 1-8-27-64. (3) 4-' —1- 4 (4)' 76- (5) 1G-32 —81. (6) 65X65 —4225 A1ns. (7) 25>XaX5 —=15625 Ans. (8) 16'-3 3 33y3 3 2-2 —-- 0 79-g As. (9) 12.,-.=. X53 Ans. (10) 13X13X13X13X13 —8561 Ans. (1), X', X 2 XSX 3 Anzs. (12) 9X9X9X9X90X9=531441 Anls. (13).025X.025X.0 ~X.095-=.000000390605 AznS (14) 143=-14X14X14 =2X4 44 Ans. (15) 194=19X19X19X19-0130321 AnIs. (16) 2", 3 X.AXn7 X X=-1 AilS. (17).09 6=.09X.09X.09X.O9X.09X.09 —.000000531441A.s. ARTICLE 286. (2) 529(20+3 —23 Ans. 400 20 1 29 2 J29 40 43 ARTICLE 287. (6) 625(25 Ans. (7) 6561i(81 An (8) 3900625(625 An. 4 64 30 45)i2 5 161)101[ 122)306 225 161 244 1245)6'25 6225 144 KEY TO RAY'S THIRD BOOK. (9) 1679616(1296 Ans. (10) 5764801(9401 Ans. ~1~~~ 4 22)67 44)176 44 176 249)'2396 4801)4801'2241 4801 2586)15516 15516 (12) 987656329(31427 Ans. 9 (11) 43046721(6561 Ans. 61)87 36 61 125)704 624)2665 625 2496 1 306)7967 6282)16963 7836 12564 13121)13121 62847)439929 13121 439929 (13) 289442169(17013 Ans. (14) 234.09(15.3 Ans. I 1 27)189 25)134 189 125 3401)4421 303)909 3401 909 34023)102069 102069 EVOL:'TION. 14i (15) 145.2'025(12.05 Ans. (16).019j6(.14 Ans. i 1 22)45 24)96 44 96 2405)120o5 12025 (.7) ].0800I6(1.004 Ans. (18).00822649(.0907 Ans. 1 81 2004)00s01o6 1807)l12649 8016 ]12649 (19) /25 —5, /729=27, I/'~ —v7 Ans. (20) u-43=, i-./121-I, /169-13, r/1 2T1= Ans, (21) 301- v 12= 1 A1n5 Aw. (22) i0(3.16227-7+ Ans. (23) 2(1.41421+- Ans. 61)1.00 24)100 61 96 G626)3900 281)400 3756 281 6322) 14400 2824)11900 63242) 175600 28282)60400 126484 565G64 632447)491 1600 282841)383600 44-27129 282841 6324547)48A44'7 100 442 71 S2 9 Key.-10O 146 KEY TO RAY'S THIIRD BOOK. (24) 2=.666666-+-(.81649+Ans. (25) 6G=-6.4(2.5.98+ Ans. 64 4 161)266 45)240 161)_61 225 1626)10566 502)1.500 9756 1004 16324)S 1066 5049)49600 65296 4.5441 163289)1577066 50588)415900 1469601 404704 (26) V60241=79 An2s. ARTICLLE 288. (1) 16-=X2X2X2: V/16=2X2= 4 An&. (2) 36=2X2X3X3: V36-=2X3= 6 Anws. (3) 100 —)X2X5X5: i/100=2X5=-10 Ans. (4) 225 —3X3X5X5: /225=3X5=15 Ans. (5) V(16X9-5)) 4X5=20 Ans. (6) 1/(36X49)= 6X7 —=4 Ans. (7) V(64X81)= 8X9=72 Ans. (8)'v(121X25)=11lX5=55 Ans, ARTICLE 290. (1) 30'- 900 (2) 812= 6561 40 2=1 600 108 —=11664 /o26500=50 Ans. V/ 18225 —135 Ans. (3) 1002=10000 (4) 452-=2025 602=- 3600 602=3600 -/6400-=80 Ans. /5s625=75 Ans. (5) 362=1296 242 — 576 V/720=26.83+ Ans. 3b — ~ — ---------------------- EVOLUTION. 1-17 (6) 60=36o00 37'=1369 1/2'31-47.2334+=width of street from foot of ladder on one side. 602=3600 2332 5229 /3071-55.4166+=width of street from foot of ladAns. t102.65 der on the other side. (7) 6002-360000 1402= 19600 V/340400=583.43+ 100 —'2= 50. Ans. 533.43+ (8) 202=400 The square root of 656, will give the 16=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 12-per.2=144 lrequired. 800 V800-=28.28+ Ans. ARiTICLE, 291. (1) 1/4096=64 Ans. (2) 4X4=16: 16X9=144: V144=12 Ans. Or, V/(16X9)=4X3=12 Ans. (3) V43560=208.71+ ft., side of acre. 43560 2=-21780: 1/21780=147.58+ ft., side of half A. 43560_ 4=10890: /10890=104.35+ ft., side of qr. A. (4) 10 A.=] 600 P. 129 A.=2-00'/3600=60 Ans. 148 KEY T9 RAY'S THIITRD BOOK. ARTICLE 295. 1728)10+2=12 root. 1000 10X1OX3=300 728 lo0X 2X3 — 60 2. 2 = 4 364 728 ARTICLE 296. (5) 91125(45 Ans. (6) 195112(5 64 125 [Ans. 271'25 170112 4X4X300=4800 55XX300=-7500 4X5X 30= 600 5XSX 30-1200 5X5 25 8X8 64 5435 71 95 8764 70112 (7) 912673(97 Ans. (8) 1i25043(107 729 1 [Ans. 183673 1X 1X300= 300 225 9X9X300=24 300 OXI10X300=30000 225043 9X7X 30= 1890 lOX 7X 30= 2100 7X7 =7X 7 X 49 26239 183673 32'149 225043 ~D-l — --- - _ C C ~~ il EVOLUTION. 149 (9) 13312053(237 Ans. 8 ~X2X300=1200 5312 2X3X 30= 180 3X3 9 1389 4167 23X93X300=s158700'1145053 23X 7X 30= 48301 7X7 = 49 163579 1145053 (10) 102503232(468 Ans. 64 4X4X300-4800 38503 4X6X 30= 720 6X6 -- 36 33336 5556 46X46X300=634800 5167232 46X 8X 30= 11040 8X 8 64 645904 5167232 (11) 5929475129(809 Ans. 512 8X 8X300= 192001 1747.5 80X80X300 —1920-000 17475129 8(X 9X 30= 216001 9X 9 - 81 1941681 17475129 150 KEY TO RAY'S THIRD BOOK. (12) 958585256(986 Ans. 729 9X9X300=24300 229585 9X 8X30= 2160 8X8 = 64 26524 212192 98X98X300=288 1200 17393256 98X 6X 30= 17640 6X 6 = 36 2898876 173932356 (13) ii7602136771(2453 Ans. 8 2X2X300=l1200 6760 2X4X 30= 240 4X4 = 16 1456 5824 24X94X300=-172800 936213 24X 5X 30= 3600 5X 5 = 25 176425 882125 2 1-5X245X300=l18007500 54088677 215X 3X 30- 22050. 3X 3 =.9 18029559 54088677 EVOLUTION. 15 (14) 128100283921(5041 Ans. 5X 5X300= 7500 3100 50X50X300=75000u 3100283 50X 4X 30= 6000 4X 4 16 756(016 3024064 504X504X300=-7604800 762109 01 504X 1X 30= 15120!X 1 = 1 76219921 70219921 (15) 53.157376)3.76 Ans. 27 3X3X300-270 0026157 3X7X 30= 630 7X7 - 491 3:379, 2 3!)3 37X37X300 —410700( 25043'76 37.X 6X 30= G660 6X 6 = 36 417396 2504376 (16).i90i76,04(.584 Ans. 125 5X5X300-75001 74176 5X8X 30=1200 8X8 = 64 876-4 I (1 12 58X58X30-l0 02 1 ()09 4064704 5SX 4X 30- 606'J~; 4X 4 = lOllT7f 4064704 152 WKEY TO RAY'S THIRD BOOK. (17) V216-6. (180) ]2744=14. Ans. q343 —7. 7ns V~6859=19. 48778 24389 V24389=29. 09 (19) Ans. ~j-9. 118638 59319' V59319=39. i(20) ~5 14_7 9 2 729=9 Ans. 95- 4 (20) 5'2-1 —5, V215=5 n. (21) 2(1.25992+ Ans. 1 X1 X300 —300__OO 1X2X 30= 60 2X;2 = 4 364 478 ] X 15 X300=43200 417 2000 1 1X 5X 30= 1800 5X 5 - 25 45)025 251253 125X125X300=4687500 4GS75000 1'5X 19X 30= 33750 9X 9 81 472.1331 42491979 1259X 1259X300=475524300 43830291000 1259X 9X 30= 339930 X 9 - 81 475864311 42982778799 12599X12599X300=47620440300 100242-_01000 12599X 2X 30= 755940 2X 2 = 4 4762G1196-244 95.24.392488 D ------------ -- ------- -------- EVOLUTION. 153 (22) 9(2.08008+ As3. 8 2X 2X300= 12001 000 20X20X300=120000 1000000 20X 8X 30 — 4800 8X8 -- 64 124864 998912 208X20SX300=- 2979200 1088000 20.80X080X300=1 2979200)0 1088o000000 20800X20800X300=129T792000000 108800000000Q 2080OX 8X 30= 49920000 8X 8 - 64 129796992064 1038375936512 (23) 200(5.84803+ Ans. 5X5X300=7500 75000 5XSX 30-1200 8X 8- 64 8764 70112 58X58X300-=109200 4888000 58X 4X 30= 6960 4X 4 — 16 1016176 4064704 584X584X300 —1 0231 680(0 8232996000 584X 8X 30= 140160 8X 8- 64 102457 024! 819656192 5848X 5848X300= 102X5973 120(0 3639808000 58480X5S480(X300:10259732120()00 3639808000000 58480X 3X 30= 5263200 3X 3:- 9 1025978383209 3077935149627 - -- - ---------— ( t15 4 ~ KEY TO RAY'S THIRD BOOK. (24) 91=91.iG666+(2.09'2-+ Ans. 2X 2X300= 1200| 1166 20o2Xo0X300=120000 1166666 20X 9X 30= 5400 9X 9= Sl 125481 112_9329 209X209X/300=13104300 37337666.209X 2X 30= 12540 2X 2' 4 13116844 26233688 (25) V1953.125=12.5 ft. AIs. (26) 3X3X3=27=cu. in. in a cube, of which each side is 3 in. 1728-7 —9764 Ans. (27) 2X2X2:-8= cubic inches in the cubical box. IX-XI=[61-= solidity of a cubical block, each side of whic- is of an inch. 8 —. =5l2 As. (28) A solid half foot is a cube, each side of which is 6 in.; it contains 6XXG6=216 cu. in.; half a cubic foot — of 1728-864 cu. in.; 864-16= —648 cu. in..Ans. cll. ft. cu. ft. ft. (29) 2S8X216X48=2985984, t29985984-144 Ans. (30) 1728X3=5184, V.5184=17.306+ inches Ans. ARTICLE 297. (31) 6 —216; 33=_27. As 216: 27:: 32 lb.: 4 lb. Ans. (32) 11XllXll=1331 Ans. ARTICLE 298. (1) 216-=2X2X2X3X3X3: V2N16 —2X3=6 Ans. (21) t (27X 64)=3X4=1S A-,ls. (3) V(l'25X343)=5X7=-35 Ains. i~ — ARITIHMETICAL PROGRESSION. 155 ARITHM DIE TI C A L PR OGRE SSIOi. ARTICLE 301.-CASE I. (3) 50-1=49; 49X3+2=149 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=-$916-$200=-$16; $16X19+$216=-$520 A4ns. (7) 99 —1-98; 98X-8 — -; 329-53 —=43 Anls. A RTICLE 302.-CASE II. (2) 300-3-=297; 10-1=_9; 297 —9=33 lis. (3) 50-5= 45; 10-1-9; 45 —9= 5 Ans. ARTICLE 303. (1) 49-5-44; 44 —4=11; 11+1=12 Ans. ARTICLE 304.-CASE III. (2) 50+ 2=52; 52'X24=1248; 1248-~-2-624 Ans. (3) 1+12=13; 13X12= 156; 156-*-2= 78 Ans. (4) 10000-1=9999; 9999X1+1=10000, last term; 1-t-10000=10001; 10001X10000-=100010000; 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=G60600; 60600 —2=30300 yd. 30300 yd.=17 mi. 380 yd. Ains. (65) 30)- 1=_9: X -} 307-=_ --- mlli. distanine travelcd last day. 30+')224=5213; 5~2X'30=158218 n i.; 1582- ). 2,- 1 —--- mlii. 4Al.s. < F (~~~~~~~~~~~~~' 156 KEY TO RAY'S TtIIRD BOOK. (7) Conmmon 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. GEOM1ETRICAL PR:EOG RESSION. AR T I C L E 307.-CASE I. (3) 919-4096; 4096X2-8192 An7s. (4) 48 =65536; 262144 —65536=4 Ans. (5) Ratio-3; 39-19683; 19683X10-IO19G6830 Ans. (6) 234=17179SG9184; 17179869184X1-171.79869184.ABs. (7) 334-16677181699666569, which, multiplied by 1, gives the slme number for the 35th term. A RTICLE 308.-CASE II. (2) 36-729; 729Xi0-=7290, last term. 7290X3=-21870; 21870-10-21860; 21860 —. 2=10930 Ans. (3) 211=2048, 2048X1=2048-last term. 204SX2=4096, 4096-1=4095, and 4095-+-1=4095 Ans. (4) 411=4194304, 4194304X1-4194304, last term. 4194304X4-16777216, 16777216-1=16777215, 4 —1=3, 16777215 3 —5592405 cts.:$55924.05 Ans. (5) tRatio=2,239-549755813888, this multiplied by 10, gives 5497558138880, for the last term. 549755813 8880X2X-109951162177760; from this subtract 10, and divide by 2-1-1, and we get for the Ans. $109951162777.50 GEO.IMETRICAL PR-OGRESSION. 157 (6) Ratio=3, 33'=-617673396283947, 617673396283947X3=18530'201818885141 3-1=") 1853020188851840 $9265100944259.20 Ans. (7).3X10=3; 10-1=9; 3-. 9=A.As. (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, X-X2-1; 2-1=1; 1 — 1=1 Ans. ARTICLE 309. —PE R TATIO N. (1) 1X2X3X4X5=120 Ans (2) 1X2X3X4X5X6X7X8X9X10X11X12 =479001600 Ains. (3) 1 X2X3X4X5X6X7X8=40320 days =110 yr. 170 da. Ans. (4) 1X2X3X4X5X6X7XSX9XOXI XI'X(12X 3X14X15 16X17Xls X19X20X21>X22X23X24X25X'26 -403'391461126605635584000000 Ans. The subject of permutations and combinations properly belongs to Algebra. (See Ray's Algebra, Part 21.) ARTICLE 311.-MIENSURATION. (1) 17X15-255 sq. ft. Ans. (2) 2 ft. 3 in.=2.25 ft., 12 ft. 6 in.-12.5 ft., 2.25X12.5 =28.1_25=28' sq. ft. Ans. (3) 15 in.=-1 ft.; 16 ft.X1 ft.=20 sq. ft. Ans. f3 -I ---- - - — 4 4 —-~ 158 IKEY TO RAY'S THIRD BOO. (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 2in.=1.25 10 2 10 6' =4.0625 4" =_'.041.6Ans. $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) 6X 5=30sq. yd., 30_. 1- =20 yd. Ans. (14) 21 ft. 3 in.=21.25 ft., 13 ft. 6 in.=13.5 ft., 21.25X13.5= 286.875 sq. ft., J4 yd.=33-33.75 ft., 286.875. 3.75= —76.5 ft=25.5 yd.=-51 yd. (15) 3Xl_=41 sq. yd., 4. —2X4 —6 yd. Ans. MENSURATION. 159 ARTICLE 312. (1) ft. in. (2) ft. in. 25 9 22 7 21 3 13 11 540 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 51 6 1 0 6 73 6=circum- 1 - 10 4 Iference e -ciln 510 4O I f~ence 333 1 6=cellng. 26 O 6=fire0 [of room. Lpl;ce. 24 6 Deductions. ft, in. 26 0 6 4 2 759 6=walls. 29 2 7 94 2 6=deduce ft. in tions. f.n 9)993 33 6. 110 sq. yd. 8 sq. ft. and 5'-=5 sq. ft. 19 6 [Ans. 2 39 0=-windows. 1G0 KEY TO RAY'S TIIRD 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 [of room. ft. in. 35 4 32 83 6900 6 2 23 282' 713 0 19 8 0 55 4=deductions. 17 4 9)29 6 9)657 8 55 4 32 sq. yd. 3sq.ft. 73 sq. yd. 8' 27 256 511 32 146 1576 19.71 3 sq. ft.-I 6 8/=27 2 6' =6 1 nAcs. $19.73 Ans. $5.83 yd. (6) 21 15 5 ft. 3 in.=5[ ft.=1l yd. 105 21 21X1, —-363 sq. yd. 315 =sq. yd. in whole. 361=sq. yd. in foot-path. 278- 36M 24 36 I11'2 216 556 108 6 27 $66.78 13.23 66.78 $80.01l Ans. MENSURATION. 161 (7) ft. in. ft. (8) ft. in. 75 6l75.5 6 iI l12 6=12.5 5 4 9)943.75 —sq. ft. 8 3 10 7=windows in a line. 40 3 1.048.6+ cts, 16 (ts. Ans. $10.4862+ 404 49 6 Arfs. $654.68 (9) ft 9)943. i ft. ft. (10) ft. in. 36 3-36.5 3 35 4 6 16) 6 —16.5 30 6 1048125 I050=sq. f 27 y0=fie-place. 21750 4 cts. eductions. 3625 362'95 X 100)9139 ft. in. j100)9130 10 6 1100)598.125 9.39squares. 8 5.98125 $5 84 O(stai,'s A3 ns. $46.9 5 7 Ans. 817.94375 II1 0 (11) ft. in, 18 6 18 6 37 0 40 1100)14SOd=sq. ft. 14.80= —squarses 83.50 74000 444(0 Arns. $51.8000 Key.-1 1 162 ~KEY TO RAY'S THIRD BOOK. ARTICiE 313. (1) 1>(15.-2=.90 sq. ft. A47s. (2) 44X].- --— 396 P.=2 A. 1 R. 36 P. An73 (3) 80X67- -2=2680 P.=16 A. 3 R1. Ans. (4) 12 ft. 6 in.=12.5 ft., 16 ft. 9 in.=16.75 in., 12.5X1 6.75 —2= ] 04.6875 sq. ft., 104.6875X12 -1256.25 cts. —$12.56 Ails. (5) 13-t-14 I15-=42, 42 —2= —21, 21 —13-8, 21 —14=7, 21 —15-6, 21X8X7X6=7056, V/7056 —84 sq. ft. Ans, (6) 2+3+4=9, 9_. 2=4.5; 4.5 —2=2.5; 4.5 —3=1.5; 4.5-4 —.5; 4.5X2.5Xl.5X.5-S.4375; /8S.4375 =-2.9047375+ sq. ft. Ans. ARTICLE 314. (1) 35+6-6-61; 61X11=671; 671 —. 2=335 sq. in. A.s. (2) 19+25_44; 44X32 —1408; 1408- 2=-704 P. — 4 A. I R. 24 P. Ans. APTT-Ic1 315. (1) 48X3.1416=150.7968 ft. Ans. (2) 4X3.1416=12.5664 ft.=12 ft. 6.7968 in. Ans. (3) 7912.4X3.1416=-24857.59584 nmi. Ans. ARTICLE 316. (.) 15-~-3.1416=4.7746.+ft.=4 ft. 9.295-+in. Ans. (2) 12 ft. 5 in.=12.4166-+; 12,4166- -3.1416-3.952338 ft =3 ft. 11.428-%in. AntS. ARTICL E 317. (1) 429 — 2=21; 21X21 —441; 3.1416X441 =l385.4456 sq. ft. Arns (2) 7; rods=:7.5; 7.5X7.5 —-56.25; 3.14X6)56.325 — i76.715 P. — A IA. 16.715 P. Ans. IIMENSURATION. 163 ARTICLE 318. (1) 962.115 —.784-25; 1/122'5-35=diamcter; 35X3.1416-109.956 -in s. (2) 160 —.7854=203.71785077t+; V/203.71785077 =14.29_ _ +; 147. 29 —2=7.1 364+ rods; or 117 ft. 94-in. Anzs. ARTICLE 320. (1) 3 ft. 3 in.=31 ft.; 4 ft. 4 iu d. —4 ft.; 12X3iX41 =I69 cu. ft. Ans. (2)') ft. 6 in f-.; ft.; 9 in.1 fft. 9 i; 6X'2X] 3I -6-4 cu. ft. Ans. (3) 3 ft. 29 in._-31 t; 2 ftft. 6 in.2-. ft. 3'X22 X2. 9 X1 — 9- 1 cu. ft. 21-, 81 cts.=-8$( 10 cts. Ans. (4) ft. in. 4 10'2 1 1 8 4 5 2 14 1 2 2 9 92S ) 4// 2 4 2 4"// Anis. 30 6/ 6" 4"'/// ARTICLE 321. ( 9) 97 ft. 5 in.=-97.416+ft.; 18 ft. 3 ia. —l8.25 ft. 2 t 3 in. =2.25 it.; 97.416(X1S.'25X*2. 5=4000.l445 cu. ft. 4000.1415 - 24.75-5161. -+P. Ans. (2) 53 ft. 6 in. —53.5 ft.; 12 ft. 3 in. —12.25 ft., 53.5 X2s'-)5X2=31(0.73 cu. ft.=O52. 0S5-P Als. e~ —_. ~"y-"n —--— "~" 1 (4 KEY TO RAY;S THIRD BOOK. (3) 53 ft. 6 in.=53.5 ft.; 12 ft. 6 in.=12.5 ft.: 53.5X12.5X2 - 1337.5 cu. ft.-54.0404- + P., 54.0404X0$9.23 4$121.59+ Ans. (4) 48 3X16 - X'l I X 13 XS-_-_..- =196_ c1. ft. 1196~}X20= 239925 bricks An1s. (5) 15 in.=l1 ft.; 320X6X11 —2400 cu. ft.; 2400><'20 _48000 bricks; 48000X<$5.875 —. 1000$282 AnZs. (6) 120X8X1,==1440 cu. ft.=2488320 cu. in. in wall. 8X4X2.25 —72 cu. in. in each brick; 2488320 -72 -34560 bricks Ans. (?7) 240X6X3=4320 cu. ft.=7464960 cu. in. in wall. 9X4X2=72 cu. in. in each brick, 7464960 729=103680 bricks. 103680-. 1000=103.68; $3.25X103.68 =$336.96 Ans. ARTICLE 322. (1) 2+2+2=6; 6 —2=3; 3-2 —1; 3 —2-=1; 3-2-1. 3X1X1X1-=3; 1/3=area of base. 14XV3-3=141/3 =area of base. 14V/3=V/196X/3V/ 588 =-24.2487+-cu. in. Aims. (2) 4X4=16;.7854X16=12.5664=-area of end. 12.5664X12=150.7963 cu. ft. Ans. (3) 181=18.5: 18.52=342.25: 34'2.25X.7854 —268.80315 cu. in.-area of end. 268.80315X8-21500.4252 cu. in. Ans. (4) 62X.7854=-2S.2744=area of end. 28.2744X8 — 26.1952 cu. ft.=390865.3056 cu. in. 390865.3056 — 2150.4 —181.764 bu. An.s. (5) 15X5X4=300 cu. ft.-518400 cu. in.; 518400 —21.50.4 =241+bu. Ants. (6) 12X3X5=1S0 cu. ft.=311040 cu. in.; 311040-21.50.4 =144.6+bu. Ais. lMENSURATION. 165 ARTICLE 323. (1) 5X5-=25 sq. ft.-area of base. 25X21-.-3 =175 cu. ft. Ans. (2) 102X.7854-78.54, area of base. 78.54X15 —3 =392.7 cu. ft. Ans. (3) 37 ft. 8 in. —452 in., 452X.7854-160460.31.l6=area of base. 79 ft. 9 in.-957 in. 160460.3616X957 3-3 _-51186855.3504 cu. in.=29622.0227 —cu. ft. AnVs. (4) 720X720-518400=area of base. 518400X477- -3 =82425600 cu. ft.=3052800 cu. yd. Ans. (5) 1=1.5: 1.52X.7854=1.76715 —area of base. 1.76715 X9-i- 35.30145=cu. in. in cup. 231 — 5.30145 =43.57+times Ans. ARTICLE 324. (1) 3X39-=area of lower base. 2X2=-4=area of upper base. 9X4=36; 1/36=6=area of mean base. 9+4+6 —19; 19XZ2-=76 cu. ft. Ans. (2) 27X27-729=area of lower base. 16X16=256=arteoa of upper base. 729X256 = 186624; /186624= 4a2 -area of mean base. 729+256+432=1417; 141.7 X — =105802.6666+cu. in.-z61.2283950+c fcut. Ans. (3) 2-=2.5: 2.52X.7854=4.90875=area of mouth. 12X.7854=.7854-area of bottom. 4.90875X.7854= 3.85533225; V/3.85533225=1].9635=:area of mean base. 4.90875+.7854-Fl.9635=-7.65765; 7.65765X-, =12.76275 cu. in. Ans. ARTICLE 325. (1) 133=2197; 21 97X.5236=1150.3492 cu. in. 15a=3375; 3375X.5236=1767$.15 cu. in. 30-==27000; 7000X.5236 =14137.2 cn. in. A.s. lb6 KEY TO PAY'S THIRD BOOK. ARTICLE 326. (1) 5 ft. 4 in.X3:-=16 ft.-circ-umference of base. 7 ft. 6 in. 4 — 7 ft., 1 6X 71 —2 —60 sq. ft. Ans. (2) 8;-\/3.1416=26.7036=oircumference of base. 26.7036X25-. 2=333.795 sq. ft. Ais. (3) 2 ft. 11 in.=35 in.; 35X3.1416G=109.956-circ. of base. 109.956X55-. 2=3023.79 sq. in.=area of conv. surface. 359 —1225 1i225X.7854-962.1i15 sq. in. —area of base. 3023.79-+969.1.1 )-3985.905 sq. in.=27.G'79S'95+ sq. ft. Ans. (4) 7912= —62599744; 62599744X3.1416 — 196663355.7504 sq. mi. Ans. ARTICLE 327. (1) 10 ft.=120 in., 5 ft.=60 in., 4 ft. —-4S in. 120XG60X48 _345600 cu. in., 345600 -231 — 1496-F-g1. Ans. (2) 12 ft.=144 in., 6 ft. —2 in., 10 ft.-=120 in., 144X72X120- -1244160 sq. in., 1244160 —2150.4 -578.57+bu. Anis. ARTICLE 328. (1) 4 ft.=48 in.; 6 ft.=72 in.; 4S2=2304; 2304 X72 =16588; 16588SX.7854-13088.-1352 cu. in.=564.01 9 -+-gal.=17.9+bl. Ans. (2) 6 ft.=72 in.; 9 ft.=108 in.; 722-=5184; 5184X108= 559872; 559872X.7854 —439723.4688 cu. in. — 1903.564d —ga1. —60.43+-,bl. Ails. ARTICLE 329. (1) 40X30=1200; 40 —30=10; 10X10 —100. 100 — 3=-33-; 1200+33: =133 3; 12333 X50 —-61666 GG 61 66b X.7854 =48433 cu. in.=A209 -'"09 B +Ggal. Ans. MENSURATION. 167 (2) 7 ft. 6 in.-90 in.; 10 ft.=120 in.; 12 ft. 6 in.=150 in.; 90X1'20-10S00; 120 —90=30; 302-900; 900 — 3 =300; 108300+300-=111100; 11100X150=1665000; 1665000X.7854 —1307691 cu. in.=-5661 gal. =-1795 bl. Ans. ARTICLLE 330a. (1) 40-31-9; 2 of 9=6; 31+6=37; 37X37=1369; 1369X50X.00C34-=-232.7 gll..Ans. (2) 32-30-=2; 2X-6-=1.2: 30+1.2=31.2; 31.2X31.2 =973.44; 973.44X40X.0034z-1 32.38+gal. Ans. ARTICLE 330Qb. ( I) 1 A.=160 P.: 160 —. 15 —10U - rd. Ans. (2) 24X1S-X10. —-=4699 cu. ft. Aiis. (3) 144-.-.7854- 183.34606569; 1/183.34606569 =13.5405+in. Anzs. (4) 1728-.-.5236=3300.229182-+; /3300.229182 =14.8884+in. Ans. (5) 30+40+50-=120; 120 — -2=60; 60-30=30; 60 —4)0 =20; 60 —50 —10. G6X30X20XO10-=360000; 1/360000 =600 sq. ft.=66- sq. yd. An.s. (6) 12 ft. 6 in.=150 in.; 15+11=26; 26X150 —2 =1950 sq. in. —13, s q. ft. Anis. (7) 162X.78542901.0624, area of greater circle. 102X.7854=78.54, area of inner circle. 201.0624-78.54 =122.52'24 sq. ft. Ans. (8) 6+6 —2+2=16=circunmference. 16X29=40=area of sides. 6X2-12=area, of bottom. 12+40-52=sq. ft. of sheet lead. 52XS-=416 lb. of lead. 4 cts.X416 -=$16.64 Ans. (9) 5 ft. —60 in.; 4 ft.=48 in.; 60X48X48=1382410 s. in.; 138240-1-. 2150.4-64.2S+bu. 25 cts.X64.28 =$16.07 Ans. 168 KEY TO RAY'S THIRD BOOK. (10) 1448- -3.] 416460.91 1G3+ ft.diameter; 460.91163+ X1448=G-67400+; 667400+ — 4=16S65)+ sq. ft. —3 A. 3 R. 12 P. 25 sq. yd. 8+sq. ft. Anls. (11) 372=1369 sq. in.= surface of one side: 1369X6 — 8214 sq. in. Anls. ARTICLE 332. (1) 1236 —3.-412 Ans. (2) 675+225=9)00: 900 _9=450 AnIs. (3) 292X3 —-876: 876 — 4 —219 Ans. (4) 5 l). 15 oz.-:-_ 5 lb.: $1.20X5 — 7.12t As. (5) 13 bu. 3 pk.=13? bu.: $.45Xl331 —$19.933 Ans. (6) 1oo+000+t$600$1: 600 o=-; ofo 32 0-200 A.-A's share': 3 of 320 A. —120 A.-B's share. Ans. 17X2 (7i) 4 a2 s 3 3 XT -X4 hr. - 11 hr. 20 min. Ans. 59X2 (8) 14= -0. X S. (9) 6 tines-3 times the number-3 times the number —81 81 — 3=27 Ans. (10) 3X4- $1 o0 $' $' - + 3o -2L23XIO,3,$23 _-33- 111 —L yd. A.ns. (11) l -1=-3 of 1 I 1 ~1 1: -=$1 ~ 125;X4 — )50() 0Ais. (12) 2i7 11+] 0-5 1T1, 10- 4 01 2 7 11I 1 41 0 5 X6' Xi-.73 _3.300781_ A15s. (13) 60-19-41 yr.= the time since marriage: 41+21= 62 yr. Alns. (14) 2)8 12 21 36 48 65)143(2 130 2)4 6 21 18 1)24 ) - 9 13)65(5 2)2 3 21 9 12 65 13=greatest common divisor. I 1 7 3 2 2X33X7X3X' 2X2'-100S, least common multiple. - t 3 - 13 1S. MISCELLANEOUS EXAMPLES. 169 (15) 3 mi. 5 fur. 110 yd.=233640 in. 233640 in. —39.371 in. — "5934.317+meters Alns. (16) 800000000 -250=3200000 min.=14 yr. 223 da. 3 hi.. 20 mnin. Ans. (17) 163-50. 12.625 —. o1262503.7575 Ans. 50 (18) yr. mon. da. 1855 5 10 1S54 5 20 11 20=l]. 2mon.=3- yr. $250X 3 $243' Ans. (19) 1 lb. of each costs 12+30+5-=47 cts.; $i7 —. 47 cts.= 100 lb. AnS. (20) - 3 —_1 _ h 7 If =35 gallons, of 35 gallons, which is 5 gallons, will be 1 and the cistern, when full, will contain 5X24-120 gallons. (21) 60X5=$300: $4X23=$9: 60 —3 —37 bl. $300+$51 =$351; $351 $92-=$259; $2590 —37-7 Aas. (22) 12 cwt. 1 qr. 7 lb.=1232 lb.' 3 qr. 13 lb.=8S lb.: 1232. 8S —=14 As. (23) 102X3.1416=314.16 sq. in. 5cts.X314.16=$15.708 Ans. (24) If 1 ox is worth S sheep, 3 oxen will be worth 24 sheep, but 2 horses are worth 3 oxen, therefore 2 horses are worth 24 sheep; hence, 1 horse is iworth 12 sheep, and at $2.50 a piece, 12 sheep are worth $30 Ans. (25) If of $1, buy a of a sheep, 1 sheep is worth 5 times $=_130, and 3 of a sheep is worth 3 of $-O=$ ~; hence, of an ox is worth $', and 1 ox is wortrtt 14 times $,0 =$20; and 10 oxen are worth 10 times $20=,$200 Ans,s (26) 19 is 21 times 9, therefore the required number must be 01 times 54: 54X2`-114 An.s. (27) ~ 1 3 of 31; l is 3 times 1 -'X3-1 As. — 24 4 -- —' —;) I 3 of3~-=1; (28) 1 3X5 1 —GS — hr. required to perform the wrok. 68- — 9 — 7 3X4 3 days of 9 hr. each. (29) $40: $100:: $15: $37.50 AnMs. (30) From Monday noon till 6 P. M. of Thursday is 78 hr. 170 KEY TO RAY'S THIRD BOOK. nwhich is 3~ days of 24 hr. eatch; hence the clock has gained 3' X7 7 min. 71 X3- 1XY 1- 3 - s __ 4 minl., therefore the time by the clock will be 6 hrs. 24 3 min. (31) If 7 men can mowv 35 A. in 4 days, 1 man can mow v of 35 A., which is 5 A, in 4 days; and in 1 day 1 man can mow T of 5 A., which is 1- A.; if 1 man can mow 1} A. in 1 day, 10 men can mow 10 timnes 14 A., which is 1i Aj, in 1 day, anid in 31 days they can mow 31- times I ). A. ~X 7 17 L -4343 A. An s. (32) 35x 6, 1 d1 1 X 6-$ 0: $20 t(;+$l? 71-oL ) 2-$4.3" 35 — 67+ An7s. (33) 11 hr.XG=66 1ir.: 10X60=660 cu. ft. 660 cu. ft.. (SX2-)=-6 ft. An,7. (34) If 4 men can do the Aork in 2:) days, 1 man can do it in 4 times 20 daysSO 0 days; hence I man can do s, and 3 men -o(j of it in one. day. IL C rwomen can do the -work in 20 days, 1 woman can do it in 6 times 20 days-_120 days; hence 1 woman can do i }, and 5 women ~~ of it in 1 day. Therefore 3 men and 5 women, in 1 day can do: 3___+ e l-U)t of the work, and therefore they can do it in 1 149_ 40-1L'2 d ays. 40O....T9 ~ (35) 40-0-GO-.100 yd.: 5 y(l.: 40:: $G: $48 B yd. 6 Oyd.:: $9: 90: 5 yd. 100 yd.:: 7: $140: $90+$4S-$13S $140 —8138 —8$2 Ans. (36) After the first drawing 4o 0 4 0 iare pure wrine, antd when 10 gal. more are drawn, 4 of the remaining 40 gnl. are pure wine; 4 of 40 gal.-32 gal. Ans. (37) In 14 days. of 10 hr. each, there are 140 hours. If 27 men caLn do the work in 140 hours, 1 man can do it in 27 times 140 lhr. =3780 h1r.; and since the work of -a tboy is half thlat of a man, it will require a 1)o? twice 37S0 lhr. —15G0 hr. to do the work. If' iit requirels I boy 7(60 dh —--- _- w3M.~ ~~ zw~-s. TMISCELLANEOUS EXAMPLES. 171 hr. to perform the work, it will take 24 boys -4 of 7560 hr.-=315 ]ir. to do it. 3!5 hr. - 45-=7 hr. Anas. (38) 16' 40"/X9=2~'30/: 2~ 30'-15 gives 10 min. A4.n1s. 10 min. before 9, (39) S,0 men: O' nmen S?O./ ft. 1. ~00 ft. 1. g;.3ft. w.. ~f.,5ft. w. -.O ft. d.: 1. ft. d. [9Xl 1=9 da. Ans., 11 hr. 9 hr. 0. da. (40) $1+$.90 —$1.20: $14 - 1. 0=-81, Ans. (41) 9 cows-3 times 3; if 3 cows=2 horses, 9 cows-3 timoes 2=6 horses: 4+6=10': 4- a! of $45_T18, A's sharec: ~3 of $45=$29-, B's share. (42) 35 mi.- -.2=14 mi. ihe sum of the rates: 14 —5 —= —9 9 *24,-l mi. one: 4A+5 —9 mi. the other. (43) S1 —.17.7-1$.85:,14.85-..892=1$8, purchase price. s18X. 1i2-$2.25: 618+,$2.2o5=300.25 Ams. qt. bu. qt. (44) 1 bu.=32 qt. 39-4-o cqt. 28: 2:: 32: 2 bu. 1 pk. I1 qct. -Ans. (45) The provisions of 32 men for 5 monr., will last 1 m:ln 160 mon.: the provisions of 1 man for 160 mon., in 8 mon. i would be consumed by as many men, as 8 mon. is contained times in 160 mon., 160- 8=20 men, hence 3220=12, the number of men that must leave. (46) Int. for 63 da. at 6 0 1$..10 $1-0105=.9895 tl009.29 —..9895 —$1020, the face of the note. Int. for 60 da. at 12 %-$.02: $1+$.02=,1.02: $1020-.-$1.02-_$1000 ATs. (47) $12 —. 20=.60 gain per yard.'T'he difference between 5 % less, and 10 a0 more is 15 %: n.60-.s-15-=W As. [ (48) 9X9X->-401 cu. ii. 14XllX —=96- cu. in.: 40-: 96:": $3.24: $7.70 Ans. (49) 120+60=180: 180: 120:: 3 mon.: 2 mon. Ans. (50) 25-2-0-=5 rd. gain in 25 rd.: 5 rd.: 47 rd.:: 25 rd.: 235 rd. Jlms.. s *M~ —----- c~Dl~crID -~P~~CI C 172 KEY TO' AY'S TI-IRD BOOK. (51) Int. of $1 for 4 mon. 3 da.=$.0205: $1- -.0205 -$.979O: $300 —.$.97790- 58306. 27+A,,s. (52) S1.50X40=$60: $60-G.8$1.0= 50 gal.: 50 gal.-40 gal. -=10 gal. Anls. (53) Int. of I1 for 63 da.=$o.0105: $1.26$.0105-$120 Ane. (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. —40 rd.,* to catch the hare. (55) 1-90: $1 -$.20 —$1.20: $1.50- -1.20=.$1,'25 the cost; $2 —$1.25-$.75: 1.25X.01-$.0125 $.75 — $.01 25-60 Anas. (56) $260+$13=$273 received fior those sold: $1.30X.40 =g.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 —=2 mi. per hour; 40 mi.. 2 mi.=-20 hr. A2es. (58) A in 1 da. mows B A.; B - A.; and both can mow - g=11 A. in 1 da.: 9 A —. 1.1 A.==6 da. Ans. (59) Both do 4 times as much as B: hence B could do it in 20X4=80 da, and A in i of 80 da.-=9-2 da, Ans. (60) 1 of the first lot cost 2A cts., and 1 of the 2d 2; cts. hence, I of each, that is, 2, cost 21_+2o:=5i cts.: If 5 sold for 14 cts., 2 sold for 2 of 14=53 cts., 5 3 3 ct.-~gain on 2 - ct.: 52 cts.:: 2: 240 As. (61).; 3= 1f6; A can do it in 5 da. and in da. he can do j,, and in 1 da.: similarly, B can do 5-1 in I da.~ -g + bothdoiwhant botIh do in 1 1 da. Ails. (62) As 2 min.: 3 min.:: 11 rods: 16, the space traveled by A in 3 minutes. HIence B, in traveling 17 rods, gains,~ rod on A; and the object bleing to find how many rounds he must make to gain half a round, we have this proportion: As, rod: 17 rods:: round: 17 rounds, the space to be gone over by B: consequently A will make 16] rounds. MISCELLANEOUS EXAMPLES. 173 (63) A does -- in 1 da., B 1 and C: all ecan do - + 7-k-o5 i in 1 da.: - 1 — + d> is. (61) From the question, it is obvious that in 1 dlay A and B do A and C. and B and C -,l' of the work. By taking, the sums of these fractions, we find lcice whlat they all do in 1 day; that is, -a- + i1 + + - v u o --:O-,= is twice the part of the work they all do in 1 dayI; here of = —o, is the part of thie Nork 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 3.0) 4 15 =.60 50 20 -10.00 109 )16.60(15To5 cts. Ans. (66) If 10 men can dig 1 acre in 11 days, then I man can dig 1 acre in 11X0 —1.1 0 days, and I man can dig T- 1 of an acre in 1 day; then 5 inen can dig 1 oX5 1 of' an acre in 1 day, and 5 5men can dir ], X6,-of an acre in 6 days. 6 -3 -6I-3 -6 3? acres to be dg bly boys. If 1S boys can dig 1 acre in 11 days, then 1. boy can dig ] acre in 11X]8-198 days, a.nd lboy can dig T00 of an acre in 1 day; then 1 bohy can dig 1 X6-1,6 of an acre in 6 days. But since 6? acres are to be dug in 6 days by boys, each of wmhom can dig ~ of an acre, it will require as many boys as.-'3 acre is contained tinies in 3 acres 3 _6 3~ X -3=63X3-189 Ans. (67) 24 in.=- ft.: 15+5++12~+12 =54 ft., the distance around; 54 ft.X10 ft.=540 sq. ft.: 540 sq. ft.. 2 ft.. 270 ft =90 yd. Ans. (683) A could do it 3 times in 1 da., B 5 times and C once' therefore, the three could together do it 9 times in 1 da., and do it once in - cda. AinS. (69) 98)=784: 784+50=834 men Ai.s. 174 KEFY TO RAY;S THIRD BOOK. (70) 40 —1600: 1600-+1600-300 sq. ft.: V'82)00==56.56 ~- + ft. A_,s. (71) Suppose the wNork to consist of 150 equal parts, then the person does' 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 I day: therefore 150. 7=214 days. (72) 1 — 1 =-4y s share; 1 a — 4 —- of the estate=$87420:!- 4~ of $7420-=$1060, and the estate —$1060X15= (73) 24,'=576: 182=324: 576-1-394=900-=square of the diagonal on the floor, this is the base in a triangle where the l-ypotenuse is sought. 1'2=144: 900-144-11044: V1.044=32.3 + ft. Ans. (74) 56 lb. tea —2X56-140 lb. coffee =3.X140- 490 lb. sugar: 10 cts.X4900-49 Ans. (75) From the question it is evident, that all three do'-7 of the work in 1 day, and that B in 1 day does i, nrd C.; hence A does I - ( 4 +-f) - -X (-078 4-0)- ig= — 3 ( —4zdq) —4-08~ in 1 day; thetefire he cal do the work alone in I -- 8 — 81 days. Ans. (76) A built 4 as much as C: B i3 of 4, that is, as mluch as C: -4 +1+1- 1l0 A built 4 outof 1P, that is, 410=: B and C each 3 out of, that is 2; - of.00 ft.=O80 ft. A -o of 200 i't.=60 ft. B and C also 60 ft. Anzs. (77) 42.43 2=1800.3049 squ'are of hypotenuse: since the two sides are equal,. of 1800.3049, or 900.15945 is th e square of 1 side, Which is the area. 900.15W215 P.=5.62+A. (78) If hle lhad wa,ked the whole time he would have earned $'1.25X" 25 —-,;1. 25. INow, each day he was idle lie lost $1.75, tha.t is, 50 cenrts whichll ie speant, and $1.25 which lie did not earn. Bi;t e lost irn all the difference between $31.25 and $19, whlich is $12.25, and he lost this at the rate of $1.75 per day tbelrefor lie e was idle as maily MISCELL ANEOUS EXAMPLES. 175 days as 1.75 is coin';ained times in $12.25, whIich is 7; therefore he was idle 7, and consequently worked 18 dst. lAns. (79) B, who works 30 days, will do, the work: hence'A must do, whllich will take him 20 days. 20-3=17 days more: 30-17-13 Anls. (80) -15z-19 3 of 3 19 4: 1-( 47 4-Cs; 4 4 - 4 _ —~$1300. Property=$l300X7=$9100: T- of $9100 -$3250=A's; 4 of $9100-$3900=B's; 1: of $9100= $1950=C's Azs. (81) By the first it receives -- in tlle ohour, by the second -; by the tthird 1te t 1 —7iid in 1 hour; 1' -7 — 52 hr. Als. (82) $1.20X60 —$72.00: $72X.10-$7.20 $72 — $7.20= $79.20 the amount to be received: $1.50X20=$30.00 $7 9.20- $30$49.20: 60 —(20+5)=35 gal.: $49.2035=$1.40; Atjis. (83) 1 apple cost 4 ets., and sold for 3 cts.: gain on 1 apple —.3 — ct.: 24 cts. —~. ct.=14i Ans. (84) $4500X.018=$S1: $500-($25+$81+$134)=$500$,240=$260, net income; $4500X.01=,$45:,260_ - 45 59 Ans. (85) 60X30=-1800 sq. ft.: 12 ft. 4 in.XS ft. 4 in.=104 sq. ft. 10': 1800 sq. ft.-D-14 sq. ft. 10'=1695 sq. ft. 2': 1695 sq. ft. 2'X4=6780 sq. ft. 8/ AnIs. (86) $700+$600-$4-00=8$1700 -7~o=-4I ~i60=- gain:,1 = of $260 —$65 and gain —$65X17= $1105 Ans. (87) 1'2X.10-12: 19+8=20: 20 —.05=4-00 Ans. (8B).$120- 4=$30 B's yearly debt: A, spending $350 less, will save $3'50 — 830=-_$30-;= his annual income; $320 X3=$960 Alns. (89) 53X.5236-65.45: 65.45X2=130. 9 53=125: 12'5X2 =250': 1o0.9+o'50-380.9.: V380.9:-7.24+in. Ans. 176 KEY TO RAY'S THIRD BOOK. (90) 1.00-$-.16=$.83:. $.83.$30 —cost of 1st.,30 — $5=$5 lost on tile first, and therefonre gained on the second. $5 i —. $.16-$31.25cost of 2nd. (91) i-1-; -24000; 4-3000 slain: of 3000-1500 lAns. (92) The 3d —- of' 3 times the 2nd —3 the 2nd. 1+3+ i+t -— 14'3, of which 4 is 4; V, 1.; and 4 F9; 4 of 95 yd.=20 yd. 1st.: {s of 95 yd.=60 yd. 2nd: and 31 of 95 yd.=15 yd. the 3d. Ans. (93) $4X60=$240: Amount of $1 for 6 mon.=$1.03; $240 —.i1.03=$233.009 —-: $250 —$233.009+-= 16.9(+ Aas. (94) As often as he buys one cow, he buys 6 sheep and 6-2 horses; 1 co -cost 30; 6 sheep $30; 2 horses $240. While buying 1 cow, he expends $30+830+$240-$300; $1500-. -$300=5 cows; 5X6=30 sheep; 30 -- 310 horses. Ans. (95) 4 polls cost $4: $o328-$4-=$3o4, the tax on his property; $324- -.016-$20250 Ans. (96) C does 1 —=-; B does 4 Ans. (97) B received X of A's+$100; both received of A's+$100,. of A's+$100=$1000; A=.$900: A's-$600, B's1.000-$600 —-- 400 Anis. (98) i of B's= — of A's+$50; -9 Of B's-: of A's —$25, and B's — of A's+$7-5, both have'17 of A's+T75=-$500: 17 of A's-$425: -$25: A's=$200 B's=8500$'200 —$300 Ans. (99), of the tilme past noon is equal to ~ of the time to midnight, and the whole time past noon — or 3 the time to midnight; the time from noon to midnight, which is 12 hr.=l+-, -4 the time to midnight: 1' - of 12-3 hr. [An.s. (100) In the 4th store $- n was her money; hence, on leaving the 3d store, she had $1; before she spent the $1, she had $2 —half what she had on entering the 3d store: hence, on leaving 2nd store, she had $4; before spending $. she had $41-1. of $9, her money on leaving 1st store; $9+$1l$10-. of $20, her money at fitst. THE METRICAL SYSTEM. 177 ARTICLE 334. (1) 30.75 M.IX100=3075 c.m. (2) 4.5 K.M.X1000=-45 00 M. (3) 75 n.m.. —1000 —-.075 M. (4) 25 D.M. X100=2500d. m. (5) 35 A. Mf.X100=3500 H.iM (6) 36.5 M.-i-10=3.65 D.M. (7) 36.5 MX10=365 d.m. ARTICLE 335. (1) 26304 d.m-. -100=263.04 M.2 (2) 1460 m.m.2 — 1000000=.00146 M.2 (3) 14 D.M2X1l00001440000 d.m.2 (d:) 20 M.M.2X100000020000000 D.DMa,.2 (5) 3294 -I 2L.. 100003294 HI-..2 (6) 14.1 H1.I.2X10000=141000 NI2 ARTICLE 336. (1) 26.25 A.X100=2625 c.a. (2) 21 H.A.X100=2100 A. (3) 3.5H.A.X10000=35000 M.2 (4) 14209 M. —. 100-142.09 A. (5) 3.8 A.X100=380M.2 (6) 27 d.m.2 -100=.27 c.a. ARTICnLE 337. (1) 9 SX1o=90 d.s. (2) 4.19 d.s —100=.0419 D.S. (3) 29 M3X1000000=29000000 c.m.3 (4) 25 D.S.X100 —2500 d.s. (5) 16.32 M.3 -10-1.632 D.S. (6) 14000 c.m.3 —1000000-.014 M.N lK cy. -1 2. 178 KEY TO RAY'S THIRLD BOOIK. ARTICLnE 338. (1) 1428.06 G. —1000=1.42086 K.G. (2) 2.8 K.G.X1000=2800 G. (3) 119 H.G.,<1000-119000d.g. (4) 171300 G —100000 —1.713 quintals, (5) 3600 G —-1000=3.6 K.G. (6) 19200m.g.-:-1000=19.2 G. (7) 4 quintalsX100000=400000 G, (8) 29 G.X100oo=2900 e.g (9) 492 D.G. —-10000 —=0492 quintal. ARTI CLE 3393 (1) 24 LX102 —400 c.l. (2) 183 H.L.X 1000183000 (c., (3).02341-5 K.L.X100023.45 L(4) 1.4 LX1000=1400 m.l. (5) 3 Ml.:X >1000=3000 L. (6) 40360 L. —100 —403.6 H.L. (7) 1400 L. —1000=1.4M. a ARTICLE 341. (1) 5 G.X10o=500 G. (2) 92KG. —100-9 92.G.,92 KG.XX1000=920 G, (3) $5,35X10 —= 3.50o (4) $0.28X10)-$2.80, (5 ) $6.25-I0 —0.625. (6) $25.10-100=~$.251. (7) $1.875X100=$187.50. (8) 880 G.X100-88000 G. 88000 G. 1000=SS K.G. (9) 1270 L.+487 L.-1563 L. —2345L.-5B65 L. 5665 L. —100 =56.65 H.L. (10) 24.67 K.G.-18.79 K.G.-5.88 K.G. (11) 135 K.M.X45O 6075 K.M{. (12) $1.16(X.135,60-$157.296. ~~~~~~~~~~~~~~~~ I ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ —— s~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ THE METRICAL SYSTEMI. 1[79 (13) 225 L.X25=625 L. $.156X5625 —$877.50. (14) 42 M.X21= —882 M. $5.69X882 —$5018.58. (15) $2.16X3.80=$8.208. (16) 5cd.l. — L. ~ of 28 cents=$.14. (17) $3.32X100 —$332, tle cost ofl hectare. $332X45-$14940. (18) $14.06 —100 —=$.1406, the price of 1 litre. $.1406X15 — $2.109. (19) $457.92-$3 —1l52.64 M. (20) 665 K.IM.-7=95 K.Mi. (21) $454.10 —380=$1.195. (22) $34.545 — 235=$.147. (23) 14-2.695 M. —453=.31 5 M. (24) $1.58X248,35=$392.393. (25) 1.25 K.G.X25.8-32.25 K.G. (26) $.56X34G.75 —$194.18. (27) 548 K.Ml. — 1439.14+KIM. (28) $3,175>X367 —$1165,225. (29).25 L. X9684-2421 L. (30) 1.6 H.L.X4685=7496 H.L. (31) 15684.08 K.G.- 4957-3.164+K.G. (32) 8 K.G., 6 D.G., and 6 G.-80.66 H.G. $32.95 —80.66 =$.408+ (33) 6.5 M..X1.25 M.-8.12D5M.2 8,125 M.2-.1,8 M.-4.39 —M. (34) In $11.50 there are 23 half-dollars. 12.441 G.X23= 286.143 G. (35) $72.10 —.$2.50-28.84 /M. (36) 12 H.A.X100=1200 A. $6.35X1200=$7620. (37) 3 A.X100=300 M. $.625X300=$187.50. (38) 3 H.L.X100=300L. $411 —300=$.137 — (39) 1.2 H.L.X792=950.4 H.L. (40) 30 M.2-~-10000=.003 H.A. 3.6 H.A.-.003 H.A.=3.597 -I. A. (41).72 M.X.48 M.X.5 M.z..1728 M.3,$.864 -.1728$5. ARTICLE 342. (1) 8 in. —.3937 in.-20.32+c.m. 180 KEY TO RAY'S THIRD BOOK. (2) 25 ft.X12=300 inl. 300 in. —. 39.37 in.=7.62+M. (3) 60 mi. —..62137 mi..=96.56+K.M. (4) 29100 ft.X12-349200 in. 349200 in. —39.37 in.=8869.68 +M. (5).62137X20-=12.4274 mi. (6) 39.37 in.X49-=1929.13 in. 1929.13 in.-+-12-9 rd. 4 yd. 0 ft. 3.13 in. (7) 39.37 in.X360 —13993.2 in. 13993.2 in. —12=1193.1 ft. 1193.1 ft. 3=393.7 yd. (8) 160 acres —2.47164.74+-11.A. (9) 1 A. or 100 3MI.2=119.6 sq. yd. J1 M.i.=1.196 sq. yd. 9 AI.X.5 M. —4.5 M.2 1.196 sq. yd.X4.5 —5.382 sq. yd. (10) 2- bu.=80 qt. 80 qt.-+-.908=88.1 -L. (11) 1.0567 qt.X32_=33.8144 qt. 33.8144 qt. —-4-8.45+gal. (12) 42 gal. —-26.417 gal.=1.5898-+H.L. (13) 4 lb. —2.2046 lb.-1.814-f-K.G. (14) 2204.6 lb.X.92=2028.132 lb. 2028.132 lb. at 1 ct. a pound =$20.28+ (15) If a pound is worth 26 ct., one kilogram, or 2.2046 lb., is worth 2.2046X26 ct.-=57-[- cents. (16) 14 ct.X2.2046-30.86+ or 31 as the nearest whole number. 16 ct.X2.2046 —35+ct. 20 ct.X2.2046-44+ct. (17) $28.80+$t.44=$30.24 the selling price. $30.24 —36= $.85 the price of one pound. $.85X2.2046=-$18.5+ (18) 15 G=1~ D.G. 1x.3527=.52905 oz. avoirdupois, = the weiglht of one letter. 41 lb. 5 oz.-8 lb. 3 oz. 15.2 dr.=33 lb. 1 oz. 0.8 dr.=529.05 oz., the total weight of the letters. 529.05 —.52905=1000 letters. NOTE.-By using different equivalents in this example, a remainder will be obtained, but the answer will be correct to the nearest entire number. (19) 1.80X253=$45.00, the cost of the tea. 25X5=125 lb., the weight of the coffee in, pounds. 125 —2.2046=:56.6996 +K.G. $,86X56.6996-$48. 76+, the selling price of the coffee. $48.76- $45.00:$3.76+, the profit. KEY TO RAY'S SECOND BOOK, CONTAINING SOLUTIONS TO THE MORE DIFFICULT QUESTIONS. ON TEACHING INTELLECTUAL ARITHMETIC. IT is to be regretted that many teachers are not acquainted with the great value of mental arithmetic. It is important, whether it be regarded as a mere mental discipline, or as the'oundation of an accurate knowledge of arithmetic, and the various branches of mathematical science. Whenever a pupil is found incapable of analyzing and 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 has 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. METIrOD OF TmE IICcG.- Classes.-Pupils study best in classes; it is almost as easy for a teacher to instruct 15 pupils in a class, as one alone. r~ -- -~ —— ~ —-------— ~ — ----------- -.1) ~ \ |182 tKEY TO RAY'S SECOND BOOK. An important punt is to keep the attention of the whole class directed 1o eaet question; one means of doing this is, for the teacher occasionally to read an example to the class, and then, having allowed sufficient 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, Second, 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. ARITTImOMETREa.-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 corn-.iencement 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 assist. ance 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 mad.e by any mechanic. The arithmometer consists of a square firame with 12 strong pieces of wi;eX passing firom one side to the other. On each piece of wire there are 12 balls, one-half black, the other half white, so _~~~ —-~SI - ------ ON TEACHING ARITHMETIC. 183 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 way 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 represenlted by the dividend, and separating them into parts, each part containing as many balls as the divisor contains units; the number of p'rts, into which the dividend is thus divided, will represent the quotient: thus, to divide 165 by 3, take 15 balls andl 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 exp!aineo I"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. ANTALnYSI.-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 wlhich 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 184 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 Model 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.-l-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 timzes 4, uwhich 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 IMENTAL ARITHIIIETIC. The following method of exercising pupils mentally, in Addition, Subtraction, Multiplication, and Division, will be found to have fl~Y~ Y ~ ~ 4 3LP~L~~IIY~Z~~-~I ON TEACHING ARITHMETIC. 185 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 sh0ould 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 incu.ect, 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,istiHnc~tly 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 8, and so on. Lastly, for the sake of brevity, the teacher says Result? instead of What 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. n o —-------------------- ---- - ---— e~ i86 Is KEY TO RAY'S SECOND BOOK. 5. Twelve, subtract 3, subtract 5, add 6G 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 6, 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, a4dd 5, multiply by 6, divide by 9, a-ld 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, multi ply by 8, add 8, divide by 8, subtract 8. Result? (1). 17. Seven, multiply by 7, add 7, divide by 7, subtract 7. Reult? (1).. & ON TEACHING ARITHMETIC. 187 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 terms square, square'oo,0 cube, aend cube root, the exercises mlay be made to embrace the operations of squaring, extlracting 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, livide 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 TEACHERS. —In the following solutions, for 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 full 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 fut!' analysis; the style and manner of doing this being a better test of scholarship than the ability to give merely a correct answer. 188 KEY TO RAY'S SECOND BOOK. SECTION 10.-LESSON IV. (6) If each man received 1-sixth, there must have been 6 imen; hence each man received 1-sixth of 42 dollars, which is 7 dollars. (10) 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. (14) 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 1-twelfth of 48 dollars, which is 4 dollars. (15) One-fifth of 25 cents is 5 cents, and 3-fifths are 3 times 5 cents, which are 15 cents: at 2 cents each, 15 cents will bI)uy as many peaches as 2 cents is contained times in 15 cents, which is 7 and 1-half. jils. 71 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. iztns. 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-seventh 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, wlhich is 3 cents. TMENTTAL EXERCISES. 189 SECTION 11.-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-seventhi is 1-half of 12, which is 6, and if 1-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, 8-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, arnd 3 times 9, which is 27, is the first cost; hence, he gained 45 less 27, w-hich 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 i-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-fiftls 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 5 and 2-fifths of 5. (8) 22. (9) 38. (10) 113. (11) 43. (12) 61. (13) 1- 28. (14) 4: 3 B (15) 4,0 (16) 7-. LESSON II. Answers. (2) 9. (3) 18. (4) 15. (5) 9. (6) 40. (7) 35. (8) 44. (9) 4. (10) 2. (11) 1. (12 (). 14) (15) 73. (16) 33. (17) 7,.,l)5 (%) ~- (07,. t) 3 -.,,5' t. ['a\r~ /'0~ 5_ {~q V /' \q_,q_ 996 ___________ -5 — 190 KEY TO RAY'~ SECOND BOOK. (23) 1-tenth of 50 is 5, and 7-tenths are 7 times 5, which are 35; if 35 is 5:ninths, 1-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 timnes 2, which are 10. 63 is as many times 10 as 10 is contained timnes in 63, which are 6 times 10 and 3-tenths of 10. Ans. 6 times and 3-tenth times. (24) Ans. 4 times and 2-sevenths of,a time. (25) 24. (26) 12. (27) 2-thirds of 21. (28) 18. (29) 40. (30) 3 times 7. SECTION 13.-LESSON I. (25) In 42 there are 6 sevens., and Joseph husks as many ttr ee,? as his father husks sevens, that is, 6 threes, wihich are IS rows. (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 number of horses has nothing to do with the solution. The question wonld be the same if the words, a certain number, 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, 4 of 72-8 horses Ans. (31) If 1 lb. of sugoar2 lb. of rice, 1 lb. of sugar and 2 lb. of rice=:4 lb. of rice —1 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 t-o gain 5 miles, it will require 8 days to gain 40 miles..Ans. 8 days. MENTAL FRACTIONS. 191 (29) 10 less 7-3 yd. gain in 1 min. The hound will gain 90 yd. in 90 —-3=30 min., duri.bg 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 Ans. (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-G6 miles Ans. (32) Since D gains 3 miles on C in traveling 5, and since 15 is 5-7hrees, it will require him to tiravel 5-fives, that is, 25 miles, to overtake C. (33) Since 8 gallons are filled, 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 hourr; hence it will require as manny 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 (36.6-66) gallons ner 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-. 2=18 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. SECTION 19. —LESSON III. (11) It requires SX11=88 hours to perform the work; 88 hours- -10 hours=8S8 days of 10 hours each. (12) 8 miles for 37 cents would be 87 cents for 1 mile, and 11 times 387 is 4-8-7=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, Mand 3 pipes 1-third as long as 1 pipe, that is, 1-third of 34 minutes-11, minutes. ~~d- Y IC -~ -------- ------— ~ —----- ~~~~i 192 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. i.-fifth of 18 bushels is -i, and 7 times is are 126, which are 251 bushels. (18) of 18- -5 feet in the earth; and - of 18 —90 1.]2 feet in the air. (19) 2 of 15=3 —~ —3, A's part. 1 of 15=5, B's part. 5+3-1-S, and 15-8 —=62, C's part. (20) 48 dollars. (21) 8 cents. (22) 12. (23) 1:a dollars. SECTION 2O.-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- days are 6` days, the time in which 1 man can do it; and - of 6: days is 3 of ~i-o days, which are 2Xo days, equal to 2- days, the time in which 3 men can do it. (14) Ans. 5` days. (15) A.-s. 9- days. SECTION 22.-LESSON I. (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 is, is 1-fourth, and 4-fourths, or the whole, are 4 times, which are,2, equal to 145 dollars. (35) Ans. 20- dollars cost; gain 4a dollars. (36) Ans. 41 dollars. (37) Ant. R3 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, ivwhich 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 21 times. Ans. 21 yards. (7) Ans. 6-} dollars. ale d MENTAL FRACTIONS. 193 (8) If 6-sevenths of a yard cost 4 dollars, 1-seventh will cost -sixtllh of 4 dollars, which is ~ or 4- of a dollar, and 7-sevenths! or a yard, will cost 7 times -, which are 1-4 dollars. 12 dollars Twill buy as many yards as " dollars are contained times in 12 dollars. But 14 is contained in 12 as often as J-4 is contained in 36; and 14 is contained in 3f6 as often as 14 is contained in 36; 14 in 36 is contained 2-8-=24 times. Ans. 24 yards. (10) Ans. 34 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 4 of 9 dollars, which is l dollars. (12) If he gained 2-fifths, he must have sold it for 7-fifths of 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 bury as many bushels as 3-fifths of a dollar is contained timles in $15, which is 25 times. Acs. 25 bu. (13) If 4 cost 3 of a cent, 5 wtill cost 4of a cent, and one whole apple lo of a cent; and 3 apples cost 3a0=3. cents. (14) 23-=4' ~ 5: ~ If 14 of a yard cost 41 dollars, -, which is twice', ill cost twice 4=39 a 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 -t the cost. SECTION 25. —LEssoN I.'3) oof 3 x- = 5X; of I —', and 2X —$5 1ns. (4): of 24 —6, 7><6-=4; - of 42=3, 5X3=15 cts. An s. (5) 4 of 15=-3. 9X3-27; of 27=$13 A Ans. (6) ~ of 30=10, SX I0=S0 i of 80=5, 7X5-35 trees. Ans. 2 7 1 (7) oi —, 4 of 142 —, 5X2=10; 2,X-l0=$25 Atis. eY.3-l3 19 - KEY TO RAY'S SECOND BOOK. (8) 1-1 2 — 5 —; ] of 2 —' -, 2 times 7 —-7 the cost of 1 ($)1 2 -—'3, 4 3 4: —~, barrel, and 2 I times,15= —,-8 dollars Ans. (9) 3-_,s10.-.j of 20 —2, and 32r times 6 — 17 cts. Ans. (1I) 3a-15, -- of 30=2, 4 times 2=8 miles, the distance traveled in 1 bour; 7' times 8=58 miles. Ans. (11) yl6' of 1-4-, 11 times -— 1=1 dollars. Ans. (12) 1 of 9-;, and - of 8_=-; 9 times and 7 times — 123 dollalrs. Anis. (13) 53=233 9_446, and,3 11 1 of 416_2,and4 tinmes __ I of: _5 s__3 ~3 5 -—; o sf -, and 11 times -'8 5 -- dolllrs. Aos. (14) 4 of 8=3,- which is - of the numlber, and 7 timles 5-= a,,nd f = V-9. - f.8.' (15) 2a-2 3 —, 4 —; 7 of -=-, and 3 tilles 7 --, or the number.'2 times ) X 3 A)n. I1a 7 I 3' (16) -1 of 4=l1, and 4 times 1 —=6 dollarrs, the pr1ice of' a: barrel; I- of 6 —- and 3 times = 18=3 dollars. A 7S. (17) of — 13o, and 3 times -o — 1 9; of 9%. —, and 5 times J. _ dollars. Ans times 0 — 4 (18) - of 30=6, and 1 of 6=2, the iwages of one man one day 4 times —:8, and 7 times 8=56 dollars. Ans (19) ~ of 36=6, and I of 6= —=.. 5 times =-4 9-, and!2 times 45=45 dollars. Ans (20) -. of 1-4, 4 of 124, 4; 5 times'-', and 3 times =74 rods. Ails (21) B of 36=6,' of 6=~; 5 times s=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 1 of 120 dawys, which are 10 days, to eat 8 acres. Ans. 10 days. (23) 400 dollars for 5 months is 80 dollars for I month, and if 8 persons spend 80 dollars in 1 month, one person will - --- ----- - - -— ~ —e MENTAL FRACTIONS. 1 9 spend 10 dollars in 1 month. Ience 11 persons wlill spend $110 in 1 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 I 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. Ams. 10 men. (2) 8 times 15=-1.0; 17w of 120 —10..Ans. 10 men. (3) 9 times 2=22); a of 2 As. 4 hr. (4) To do the -work in ~ of the time will require 5 times as mnany men; 5 times 5 men are 25 men; twice as lLarge will require twice as many men; twice 25 men are 50 men. Ans. 50 men. (5) 8 times 5=40; - 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 6i 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 days, or 13 days. Hence the whole time is 2'+12:=44 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 1 of 14 days, or 3' days. IHence the whole time is 2+3-=51 days. (8) 5 times 8-40, the number of cent loaves. I of 40=10, the number of 4 cent loaves. To of 40_4, the number of 10 cent loaves. O10) 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. ~ I -' -- _- -.,. --— 0 --- - 196 KEY TO RAY'S SECOND BOOK. (11) 3 times 8=24, the weight in oz., when flour is $1 a barrel. of 241 of 24=6, the weight respectively, when flour is 2 and 4 dollars a barrel. (13) $3.=$7: $2-=$5: 7 times 7=49 oz when flour is ${ per barrel: s of 49-=9` oz., the weight; when flour is.$ per barrel. Ans. 9' oz. (14) 1 man could mow it in 6 times 5V=33 days, and 10 men in'-' of 33=3- days 5 —33 -, 21 days. Ans. LESSON III. (2) 3+4+5=12. TS. of 36=9, W's; 7, of 36=12,T's; 7 of 36-15, John's share. (3) 2 of 7V=5, share of 1st; 1 of 7V1-1, shale of 2nd. (4) 40-25=15, what B paid. 2s=_, and -8 of 56=35 dolD ~, lars, A's share. 1 — 8 and of 56-=21 dollars, B's share. (5), and 1 -an, C's part. $860-$500= g360: of 360=45 dollars, A's loss;, of 360=90 dollars, B's loss; 5 of 360=225 dollars, C's loss. (6) 72_ —4 and 4-.94; - of 21=7- dollars, share of 1st, and 9 of 21=13A dollars, share of 2nd. (8) %0o-=; 3 of 25=15 dollars. Ants. (9) 1-3; A. 3 of 80=:12 dollars. Alns. (10) 180 sheep eat as much as 18 oxen. 27+18=45. 4 _3=o a of 25=15 dollars, what A pays. a;-7, and - 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. 47a, and -; 1 of 60=5, of 60=35, and of 60=25. HIence A should pay $35, and B $25. (12) 15 oxen- are 5X3 oxen, and 5X2 horses are 10 horses; hence 10 horses eat as much as 15 oxen. Since 1 horse eat as much as 20 sheep, 6 horses eat as much as 120 sheep; hence 15 oxen and 120 sheep eat as much as 16 horses. 8-+-16=24, 8 =s l a I of 75=$ 5, A should pay; and i- of 75-=$50, B should pay. ~,l~m ew~m~~a~r~a~an( 3 a~~~ MENTAL FRACTIONS. 197 (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 +300=500; aoQ=:, A's share, and 3Q-Q=3, 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=1 man for 15 days. 121-15=27; 1-4, -15-; 4 of $81-$36, and g 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+12= 22, - = --- JTi1; of $55=$25, A's share, and -fof $55=$30, B's share. (17) 4 horses for 5 mon. is the same as I horse for 20 mon., and 10 cows for 6 mon., as 1 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 1 horse for 40 mon. 20+40-60, 20;-, and 4o -3 - of $27 -$9, what E should pay; and 2 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 wlhich 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 each 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. 1.98 KEY TO RAY'S SECOND BOOK. (7) The number and 3 times the number, that is 4 times the nunber, is 48, hence once the number is 4 of 48, wLich is 1'2. (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 321, C's plums. (9) 1+4 —5; of 25=5; 4of 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; 2 of 25 cts. —10 cts., John's; 3 of 25=15 cts., George's. (13) 5+7=12; -5 of 48=20, tile 1st. -7 of 48=28, the 2d. (14) 3+4+5=12; -3 of 60=15,? of 60=_-~0, and l of 60 -z25. (15) 1+2+3+4=10; 1-' of 70=7, - of 70=14, 3 of 70= 21, and -oj of 70 —28. (1 7) 1. ++2.-=4. 4 in 16 is contained 4 times; 4 times 1 -= 6, Thomas' share; and 4 times 2-`-10, Moses' share. (18) 1+-!4=I3. 13 in 39 is contained 36 times; 36X;-!18, 36X=-12, and 36X4=9. Ans. 18, 12, and 9. (20) 1+1-3-=t. in 33 is contained ]2 times; 12 times 1 =12, and 12 times 143-21. Ans. John 12, James 21. (21) 1+1V-)-. 8 in 32 is contained 12 times; 12X]-12, age of Frank, and 12X1-2=00, age of William. (22) 1+21 —31. 3~ in 30 is contained 9 times; 9 times 1=9, the number of unsound, and 9 times 2s-'-l1, the number of sound apples. (23) 1+-9'. 9 in 27 is contained 15 times; 15 times 1=15,, 6 —~ ——. - — 1"I —.C MENTAL FRACTIONS. 199 the number of feet the 2d can build, and 15 times -=-12j, the number the 1st can build. (24) Since B has ~ as many as A, and C' as many as B, therefore C has 4 of 1, or I as many as A. 1+-t-+ -7. 7 in 42 is contained 24 times; 24 times 1=24, A's share; 24 times'=12, B's share; and 24 times 1G6, C's share. (25) Since there are 2 as many sheep as hogs, and - as many cows as sheep, for each hog there is - of a sheep, and 4 of.=1-I'=~ of a cow. 1++ 1 3, 1 13 in 104=48; 48X1=48, the number of hogs; 4SX-3=32, the number of sheep?; and 48X-=24, the number of cows. LESSON VI. (2) 19-5=14; 3 of 14=7, the number of apples James has; and 7+5=1-2, the number Thomas has. (3) 31 -7-24; -1 of 24=12, the less number; and 12+7= 19, the greater number. (4) 32 —8=24; 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 left. (6) 24+7+5=36. Since nMary bought twice as many as Sarah, they both bought 3 times as many as Sarah, therefore Sarah bought -i 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 3 of 45=15, the required number. (8) 31 —410=21, which is - of the number; hence ~ of 21, or 7, is - of the number, and 4- is 4 times 7, or 28. (9) 21+7= —8, -1 of 2.8=7 is, and n=35, the number. (10) 4+3=7. 37-7=30. ~ of 30-10, Ilenry's age; 10 +3=13, Oliver's age; and 10+4-14, James' age. (11) Since Sarah has 3 cents less than rMary, she has only 5 mole than Jane. 8S+513, 43-13=30, 1 of 30=10, Jane's; 10+8=18, Malry's; 18-3=-15, S.arah's. 200 KEY TO RAY'S SECOND BOOK. (12) Twr;ce 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) Tile 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. 3 of 45-$15, the cost of the chain; $15+5=$20, the cost of the ring; and $15+12=$27, the cost of the watch. (14) 30 —H6-=36, which is a of his money. of 36-18, which is a 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. $ of 42-$7, James' money; 7+7+3=$17, John's, and 7+17+7=$31, Frank's. LESSON VII. (2) q-+-=-. m1 of 60=6, which is -. 7-=42, and 3= 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 the time, or 4 of the time to midnight=the time from noon to midnight, 12 hours. If a is 12 hours, -1 is l 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. IIence the time was 4 o'clock, afternQon. (4) 1+- =, = of 12 hours=l1 hours, which is and 3- =4 hours; hence the hour was half past 4 P. M. (5) 1 +-. of 56 ——; 432, and 3-4, the distance traveled each day respectively. (6) Since the first, plus q of the first less 8, that is ]92 of the first less 8 —100,`l of the first is 108. -1- of 108-9, is 7 of the first part.`-63, the 1st; -45, and 45-837, the 2nd part. 8- --- ~- - --- - ---- MENTAL FRACTIONS. 201 (7) The first+'+', that is 9 of the first=45. - of 45=5, 4 of the first. 4=20, the first part; I of 20=10, the second; and I of 20=15, the third. 4 (8); of a —-twice X of 5 —x; hence C has ~ as iany as A. 3 55 -a s~,_ 1+a'1_ -; logo of 40=4, which is D1 5,=20, A's; =12, B's; and'-=S, C's. (9) of 3 times I o f 5-s_1 1+l+1=+- + 1.-4 hence'4 of the top part=70; -j/~ of 70=5, which is.; =-30, top; - =25, middle; and 3=15, the lower part. (10)' of I-; = 3a 1-{ — K - -0T +-t-+ -o='~iS; hence 17 of the cost was $34.,7 of 34 is 2, which is'; T- $20, the cost of the coat; =40-$8, the cost of the hat; and -— i$6, the cost of the vest. (11) If 1 thle cows=4 the sheep, the cows-=4 of the sheep. 1+4=_1; hence 17- of the sheep is 55. -L of 55=5, which is 4-; 7=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 32 of the greater. 1+2 —=5; lIonee 5 of the greater=100;. of 10 —20 is of the greater. 3-=60, the greater number. =-40, the less. (13) If 4 of Mary's age=3 of Sarah's, -4 of Mary's=4 of Sarah's. 1A-4I =. a of 1.4=2, is - of Sarah's. 3-= 6, Sarah's age; and 1=8, Mary's age. (15) If 3 of the first=.3 of the second, of or - of the 4 second, and -, or the whole of the first=g of the second. Hence the second plus 9 of the second, or ]7 of the seeond=51. -j' of 51-3 is - of the second. 8=24, the second part; and 8 —27, the first part. (16) If g of the apple trees=, and., or all the apple trees- of the peach trees. ITenee the peach trees and 7 of the peach trees, that is, 7-3=5. 1T of 65=5, is 7 735, the peach trees; and -=30, the apple trees. (17) Since A's distance=- B's, - A's=- of, or TO of B's, and 3 of A's -3X, or ] of B's; hence B's and of B's, that is', is 66 miles. o- f 66 is 6 miles, L of -- _ --- -.-. _-. _ 202 KEY TO RAY'S SECOND BOOK. B's, and since the difference between B's and ~ of B's is I of B's, B traveled 6 miles further than A. (18) Since once the time past noon+3 hours is. the time to midnight, twice the time past noon+6 hours-=the whole time to midnight; but the time past noon+the time to midnilght is 12' hr., hence the time past noon, with twice the time past noon + 6 hr. =12 hr.; hence 3 times the time past noon is 6 hours, and the time past noon is X of 6 hr., or 2 hr. (19) See the " Explanation." If 12 hours is A, 4 of 12, which is 3, is a; hence the hour is 3 in the afternoon. (20) IHere 12 hours must be 4 of the whole time; hence -: mnust be. of 12 hours, which is 4 hours; that is, the hour is 4 P'. iM. (21) If. the time past noon is 1 of the time past midnight, the time past noon must be T-~- of the time past midnight; then the time past noon is A-j of the whole time from midnight, and the time from midnight to noon must be -0 of the whole time. 4 of 12 hours is 1 hours, that is, the hour is 1' P. AI., or 20 minutes past 1 o'clock. ILESSON VIII. (2) 1+ —s. 1- of 20=4, is - of the number; 3-12, the n umber. (3) 1-] of 21=3, is I of the age; — =15, her age. (4) Twice the number and a of the number equals 13 of it. of 52 is 4, which is a; -920, the number. (5) -Twice the number less its 4, is 14 less 4, or Lo; if lo is 40, l of 40, which is 4, is 7, and 4, or the number is 28. (6) Three times the number less its is lj less, or 1:; if is 48, -' of 48, 4, is a and a is -90. (7) lA "1 -a+4= 3.'1 of 6=, is, and 6 —1 his aie. (8) lter oe, and of her age are 11-:+ L or. but twie her at.e is |; hence 1'lg of her age plus 10 years,f4. - of her age is 10 years. and her agoe, 24 years. MENTAL FRACTIONS. 203 (9) 1- e3 33 is 30 cents; hence'5 is 50 cents. (10) 1+-+t- -- g q- 7% —+7; treble the number is a3, hence,-'9 of the number is 27, and the number is 30. (11) 1 —f3-= -. 4 of 40-5, is -q of the father's age, or 55 years; hence the son's=15 years. (12) 1+ — ~. 3 times her age is 15 IHence 6 of her age is 18 years, and her age is 15 years. (13) 1 —,-,. 4 of 28=4 yards is 3; -36 yards. AnLs. (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, tend fiom C to D is 30 miles. (15) 1+4+ +-I-+''4 5 )_ 4a of 94 is 1, which is;'~i; hence the age is 45 years. LESSON IX. (1) AnS. 3. (2) Ans.. (3) 1Ins. 2. (4) A zs. as mnany as 3 of a day is contained times in 1 day, tihat is, 2'. (6) IIe could dig 2 trenches in 7 da., or q of 1 trench in 1 da. (7) tIe could vwalk 5 times the distance in 10 days, or -5- of the distance in 1 day: and in 2 days -o_ -. -(8) +' which are 4. aAn,:2 (9) A+l'.s. --'1 + which t'e 1 (11) 1 + — 13 -; 4 in 1 is contained 4 times. Ans. 4 da. (12) a+-j,; in 1, 21] times. Ans. 24 days. (-;; - ~ ays.5 3 I (13) 1, 4lIl- — 1; hence all three can do it in 1 day. (14) +, 3o _ 7; o7 in 1, 1- times. Ans. 1$ hr. (15)' 1 1;' in 1 is contained 20 times. Ans. 20 da. 17) A a~nd~ B caln do-' +1 in 1 da..,- 1, th e part thalt C can do in a dca.; lhence 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 I hour; hence both will empty it 8 tinmes in 1 hour, or once in 4 hour. - ~ -- - -e 204 KEY TO RAY'S SECOND BOOK. (20) T70 days in 1 day is contained 1-$ times. Ans. 14 days. (21) C can reap - of the field in 1 day, and since B can reap it in 3- days, he can reap 3 such fields in 10 days, or y3ua of a field in 1 day. +1 _+ ao; hence both can reap F the field in I day, or the whole field in 2 days. (22) A can do 3 of his piece in 1 da. If B can do a piece 3 times as large in 7 da., he can do 4 of such a piece in 1 da.; hence both can do 1+_- 3_ of it in 1 da., or the whole in as many da. as 44 is contained times in 1. in 1, 1 times. To do a piece of work 5 times as large will require 5 times as long. 5XI]Ta da.=6-9- da. Ans. (23) -2 of 100 gallons5 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 1st and 2d empty 40 gallons in 20 minutes and the 3d must empty 100-40=60 gallons. LESSON X. (2) As. 3Os. (3) Ans. 51 (4) Ans. 5. (5) Ans. 13 (6) 4:.7t8. -17 ( 7) Xns. 2 5o. (8) Ans. 11. (9) A4ns. 20 Thus, if she gave away 4 of what she received, she had -a left; 4 of =-4, and I of 60 is 20. (10) Since: remains, 35 is s, and 4 is 14 miles. - of 14 is 6, and 14 less 6 is 8 miles, the required distance. (11) Anis. 25. (12) of 27 is 12, of 8 is 6; 12 and 6 arc 18, which is - of 30. Airs. $30. (13) After losing she had left, and after giving of to Sarah, she had - of leIft. 3 of ---. If 3 is 6, -7 is 14. Abls. (14) Ans. 20 cents. (15) 4 of 36 is 45, and 45 is 9 of 35. Ans. (16) If of B's= of A's, then all of B' of A's: -- of A's-12: 4 -4 7 —.28 cts., A's; and 4 —16 cts., B's. (17) - +T -i' --; hence 32 is 4, and the whole is 96, M ~ L~T-j~~ _Mi~ 3 MENTAL FRACTIONS. 205 of whicli 32 are apple trees, 24 pear trees, and 8 plum trees. (18) The whole less g is a, and the difference between -s and - is -; if " are 25, ~- is 5, and - are 10, the pear trees, and are 35, the apple trees. (19) Since the plum trees are - of the apple trees,' of the plum trees=A.~ of the apple trees; hence the cherry trees =-( and -J~, that is, 7Z of the apple trees; and the whole number of trees-once the apple trees, 3 of the apple trees, and 72 of the apple trees; that is, 1+,-+ o2', or.2e, of the apple trees=69. 13 of 69 is 3, which is Tl. {2=36, the apple trees; 1=12, the plum trees; and -7-=-21, the cherry trees. (20) 5 of 12 years is 20 years, which is - of the ages of Jane and Sarah; if 20, is, 1 of 20, or 5, is, anl -45 9 ~,-4 years, the sum of their ages. Then Sarah's age and, of her age, or U of her age, are 45 years; A of 45 is 3, which is - of her age, and A, or her age, is 24 years; of 24 years is 21 years, Jane's age. (21) -3 of 55-15; - of 30, or 24, is. of 54: twice 54 —108; 15 in 108, 7-5 times. Ans. (22) -3 of John's money=3 of -of C of Charles'.$33=Charles' money; hence $33 is U-, 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 1-4 of Henry's, and 5 of John's to 61 of Henry's; but while John takes 5 steps, Henry takes 6. Hence John takes 5 steps to gain I of one of hIenry's steps on him, and therefore must take 20 steps to gain 1 of IIenry'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. Aims. (5) Since 3 leaps of the hound are equal to 10 of the hare, 206 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, he 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. Ans. (6) Since 4 of M's steps=7 of N's, 1 of M's:=1 3 of N's, and 5 of M's =S- of N's; hence while M takes 5 steps he gains 1:3 of N's steps on him, and to gain 35 steps, he must take as many times 5 steps as 1 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 from it. (9) Since the tail is as long as the head and -1 of the body, therefore the tail is 4 inches plus J1 the body, and the body is equal to 4 inches, plus 4 inches, plus I of the body; hence ] of the body is 8 inches, and -1 is 4 inches. 3-=12 inches, the length of the body; 4+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-4- of B's; but B's money —A's and C's; hence B's=10 cents+10 cents+-. of B's; or 1 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 - of the horse, therefore the cow cost 8 dollars and 4 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. IIence the horse cost $64, and the cow $24. (12) Since the tail weighs 3 pounds more than the head and a the body, therefore the tail weighs 11 pounds plus 1 the body; but the body weighs as much as the head andl tail; hence the body is equal to 8 pounds plus 11 pounds MENTAL FRACTIONS. 207 plus the body; therefore l 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, i 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 givenby 6. HIence he has as many children as 2 peaches is contained times in 6 peaches. 2 in 6 is contained 3 times. Ains. 3 children. 3 times 2-6, and 6+9-15, the number of peaches he had. (3) The difference bletween 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 1i dollars a yard is 4 a dollar, and the difference between having 5 dollars left and nothing left is 5 dollars. hIence 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 —=50 cts. Ans. (6) 12 —10=9-,; 24 in 40, 16 times. Alns. 16 yards. (8) 3 times the age of James less his age is twice his age, or 10 years; hence his age is 4 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. IIence C's age is 5 years; B's 10 years; and A's 50 years. (10) Since A's money - B's, twice A's=-B's; and since bA - --- - ----- 208 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 $15; 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. (11) 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 afnd 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 timecs the first are 70 sheep, -and the first contains 10 sheep, she second 40, and the third 120. (12) If' 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 as many days idle as 50 cents is contained times in 4 dollars, which is 8. 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-a- of (twice A's+10 years)-=-~ of A's+6 years: hence a 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 ~ the second; 3 MENTAL FRACTIONS. 209 times the first horse and $36 are wnorth 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 (21) If - cost. pence,' will cost 6, and 1, penny: hence, for 9 pence, you can buy as many as 4 are contained times in 9 or 27, which is 54 Ans. LESSON XIII.-M-ISCELLANEOUS EXAMPLES. (1) Ass. 105. (2) o.. $60. (3) If A N7walks 10 miles in 1 days, he walks 8 miles in 1 day, and if B wv-lks 8 miles in 1 3 days, he walks 5 miles a day. It tal eis At o24 days to travel 20 miles; and in 2~, daLys B will travel 124 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-t cents, the rate of selling; hence on each apple sold he gained 4 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.='2- cts. a piece. By selling them at 3 for 10 cts. he sold themrn at 3 cts. a piece; lhence on eachl egg he gained 3 —2s, or, cts., and on each dozen 8 cts. Had lhe sold them at 4 for 10 cts., or 2- cts. a piece he would have lost 2~ —Q2, or - cts. a piece, or 2 cts. per dozen. (6) 2 for 1 cent is - cent a piece; 4 for 1 cent is 4 cent a piece. Ifence each 2 pears cost -+4, or a cent, and each 3 cent. IIe sold them at 4 cent a piece; 3 less 3 h.; hece lie gLined Ti% of a cent on each pear sold; an d to gain 18 cents lie must sell as many pears as 49 is corntained times in IS cents, which is 80. A7Is. 80 pears. (7) Each duck cost $}, and each 2 chickens $; therefore 2 chickens and a duck together cost 1 — t. $4 —f 210 KEY TO RAY'S SECOND BOOK. 210 iJ what he gained on each lot. HIence he sold as many lots as yl is contained ties in ic is 30; therefore he sold 30 ducks, and 60 chickens. (8) 10X4-40, and 3X40=1 20, the hours wNork of 1 man; of 120-=15, the hours work of 8 men; and 15 hours —21 days of 6 hours each. Ans. 2. days. (9) IRer —rving 2 for MIary and 3 for Lucy, there are left 27 to be divided eqcially. J of 27 is 9, James' slare. 9+ 2-11, Mary's share; and 9+3-=12, Lucy's share. (10) Ten gal. in I hr. is 20 gal. in 2 hr., and 20 —173, the number of gal. left in each 2 hr.; hence it will require as many times 2 hlr. to fill the vessel as 3 gal. is contained times in 15 gal. Alns. 10 hr. (11) If A can do the work in 4 — dtays he can do twice the work in 9 days, or 4 of the work in 1 day. If A and B together can do the work in 24 days, they can do 7 times the work in 18 days, or -7 of the work in 1 day. T7 21 s- - the part of the work B can do in 1 day. IIence B can do the -work in 6 days. (12) A. B, and C can do 5 of the work in I day; A and 13 do 4 of it in 1 day; and B and C a of it in 1 day. ience --' —'4-(, the part C does in 1 day, therefore lie can do it in 13' days..-,_4 the part A does in 1 day, a _3 I therefore lie can do it in 114 days. I - 3 —, 3 a, the part B does in 1 day, therefore lie can do it in 27-s9, days. (13) Since 5 men, or'7 women, can do the work in the sanme time, both together can do it in half the time of either; that is, 4 of 35 days, or 174 days. (14) Since 1 woman can do 3 of a man's iwvork, 4 womnen can do 3 men's work; hence the work of 2 men1 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 J1' men; hence it will take 1 man and I woman as many times'28 days as 13 is contained times in 5. 13 in 5-o2 times, and 2qX258 dlays are 80 days. As. (15) Both consumed., in 1 day, and Ad or in 6 days. After --.-. _- - - ------— ~ —--------— ~ ~ s4 ------ -, 1-j - - MENTAL FRACTIONS. 211 were consunmed 3- were left, and if the woman consumed -, in 30 days, she consumed d in 10 days, or the whole in 50 days. 1 the pat the man consumed in 1 day; hence it would last him 1 o:=91. 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 eggs at these rates bring 48+-18t-34-=100 cents. 80 eggs for 100 cents1 cent a. piece, or 15 cents a dozen. (17) James' money+i of James' money+$6, or 4 James' noney+$86-$34. Hence,- is 28, and - is $4; 4 —$16, James' money; and $18=-Thomas' money. (18) Iere A's money plus s of A's money less $5 is equal to $51; or,'- of A's money is equal to $56. Hence 4 is - of $56, which is $S, and 4, or A's money, is $40; and B's money is $11. (19) Since A's age is twrice B's, therefore 21 times A's must be equal to 5 timnes B3's age; but 3 of B's age plus 44 years is equal to 2-14 times A's age; hence 3 of B's age. plus 44 years is equal to 5 times B's age; therefore 4B's age is 44 years; or %- B's ag-e=44 years; hence _-o=9 years, and his age is 10 years; and A's age is 20 years. (20) As many, and half as many more, are 1.1 times as many dollars as he has; hence 141 times A's money plus $2- is $40, or 14 times A's money —$37l-, or 3 of A's moneyz$7,5; hence, is $2, or A's money is $25. (21) 6 of the number-100 less 21 07 1 95; of the numnber=3_ and the number=39. 3 is 5 5 5.o45 5 (22) The,whole minus 4 is. X of t and 4-+ 5 =-4 + eo~ —.; hence n4 of the number at first is 65; therefore -, is 1 sheep, and -,, or the whole flock, was 72 ssheep. (23) Since 5 of the silver+$10-X of the gold, - of the silver +$82= — of the g(old, or the silver+$ >12 —4, or of the gold, or the silver -- of the gold —12. But the gold and sil-ere=-$50, and the gold+2- of the gold ~O~c~.~~ rrmD~.~, ~ u1PE~a5 —- _T 6 212 KEY TO RAY'S SECOND BOOK. -$12=;$50, or ~3 of the goldH-GG. Ilence o=2-$_2, and 10, or the gold:$20; therefore the silver is $30. (24-) Since 4 of VA's —$3=- of B's+$-, therefore. of A'si of B's+$8, or the whole of A's-2 of B's+$16. But since both have $56, therefore B's and - of B's+$16= $56; or or of B's+$16 is $56; therefore a of B's is $40. - of $40 is $8, which is I of B's, and -3 that is, the 3, ~3, whole of B's, is $24. $56-$24=$32, A's. (25) Since 7 of A's distance=-4 of B's, therefore 7 of A's distancej of = 4, or -4 of B's, and 7, or the whole of' A's distance=7XF-4-, or, i of B's. 8 Of B's distance and B's distance, or 4 3 of B's distance-86 miles. HIence 5 of B's distance is -1 of 86 miles, or 2 miles, and s' is 30 miles. A's distance is _8s, or 56 miles. (26) Since 2 of A's is equal to -4 of B's, 3 of A's is equal to 5of B's, and 3, or the whole of A's, is equal to 6 of B's. HIence their difference is - of B's, and - of -} of B's is $15, that is, 6 of B's is $15; hence wo2 i"s of $:15, which is $5, and 20, or the whole of B's, is 20 times $5, or $100; and A's is - of B's, which is $120. (27) Since the watch and chain are 9- of the wrntch, therefore the chain is ~ of the watch. 3 times i are, and twice the price of the watch plus -v of its price are 7O, which are $100. Hence ~ is 210 of $100, or $5; and n-, or the price of the watch is $35, and the chain is 7, 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 1, 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 monthsG6 horses for 3 months-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 iiuch as 2 horses, and 20 sheep as much as 4 horses; hence 20 sheep for 5 months —1 horses for 5 months=20 horses for 1 month. Htence, A has the same as 8 horses 1 month, B the same as 18 MENTAL FRACTIONS. 213 1 _ I horses 1 month, and C the same as 20 horses 1 month, in all 8+18-t20-46 horses 1 month. 46 for $99 is $2 for I 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 2_ melons: James eats 2` melons out of the 3 melons he furnished. Of the 2. melons eaten by Thomas, James furnished 1, and John 2 —, that is, James furnished 3, and John -7, both together making 8, Therefore, since James furnished a of the a,,nd John - of the J, James should have g of the 8 cents, which is 1 cent; and John 7 of the 8 cents, which are 7 cents. (31) 2+ l i-+Ti-3T —-; and 1 3 C's part. k -2 -=-;'hence ft is $160, and 2' is. of $160=$80; I =6 times $80=$480, A's share; 7 times $80=v5M0, B's share; and sIl8 times $80:=$640, C's share. (32) Since A's age is -5 of B's, twice A's age is Lo of B's; hence B B's ag B's ge+-' B's age= B's age-2 years; or ]3 of B's age=2~0 of B's age-2 years; therefo1e of B's age is 2 years, and his age is 12 years; and -5 of B's age, or A's age, is 20 years. (33) A and B can do Tie in 1 day, hence in 4 days they can do - or -I of it, and' remains. If B does -- in 36 days, he does 3 of 4 —-, in 12 days, or 4~ of |-'1, in 1 day, or the whole in 48 days. Since both can do Q16 in 1 day, and B can do - in 1 day, A can do 1 in 1 day, and the whole in 24 days. (34) Since 2 of B's and C's is $12, a is $6, and 7, or the sum of their shares, is $42. Again, since I of C's=-3? of A's and B's, 8 of C's — Il of A's and B's; or'0, that is, 5 of C's=A's and B's; but since A's is $12, therefore 45of C's $12 and B's, or 4 of C's less $12=B's. Therefore C's and,- of C's less $12 —$42, or 9 of C's=$54; hence l of $54, which is $6, is 4 of C's, and 4, or the whole of C's, is $24; and $42 less $24 —$18, B's share. (35) To mow ~ in 12 days would require 3 men; to do it in 4 the time, 12 men; in 3 the time, 4 men. Ans. 214 KEY TO RAY'S SECOND BOOK. (36) During the first 6 months the second has $100 more than the first;,o equalize it, then, the first must during the last 6 months have $100 more than the second, that is $300, and therefore must put in $200 Arts. (37) After A had given B 2, they each had ~ of 24=12 cts.; hence - had 12+2=14 cts., and B 12-2 or 10 cts. (38) 5 of 50=-30. If 30 exceeds the age as much as 10 is less, 30-1.0=20 must be twice the excess, therefore the age is 30 —1020 years. Anzs. (39) The difference of 4 cts. in price makes 30 cts. difference in amount; hence there were 34~=74 dozen; 12 cts.-t — =1~ cts. loss per dozen; 6+1 -=77 cts. cost per dozen. (40) A difference of 3 cts. in price makes $1.50 cents difference in amount; hence there were $1.50 —. 3=50 lb. Ans. (41) 12 boys=4 women=2 men. If 12 boys or 2 men can do the work in 6 days, it would require 6 times 2 men to do it in 1 day. Ans. 12 men. (42) 2 lb. would be worth 31+6 —10 ets., or 5 cts. a lb.; -1 lb. is worth 21 cts., and for $1 we must give 20 lb. (43) If - of the gain=-4 — of the selling price, - will be 2 and the gain of the selling price; hence the cost will be 1- 2-3- of the selling price. 33 yards at $4 cost $15 nwhich is -of $25. 4ns. (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., and must be sold for 48+5 =53 cts. SECTION 26.-PERCENTAGE. NOTE.-The greater part of the questions in this section are of such easy solution, that it is deemed unnecessary to present either their answers, or the method of solving them. LESSON'[I. (2) Abns. 25. (3) Ans. 20. (4) Atns. 50. (5) Ans. 60. (6) Ans. 30. ~~ — -. ---—;~ —------- -------- --- -- --—. — 11 PERCENTAGE. 215 (7) If he clearced i he must have sold them for C of the cost. If 6 cents wvas i, h is 1 cent, and or the first cost, was 5 cents. IHence, in selling them for 8 cents, he cleared 3 cents on each; 3-60 %. (8) 6 yards for 83 is $, a yard, and 5 yards for $4 is $4 3'r yard. * p, z is - a4=nd -I is 3 of I 3 is (9) If he lost 1, 8 cents are - of the first cost, and the first cost was 10 cents. 3 for 25 cents is 81 cents a piece. 10 cents —8S cents is 1 W cents. t1 cents is B of 10 cents; therefore by selling them at 3 for 25 cents, he would have lost i1 of the cost, or 16- per cent. Again: 2 for 25 cents is 14 cents a piece, a gain of 24 cents. 21 is I of 10; hence by selling them at 2 for 25 cents, he would have gained - of the first cost, or 25 %. (10) 2 for 3 cents is 1V cents a piece; 3 for 2 cents is' cent a piece. ~1 -- is;5 loss on each. 1-1 is -, therefore - is s of 1V cents, or the first cost; hence he lost 5 of the first cost, or 555 ~. (12) Ans. 20. (13) Ans. 10. (14) Ans. 40. (15) A1ns. 50. (16) Ans. 12. (17) Ans. 3. (18) Adns. 25. (19) 1- -- =40 %. At 7 the loss is'=-12, -.o At - the loss is T t.10 %. At 27 the loss is T3W15 O. PROPIT AND Loss.-LESSON III. (1) 20 %s *0. Adnss. 75 Y-7GO4.Ans. (2) 100 1 -X 1=, th at is, the cost and gain are equal. 150 %o-,= 00 1' or 1; the gain is 1 times the cost. (3) 25 no = -- As. 35 7~ =..d1s. (4) The whole, or 1_ 1A= Ans. 100 %. 3 —0j An.s. 300 %. (6) If - of the gain eqluals T-4 of the cost,, equals and the whole gain equals 9 T3 O._36 co. (7) rIf a. 3 - o; and, or the -whole of the gain -6 or _ of the cost. -- 7v=75 % Ans. 216 KEY TO RAY'S SECOND BOOK. (8) t _ 25 =-T~'s-,1; hence the watch sold for 5- of its cost: if $10 is i, o, or the first cost is $8. When he sold it for $12, lie ained $4, and - --- O. Ans. (9) 12- 7:, vwshen A is lost, the muslin is sold for a of its cost. Hence 7 cents is 7 of the cost, and the first cost was 8 cents. By selling it at 6 cents per yard he lost 2 cents.,-=-=25 o. Ans. o10) 163 s =. If nwas lost the horse'was sold for ~ of his cost; if $35 are,, 1 is $7, and c- are $42, the cost. $63 — 4'=$21, the gain; 4- 121 -% —I III 0. 1AnS. --- — ~0 0 —'-V I _ (11) 20 l hence $1 S are t of the first cost, 01or $15. 10 o= and i of $15 is $.1i. 15 —1 =$13, 3watch sold for. IIe lost $18-1 3 =$41.- Ans. (12 ) 2 0 So=-00-; hence.60 ar e ~ of the first cost, and &of $60 is $10, which is -, olr A's gain. B lost 20 %, or, of what it cost him:, of 860 is $12, B's loss. IHence B lost $2 more than A gained. (13) Since 25 % is 4, he sold one watch for I of its cost, and the other for 4 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 Ais. (14) 4 apples for 3 cents is 3 of a, cent each. 50 % is 1; hence 3 is 3 of the first cost, and the first cost is l cent. 5 apples for 4 cents is 4 of a cent each; 4-_ 3-i the gain on each apple. I1 h~ence 3 is _ of the filst cost; ~-60 7%, the gain o. (15) 5 lemons for 4 cents is 4 of a cent each. 20 % is -, and he sold the lemons at 4 of their cost. If 4 of a cent is 4 of the first cost, s of a cent is a of the cost, and the cost was 1 cent each. 6 lemons for 5 cents is - of a cent each; 1 —-a _-=iGf So, the o of loss. (16) 3 of 10 %o of 60= of of 60-4; 4 is A of 8; 8 is of 40; -=20 S /o Ans. C~~- --- - - ----------- Cgl~~~~~~~A PERCENTAGE. 217 (17) of of 50,% of 120=,o of of 120=18; 18 is 10 less than 28; 28 is 20 %, or } of 5 times 28, which is 140. AnIs. LESSON IV. (3) Ans. GC cts. (4) Anbs. $2. (5) Anls. 84.80 (6) Arc'. $6. (7) Aiis. $8. (8) A.is. $9. (9) Ans. $8.40 (10) Ais. $9. LESSON V. (2) Ans. 81 (3) A77s. $2.80 (4) Aibs. $2.40 (5) Ans. $4.50 (6) 4ns. $3.25 (7) A72s. $5.162 (8) Ans. $3.53 (9) Ans. $4.80 (10) Ans. 819.20 (11) Ans. $8.45 (12) Ans. $4.24 (13) A7s. $3.20 (14) Ais. $6.75 (15) Aibs. $3.80 (16) A4s. $28. (17) Ans. $44. (18) Als. $68.20 (19) iAbs. 832.80 (20) Ans. $56.80 (21) Anis. $99.112 LESSON VI. (2) Ans. $50. (3) AiS. $60. (4) Ans. $75. (5) Aiis. $140. (6) Ans. $240. (7) Ans. $350. (8) Ans. $4000. LESSON VII. (2) iAis. $200. (3) Ans. $50o. (4) AIs. $250. (5) Aibs. $300. (6) ibns. $25. (7) Ass. $750. LESSON VIII. (2) Ans. 4 years. (3) Aibs. 2.1 years. (4) Aibs.,2- years. (5) Ais. 3 years (6) Anbs. 62 years. (8) Ais. 50, 331, 20, 16, 14-W 124, 10, and 81 years respectively. (9) Aibs. 40 years. (10) Aibs. 25 years, and 20 years. 218 KEY TO RAY'S SECOND BOOK. LrEssON IX. (2) Ans. s. (3) Ann. 5. (4) A)ns. 7. (5) Ans. 8. (6) Ans. 7. (7) Ans. 5. (8) Ans. G. (10) Anss. S, (11) Ans. 10. (12) Anis. 12 9, 20, 25, and 50, respectively. LESSON X. (2) Ans. $400. (3) ABs. $25. (4) Ants. $500. (5) Ans. 8300. (6) The amount of $1 for 4 years, at 6 %, is $1.24; and the required principal will be as many dollars as $1.24 is contained times in $496: 496 —8 $1.4-=400. Ans. $8400. (7) $81, at 5 ~%, for 4 years, will amnolnt to $].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-=0. 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-$15. (9) The amount of $1 for 5 years, at 5 %, is $1.25; hence the present worth of each $1.25 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 6a years it will lbe 40 cts.= $2; hence the amount of $1 for 63 years, at 6 %, is $1 —. Terefore the present worth of $77 will be as many dollars Is is contained times in 77, which is 55. Hence the present w-orth is $55, and the gain by paying it now 877-$55=$ 22. I-MENTAL FRACTIONS. 219 LESSON XI. MIISCELLANEOUS EXAMIPLES IN INTEREST. (1) 6X4 5;1 = -521 A 2 1. (2) 5X5 —25; 0- I-4. +4 4 and.- is ~ of ~. Ans. (3) 1 of hn1c; hence the interest of 1 year is 1 of the principal; Tj — o0 i.As. 10 2. (4) 2 years 6 months —~-j years. f of and 2 tines' TO-' -o the part the interest is of the principal for 1 year; 0=-T 0~-10 %f. Ans. (5) 10 =1 =- 1o; hence for 1 year the interest is'T of the principal, and at the same rate eo, the time for which the interest is 3 of the principal, will be as many years'as jl is contained times in 1 in i contained 6 times. Ans. 6 years. (6) If 3 timee the yearly interest is equal to ~; of the principal, once the yearly interest is equal to 3 of the principal. 3 — 1o-2 7o. Anis. (7), of the interest for 2 years is the same as - of the interest for 1 year. If - of the interest for 1 year equals 4 o' the principal, and, of the princi-pl, 4 of the interest for 1 year equals of the principal, and, or the interest for 1 year, equals of the principal. o_ 4o_ an pal. 251 0- 40 %. Ans. (8) 4 for 3 years is the same as 15 for 1 year. If 5 of the interest for 1 year equals 9 of the principal, 4 equals Qi of =4 3 and - equals, or TRo of the principal..x8 equals 6 1. Anis. (9) 1 year 4 months-16 months, which is twice 8 months; hence the interest for 1 year 4 months will be a of the principal; - of $200=$16. Arns, (10) 1 year 4 mon. are 1, or 4 yr. If tile interest of 4 yr. is 3 Of the principal, for 4 year it is 4 of':- -io and for 1 year it is -0i iof the pr-incipal; that is, the rate eA _ _ _ 220 KEY TO RAY'S SECOND 1BOOK. is (9. HIence for 1 year the interest is $9; for I month the interest is $4, and for 8 mon. $6; folr S days the interest is o-, or r of 3 or 45 cents. Therefore, the whole interest is $9+$6J=$150, or $15.45 (11) 4 years at 10 % is the same as 1 year at 40 %, and 1 year at 40 0o is the same as 8 years at 5 %. Ans. 8 years. (12) 31 years are 1o years. $40 for LO years is $4 for A of a, 3 ye3u', 3 $2 ea. year, or $12 a year. 5 % is % oi, hence $12 are r of the principal, which must therefore be 12 times $20-= $240. IIence B's money, and twice B's'money, or 3 times B's money is $240; therefore B's money is $50, and A's twice $SO0, or $160. (13) 1 years are 7 years. $49 for.7 years are $7 for - year; or $35 a year. 7 %o is Wi', hence $35 are T~-~ of the principal, and $5 are T 0o, and 1Qo 0 are $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 14 B's; therefore B's money and.1i tinmes B's money, that is, - are equal to 500; hence 4 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 9 yrs.=$2 for year, or $S for 1 year. 4 %4 is T _0_= enc $Sn - of the principal, and., or the whole principal, is $200, which is 4 of A's, and 4 of B's money. Since 1 A's = B's money, I A's — B's, and - A's=B's money; hence B's money and 4 B's, or B's-=$200; of $200 — G:66, which is I B's, and $133-=B's money. Since 8 A's=` B's money, A's money-4 B's, and 4 of $1353 =$177', A's money. W/ILSON, HINKLE & Co's NEWV PUBLICAITIONS. Yenable's United States: A SCHOOL HISTORY OF THE UNITED STATES, by W. H. VENABLE. 12110., 288 pp., finely illustrated and accompanied with carefully drawn Maps and Charts. Contlins VALUABLEn FOOT-NOTES, referring to literary matter relating to subjects discussed in the text; an original system of General Questions; an original systemn of Biographical Reviews, illustrated iwith authentic portl'aits; lnulnerous MIaps,'itmCeq.(tled for illustrative -value, accuracy, and beauty; and sustains throughout a unity of design anii execution, presenting a complete, though concise, artistic, anld pleasing narratioll of the leading facts of the history of our country, in an attractive form. Thallheilmer's Ancient [istory: A MANUAL OF ANCIENT HISTORY, fiom the earliest times to the fall of the Western Empire, A. D. 476. By AM. E. THALHETMER, formerly Teacher of History and Composition in Packer Collegiate Institute, Brooklyn, N. Y. 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