^Educational HAMBLIN^SMITH'S ARITHMETIC Kirkland & Scott, LIBRARY OF THE UNIVERSITY OF CALIFORNIA. GIFT OF J*^ Class & A TREATISE ON ARITHMETIC, J. HAMBLIN SMITH, M:A, Of Gonville and Cains College, and late Lecturer at St. Peter's CoUetjc, Cambridge. ADAPTED TO CANADIAN SCHOOLS BY THOMAS KIRELAXD. M.A., SCIENCE MASTER, XOBMAL SCHOOL, TOI?0>XTO. AND WILLIAM SCOTT, B.A., MATHEMATICAL, MAIbTEtt, NORSIAI, SGHOOIi, OTTAWA. Prescribed by the Council of Public Instruction for use in Novz Scotia. A uthorized for use in the Schools of Ai atnioba. Recommended by the University of Halifax, Nova Scotia. Recommended by the Council of Public Instruction, Quebec. 'Authorized by th-e Education )e.<,rtment, Ontario. FOURTEENTH EDITION. PRICE 75 CENTS. T. Toronto and Winnipeg /t/ Entered according to the Act f Parliament of thr Dominion d Canada, in the year one thousand eight hundred and seventy* seven, by ADAM MILLER & Co., in the Office of the Mit* (tfer of Agriculture. PREFACE TO THE CANADIAN EDITION. THE present edition of Hamblin Smith's Arithmetic is not simply a reprint of the English work. Several articles have been introduced, which, it is hoped, will render the book mor, useful. Some of these will be found on pp. 88-40, 45, 48, 64, 92-95, 115. From Simple Interest to Proportion the work has almost altogether been re- written. Important subjects, such as Discount, Stocks and Shares, Exchange, &c., have been treated at much greater length than in our ordinary text-books. Some articles of a practical business nature, not usually found in school arithmetics, such as Equation of Accounts, p. 188 ; Partnership Settle- ments, p. 224 ; have been introduced. For these we are indebted to Mr. S. G. Beatty, late Principal of Ontario Commercial College. Special care has been taken to adapt the work to the wants of the business community. Examination Papers of a somewhat diffi- cult character have been added to each chapter. These are designed to stimulate the student to think for him- self, and to assist him in preparing for the different official examinations. In order to render the work i at- complete as possible, an Appendix has been added, in which the subjects of Interest, Discount, and Annuities have been treated algebraically. We have to thank Dr. McLellan for hints and advice during the progress of the work. TORONTO, August, 1877. 184004 CONTENTS. PUKE AEITHMETIO. PAGB, I. ON THE METHOD OF REPRESENTING NUMBERS BT FIGURES 1 n. ADDITION 9 III. SUBTRACTION 18 IV. MULTIPLICATION , . . 15 V. DIVISION 22 VI. ON THE RESOLUTION OF NUMBERS INTO FACTORS .. 27 VTL INEXACT DIVISION . . 80 VIII. METHODS OP VERIFYING THE OPERATIONS AND BOMB PRACTICAL METHODS OF SHORTENING LABOR IN THE FUNDAMENTAL RULES .. .. 83 X. HIGHEST COMMON FACTOR 48 X. LOWEST COMMON MULTIPLE .. .. 46 XI. FRACTIONS , 61 XII. DECIMAL FRACTIONS 74 Xm. SQUARE ROOT 103 XIV. CUBE HOOT 109 COMMERCIAL ARITHMETIC. XT. ON ENGLISH, CANADIAN, AND UNITED STATES Omu BENCIES 114 XVI. ON MEASURES OF TIME, LENGTH, SURFACE, SOLIDITY, CAPACITY AND WEIGHT 129 XVEt. FRACTIONAL MEASURES 140 VTn. DECIMAL MEASURES.. .. 142 XIX. PRACTICE . . 148 Vlll. CONTENTS. COMMERCIAL ARITHMETIC. Continued. AECT. PAGE. XX. PROBLEMS 154 XXI. SIMPLE INTEREST 166 XXII. COMPOUND INTEREST.. 174 XXIII. PRESENT WORTH AND DISCOUNT 178 XXIV. EQUATION OF PAYMKNTS 180 XXV. AVERAGES AND PERCENTAGES 191 XXVI. PROFIT AND Loss 200 XXVIE. STOCKS AND SHARES 204 &XVILL DIVISION INTO PROPORTIONAL PARTS 218 XXIX. ALLIGATION . . 226 XXX. EXCHANGE .. .. 280 XXXI. RATIO AND PROPORTION 241 XXXII. THE METRIC SYSTEM 250 KXXJII. MEASUREMENT OF AREA 253 EXAMINATION PAPERS 263 ANSWERS '. 294 APPENDIX I 326 OF THE UNIVERSITY } OF ARITHMETIC. L On the Method of Representing Numbers by Figures. 1. ARITHMETIC is the science which teaches the use oJ numbers. * 2. The number on*, or unity, is taken as the founda- tion of all numbers, and all other numbers are derived from it by the process of addition. Thus: Two is the number that results from adding one to one ; Three is the number that results from adding one to two ; Foar ia the nnmber that results from adding MM to three ; and so on. 3. By means of the symbols or figures 123456789, called the NINE SIGNIFICANT DIGITS, together with the symbol or figure 0, called ZEKO, we can represent num- bers of any magnitude. 4. First, each of the significant digits, standing by itself, represents a number greater by one than the number represented by the digit that immediately pre- cedes it in the list uf digits. Thus 7 represents a number greater by one than the Dumber represented by 6, REPRESENTATION OF 5- The symbol -f, read PLUS, is used to denote UK operation of ADDITION. The symbol *= stands for the words "is equal to," or "the result is." Since 2=1 + 1, where unity ig written twice, 8=2 + 1=1 -}-l-f 1, where unity is written three times. 4=3 -f 1=1 -j- 1 + 1 -f- 1, where unity is written four times and so on. 6. Numbers between nine and a hundred are repre- sented by two figures, the one on the left-hand signifying how many groups of ten units are contained in the num- ber represented, and the one on the right-hand signify- ing how many single units are contained in the number, in addition to the groups of ten units. Thus, in the expression 69, the figure 6 represents six groups of ten units, the figure 9 represents nine single units. These groups of ten units are for brevity called Tens, and the single units are for brevity called Units. Numbers between ninety-nine and a thousand are represented by three figures. In the expression 745, the figure 7 represents seven groups of a hundred units, the figure 4 represents four groups of ten units, the figure 5 represents five single units. In the expression 8475, the figure 8 represents three groups of a, thousand units. In the expression 23475, the figure 2 represents two groups of ten thousand unite. In the expression 128475, the figure 1 represents one group of a hundred thousand units. In the expression 9123475, the figure 9 represents nine groups of a million units ; and so on. HUMB8BS fi* KlG-U&ES. 9 7- To put the matter briefly : when we express a number in figures, and tell off the figures from rigid to kfa the Jirst figure represents a number of units, \ the seeond figure represents a number of tens, the third figure represents a number of hutidreds, ) the fourth figure represents a number of thousands, . the fifth figure represents a number of tens of thou- sands, the sixth figure represents a number of hundreds of thousands, } the seventh figure represents a nnmber of millions, \ the eighth figure represents a number of tens of millions, I the nvtdh figure represents a number of hundreds of miUiona, ) the tenth figure represents a number of billions, \ the eleventh figure represents a number of tens of billions, I the twelfth figure represents a number of hundreds of billiwis, ) the thirteenth figure represents a number of trillions. 8. When the symbol appears in an expression, it shows that the number, represented by the expression, contains no single units, tens, hundreds, etc., according as the is placed in the first, second, third place, the order of place being reckoned from right to left. Thus: 20 represents the number which contains two groups of ten units and no single units ; 800 represents the number which contains three groups' of a hundred units, and no group of ten, and np sin- gle units ; 4007 represents the number which contains four groups of a thousand units, and no group of a hundred, and no group of ten, and seven single units. NUMERATION. & To write in words the moaning of a mimbei expressed in figures, is called NUMERATION. The remarks, which we have already made, 4 NUMEKATION. to enable the learner to write in words all numbers expressed by ONE, TWO, or THBEE figures. Thus: the number expressed by 8 is written EIGHT ; the number expressed by 27 is written TWENTY-SEVEN ; the number expressed by 304 is written THEKK HUNj>jtai> AND FOUR. 10- Next take the case of numbers expressed by POUR, FIVE, or six figures, as 4287, 23509, 402675. Draw a line, separating the three figures on tJie right of each expression from the rest of the expression, and over the figure or figures on the left of the line write the word Thousand, thus : Thousand I Thousand j Thousand 4 237 23 509 ; 402 675. Then the meaning of each expression can be written at once in words, thus : Four thousand, two hundred and thirty-seven ; Twenty-three thousand, five hundred and nine ; Four hundred and two thousand, six hundred and seventy-five. 11. Next take the case of numbers expressed by SEVEN, EIGHT, or NINE figures, as, for instance, the num- ber expressed by 847295328. Draw a line, separating the three figures on the right from the rest of the expression, and a second line, marking oif the next three figures. Over these write fehe word T/wusand, and over the figures on the left of this second line the word Millions, thus : Million* f Thousand I 847 | 295 1 328. Then we can write the meaning in words, thus : Three hundred and forty-seven millions, two hundred and ninety-five tkous&nd, three hundred twen'v-eight. NUMERATION, Again, to express in words 20040030, write it thus : Millions Thousands I 20 ! 040 080 and the number expressed iu words ii Twenty millions, forty thousand and thirty. 12. If more than nine figures are in the given num- ber, mark off the figures by threes, as before, and over the fourth parcel write the word billions, over the fifth parcel write the word trillions. Thus, to express in words 24003269407082, proceed thus: Trillions I Billions I Millions I Thousands j 24 I 008 I 269 | 407 | 082 and the number expressed in words is Twenty-four trillions, three billions, two hundred and sixty-nine millions, four hundred and seven thousand and thirty-two. Note. 1 followed by three zeros, 1000, represents a thousand. 1 followed by six zeros, 1000000, represents a million. 1 followed by nine zeros, 1000000000, represents a bil- lion. Examples- (i) Write in words the numbers expressed by the following figures : (1) 7, 18, 45, 59, 326, 4578. (2) 90, 110, 207, 4300, 4036, 4306. 780, 609, 5360, 2020, 1101. 86497, 49532, 654321, 743269. 45000, 32600, 75230, 600000. 8572914, 3469218, 4629817. 7) 9000000, 29000000, 715000000. 8) 910807240, 307004205, 380503040. 9) 243759268842, 307405006270. (10) 417235682719435, 203056300072010. \ / s 6' NOTATION. DOTATION. 13. To represent by figures a number, expressed in words, is called NOTATION. The method to be employed is this : Prepare the divisions in which the figures represent- ing thousands, millions. &c., are to be placed, thus : Trillions Billion* vlillioaa Thousand and place in each division, as well on the right and left of the outermost lines, the figures required. Thus, to represent by figures forty-seven thousand, three hundred and nine, we proceed thus Thousand I 47 I 809 and tae number expressed in figures is 47309. Again, to represent by figures four billions three hundred and two millions, eighteen thousand and fifty- three, we proceed thus : Billions I Millions } Thousand I 4 'I 802 I 018 I 058 and the number expressed in figures is 48320180&I. Examples- (ii) Express in figures the following numbers : (1) Nine ; twelve ; seventeen ; nineteen ; thirteen ; six* teen; eleven. (2) Twenty-three ; twenty-seven ; thirty-five ; thirty- eight ; forty-four ; forty ; twenty-six ; thirty-four. (3) Sixty-seven ; seventy-five ; sixty-two ; eighty-three ; seventy-four ; ninety-two ; sixty-eight ; ninety-five. (4) Seventy-six ; twenty-two ; fifty ; fifteen ; twenty- eight ; sixty-one ; forty-nine ; eighteen ; ninety ; seventy- three. (6) One hundred and seven; one hundred and thirty; two hundred and forty-six; three hundred and seventy-two ; aix hundred and sight ; seven handled and forty ; nin hundred and ninety. (6) Eight hundred and thirty-si* ; seven hundred and forty-seven ; four hundred and ten ; nine hundred and thirteen ; seven hundred and fifty ; three hundred and eighty-four. (7) Eight hundred and eighteen; eight hundred and eight ; two hundred and six ; four hundred and thirty ; five hundred and twelve ; seven hundred and eighty-seven. (8) Seven thousand eight hundred and foi"ty-fiv ; nine thousand **ix hundred and thirty-seven ; twelve thousand r eight thousand four hundred ; sis thousand and three ; eighty-five thousand and forty. (9) Five thousand four hundred and seventy ; three thou- sand six hundred and fifty ; eight thousand seven hundred and eighty ; one thousand two hundred and forty-seven ; four thousand eight hundred and eight. (10) Six thousand and four ; seven thousand and twenty- two ; three thousand five hundred ; nine thousand and fort'y- seven : two thousand and seventeen ; nineteen thousand four hundred and two. (11 ) Seventy thousand and seven; sixty thousand and sixty, fourteen thousand and fourteen ; seventy thousand and seventeen ; twelve thousand three hundred and three ; six- teen thousand and five. JL2) Three hundred and fifty-six thousand seven hundred twenty-eight ; six hundred and forty thousand eight hundred and forty-two ; nine hundred thousand ; eight hun dred thousand and forty. (13) Seven millions; four millions five hundred and seventy-six thousand eight hundred and sixty-five ; seventy- five millions eight hundred and six thousand, nine hundred and forty. f (14) Three hundred and fifteen millions ; five millions and forty thousand ; eight millions and seven hundred ; eighteen millions and twenty ; seven hundred millions and two. (15) Three hundred and fifteen billions six hundred and seventy-four millions, eighteen thousand and .three ; thirty- *\ve billions six hundred millions, five hundred and twenty. (16) Seven billions ; five trillions, eight hundred billions, six hundred thousand and forty-seven ; eight trillions, forty-three thousand and seven. (17) Three hundred and five trillions, five billions, four millions, six thousand and three ; fifty-three trillions, fifty- threo millions, fifty-three thousand and fifty-three. 8 BOMAN NUMERALS. (18) Nine trillions and nine ; ninety trillions and nin hundred ; nineteen trillions and nineteen thousand ; one trillion, one million, one thousand, one hundred and one. ROMAN NUMERALS. 14. In the Roman system of Notation, which is still used frequently in inscriptions, in references to chapters of books, and for other purposes, the symbols chiefly employed were I, V, X, L, C, D, M. These symbols, standing by themselves, represented respectively the numbers one, five, ten, fifty, a hundred, five hundred, and a thousand. Intermediate numbers were represented by means of an arrangement that the numbers represented by the symbols I and X when standing on the right of a higher symbol were to be added to the number represented by that symbol, and when standing on the left were to be subtracted from it. Thus: VI represented the number x, IV represented the number four, and LX represented the number sixty, XL represented the number forty. The following table will explain the method for num bers up to a thousand : * 1 1. 11 XI. 81 XXI. 110 CX. 2 II. 12 XII. 80 XXX. 150 CL. 8 III. 18 XIII. 40 XL. 188 CLXXXVII1 4 IV. 14 XIV. 44 XLIV. 200 CO. 5 V. 15 XV. 60 L. 800 CCC. 6 VI. 16 XVI. 60 LX. 400 CCCO. 7 VII. 17 XVII. 70 LXX. 600 D. 8 VIII. 18 XVIII. 80 LXXX. 600 DC. 9 IX. 19 XIX. 90 XG. 900 DCCCC. io x. ao xx. 100 c. 1000 M, ROMAN NUMERALS. Examples- (ii Write in words : (1) XXVII. ' (2) XLIX. (4) LXXIII. (5) XCII. (7) CLXIII. (8) CXCIX. (10) MDCCOLXXII. Write in Koman Numerals : (1) 37. (6) 139. (2) 59. (7) 145. (3) 62. (8) 179. (3) LXVIII (6) CXLIV. (9) DCLXIV. (4) 87. (9) 846. (5) 95. (10) 1763. II. Addition- 15. If we- combine two or more groups of units, so as to make one group, the number of units in this sin- gle group is called the SUM of the numbers of units in the original groups. To find the sum of 5 and 3, we reason thus : Since 3 = 1 + 1 -f- 1, (Art. 5} 5 + 3 = 5+1+1+ 1 (Art. 4) = 8. 16. By practice we become able to express the result of adding a, number less than ton to another number, without breaking up the number, which we have to add, into units. Thus we say 7 and 5 make 12, 15 and 8 make 23 ; and so on. Again, if we have three or four numbers, each less than ten, to add together, we perform the process men- tally ; thus, to add 4, 7, 9, and 6 together we say 4, 11, 20, 26. 17. We now proceed to explain the process of addi- tion in the case of higher numbers. 10 ADDITION. Suppose we have to add together the four numbers 2475, 397, 486, and 3007. We arrange them thus : 2475 397 486 8007 6366 placing the figures that represent units in each number in the same vertical line, and those that represent tens in the same vertical line, and similarly for those that represent hundreds and thousands. We then draw a horizontal line under the last number, and under this line we place the number representing the sum of the given numbers, which is found in the following way : Adding 7, 6, 7 and 5 units, the sum is twenty-five units, that is 2 tens and 5 units : we place the five under the line of units, and carry on the 2 tens for addition to the line of tens. Adding 2, 0, 8, 9 and 7 tens, the sum is twenty-six tens, that is two hundreds and 6 tens : we place the 6 under the line of tens, and carry on the 2 hundreds for addition to the line of hundreds. Adding 2, 0, 4, 8 and 4 hundreds, the sum is thirteen hundreds, .that is 1 thousand and three hundreds : we place the 3 under the line of hundreds, and carry on the 1 thousand for addition to the line of thousands. Adding 1, 3 and 2 thousands, the sum is six thou- sands, and we place 6 under the line of thousands. Examples. (i*> Add together (1) 4 and 7, 3 and 1 B, 5 and 15, 9 and 27. (2) 62 (8) 40 (4^ 36 86 27 24 11 237 349 823 (6) 209 140 600 (7) 562 70 106 (8) 459 6 237 4269 (9) 5462 723 8004 9217 (10) 24609 3470 40052 6207 UL1) 429 347 425 269 538 (12) 3(54 629 488 976 853 (18) 253 189 567 278 384 (14) 140 49 257 6 428 6842 (16) 8750 (17; 8604 (18 6848 5679 4623 4007 4297 8526 7988 5290 326 5037 6543 8046 52 2409 5729 7259 7008 (19) 64+43+74-85+9. (20) 247+856+28+423+97+12. (21) 425+3742+4236+39+847. (22) 7288+976+45+623+4000. (23) 8+97623+3407+5260+86. (24) 41537+9215+48+6077+23+2418. (25) 275413+3126+725+5007. (26) 74259+346274+30000+1000001 + 207. (27) 4692+72430+80000729+40+600000000. (28) 46243 85297 825649 246728 815 42376 645980 (29) 748325 54297 532684 20047 4207 617048 3025 (30) 5629 426580 87259 506 670492 87987 6493 SUBTRACTION. (31) 256497 648098 720480 630689 407246 864928 254884 (34) 7462594 8625837 4398025 6702403 5124917 6219806 4390143 7409425 (32) 654297 248643 380469 472586 582987 639458 498468 (35) 4697498 527 4307046 27209 152372 4058 7265204 4372943 (38) 625498 75862 5436 87294 4859 862 13 (36) 6572043 2869257 436 698206 45297 3526084 67002 852968 (87) Seven hundred and forty ; forty thousand and fifteen ; six hundred and forty-seven ; fifty three thousand three hundred and three ; seventeen thousand five hundred and forty-six. (88) Five hundred and eight; six thousand and nine; fifty-five thousand and fourteen; eight hundred and nine- teen ; seven hundred thousand and six ; two thousand and twelve. (89) Six hundred and forty-five thousand, eight hundred and forty-five ; seventy thousand and forty-seven ; sixty thousand and forty ; seven hundred and fifty thousand ; three hundred thousand and fifteen. (0) Two hundred and one millions, ninety-si* thousand, three hundred and forty-two ; fifty-four thousand three hun- dred and four; eighteen millions, six thousand and three; five hundred thousand and forty ; eight millions aud^ eight. III. Subtraction. V 18. If from a number we take away a smaller num- ber, the process is called Subtraction. Strictly we ought to take away each of the units, of which the smaller number is composed, separately from the larger number : thus, to subtract 8 from 5, we rea- son thus 18 if we take away one of these units from 5, we have 4 left ; if we take away the second unit from 4, we have 8 left ; if we take away the third unit from 3, we have 2 left. The Symbol , read minus, is used to denote the operation of Subtraction. Thus the operation of sub- tracting 8 from 5, and its connection with the result, may be briefly expressed thus : 5-3=2. 19. By practice we become able to subtract a num- ber, less than ten, from another number, without breaking up the smaller number into units ; thus we say, 7-4=3, 18 5=13, 49 8=41; and so on. 20. Before we proceed to explain the process of Subtraction in the case of higher numbers, we must notice the principle on which a certain step in the process is founded. If we are comparing two numbers, with a view to discover &e number, by which one exceeds the other, we may add ten single units to the greater, if we also add one group of ten units to the less, and we may add ten groups of ten units to the greater, if we also add one group of a hundred units to the less ; and so on. Suppose, for example, we want to find the number by which 56 exceeds 29, we might reason thus : 56= five tens together with six units. 29 = two tens together with nine units. To the former add ten single units, and to the latter add one group of ten units. Then the resulting numbers will be, in the first case, five tens together with sixteen units, in the second case, three tens together with nine units. Hence the excess of the former over the latter will be the number, made up of two tens together with seven units, and will therefore be represented by 27. 14 StJBTfcAOTION. Let us now take an example, to show the practical way of performing the operation of substraction, accom- panied by a complete explanation of the process. Suppose we have to take 589 from 926 ; From 926 Take 589 Eemainder 837 We arrange the numbers, placing the figures that represent units in each in the same vertical line, and doing the same with those that represent tens and hundreds. We then reason thus : we cannot take 9 units from 6 units ; we therefore add ten units to the 6 units, making sixteen units, and we take 9 units from the sixteen units, and set down the result, which is 7 units, under the line of units. Having increased the upper number by ten units, we add, by way of compensation, 1 ten to the lower number, changing 8 tens into 9 tens. We proceed thus : we can- not take 9 tens from 2 tens ; we therefore add ten tens to the 2 tens, making twelve tens, and from these we take 9 tens, and set down the result, which is 8 tens, under the line of tens. Having increased the upper number by ten tens, we add, by way of compensation, 1 hundred to the lower number-, changing 5 hundreds into 6 hundreds. We then take 6 hundreds from 9 hundreds, and set down the result, which is 8 hundreds, under the line of hundreds. Examples- (7) Find the difference between the following pairs oi numbers : (1) 13 and 6. (2) 15 and 7. (8) . 28 and 4. (4) 8 and 32. (5) 57 (6) 96 (7) 74 (8) 87 (9) 92 2i> 42 89 58 47 MULTIPLICATION . (10) 813 (13) 704 (12) 630 (18) 7426 247 195 548 8618 (14) 6239 (15) 4729 (16) 6258 (17) 65472 4127 501 86 4001 (18) 857 (19) 4G25 (20)72649 (21) 20004 249 1846 43821 17243 (22) 437 56 (23) 529 483 (24) 827 795 (25) 3000 958 (26) 7040 583 (27) 6259 479 (28) 586237428 (29)6429553296(30)7000068904 (31) 52764 and 34297. (32) 42456 and 102479. (83) 824300 and 14000702. (34) 99999 and 100000. (35) A million and a thousand. (36) A hundred millions and a hundred thousand. (;)7) Ten billions and a thousand and one. (38) What number must be taken from 26 to leave 18? (39) "What number* must be taken from 427 to leave 401? (40) What number must be taken from three thousand an In this Chapter we shall treat only of cases in which the Dividend contains the Divisor an exact number of times. 30. For small numbers, the Multiplication Table affords the means of solving questions in Division, For instance, since 12 =4x3, 12 -i- 4 = 8, and 12 -f- 3 = 4 ; and since 96 = 12 X 8, 96 -f- 12 = 8, and 96 8 =12. 31. When we divide one number by another, we find how many times the latter is contained in the former, and therefore any process by which we can discover how many times one number is contained in another will furnish a rule for division. Such a process is explained by the examples, which we shall now give. Ex. (I). Divide 408 by 17. Since 17x20-= 840, and 17X30 = 610, it is plain that 17 is contained in 408 more than twenty times, and less than thirty times. If then we take away 840 from 408, and find how many times 17 is contained in the number that remains, Division. 28 we shall find how many times, more than twenty, the Divisor is contained in the Dividend 408. Now 408 340 = 68, and this number contains 17 just/ow times. Hence 17 is contained in 408 twenty times, and also four times, that is, the Quotient resulting from the div ision of 408 by 17 is 24. This process is represented more briefly thus : 17 ) 408 ( 20+4 840 68 68 Hence 408 -17 -24. And yet more briefly, availing ourselves of the notation by which the local value of digits is represented, and we are enabled to omit zeros, 17)408(24 34 68 Ex, (2). Suppose we have to divide 89012 by 10 Divior Dividend Quotient 17) 89012 (5236 85 40 84 61 51 102 We nrst find how often 17 is contained in 89, and as it is contained five times, we set down 6 as the first figure in the quotient, then multiply 17 by 5, and subskact 24 DIVISION. the result 85 from the 89 : to the remainder 4 we annex the next figure in the dividend ; then as 17 is contained in 40 twice, we set down 2 as the second figure in the quotient, then multiply 17 by 2, and substract the result 34 f * the 40 ; and proceed by similar steps to t)^ end of the operation. Ex. (3). Divide 920575 by 23. 23) 920575 (40025 92 057 46 115 115 Here, when we bring down 0, the third figure of the dividend, 23 is not contained in it ; we therefore set down as the second figure of the quotient, and when we bring down 5, the fourth figure of the dividend, 23 is not contained in 5 ; we therefore set down another as the third figure of the quotient. When we then bring down 7, the next figure of the dividend, 23 is contained in 57 twice, ; and the operation proceeds easily. Divide Examples. (1) .18 by 6. (2) (3) 84 by 7. (4) (5) 182 by 18. (6) (7) 456 by 19. (8) (9) 3996 by 37. (10) (11) 431376 by 817. (12) (13) 19249470 by 342. (14) (15) 224009433 by 489. (16) (17) 2680376 by 1369. (18) (19) 98955005667 by 4123. (20) (21) 13312053 by 237. (22) (28) 360919856 by 83. (24) (25) 218860161 by 689. (26) (27) 39916424548 by 1001. (28) (29) 26540538445 by 7649. 27 by 9. 132 by 12. 238 by 17. 3708 by. 86. 6499 by 493. 976272 by 946. 86366784 by 358. 4690325214 by 618. 10781526 by 6142. 4076361 by 2019. 505350366 by 89. 4600304 by 907. 337103025 by 861. 152847420 by 5060. DIVISION. 26 1165584898000 by 17072. 35088008823434 by 74291. 369187022085112 by 65432. 837741356152459 by 98989. 58376823669 by 642867. 2959990965442 by 9864302. 261449109180 by 8723694. 32t If any two of the three numbers that form the Divisor, Dividend, and Quotient be given, we can find the third. For Dividend-$-Divisor= Quotient. Dividend-i-Quotient = Divisor. Divisor x Quotient = Dividend. Examples, (rii) (1 The Dividend is 1171692, the Divisor 842. Find the Quotient. (2) The Dividend is 149201, the Quotient 23. Find the Divisor. (8) The Divisor is 987, the Quotient 64852. Find the Dividend. SHORT DIVISION. 33- When the Divisor is not greater than 12, the process of division may be greatly abridged. Suppose we have to divide 92368 by 8. The operation is set down in the following form : 892368 11646 Quotient. The following is the process : Since 8 is contained once in 9, with 1 as remainder, we set down 1 under the 9, and mentally prefix the remainder 1 to the 2, reading the result as 12 : then since 8 is contained once in 12, with 4 as remainder, we set down 1 under the 2, and prefix 4 to the 8, reading the result as 43 ; then since 8 is contained Jive times in 43, with 3 as remainder, we set down 5 under the three, and prefix 3 to the 6, reading the 26 DIVISION. res^l as 86 : then since 8 is contained foivr times in 86, with 4 as remainder, we set down 4 under the 6, and prefix 4 to the 8, reading the result as 48 : then since 8 is contained six times in 48, with no remainder, we set down 6 under the 8, and our operation is completed. Next, suppose we have to divide 11042304 by 12* The operation is set down thus : 12111042804 920192 Quotient The following is the process : We must take three figures before we obtain a number which contains 12 ; then we say 12 is contained nine times m 110, with 2 to carry on ; then 12 is contained ticice in 24, and there is nothing to carry on ; then 12 is not contained at all in 2, we therefore set down under the 2, and carry on 2 ; then 12 is contained in 23 once, with 11 to carry on ; then 12 is contained in 110 nine times, with 2 to carry on : lastly, 12 is contained in 24 twice exactly. Examples- (xiii) Divide (1) 7652 by 2. (2) 725961 by 8. (3) 8650232 by 4. (4) 8749320 by 5. (5) 7463424 by 6. (6) 8504221 by 7. (7) 713406960 by 9. (8) 4862017 by 11. (9) 7912464 by 12. (10) 4000623070905 by 9. (11) 7642300721 by 11. (12) 86089882405604 by 12. Divide each of the following numbers by 2, 3, and 4 separately: (18) 4268924. (14) 620437548. (15) 27540918264. Divide each of the following numbers by 5, 8, and 9 separately : (16) 46528920. (17) 981754200 (18) 284567000. Divide each of the following numbers by 7, 11, and 12 separately : (19) 7971348. (20) 29574468. (21) 6736387812. BKSOLUTION OF NUMBEBS INTO 1 ACTORS. 27 VI. On the Resolution of Numbers into Factors. 34. We shall discuss in this section an operation, which is the opposite of that which we call multiplication. In multiplication we determine the product of too given factors : in the operation, of which we have now to treat, the product is given, and the factors have to be found. 35. For small numbers the factors may be determined by inspection : thus, the factors of 21 are 3 and 7, the factors of 55 are 5 and 11. 36. When we have found two factors that make up a product, one or both of these factors may be themselves reducible to simpler factors. Thus 9 and 6 are factors of 54 : and the factors of 9 being 3 and 8, and the factors of 6 being 2 and 3, the number 54 can be split up into four factors, 2, 8, 8> 8. 37- Prime numbers are those, which have no exact devisor but themselves and unity. Thus 2, 3, 5, 7, 11, 13, 17, 19 are Prime Numbers. Composite numbers are those, which can be resolved into factors, each of which is greater than 1. Thus 4, 6, 8, 9, 10, 12, 14, 16, 16, 18 are Composite Numbers. 38. Every composite number can be resolved into factors which are prime numbers : thus 4 = 2x2; 6 = 2x3; 8 = 2x2x2; 9 = 3X3. Hence, in resolving a large number into factors, we divide it by any small prime number, by which we know it is exactly divisible, and then divide the quotient by any small prime number by which it is exactly divisible, and proceed in this way, til the quotient is I ; theo the divisors* are the factors required. 28 RESOLUTION OF NUMBERS INTO FACTORS. Thus, to find the factors of 2520 : 2 | 2520 2 1260 630 315 105 35 7 1 Hence 2520 = 2x2x2x3x3x5x7. In practical arthinetic we seldom require to find all the factors of a composite number, but very frequently we want to know whether a number is exactly divisible by a particular number. The student will find it of use to remember the follow- ing properties of numbers. A number is exactly divisible by 2 when its last figure is or an even digit, as 426 ; 3 when the sum of its digits is divisible by 3, as 579 ; 4 when its last two figures are divisible by 4, as 2364 ; 8 when its last three figures are divisible by 8, as 25256 ; 5 when its last figure is or 5, as 30 and 135 ; 9 when the sum of its digits is divisible by 9, as 275265 ; 10 when its last figure is ; 11 when the difference between the sum of the digits in the odd places (reckoning from the right) and the sum of the digits in the even places is either 01 divisible by 11. Thus 24794 and 829191 are divisible by 11. Examples- (xv) Find whether the following numbers be exactly divis- ible by 2, 3, 4, 5, 8, 9, 10 or 11. RESOLUTION OF NUMBERS INTO FACTORS. 28 495. (6) 42345. (9) (1) 117. (2) 288. (8) (4) 1050. (5) 23472. (6) (7) 27464. (8) 32495, (9) 84782. (10) 6480. (11) 619182718. NOTE. We have inserted these remarks at this point, because, in attempting to resolve a large number into factors, it is well to know whether the attempt to divide it by 2 or 8 or 5, &c., will be successful. The student may now, following the instructions given in Art. 38, work another set of Examples. Examples, (xvi) Resolve into prime factors : (1) (5) (9) (13) (17) (21) (25) 18. 36. 54. 91. 108. 288. 729. 8 (10 (14 (18 (22 (26, 24. 39. 67. 99. 112. 432. ) 999. (3) (7) (11) (15) (19) (23) (27) 27. 42. 72. 100. 132. 525. 1296. (12) (16) (20) (24) (28) 82. 51. 86. 105. 176. 625. 1760. (29) 5760. 39. The process of Multiplication may often be made shorter when the Multiplier is a composite number, by resolving it into two or more factors. Thus if we have to multiply 2579825 by 66, we may resolve 66 into the factors 8 and 7, and proceed thus, 2679825 8 20638600 7 144470200 The advantage of this method will be more apparent when we come to multiplication of sums of money, weights, and measures. Examples, (xvii) Multiply, after resolving the multiplier into factors not greater than 12, 30 INEXACT DIVISION. (1) 847 by 14. (2) 423 by 22. (8) 5462 by 27 (4) 8497 by 36. (5) 8573 by 49. (6) 28472 by 56. (7) 49273 by 63. (8) 90728 by 132. (9) 90725 by 360. (10) 40207 by 108. (11) 36729. by 1320. (12) 704075 by 14400. 40- So also we may often simplify the process of division, when the Divisor, though greater than 12, can be made up by factors each not greater than 12. For we can divide the Dividend first by one of these factors, and then divide the Quotient by a second factor, and so on. Suppose we have to divide 47268540 by 45. Here 45 can be made up of the factors 9 and 6. 91 47268540 5 5252060 1050412 Examples- (xviii) Apply the process just explained in the division of (1) 34608 by 14. (2) 6791040 by 15. (8) 752364576 by 18. (4) 1143995886 by 27. (5) 285216822 by 33. (6) 2095501072 by 49. (7) 41 57028792 by 56. (8) 1200130008 by 84. (9) 22039992 by 108. (10) 57667632 by 132. (11) 472634500 by 125. (12) 565184160 by 720. (13) 537062400 by 14400. VII. Inexact Division. 41. Hitherto we have chosen Examples, in which the Divisor is contained an exact number of times in the Dividend. Now suppose we have to divide 23 by 7. Since 3x7 = 21, it follows that we can divide 23 units into 3 parcels, each containing 7 units, and when we have done this, 2 units out of the 23 remain over. In such a case we call 3 the Quotient, and 2 the INEXACT DIVISION. 81 Again, if we have to divide 72469 by 63, we proceed thus, 63) 72469 (1367 63 194 159 856 818 889 871 18 Hence the Quotient is 1367, and the Kemainder 18. NOTE. If we multiply the Quotient by the Divisor, and add the Remainder to the product, the sum must be equal to the Dividend. Examples* (xix) Divide (1) 3492 by 37. (2) 486296 by 41. (3) 872968 by 47. (4) 67092 by 65. (5) 7492736 by 71. (6) 82749325 by 98. (7) 87467 by 103. (8) 978462 by 409. (9) 8276253 by 728. (10) 974004562 by 1009. (11) 48237654 by 4821. (12) 68725642903 by ($871. 42. When we employ, in cases of inexact division, the method of short division, after breaking np the divisor, into component factors, as in Art. 40, the re- mainder will be found by a process now to be explained. Ex. (1). Divide 48276 by 21. 43276 f 8 ill 7 14425 and 1 unit over, 2060 and 5 parcels of 3 units, or 15 units over. Whence the Quotient is 2060, and the .Remainder is 16 + Lor 16. 82 INEXACT DIVISION. Ex. (2). Divide 672948 by 125. 5 572948 126 J 5 114589 and 3 units over, 22917 and 4 parcels of 5 units, or 20 units over. 4583 and 2 parcels of 25 units, or 50 units over Whence the Quotient is 4683, and the Remainder if 50 + 20 + 8, or 73. Examples- (xx) Divide, employing Short Division, (I) 4153 by 15. (2) 587595 by 16. (3) 42818 by 18. (4) 423672 by 21. (5) 724972 by 25. (6) 569024971 by 27. (7) 2825780 by 33. (8) 8642396 by 35. (9) 356599 by 48. (10) 8274913 by 64. (11) 230047914 by 77. (12) 419421 by 99. (13) 44487 by 105. (14) 95379 by 189. (15) 1194477 by 210. 43. In dividing a number by 10, we have merely to mark off the last figure, the other figures giving the quotient, and the figure marked off the remainder. Thus 2460197-J-10= 246019 with remainder 7. Again, to divide 42395675 by 20, we might proceed thus, 1042395675 2 4239567 and 5 units over, 2119783 and 1 parcel of 10 units over; vrhence the Quotient is 2119788, and Remainder 10 + 5, or 15. But the operation is written more briefly thus : 2,014239567/5 2119783 and 15 remainder. Again, in dividing by 100, we mark off the last two figures, in dividing by 1000, we mark of the last tnree figures PRACTICAL METHODS OF SHORTENING LABOR. 33 from divisor and dividend, and find the quotient and remainder by a similar process. 44. If any three of the four numbers, that form the Divisor, Dividend, Quotient and Remainder be given, we can find the fourth. 1. Let Divisor, Dividend, and Quotient be given. Multiply the Divisor by the Quotient, subtract the result from the Dividend, and you have the Remainder. 2. Let Divisor, Quotient, and Remainder be given. Multiply the Divisor by the Quotient, add the Remainder to the result, and you have the Dividend. 8. Let Divisor, Dividend, and Remainder be given. Substract the Remainder from the Dividend, divide the result by the Divisor, and you have the Quotient. 4. Let Quotient, Dividend, and Remainder be given. Substract the Remainder from the Dividend, divide the result by the Quotient, and you have the Divisor. Examples- (xri) (1) The Divisor is 25, the Dividend 4276, the Quotient 171. Find the Remainder. (2) The Divisor is 842, the Quotient 1381, the Remainder 67. Find the Dividend. (3) The Divisor is 696, the Dividend 372149, the Remain- der 245. Find the Quotient. (4) The Quotient is 2910, the Dividend 8765237, the Re- mainder S17. Find the Divisor. VHL Methods of Verifying the Operations and some Practical Methods of Shortening Labor in the Fundamental Rules. 45. ADDITION. The usual verification is to add both upwards and downwards, and see if the sums agree. This is generally sufficient. Another method is to draw a horizontal line across the middle of the sum and add it in two separate parts, then find the sum of 84 PRACTICAL METHODS OF SHORTENING LABOR. the two answers, which must agree with the work it is to verify. If it be a very long sum, it may be divided into three parts by two horizontal lines, and the three separate sums found, &c. 46. SUBTRACTION. The correctness of the result in subtraction may be tested by adding the remainder or difference to the subtrahend, when the result ought to be the same as the top line or minuend. 47- MULTIPLICATION. The proof of multiplication by casting out the nines depends on the following property of numbers : Any number divided by nine will leave the seme re- mainder as the sum of its digits divided by nine. This will be evident from the following example : 6783 _ 6000 700 80_ j$_ 9 ~ = ~9~ ~*" 9 "*" 9 "*" 9 = (666 + f ) '+ (77+f)+ (8 + f+f ) = 666+77+8+ -- -+-- . Hence it is clearly seen that the remainder, arising from the division of 6783 by 9 is the same as that aris- ing from the division of the sum of the digits by 9. This test may be given in the form of the following rule: Divide the mm of the digits in the Multiplicand by 9, and set down the remainder. Divide tJie sum of the digits in the Multiplier by 9, and set down the remainder. Mul- tiply the two remainders together, divide the result by 9, and get down the remainder. If the process be correct, this remainder will be the same as the remainder obtained by taking the sum of the digits in the Product and divid- ing it by 9. For example, if w wittj>ly 708*1 by 854 tiw pro- duct IB PRACTICAL METHODS OF SHORTENING LABOR. 35 Sam of digits in Multiplicand = 24, and 24---9 gives remainder Sum of digits in Multiplier = 17, and 17-7-9 gives remainder First remainder X second remainder = 48, and 48-4-9 gives remainder 8. Sum of digits in the Product = 30, and 80-7-9 gives remainder 3. This so-called proof is defective as a proof in the following, as it fails to detect errors in the product 1. If the order of figures in the product be misplaced, as 87 for 78. 2. If errors be made which counterbalance each other, as 85 written for 62, the sum of digits in each case being the same. 8. If 9 be written for 0, or for 9, or either be omitted or inserted too often. 48. DIVISION. To prove division, multiply the divisor by the quotient, and add the remainder, if there is one, to the product. If the result is equal to the dividend, we have a verification of the first operation. Division may also be proved by casting out the nines, but the proof is less direct than in multiplication. For instance, if we divided 417 by 29 the quotient is 14 with remainder 11. The most convenient form in which to apply the proof of nines is to write this in the form of 29 x 14 + 11 = 417. The remainder gives 2x5 + 2 or 12. This remainder and the dividend, 417, divided by 9, give a remainder 8, which therefore proves the work. 49. ARITHMETICAL COMPLEMENT. The arithmetical complement of a number is defined to be the difference between any given number and the unit of the next superior order ; thus 6 is the arithmetical complement of 4, 47 of 53, 8468 of 1532, and so on, being the differ- ences respectively of 4, 58, 1582, and 10, 100, 10000, the next superior units of these numbers. Conversely, also, 4, 53, 1532 are the arithmetical complements of 6, 47, 8468 respectively. 36 PRACTICAL METHODS OF SHORTENING LABOB. The arithmetical complement of a number may b6 found by the following rule : Begin at the left hand and subtract every figure from 9 until the last ; subtract that from 10. The arithmetical complement may be used to find the difference between two numbers, thus : if 239 be sub- tracted from 576 the remainder is 337. But if 761, the arithemetical complement of 239, the less number, be added to 576, the greater, the sum will be 1337, one unit (1000 in this case) of the next superior order greater than the difference of the two numbers. By removing this unit, the number will be left equal to the difference of 239 and 576 ; so that the difference of the two numbers can be found by addition. The arithmetical complement may be written thus 1761, with the sub- tractive unit on the left, which when added to 576, the sum will be 337, the additive and subtractive units being together equal to zero. This . method is employed with great advantage to find the aggregate of several numbers when some of them are additive and some subtractive. Thus, if we have 3795 - 1532 - 2019+8759 - 5104 We arrange them as follows : 3795 A. 0. of 1532 is 18468 2019 " 17981 8759 5104 " 14896 3899 the aggregate required. 50- CONTBACTIONS IN MULTIPLICATION. The multipli- cation by any number from 12 to 19 inclusive, may be effected as follows : Multiply by the figure r/ the Multiplier in the unib' place, and to the number to be carried add the figure of the Multiplicand just multiplied. PKACTIOAL METHODS OF SHOBTENINO LABOB. ft? 1. Multiply 2384 by 19. 2384 19 45296 4X9 = 86 ; set down 6 and carry 8. 8x9+3 carried-|-4, the units figure of the multiplicand = 79 ; set down 9 and carry 7. 8x9+7 eamed-f-8, the tens' figure of the multiplicand = 42 ; set down 2 and carry 4. 2x9+4 carried+3, the hundreds' figure of the multipli- cand = 25 ; set down 5 and carry 2. 2 carried+2, the thousands' figure of the multiplicand = 4 ; set down 4. The backfigwrt. system, as it is sometimes called, may be extended to numbers between 20 and 80, and be- tween 80 and 40, by adding to the number to be carried the double or the treble of the figure of the multiplicand just multiplied. Ex. 2. Multiply 84578 by 999. Here 34578000 = 1000 times 84578. and 34678= 1 84548422 = 999 times 34578. Ex. 8. Find the product of 84578 by 699. Here 699 = 700-1 And 24204600 = 700 times 84578- 84578 = 1 * 24170022 = 699 times 8457*. Hence, any number can be multiplied by 99, 999, &e., by annexing 2, 8, 4, &c., ciphers to the multiplicand, and subtracting the multiplicand from this product. And in a similar way any number can oe multiplied by another composed of a repetition of the figure with any other figure in the highest place. 88 PRACTICAL METHODS OF SHORTENING LABOR. Ex. 4. Multiply 9648287 by 878427. 9643287 (878)(42)(7) 7 times the mnltipland = 67603009 42 times the multiplicand 6 ] times 7 times multiplicand = 6 [ = 405018054 times 67508009 j 878 times the multiplicand] = 9times 42 times the I mulultiplicand = 9 times f 405018054 8649280169549 To SQUARE ANY NUMBER ENDING IN 5. Square the 5 and write down the result ; then in- crease the number to the left of 5 by 1, and multiply this sum by the number to which the 1 was added. Set this product to the left of the 25 and the number thus formed will be the result required. Ex. Find the square of 75. 5 squared = 25. Add 1 to 7 and multiply by 7 and place the 56 to the left of the 25. 5625 is the result required. 51. ABBREVIATIONS IN DIVISION. Since 4 x 25 is 100, and 8 x 125 is 1000, the division by 25 will be effected by multiplying the dividend by 4, and cutting off the last two figures from the product. The division by 125 will be effected by multiplying the dividend by 8, and cut- ting off the last three figures from the product. In each case the figures cut off, when divided respectively by 4 or by 8, will be the remainder, and those left will be the quotient. Any number can be divided by 9, 99, 999, &c., by successively dividing the given number by 10, 100, 1000, &c., respectively, and taking the sum of the successive remainders for the true remainder; except when the sum of the latter exceeds the next higher unit ; in that case both the quotient and remainder must be increased by unity. PRACTICAL METHODS OF SHORTENING LALOK. 39 Ex. Divide 65874 by 99. 100 ) 658,74 6,58 6 665,39 Here the sum of the partial remainder is 188, and both the quotient and remainder must be increased by unity. The reason of this we leave as an exercise for the student. There is a method of dividing one number by an- other, termed the Italian method, which materially shortens the process. In this method all the partial subtrahends are omitted, and only the partial remain- ders retained in the working. Ex. Divide 108419716121 by 5788. 6783 ) 108419716121 ( 18748006 50589 46296 2121 3206 final rem. The first step is simply subtraction, giving 5058 for remainder. The work of the next step is as follows : 8 times 3 is 24 : 4 from 9 gives 5 (which put uo>,v_:> and carry 2. 8 times 8 and 2 give 66 : 6 from 8 gives 2 (which put down) and carry 61 8 times 7 and 6 give 62 : 2 from 5 gives 3 (put down) and carry 6. 8 times 5 and 6 give 46 : 46 from 50 gives 4 (put down.) It sometimes happens that one has also to be carried from the subtraction. For instance, in this case 0783)50581(8 4*17 40 EXAMINATION PAJPEBS. We say : 8 times 3 is 24 : 4 from 11 gives 7 (pufc down) and carry 3 (instead of 2). Then 8 times 8 and 3 give 67 : 7 from 8 gives 1 (put down) and carry 6, &c. Examination Papers. I. (1) Express in words, 4237496 ; and in figures, six hun- dred and fifty-three thousand eight hundred and twelve. (2) Find the sum of 24753, 88729, 4237, and 80462. (3) Find the difference between 86293 and 78464. (4) Multiply 8627 by 493, and 50042 by 307. (5) Divide 8423793 by 9, and 2659582 by 358. II. (1) Write in figures, twenty-five millions two hundred and fifty-seven thousand six hundred and thirty : and in words, 402050407. (2) From seventeen millions and seventeen take eight thousand and eight. (3) Multiply 6549 by 4037, and 27004 by 3700. (4) Divide 32456789 by 96, first by long division and then by short division, and show that the results agree. (5) Find the sum of one million and six, fifteen thousand and eleven, one hundred thousand and ten, and sixty thou- sand four hundred ; and divide the result by 9. III. (1) Write in words, 10010201401 ; and in figures, one million twenty-three thousand and one. Add together the two numbers, and from the sum subtract their difference. (2) Multiply 740296 by 2089, and 426004 by 3704. (3) Divide 78297426 by 35, employing short division. (4) From one hundred and twenty-six millions four hun- dred and six thousand and three take ninety-five millions and four. . (5) Divide the product of 723 and 347 by 48 IV. (1) Express in figures the number represented by MDCCCLXXXVIII. (2) Divide 987654321 by 132, using short division. EXAMINATION PAPERS. 4l (3) Berlnce to prime factors 56, 78, and 114. (4) Multiply the sum of 86297 and 40025 by the difference between 789 and 694. (5) By how many does one million exceed one hundred and one ? V. (1) Divide three hundred and fifty-three billions eighl millions nine hundred and seventy-two thousand six hun- dred and two by 5406. (2) Multiply 8976589 by 9876. (3) Resolve into elementary factors (L e. prime numbers' 40, 90, and 126. (4) Express in Roman Notation 24, 47, and 178. (5) How many bricks may be taken away in 24 carts, each taking 500 bricks ? VI. (1) Explain the method for the multiplication of two num- bers, each consisting of several figures, and multiply 30071 by 20590, explaining the reason for each step of the process, (2) Multiply 76894754 by 112756 in three lines of partial products. (3) By what number must the product of the sum and difference of 8376 and 5684 be increased so that the result may be exactly divisible by 7859 ? (4) A drover bought 527 sheep at $2 per head ; twice as many calves at thrice as much per head, 19 cows at $29 per head, and thrice as many horses as cows at four times as much a piece. How much did the whole drove cost him ? (5) One-half the sum of two numbers is 4331, and one-half their difference is 3353. Find the numbers. VII. (1) Bight head of cattle at $23 each, and 7 horses at $89 each, were given for 3 acres of land. What was the land worth per acre ? (2) It 18 men can reap a field in 76 days, how long will it take 19 men to reap the same field ? (3) A man bought an equal number of sheep and eows for $6300. Each sheep cost $3.50, and each cow $21.50. How many of each did he buy ? 42 EXAMINATION PAPERS. (4) It was found that after 789 had been subtracted 375 times irom a certain number that the remainder was 362, Find the number. (5) The ages of three brothers are 19, 17, and 15 years, and their father wills them his property worth $35,700 according to their ages. What does each get ? VIII. (1) There is a number which, when divided by 4 and the quotient diminished by 35 2 and the result multiplied by 10, and the product decreased by the difference between the arithmetical complements of 7846 and 3479 gives 883. Find the number. (2) If 5 Ibs. of tea are worth 15 Ibs. of coffee, and 4 Ibs. of coffee are worth 8 Ibs. of sugar, how many pounds of sugar are worth 75 Ibs. of tea ? (8) Find the number from which if 13675 be taken the remainder will be 45209 less 27045. (4) A horse is worth 8 times as* much as a saddle, and both together are worth $2t>l. Find the value of the horse. (5) A dealer in cattle gave $6400 for a certain number,, and sold a part of them for $3600 at $18 each, and by so doing lost $2 per head. For how much a head must he sell the remainder to gain $800 on the whole ? IX. (1) Any number may be multiplied by 5, 25, 125, &c., by annexing 1, 2, 3, &c., ciphers respectively to the number, and then dividing it by 2, 4, 8, &c. Explain the reason of this rule. (2) Of what number is 99995 both divisor and quotient? (3) A person bequeathed his property to his 3 sons. To the youngest he gave $1789; to the secund 5 times as much as to the youngest ; and to the eldest 3 times as much as to the second ; find the value of the property. (4) In walking a certain distance John takes 17694 steps; how many steps will James take in walking halt the dis- tance, John taking 3 steps for every four of James's ? (5) A merchant failed and his goods were worth $7770. Out of this he can pay his creditors 37 cents on the dollar. One of his creditors got $1998 as his share. Find the mej> chiont f B indebfceda*s, and what he awed tire one oreditot* HIGHEST COMMON FACTOR. 42 X. (1) In the multiplication of numbers, how do yon prove the correctness of the operation hy casting out the nines ? Explain and give reasons for the rule, and show the errors to which it is liable. (2) Multiply together 172814412 and 987654321 in three lines of partial products. (3) Simplify 1-2+ 4- 8 + 16- 82-f 64-128+256-512+ 1024 - 2048+4096 - 8 L9 2 + 16384 - 32768+65536 - 131072+ 262144 - 524288+1048576 - 2097152+4194304. (4) Divide 7864643457 by 9999. (6) The quotient is equal 6 times the divisor, and the divisor to 6 times the remainder, and the three together amount to 516 ; find the dividend. IX. On the Method of Finding the Highest Common Factor of Two or more Numbers. 52. A number is said to be a Factor of another num- ber, when the latter is exactly divisible by the former. Thus 8 is a factor of 12. A number is said to be a Common Factor of two or more numbers, when each of the latter is exactly divis- ible by the former. Tims 3 is a Common Factor of 9, 12, and 15. The Highest Common Factor of two or more numbers is the highest number which will exactly divide each of them. Thus 6 is the highest Common Factor of 6, 12, and 18, and 9 is the Highest Common Factor of 27, 36 and 108. The words Highest Common Factor we shall write briefly H. c. F. For small numbers the H. o. p. may be found by in- spection, and by way of practice the student may work the following examples, applying the tests of divisibility given in Art. 38. 44 HIGHEST COMMON FACTOR. Examples- Find the H. c. F. of (1) 8 and 14. (3) 40 and 60. (5) 48 and 144. (7) 15, 27, 105. (9) 16, 64, 256, 1024. (2) 12 and 30. (4) 36 and 90. (6) 7,14,21. (8) 32, 48, 128. (10) 24, 51, 105, 729. 53. In large numbers, the factors cannot often be determined by inspection, and if we have to find the H. c. P. of two such numbers, we have recourse to the following Kule : Divide the greater of the two numbers by the less, and the Divisor by the remainder, repeating the process until no remainder is left: the last Divisor is the H. c. P. required. Thus, to find the H. c. F. of 689 and 1573, we proceed thus: 689)1573(2 1378 195 ) 689 ( 3 585 104)195(1 104 91 ) 104 ( 1 91 13)91(7 91 Hence 13 is the H. c. F. of 689 and 1573. The reason of the above process depends upon the following proposition : A common factor of any two numbers is also a factor of their sum, of their difference, and of any multiples oj eitner of them. Thus, 7 is a common factor of 28 and 91 ; 7 is also a factor of their sum, 28-j-fll, or 119; 7 is also a factor of their difference, 91 28 or 63. HIGHEST COMMON FAOTOB. 46 Also, 7 is a factor of 5 times 91, and of any other multiple of 91. And 7 is a factor of 8 times 28, and of any other multiple of 28. Any number which is a factor of 689 and 1573 is a. factor also of their difference 195, and is therefore a factor of any multiple of 195 e. g. t 585, and therefore of 585 and 689, and therefore of their difference, 104, and therefore of 104 and 195, and therefore of their difference, 91, and therefore of 91 and 104, and therefore of their difference, 13, and therefore of 91 and 13, and therefore since 13 is a, factor of itself and 91, it is a factor of the given numbers 689 and 1573. Also, 13 is the Highest Common Factor of the given numbers, for it has been shown that any number which is a factor of 689 and 1573 is also a factor of 13, and since 13 is the highest factor of itself, it is the Highest Common Factor of 689 and 1573. In the preceding proof it may be observed that the quotient* are of no importance to the result. We are simply finding the difference between a certain number used as a dividend and a multiple of another number used as a divisor. This multiple, therefore, need not always be less than the dividend, and it will be sufficient to find the difference between the dividend and the nearest multiple of the divisor. Attention to this will some times shorten the labor. Thus in the preceding example, 195)689(4 780 91)195(8 182 13)91(7 91 Examples- (iii) Find the H. o. F. of (1) 884 and 1296. (2) 2272 and 8552. (3) 7455 and 47223. (4) 12821 and 54345. (5) 6906 and 10359. (6) 1908 and 2736. (7) 49608 and 169416. (8) 126025 and 40115. (9) 1581227 and 16758766. (10) 35175 and 236845. 54. If the H. c. v. of three numbers be required, we first find the H. o. p. of two of the numbers. Then the H. c. P. of this result and the third number will be the H. o. r. required. 46 LOWEST COMMON MULTIPLE. For example, if we require the H. o. p. of 851, 459, and 1017, we first find the H. c. F. of 351 and 459 to be 27, and then we find the H. o. p. of 27 and 1017 to be 9, which is therefore the H. o. p. required. Examples, (xxiv) Find the H. o. P. of (1) 16, 20, 28. (2) 14, 42, 56, 138. (3) 365, 511, 803. (4) 232, 290, 493. (5) 492, 1476, 1763. (6) 148, 444, 592, 703. X -Lowest Common Multiple. 55. A number is called the Multiple of another num her, when the former is exactly divisible by the latter. Thus 12 is a multiple of 8. A number is said to be a Common Multiple of two or more numbers, when the former is exactly divisible by each of the latter. Thus 12 is a Common Multiple of 2, 8 and 4. The Lowest Common Multiple of two or more num- bers is the lowest number, which is exactly divisible by each of them. Thus 12 is the Lowest Common Multiple of 4, 6 and 12, and 60 is the Lowest Common Multiple of 15, 20 and 80. The words Lowest Common Multiple we shall write briefly L. c. M. 56. To find the L. o. M. of two numbers we have the following Rule. Divide one of the numbers by the H. o. p. and multiply the quotient by the other number. The result is the L. o. M. For example, to find the L. o. M. of 24 and 86. The H. o. P. of 24 and 86 is 12. Now 24-J-12 =2. .*. L. o. M. of 24 and 86 = 36x272. NOTE. The symbol .'. stands for the word therefore Since 12 is then. o. F. of 24 and 36, then 24 == 12X2 and 36 = 12X3, also 24X36 = 12X2X12X3, and obviously the least com- LOWEST COMMON MULTIPLE. 47 mon multiple of the two numbers will consist of the product of all the prime factors in the two numbers ; or the least common 24X3fi multiple of 24 and 36 = 12X3X2 or ^~ = 72 ' - And there is no integral number less than 72, which is a mul- tiple of 24 and 36. For 72 contains 24, 3 times, 36, 2 times, and 3 and 2 being prime to each other : Wherefore the L. o. M. of 24 and 36 = ^1 or the least common multiple of two numbers, is equal to their product divided by their highest common factor. The following form is perhaps, more convenient in practice. 24X36 36 L. C. M. of 24 and 36 =~Y2~ = 24x The i. o. M . of two numbers is equal to the product of either of the numbers multipl ed by the quotient arising from dividing the other by their highest common factor. Examples, (xxv) Find the L. C. M. of (1) 27 and 54. (3) 633 and 844. [5) 1000 and 213*. 7) 936 and 2925. [9) 2443 and 4537. (2) 88 and 108. (4) 195 and 735. (6) 3432 and 3575. (8) 2304 and 4032. 57. To find the L. o. M. of three or more numbers, we might find the L. c. M. of any two, and then find the L. o. M. of the resulting number and of a third of the original numbers, and so on, tjje final result being the L, c. M. required. T?MIS to find the L. o. M. of 12, 20, 36 and 54, we mig- nroceedthua: the L. o. M. of 12 and 20 is 60, of 60 and 86 is 180, of 180 and 54 is 540; . . the L. o. M. of 12, 20, 36 and 54 is 540. But in practice it is generally more convenient to pro- ceed by the following Rule. Set down the given numbers side by side ; divide by any number, commencing mth 2, 3, 5 ... which will exactly tW9 M tout (6) < 8 > (9) MHVy^Wr 74. Having now established the elementary rules for operations performed with fractions, we proceed to notice some other points belonging to this branch of Arithmetic. 75. A whole number, or integer, can be written as a fraction, by putting 1 beneath the number as a denomi- nator : thus 6 may be written as a fraction, thus -f. Also, since f = ^ = \ 5 = 2 and so on, it is clear that we can represent a whole number by a fraction, whose denominator is any whole number we please to select. 76. A Mixed Number is a number made up of an integer and a fraction, as 4. This may be read thus, four arid two-sevenths, and must be regarded as the mm of 4 and f . A mixed number can be brought into the form of an improper fraction, by multiplying the integer by the denominator of the fraction, adding to the product the numerator of the fraction, and making the sum the numerator of a fraction, of which the denominator is the denominator of the original fraction. Thus 4 = 3o Conversely, an improper fraction can be reduced to a mixed number, by dividing the numerator by the de- nominator, sotting down the quotient as the integral part, and making the remainder the numerator of the fractional par' of the mixed number, the denominator being the den:, ainator of the original fraction. Thus V = ^ FKACTION3. 61 For y = JLL* Examples- Convert into improper fractions (1) 7 (2) 23^ (3) 216TJ (4) and into mixed numbers 1*8* 77. The rules for the Addition, Subtraction, Multi- plication, and Division of Fractions are applicable to Improper Fractions. 13 v a 7 _ 1 3_* 2JT _ 13X9X3 _ _ 11 ? * TJ 8 9x26 - 9X13X2 7 ~ ' X T 1= m x 78. In the application of the rules to Mixed Numbers, we ?.ay in all cases change the Mixed Numbers into Improper Fractions, and proceed as in the foregoing Examples. In Division we must proceed thus. For example, 16*12*= V + V = V x A =i = ij. In Multiplication it is usually the best course : thng 7| x 5 1 == y x 3 7 9 = 2 A X J_3 __ 2 9_9 __ 42 s In Addition it is often advantageous to proceed thus 025 FRACTIONS. and, similarly, when three or more numbers are to be added, we may separate the fractions from the integers, and make a distinct operation for each class. In Subtraction we can employ the same method, but a little care is necessary. Suppose we have to take 3f from 4| Keducing the fractional parts of the numbers to equi- valent fractions with a common denominator, we have 8tf and 4ft We can now take the integral part of the first num- ber from the integral part of the second, and the frac- tional part of the first from the fractional part of the second, and we have 4tt-8Jf=l*- But suppose we have to take 3-f from 10J Since f = ff and f =& f is greater than ^ and we cannot take away the fractional part of 3f | from the fractional part of 1 ()-$. We escape from the difficulty by the device of adding unity to each expression, to 3f j[ in the form of 1, and to 10f in the form of . Thus 10J*-8ffc = lOjf -4|* = 6|*. Take another illustration of a practical nature. From 5jd. take away 3fcZ. "We add four farthings, i.e., f df a penny, to the for- mer sum, and 1 penny to the latter, and reason thus : 6d. -3fd. = ofd. -4fd. = Ifd. = IJd. Examples- (xxxv Simplify the following fractions : (1) 4f*8J V 2) 8J-S-.6I (3) (4) 6fx9| (5) 14x8/r (6) (7) 2i + 8J (8) (9) i6{+4J + l7i| (1C) (11) 14|-6| (12) FRACTIONS. 63 The following examples should be carefully noticed. I. From 17 take 4-/J-. 17-4^=16 + 1 4/ r = 16 4 + 1 ^i. = 12 + H II. From 317 take -A, 317 -A- = 310 -f 1 -A = 316 + U = 3 III. Multiply ^ by 397. Since T \/ = 1 ToW N 397 X -rYA = 3 97 - tWo = 396 + 1 = 396 + AV-o = SOGrV^. 79. A COMPOUND FRACTION is defined to be the frac tion of a fraction. Thus | of f , and | of 2 of 5f are compound fractions They are reduced to simple fractions by the process of Multiplication. Thus f of 2i of 5f = | x f x ^ = l$im = f ? = 81?|, 80. A COMPLEX FRACTION is one, of which the Nume- rator or Denominator is itself a fraction or mixed number. 32 4 Thus - 3; and ~j-g are complex fractions. They are reduced to simple fractions by the process of Division. I I Thus f = J ^ 7tf + * = | X * A and | = 2 -- = i x I - V- = 3f.^ t Examples, (xxxvi). Simplify the following fractions : (1) f of 5 J- of 7^, (2) 4f of 11| of 13. (3) I of 21- of 2^ of 90. (4)| (5) -|| f *> % > i? 64: FRACTION'S. THE HIGHEST COMMON FACTOR AXD THE LEAST COMMON MULT.PLE OF FRACTIONS. 81. The H.C.F. or L.O.M. of fractions can be readily found by considering that the denominator is simply the name of so many units represented by the numerator. No difficulty is ever experienced in finding the H. C. F. or L. C. M. of $12 or $16, or of 12 apples and 1G apples. In fractions the name is written under the number representing the collection of units of that name. Thus to find the H. C. F. of ^ ;- acd J-J, proceed as in whole numbers ; find the H. C. F. of 12 and 16, which is 4, and call it by its name, which in this case is thirty- sixths. Hence the H. C. F. is ^ Similarly to find the L. C. M. of J* and JJ, find the L C. M. of 12 and 16, which is 48, and call it by its proper name. Hence the L. C. M. is J|. Hence to find the H. C. F. of fractions we have the following rule : Change them to others having the same naive or denoini- inalor, and find the II. C. F. of their numerators. This placed orer t1u>, common denominator iriU be llic Li. C. I 1 ', of the f factions. To find ilie L. C. M. of fractions : Change them tc others harimj a common denominator, an- 1 find the L. C. AT, of the numerators. '1 ILLS ])l<-e.d orcr the common denomina- tor wi.'t be the L. C. M. of the fractions. The foyowing is somewhat shorter : Find the L. C. H. of the iimiirnttors, and under thin place the II. C. F. of tJte denominators of the /ructions. Hie resulting fraction will be the L. C. 31. required. Examples (xxxvii). Find the n. c. F. of the following' fractions : (1) I find f . (2) ft and (ZytSk 4fr and CJ. (4) | ? -H 41 and 5J. ON THE USE OF BRACKETS. 66 Find the L. o. M. of the following fractions : (5) | and | (6) 2* and 7* (7) 4* 5| and 8& (8) i of 2f of ~ and f of-^ of 2* ON THE USB OF BRACKETS. 82. When an expression is inclosed in a bracket ( ), it is intended to show that the whole of the expression is affected by some symbol, which precedes or follows the bracket. Thus 24 X (3$ -J-7J) means, that 24 times the sum of the numbers 8^ and 1\ is to be taken, which we may effect by combining 3 and 7^ by addition, and multiplying the result by 24. Again, 2| -i-(4f-2) signifies, that 2 is to be divided by the difference between 41 and 2 ; and therefore the result will be 2* -2* orV-l oryxf or -J And, generally, we may say, that when numbers are included in a bracket, the expression, within the bracket, must be Brought into the simplest form, before combining it with expressions, not in the bracket. 83. The methods of denoting a bracket are various; thus, the marks [ ] and { } are oltea employed. Brackets are made to iuclooe one anoUicr, as in the ex- pression BHK [24-8-*- { 4+^(24- j)[] In removing such brackets it is best to commence with the initertiiutt, and to remove the brackets onti by one, thus =8-*- [2 +3-5- V 3 I ON THE USE OF BRACKETS. We have worked out this example at length because it will teach the learner how to simplify with neatness a peculiar class of fractions called Continued Fractions, which appear in a form like the following : 1 This fraction, by the aid of brackets, may be repre- sented thus, and then we can simplify it by the gradual removal of the brackets, the final result being / T . 84. There is another method of simplifying Complex and Continued Fractions, which we may explain by the following examples : Ex. (1) To simplify JL. Multiply all the terms of the fraction by 7, and it becomes T4+T or Y7- Ex. (2) To simplify ,JL-. Multiply the terms by 30, and we get 20 nr oo 150 + 9 r T5?' Ex. (3) To simplify f-^-|- Multiply all the terms by 42,. and we get Ex. (4) To simplify _ 195 195 + 28 "~ 2-3 ON THE USE OF BRACKETS. 6? To simplify 1 + 1 5 TTT Examples (xxxviii). Simplify tlie following fractions : 2 -4- 85. If two brackets stand side by side, with no sign between them, as (|4-|) (| |) it is implied that the c intents of one bracket are to be multiplied by the con- tents of the other. The following examples are selected as containing peculiar forms of symbolic' representation, which should be carefully noticed. (1) ! + I of 1 - T V The first step here is to take the product of $ and J, so that the expression bacomes | 4- 2 8 i T 8 o i then add the first two fractious together, aud take T * B from the urn. 68 MISCELLANEOUS EXAMPLES IN FRACTIONS. (2) ! + t of (i - T " ). l r ir.st ttke the difference of | and , s d , multiply the ro~ult by t, and add the product to |. (3) 5XS-5-? --. First simplify X 5 , the result being 4 . Then divide I s * by 5, the result boiug^xj or ||. ) i -MX?. l r ir.st simplify S -f- 2, the result being ! XI, or {. Then multiply j by ?. tlie result being $* Examples, (xxxix). Simplify the followiug expressions: (1) 81 ^-(2^4- If) (2) (4 T 3 r (3) 1+ 6 (4) (v) iofS + 2-^-J (8) (9) (S-i s i)(2?+31) (10) ( 88. We shall conclude this Chapter with a S3fc of Miscellaneous Examples oil Fi'actions. Examples, (xl). (1) Add together I 7 "5 8 Z^I v l !> S3 If -4? r 2S iff (2) Add | of f to | o and multiply the result by (3) Subtract | of f from 1 J of J and divide the result by MISCELLANEOUS EXAMPLES* IN FRACTIONS. 69 (4) Simplify the fractions l^ 7 g 2 sV/ and fiud tbeir product. (5) Divide the product of 3| and 32 by the produ of 14 aud If?. (C) Multiply together the fractions of 4 2| and add the result to (7) Multiply the difference between |j and|gj- by the sum of 4 T 7 y and 1 1 ; and multiply the result by the difference between 10 and 51 . (8) Simplify 20} (i+) (9) Simplify divided by . (10) Simplify *+*?) (11) Simplify divided by (*I (12) Simplify If (18) Simplify -i 4 1 ~ '* i- 2 -T^r 70 MISCJELtANEOUS EXAMPLES IN F&ACT1ON& (14) Simplify (15) Simplify (16) Simplify 1 1 5 ~5^ 6 +6^1 , x 2 9 g- of 7 and - ~ + lOf 3- L 4- -J_ -* 4-i (17) Simplify (18) Simplify jTX5Hx6A+6j 9 rXl?J4.2^ + Ug l--5 3AA-- Simplify A of 6|f of 24il-4|| x 3^ ^ 337 (20) Simplify 19 7 x s - if (21) Simplify (22) Simplify 1 -18 "~t ' ' 1-1 6-4 |-|: 2 x EXAMINATION PAPERS. 71 (23) Simplify 2 Examination Paper- L ^1) Explain how to reduce a mixed number to an im- proper fraction and show the reason for each step. (2) Bought 18| yards of silk at $2 a yard, and 27 Ibs. of cheese at $/o per Ib. ; how much money did I spend ? (3) How many times does the sum of 12*- and 8| contain their difference ? (4) B who owns ft of a ship, sells f of his share for $34300 ; what is the ship worth ? (5) There are two numbers whose sum is 4|- and whose difference is 2f ; find the numbers. II. (1) What is meant by expressing one number as the frac- tion of another ? Explain how to express 3^- as the fraction (2) How may the relative magnitude of two or more frac- tions be compared? Arrange the fractions fj, -ft-, 3$, 3$, in the order of magnitude. (3) Add together f> -fa, and -Hh^ ftd nn( l what is the least fraction with denominator 1000, which must be added in order that the sum may be greater than unity. 2 -4- 5 ' (4) Show that the value of +~f lies "between f and f. (5) A ship and her cargo are valued at $60,000, and f of the value of the ship is equal to of the value of the cargo ; find the value of each. III. (1) Define Numerator and Denominator, and explain why they are appropriately applied to the terms of a fraction. (2) If f of f of 2 bbls. of flour is worth $7, what is the value of 2-. 2 t - bbls. ? (3) If any number of fractions be equal, then any of thso is equal to the fraction whose numerator is equal to the sum 72 EXAMINATION PAPERS. of all the numerators, and whose denominator is equal to the siun of all the denominators. Exemplify this in the case of six equal fractions. (4) Add together $, , |, and , and subtract the sum from 2 ; multiply the difference by $ of of 88, and find what fraction the product is of 999. (5) A's age is ^ of B's, and B's is -f of C's, and C 12 years ago was 72 ; what are their respective age* ? IV. (1) Before adding fractions together, why is it necessary to change them to others having the same denominator ? (2) What number must be taken from 17. V so that it may contain 3f an exact number of times ? (3) There is a number which divided by 8yV and the 2i quotient increased by 2$ and the sum multiplied by ~o~, and the result diminished by of f of 14,} gives 2f. Find the number. (4) A bought a horse and carnage for $225, and paid for the harness y^- of what he paid for the horse. The carnage cost I of the value of the horse. What was the price of each? (5) Divide $8888 among A, B, and C, so that A may re- ceive $88 less than 3 times B's share and C $176 more than one half of A and B's shares. V. (1) Explain each step in the process of reducing a complex fraction to a simple one. (2) Simplify 3 X 3 X 3^ - 1 divided by 3>- X 3 J- - 1. (3) What is the smallest sum of money with which A can purchase sheep at $4^- each, calves at $5 each, or pigs at $2. reach; and how many* of each can be bought with this sum? (4) John spent $80 less than of his money at one time, and at another $40 more than of the remainder and now has $40 left. How much had he at first ? 2 3 (5) One fourth of -=- of the length of a pole is in the mud ; two-thirds of the remainder is in the water and there are 5 feet in the air ; what is the length of the pole ? EXAMINATION PAPERS. 73 VI. (1) Show that if-f-f = X (2) Find three fractions whose numerators shall be 3, 5, 7, respectively, and their sum equal to unity. (3) From the sum of 3ij and 4^ subtract 6, multiply the difference by 2, and divide the product by 4}. (4) A sold a watch for -J more than it cost him to B, who sold it to C for $33, which was less than it cost him. What did the watch cost A ? (5) There are three rooms 21 f, 1SJ, and 111 feet long respectively. Find the longest plain ruler with which the three rooms can be measured. VII. (1) Give a definition of multiplication that will apply to fractions. (2) A person dies worth $40000, and leaves i of his property to his wife, to his son, and the rest to his daugh- ter.* The wife at her death leaves f of her legacy to the son and the rest to the daughter ; but the son adds his for- tune to his sister's and gives her ^ of the whole. How much will the sister gain by this ? and what fraction will her gain ba of the whole ? .(3) One half of a population can read ; I J- of the remain- der can read and write ; -& of tha remainder can read, write aiivl, cipher, while the rest, 24'5G')0, can neither read, write nor cipher ; whut is the population. (4) Three men, A, B, C, run round a circle in 5, 6, and 1\ minutes, respectively. If they start from the same point at the sam.3 time and run in the same direction, how long will they run before they are all together again ? and how often will each havo gone round it ? (5) A owned of a ship, and sold | of hn share to B, who sold 4 of what he bought to C, who sold -j 9 ,- of what he bought to D ; what part of the whole ship did each now own ? VIII. (1) What lira the advantages in arithmetical operations of employing fractions expressed by the smallest number possible? State how fractions expresse 1 by large numbers may be reduced to equivalent fractions expressed by smaller auinbers. Is this always possible ? 74 DECIMAL FRACTIONS. (2) Is -fa more nearly equal to or to o\ - 2S 4- gj- of 2 J- - 1-f and by liow much ? (3) Of the sovereigns who have reigned in England since the Norman conquest, there are -&th of one name, -jths of another, ft of another, of each of two others, and ft of each of tliree otliers, and there are 5 besides; find how many sovereigns have reigned in England since the con- . quest. (4) Three horses start from the same point, and at the same time, upon a race course 300 rods in circuit; the first- horse passing over ^ the circuit, the second f , the third -, in a minute. In how many minutes will they all be together again, and how far will each have travelled ? (5) Divide the difference of 13.\ ~ {(2f-2ft) X If } and 13&-5- (2f-2ft) X If by 18fr -5-2*- 2ft X If. . XII. Decimal Fractions. 87. The multiples of 10 are 10, 20, 30, 40, 50, and so on. (Art. 89.) The POWERS of 10 are 10, 100, 1000, 10000, and so on, and these are called the first, second, third, fourth powers of 10. (Art. 27). 88. A Fraction, which has for its denominator one of the POWERS of 10, is called a DECIMAL FRACTION, or for shortness sake, a DECIMAL. All other fractions are, by way of distinction, called VULGAR FRACTIONS. 89. To save the trouble of writing the denominators of decimal fractions, a rneth >d. of notation is used, by which we can express the value of the denominator in every case. This method will be best explained by the following examples : 3 stands for ft, and is read thus, three-tenths. 25 stands for ft%-, and is read thus, twenty-jive hundretlis. 347 stands for -ftV?r an d is read thus, three-hundred and forty-seven thousandths. The figures which follow the Point are those which form the Numerator of the fraction in each DECIMAL FACTIONS. 75 The number of the figures, which follow the Point, corresponds to the nurnher denoting the particular Power of 10, which forms the Denominator of the frac- tion in each case. Now, as the first power of 10 is 1 followed by one zero, and the second power of 10 is 1 followed by two zeros, and the third power of 10 is 1 followed by three zeros, and so on, we can in every case write the denomi- nator, by affixing to 1 a number of zeros, equal to the number of figures that follow the Poiut. Thus, -426789 stands for - f VoVo a oV six zeros being affixed to the 1, because the number of figures that follow the Point is in this case six. Again, 07 stands for r ^ 005 stands for 00025 stands for the zeros, which come between the Point and the fig- ures 7, 5, and 25, not being set down in the numerators of the fraction, as having no effect on the value of the numerators, seeing that 07 arid 7 stand for the same number, and.that 005 and 5 stand for the same number. But these zeros affect the value of the denominators, as for instance .7 = T V while -07 = ^ and .007 = T ^ 90. Zeros affixed to a decimal have no effect on its value : that is, 7, -70, -700 are all equal : . for -7 = T V 7O 70 _ 7 '" T0-793 CIQ} \ '/ T"D"oo"oiJ7 \- LU / Tvinytro \ x ^/ 93. We call 5, 3'7, 15-9 decimal expressions of the first order, -25, 4-39, 143-73 decimal expressions of the second order, 043, 5-006, 27-009 decimal expressions of the third order, the number of the order depending on the number of figures that follow the point. ADDITION OF DECIMAL FRACTION*. 77 The number denoting the order we call tlie INDEX oJ the order : thus 1 is the index of the first order, 2 of the second order, and so on. 94. From what if stated in Art. 90 we learn that a decimal of any order may be made into an equivalent decimal of a" higlwr order, by affixing one, two, three zeros according as the index of the higher exceeds the index of the lower by 1, 2, 3. Thus -43 may be made into an equivalent decimal 01 the fifth order, by affixing three zeros, thus, -43000, and '047 may be made into an equivalent decimal of the order, by affixing four zeros, thus, '0470000. ADDITION' OF DECIMAL FRACTIONS. 95. To Add -27 to -45 we might proceed thus, But we obtain the same result, if we set down the decimals one under another, Point under Point, add the figures as if they stood for whole numbers, and place the Point in the result under the other Points, thus, 27 45 96. IT the decimals to be added be not of the same order, as for instance -87 and '049, we reason thus : 049 is a decimal of the third order, 87 is a decimal of the second order, but it can be made into an equivalent decimal of the third order, by affix- ing a cipher, thus, -8?0. Then we proceed to add the decimals thus : 370 049 419 78 ADDITION OP DECIMAL. FRACTIONS, Now suppose we have to add more than two decimal expressions, as -0074, -72, -05, and -123456. Of these four expressions the last is of the sixth order, and we may make the other three into equivalent deci- mals of the sixth order, and set them down thus : -007400 720000 050000 123456 900856 When the learner is thoroughly acquainted with the principle, on which this process of addition depends, he may omit the affixed zeros, since they have no effect on the result, and may write the sum just worked out in the lollowing way : 0074 72 05 123456 900856 If the numbers to be added be made up of integers combined with decimals, we keep the Points in a verti- cal line, and proceed as in addition of integers. Thus to add 4-27, 15*004, -9007, and 23, we proceed thus : 4-2700 or thus, 4 '27 15-0040 15-004 9007 '9007 28-0000 23- 48-1747 48-1747 Examples* (xlii.) Find the sum of (1) -275 and -425. (2) '007 and -2894. (3) -001 and '0002. (4) 13-279, 8-00046, 742-000372. (6) '000493, 8-24, 15, 42'6, 824-42037- SUBTEACTION OP DECIMAL FBACTIONS. 79 (6) 49-327, '458, 8317-05, 341-875, 82-4962. (7) 700-372, 894-0009, -347, -00082, 5370-006. .(8) 660-379, -45687, 350-0036, 7'074, 52*257. SUBTRACTION OF DECIMAL FRACTIONS. 97. H we have to find the difference between -47 and *85, where both decimals are of the same order, and 47 is the larger of the two, we proceed thus : From -47 Take -85 Result -12 performing an operation like that of Subtraction of In- tegers, and keeping the Points in a vertical line. That this method gives the correct result is evident, for 98. If we have to find the difference between -888 and -9, we may make the latter into a decimal of the third order, thus, -900, and since this is larger than 888, we proceed thus : From -900 Take -888 Result -012 If we have to find the difference between *998, and 1, we ebserve that 1, being an integer, must be greater than -998, which is a Proper Fraction, i. e. we proceed thus : From 1-000 Take -998 Result -002 Examples, (xliii) Find the difference between (1) 66-429 and 5-218. (2) 9.005 and 7 '462. (3) 68-316 and 5*0867. (4) '799 and -8. (6) 6-047 and 5 '9863. (6j, 850-007 and 270-8796. (7) -0000086 and -00001. (8) -00537 and -000985. (9) 10 and -0002. (10) -09999 and- 101. SO MXTLTlPLIOA-nON OF DECIMALS. MULTIPLCATION OF DECIMALS. 99. In finding the product of -12 and -11, we might proceed thus, = -0132, the result being a decimal of the fourth order. Again if we have to find the product of 4*32 and 00012, 4-32 x -00012 = *8t x ^5^= yw*i**inr = '0005184, the result heing a decimal of the seventh order. And, generally, the product of any two decimal ex- pressions is a decimal expression of an order, whose index is the sum of the indices of the orders of the two expressions. Hence we deduce the following rule for Multiplication of Decimals : Multiply as in the case of integers, and mark off in the product a num/ier of decimal places equal to the sum of the number of decimal places in the two factors. For examole, to multiply 2-4327 by 4-23. 2-4827 4-23 72981 48654 97308 10-290321 , to multiply 43-672 by -00000047. 48-672 -00000047 305704 174G88 2052584 We have now to mark off eleven decimal places from this product, and as the product contains only seven figures, we must prefix, four zeros, and put the Point on DIVISION OP DECIMALS. 81 che left of these, thus -00002052584 and this will be the required product. One more case must be considered. Suppose we have to multiply -235 by '48 ; 235 43 1S30 940 11280 This decimal of the fifth order is equivalent to a deci- mal of the fourth order -1128 (Art. QO), and this is the simplest form of the result. Examples, (xliv). Multiply (1) 7-5 by 4-7. (2) 3-G2 by 5-23. (3) -427 by -235. (4) -502 by -00074. (5) 3-00704 b>r 4.0205. (6) -GOOD by 1000. (7) 623-4075 by 24-0259. (8) -00740 by -OOG23r,. (9) 1432-0749 by -00004030705. (10) 50704-042 by -004007090061. Find the value of the following : (11) -407 X 4-03 X -006. (12) 1-01 X 1000 x -001. (13) -52 X .007 X 4-3 X '02. Find the continued product of (14) -07, 4-6, -009 an'l 52-47. . (15) 42-3, -7%, 4-03 an.l -00074. (16) What is the C'ibe of 2-74 ? (17) liaise 3'5 to the fourth power. X OF DECIMALS. 100. If we have to divide -27 by 8, we might proceed thus, 27 -4- 3 = -n/0- ~- 3 = TO T = '09. As;ain, if we have to divide -OOG25 by 25, we might proceed thus, 00625 -s- 25 - TT&8 OTT -5- 25 - -nrJSrnr - '00025. 82 DIVISION OP DECIMALS. In both cases the Quotient is a decimal of the same order as the Dividend. Hence we derive the following Kule : If the Divisor be an integer, perform the operation oj Division as if the Dividend were also an integer, and 'mark off in the Quotient as many decimal places as there are Decimal places in the Dividend. For example, suppose we have to divide '0086751 b\ 243. 243) -0086751 (357 729 1385 1215 1701 1701 The Quotient is to be a decimal of the eighth order, .-. the result is -00000357. 101. Next observe that, if the Divisor be a decimal ex- pression, we can in every case change it into an Integer, by a process which we shall now explain. If we multiply a decimal expression by 10, the effect is to move the Point one place to the right, by 100, the effect is to move the Point two places to the right, by 10000, the effect is to move the Point three places to the right, and so on. For instance, 128-456x10 = 1234 '56, and 123-456x100 = 12345.6. The reason is obvious, for 123 -456 X 10='?o?o* x 10 = L T-^ = 1234-56, and 123-456 X 100 = y- X 100 = = 12345-6. Hence we can transform any Divisor into an Integer, by multiplying it by 10, 100, 1000, .... according as the Divisor is a decimal of the first, second, third .... order DIVISION OF DECIMALS. 88 For example, if the Divisor be '000492, and we mul- tiply it by 1000000, we transform it into the Integer 492. Now we may multiply a Divisor by any number, if we multiply the Dividend by the same number. For instance, if the Divisor be 8 and the Dividend 82, we may multiply each by 10, so that the Divisor becomes 80, and the Dividend 820 ; and whether we divide 82 by 8, or 820 by 80, the Quo- tient will be the same number, that is, 4. 102. We can now lay down a general Kule for Divi- sion of Decimals. If the, Divisor be a decimal, change it into an Integer by removing the Point a sufficient numbw of places to t/ie right, and also remove the Point in the Dividend ilie same .number of places to the right. Divide as in tlie case oj integers. T/ten, if tJie dividend be an integer, the quotient will be an integer, ar 2*9 by '007. DIVISION OF DECIMALS. 89 106. If we continue the division further in the Exam- ple given in Art. 105, we find the figures 756 coming again and again in the same order in the Quotient, so that the Quotient is 6-6756756756 . . . without any ter- mination. Let us now take this example. Divide 90 by -0011. Here 90 + -0011 == ^ = ^fj-^ 11)^00000 81818 Up to this point the Quotient is an Integer : bnt, if we proceed further with the division, we shall obtain a decimal expression : thus, if we affix two more zeros, preceded by a decimal point, to the dividend, we shall have ii)_90oqqo;oo 8181818 If we carry on the division to any extent we shall have the two figures 18 coming again and again in the same order. A decimal of this kind is called Periodic, Circulating, or Recurring. 107. The extent of the Period is denoted, by placing a dot over the./w*s$, and another dot over the last of the figures in it. Thus -i8 denotes a decimal of an order such that it can be represented by no finite index, since it runs on 18181818 ... to an infinite number of figures. So also, 6-756 stands for 6-766756756 .. ., 047 stands for -047047047 4872 stands for -43728^2372 26-0479 stands for 26'04797979 00026 stands for '000266666 108. A Vulgar Fraction may be converted into a Decimal Fraction by the following process Reduce the fraction to its lowest terms, and then find the Quotient resulting from the division of the numera- 9O DIVISION OF DECIMALS. tor by the denominator, by the rule for division of deci- mals. Thus to reduce to a decimal, we proceed thus : 8) 3-000 ~W5 .: f = -375 Again to reduce ^ to a decimal, we proceed thus : 82)47-00000(1-46875 32 150 128 220 192 280 256 240 224 160 160 ... ii = 1-46875 Or we might work by Short Di^isio^ thus : 47-00 11-75 1-46875 Again, to reduce | to a decimal, we prtMseed thus - 7 I 1-00000000 , -14285714 . .-. | =-142857. 109,. To shew that, when a Vulgar Fraction ia reduced to a decimal, either the operation must terminate or tht ngures of tfte Quotient must recur in the same order. Consider the operation by which su*U a fraction as | is reduced to a decimal. The only remainders that can DIVISION OF DECIMALS. 91 occur are 0, 1, 2, 3, 4, 5, 6. If the remainder should occur, the division terminates : if not, we can only have six different remainders, and when any of these occurs a second time, we must have a recurrence of the former remainders in the same order. When a fraction in its lowest terms is reduced to a decimal and produces a recurring decimal, the extreme limit of the number of places in the period of the recur- ring decimal is one less than the denominator. Thus -f produces a recurring decimal of 6 places. !~ 9 produces a recurring decimal of 18 places. 2^9 produces a recurring decimal of 28 places. 110. When a Vulgar Fraction is in its lowest terms it can only be expressed as an Exact Decimal when the denominator is composed of factors each of which is one of the numbers 2 and 5. Thus f can be expressed as an exact decimal because 8 = 2x2x2. 2 3 o can be expressed as an exact decimal because 20 = 2x2x5. y-f-g- can be expressed as an exact decimal because 125 = 5 X 5~x 5. The reason for this is, that no vulgar fraction can be expressed as an Exact Decimal, unless it can be trans- formed to one which has 10, or some power of 10, for its denominator. Now no number can by multiplication be made a power of 10 unless it be composed of factors each of which is 2 or 5. Thus 8 can be made into a power of 10 by multiplying it by 5 x 5 X 5. 125 can be made into a power of 10 by multiplvin it by 2 x 2 x 2. 40 can be made into a power of 10 by multiplying it by 5 x 5= Hence f = ^r 2 = ~^|^|_ = ^ = , 375 7 7 7x2x2x2 __ f\ f* T25~ 5 X 5 X 5 = 5 x 5 X5X2X2X2 T"0~0~0~ == "U>O 9 _ 9 9X5X6 ,,, oor 1 **-2X2xlO = 3x2xl0x5X 5 = T Wtf= <22 5 92 CONTRACTIONS OP DECIMALS. But such numbers as 7, 12, 80, cannot be made into powers of 10 by multiplication, and hence ? & ^J can- not be reduced to exact decimals. It may also be remarked that, when a Vulgar Frac- tion in its lowest terms is reduced to an exact decimal, the order of that decimal is expressed by the greatest number of times that either of the factors 2 or 5 occurs in the denominator. Examples, (xlix) Convert into decimals the following vulgar fractions : (!) A ( 2 ) U W * (4) & (5) F^ (6) A (7) H (8) & (9) CONTRACTIONS IN MULTIPLICATION AND DIVISION OF DECIMALS. 111. When the number of decimal places is great the .figures obtained by the ordinary mode of multiplica- tion are often unnecessarily numerous. Thus, in mul- tiplying 62-37416 by 2-34169 by the oijdinary method, there would be ten places of decimals in the product, while for all practical purposes three or four are quite snough. Ex. Multiply 62-37416 by 2-34169 so as to retain >nly 4 places of decimals. ORDINARY METHOD. CONTRACTED METHOD. 62-37416 62-37416 2-34169 96 143-2 56 136744 1247483 = 623741 X 2 -f 1 374,24496 187122 = 62374 X 3 6237416 24950 = 6237 X 4 -J- 2 24949664 624= 623x1 + 1 187122 48 374 . 62 X 6 -j- 2 12474832 56= 6X9 + * 146 0609J467304 146-0609 OONTBACTIONS OF DECIMALS. By comparing the contracted method with the ordi- nary method, the reason of the preceding operation will be readily understood. Since the product of any order of units hy units is of the the same order as the figure multiplied, the units figure of the multiplier is written under the place to be retained. For convenience, the other figures are written in an in- verted order. Now (Art. 99) 4, a decimal of the third order multiplied by 8, a decimal of the first order, will give a decimal of the fourth order ; also, 7, a decimal of the second order, multiplied by 4, a decimal of the second order, will give a decimal of the fourth order, etc., etc. Now, to the product of 2 and 1, one must be added: since, if 6 had not been rejected, there would have been one to carry ; then the other figures are multiplied in the usual way. Next, multiply 4 by 8 and set down 2 under the 8, and multiply the other figures by 8 in the usual way. Next, multiply 7 by 4 and to the product add 2 : since, if 416 had not been rejected the product would have approximated to 2 thousand, etc. Hence we have the following rule : Write the Multiplier with the order of its figures reversed under the Multiplicand, so that the unite figure may be under that figure of the Multiplicand which is the lowest decimal to be retained in the Product. Then mul* tiply by each figure of the Multiplier, neglecting all the figures of the Multiplicand to the riyht of it, eoccept to find what is to be carried, and carrying one more when the re- jected part of any product is 6 or greater than 5. Arrange the partial products so that their right hand figures may stand in the same vertical column. Their sum will be the product required. Ftom this product cut o/ the desired number of decimal places. 112. When the divisor consists of several figures, the work will be much shortened by cutting off a figure from the divisor at each successive step of the division t instead of annexing a figure to the dividend. Care must be taken to increase each product by what would 94 CONTRACTIONS OF DECIMALS. have been carried if the figure or figures had not been cut off. Ex. (1). Divide 8-784169 by 2-716418 correct to three places of decimals. 2716418) 3784169 ( 1893 2716418 10(37751 814925 252826 244478 8348 8149 199 By comparing the units of the highest order in the divisor with the units of the same order in the dividend, it is evident that there must be one figure to the left of the point in the quotient ; hence the answer is 1'393. Ex. (2). Divide 763-14163 by 21-3642 correct to four places of decimals. ORDINARY METHOD. CONTRACTED METHOD. 213642 ) 76314163 ( 357205 21 ??* 2 ) 76314163 ( 357205 640926 640926 122215 6 122215 100821 439 427 106821 15394 63 15394 1495494 14955 690 439 284 427 12 40600 12 1068210 11 1172390 1 Here the figures of the quotient are 857205, and Joy jomparing the 2 tens of the divisor with the 76 teaa of BBOUBBINO DECIMALS. 96 the dividend, it is plain there must be 2 places to th left of the point ; hence the quotient is 85-7205. From considering these cases we have the following rule : Compare the left-hand figure of the divisor with the unite of the same order in the dividend, and thus deter- mine the position of the point in the quotient. Then divide as in Ex. (1 ), dropping a figure from the right oj the divisor at each step of the division. NOTE. Care should be taken to mark the figures dropped by placing a dot or other mark beneath them. Examples- (1) (1) -863541 X '10983 to five places of decimals. (2) -053407 X '047126 to six places of decimals. (3) -3-141592 X 52-7438 to four places of decimals. (4) 325-701428 X "7218393 to three places of decimals. (5) 8-1729432x8-316259 to four places of decimals. (6) 2-3748-5-1-4736 to three places of decimals. (7) 31-47-7-839-27656 to four places of decimals (8) 252070-520751-J-591-57 to three places of decimals. (9) 73-64-T--43232 to four places of decimals. (10) G-5555-7-7'06249 to three places of decimals. REGURHim DECIMALS 113. Pure recurring decimal fractions are those in which the period commences immediately after the deci- mal point; thus -3, -27, '0429 are pure recurring decimals. Mixed recurring decimal fractions are those in which one or more figures precede the period : thus *23, -2427, '850429 are mixed recurring decimals. 114. To find the Vulgar Fraction which is equivalen* to a given pure recurring Decimal. Ex. (1). Find the Vulgar Fraction equivalent to -8. The decimal = '838... 9 RECURBING DECIMALS. From 10 times the decimal, or 3'333 ..... * Take the decimal, or 333 ...... Then 9 times the decim%l=3'000 ...... .*. the decimal =f =. Ex. (2) Find the Vulgar Fraction equivalent to 247. The decimal =s -247247 ...... From 1000 times the decimal, or 247'247 ...... Take the decimal, or '247 ..... Then 999 times the decimal = 247'000 ...... .-. the decimal = f f J. Ex. (3). Find the Vulgar fraction equivalent to 0423. The decimal = -04230423. . . From 10900 times the decimal, or 423-0423... ; Take the decimal, or -0423 ...... Then 9999 times the decimal = 423-0000 ..... .. the decimal Examples* fli.) Convert into Vulgar Fractions in their lowest terms : (1) -6 (2) -27 (3) -045 (4) -3123 (5) -0072 (6) -4023 (7) '00054 (8) -00009 Hence we deduce the following rule for reduc- ing a pure recurring decimal to a Vulgar Fraction : Take one of the periods to form the numerator, and for the denominator the number formed by repeating 9 as many times as there are figures in the period. Tmn '7 = 5 06 = A 4827 = tni 116 To find the Vulgar Fraction, whieJi is equivalent to a given mixed recurring decimal. RECURRING DECIMALS. 97 Ex. (1). Find the Viflgar Fraction equivalent to 237. The decimal = -23737 ...... From 1000 times the decimal, or 237'37 ...... Take 10 times the decimal, or 2-37 ...... Then 990 times the decimal = 235, .-. the decimal = |M Ex. (2). Find the Vulgar Fraction equivalent to 0-4726. The decimal = -04726726. From 100000 times the decimal, or 4726*726 ...... Take 100 times the decimal, or 4-726 ...... Then 99900 times the decimal = 4722, .-. the demand = ^VgVv = rl Ilir Ex. (3) Find the Vulgar Fractions equivalent to 3-14. The decimal = 3-1444 ...... From 100 times the decimal, or 314-44 ...... Take 10 times the decimal, or 31-44 ...... Then 90 times the decimal = 283, .-. the decimal = -g^ 3 . Examples, (lii). Convert into Vulgar Fractions in their lowest terms : (1) -425 (2) -4759 (3) 4*253 (4) -00426 (5) 53/00243 (6) 7-2011 (7) 2-5306. 117. Hence we deduce the following rule for reduc- ing a Mixed recurring decimal to a vulgar fraction : Form the Numerator by taking from the figures up to the end of the first period the figures tJiat precede the first period; and form the Denominator by setting down 9 as many times as there are figures in the period, and affix- ing as many times as there are figures between the deci- mal point and the first period. 98 RECURRING DECIMALS. THUS . 2 i5 A L 45-4 41 4-5 = 7345 - 734 _ 6^6 11 900 "900 118. The method of performing arithmetical opera- tions with recurring decimals will be best explained by taking the operations separately. I. Addition. Find the sum of 3-49, 4-047, and -1463. First make the decimals all of the same order, thus 3.4999, 4-0470, -1463. Then, since the periods consist of 1, 3, 2 figures respect- ively, and the L. c. M. of 1, 3 and/ 2 is 6, carry on all the decimals six places further, thus 3-4999999999 4-0470470470 1463636363 7-69341068 II. Subtraction. Here we proceed on the same principle as in Addition. Thus to subtract 5-247 from 8-059, 8-059059 5-247777 2-8li28 In both operations some care is requisite in observing what figure would be carried on if the columns omitted were taken into account. III. In Multiplication and Division the recurring decimals should be converted into vulgar fractions, and when the Product or Quotient of these fractions has been found, it may be converted into a decimal. TVmQ 4-^ V q-7 45-4 ^, 37-3 4 i v 3 4 _ 1 3 9_4 < X > ( - X = ff X ^ g-j ' on f\ -l\t^ -A/l6 A 38 r> N--900 5xl _25 and Oo- 042 - go --^- 1JO --- 50 -X 38 lias -T- We may then, if it be required, convert -HH A and f| into decicuals, by the process explained in Art. 108. EXAMINATION PAPERS. Examples- Find the value of the following expression* : (1) 2-57 + '043 + 13-2. (2) 14'762 + 3-549 + 2-204. (3) 15-025 -13-247. (4) -0246- -00897. (5) 8-7 X 5-49. (6) -0072 X '45. (7) 3-4 ~- 4 -09. (8) -074 -r- '59. 119. When vulgar and decimal fractions ase com- bined in the same expression, it may usually be simpli- fied in the neatest and easiest way by reducing the vul- gar fractions to a decimal form. Thus, if we have to find the sum of 476, 13|, and 10-375, we should proceed thus : 476^=476-25 13f= 13-375 10-375 Sum 500-000 Examination Papers. I. (1) Show that any decimal is multiplied by 1000 by re- moving the decimal point in the multiplicand three places towards the right. (2) Enunciate the general rules for the divsion of deci- mals. In cases when the division does not terminate, explain how to determine the place of the decimal point in the quotient. (3) Which of the following statements is more nearly cor- rect? (4) How many times can *0087 be taken from 2*291 ? What fraction will the remainder be of the former ? (6) Whence does it appear that a vulgar fraction may always be reduced either to a terminated or a circulating decimal ? Calculate the limits of the error made in taking fff as au approximate value of 8-1415926 to seven places of decimals. V 100 EXAMINATION PAPKBS. II. (I) Explain what vulgar fractions can be expressed BA finite decimals. Which of the following fractions can be thus expressed ? (2) If a pound of sugar cost -0093125 of $8, find the value of -0625 of 16 barrels of 200 pounds each. (8) Whether is 3 4 714535 more accurately represented by 8-715 or 3-714, and why ? (4) What vulgar fraction is equivalent to the sum of 14'4 and 1'44 divided by their difference ? (5) Find a decimal which shall not differ from f by a ten- thousandth. Ill (1) What are the advantages and disadvantages of work- ing with decimals instead of vulgar fractions ? (2) If a business produces an annual return of $6,000, and of three partners one has '475 and another '88- share of the profits, how much money falls to th share of the third partner ? (3) A man who owns f of a steamboat sells -7 of his share for $1,400 ; what decimal part of the boat does he still own, and what was the boat worth ? (4) A man paid $120 for a horse ; for a buggy $86^ more than -8 of the cost of the horse ; for harness -185 of the cost of horse and buggy. Find his entire outlay. (5) The product of three vulgar fractions is f ; two of them are expressed by the decimals. *63 and *136 ; by what fraction will the third one be expressed ? rv. (1) How do the Decimals differ from Vulgar Fractions ? (2) A storekeeper buys 140 yards of cloth at $'36 per yard. In selling he uses a measure which is ^ 8 of a yard too short, and charges $'60 per yard. What is his net gain ? (8) One vessel contains a mixture of 18 pints of brandy and 7 of water ; another contains 84 pints of brandy and 13 of water. If the strength of the first mixture is represented by 428, what number will represent that of the second ? EXAMINATION PAPERS. 101 (4) Write in figures four millions and four, and ten oillions ninety thousand and seven hundred quadrillionths. Ex- press in words 74000306-000060000007. (5) A piece of cloth was said to contain 84 yards, but it was found that the so-called yard measure with which it was measured was -02088 of a yard too short ; what was the correct length of the cloth ? (1) When a vulgar fraction is changed to a decimal, ex- plain how many figures there will be in the decimal if it does not repeat ; if it is a repeating decimal explain when it will consist of a part which does not repeat, and show how many figures there will be in this part. (2) The French metre is 89 '371 inches in length. Ex- press the length of 25 metres as a fraction of an English mile, there being 5280 feet in it and 12 inches in a foot. (8) If a Bteamer makes a passage from New York to Liverpool (say 2700 miles) in 230 hours, and a train goes from London to Edinburgh (say 405 miles) in 18 hours ; how much does the one go faster than the other ? Give answer in miles and decimal of a mile. Jl) Given that the sum of the divisor and quotient is 7*5 ; that the divisor is | of the quotient ; also that the remainder is f^ of the divisor. Find the dividend. (5) Divide $448.71$ among A, B, and C, so as to give B $46.70 less than A, and $34.59 more than C. VI. (1) What vulgar fractions must be represented by mixed repetends, and what by pure repetends ? (2) Show that no recurring decimal can have more places in the period than there are units in the denominator less one. (8) A man spent $2.50 more than .79 of his money at one time and $1-15 less than fffa of the remainder at another, and now has $2*609 ; how much had he at first ? (4) Simplify (6) Simplify 102 SQUARE BOOT. XIII. Square Root. 120. When a Number is multiplied by itself, the result is called the SQUARE of the number. Thus 144 is the square of 12, and 225 is the square of 15. The symbol 9 placed over a number expresses the square of the number : thus 5 a denotes the square of 5. 121. The SQUABB ROOT of a given number is that number, whose square is equal to the given number. Thus the square root of 144 is 12, because the square of 12 is 144. The symbol y, placed before a number, denotes that the square root of that number is to be taken : thus v/25 is read " the square root of 25." 122. A number which has an integer for its square root is called a PBBFBOT SQUARE. 123. For Perfect Squares not greater than 100 we know the square roots, thus we know that the square root of 81 is 9 ; and for many Perfect Squares greater than 100 we know the square roots by experience, as, for instance, we know that the square root of 169 is 18, and the square root of 400 is 20, and the square root .of 10000 is 100. But we have rules for finding the Square Root of any Number, as we shall now explain First, suppose we have to find the Square Root of 1225. We draw a line separating the two figures on the right from the other two ; thus 12125. The figures 12 make what is called the first period. The figures 25 make what is called the second period. We then take the nearest perfect square not greater than 12, that is 9, and place it under the 12 and pat its square root, that is 3, as the first figure of the square root we have to find, thus 12125 ( 8 UNIVERSITY OF SQUARE BOOT. 108 We subtract 9 from 12, and annex to the remainder 3 the second period 25, to make a dividend, aud we double the first figure of the root, and set down the result as the first term of a divisor ; thus our process up to this point will stand thus : 12125 ( 3 9 325 Now we shall have to annex another figure to the 6, and we must therefore reckon the 6 as six tens, or 60, and then we seek the number of times 60 is contained in 325, and this being five times, we set down 5 as the second figure of the root, and annex 5 to the 6, so that our process up to this point will stand thus : 12125 ( 35 9 65 I 325 We then multiply 65 by 5, and set the product down under the 325 ; and subtracting the product from the 325, we have no remainder, and we conclude that 35 is the square root of 1225, the full process being 12125 ( 35 9 65 825 325 Next, to find the square root of 622521. Drawing a line to mark off the two figures on the right, and another line to mark off the next two figures our process for finding the first two figures of the root will be the same as that explained in the first example, and it will stand thus : 62|25I21(7 49 104 BQOABE ROOT, We now annex to the remainder the third period 21, and we double the part of the root already found, 78, and set down the result 156 as a partial divisor, and proceed, as before, to divide 14121 by 1560, and annex the quotient; 9 to the root and to the divisor; and multiplying 1569 by 9 we set the product under the 14121 ; thus process in full will be 62125121(789 49 148 1325 1184 our 1569 14121 14121 .'. 789 is the root required. NOTE. In practice, instead of dividing 1825 by 140, it is usual to divide 132 by 14, and instead of dividing 14121 by 1560, to divide 1412 by 156. The quotient fchus obtained is, however, sometimes too great, as will be seen in the next Examples. We now give two Examples in which the first period has only one figure, which must always be the case when the proposed square has an odd number of figures in it. To find the square root of 189475225. Marking off the figures by pairs, commencing from the right, we have 1189147:52:25 ( 18765 1 23 89 69 267 2047 1869 2746 27525 17852 16476 137625 137625 SQUARE BOOT. 105 NOTE. In dividing 89 by 20 the quotient is 4, but if we added this to complete the divisor, it would become 24, which being multiplied by 4, would give 96, a num- ber larger than u To find the square root of 39601. 8196101 ( 199 1 23 296 261 8501 8501 I. The division of 296 by 20 illustrates the remarks made on the last example. NOTE II. The second remainder, 35, is greater than the divisor, 29, a result not uncommon in this operation. Examples. Find the square roots of 196. 1024 88209. 106929. 193600. 36372^61. 550183936. 4124961. 82239684. (5) 8 SI S (2) 529. (4) 5625. (6) 119025. (8) 751689. (10) 697225. (12) 22071204. (14) 5256250000. (16) 546121000000. (18) 191810713444. 124. To find the square root of a Decimal Fraction. When the given number has an even number of deci- mal places, we proceed to find the square root as if the number were an integer, and mark off in the root a number of decimal places equal to half the number in tht tquare. Thus, if the square be a decknal of the sixth order, the root will be a decimal of the third order, For example, to find the square root of 6*822249. 106 SQUARE ROOT 5-!32!22'49 (2-307 4 46 132 129 322 Since 46 is not contained in 32, we annex an to the divisor, and also to the root, and bring down the next period thus, 4607 I 32249 | 32249 Examples, (lvi). Find the square roots of (1) 16-81. (2) 281-9041. (3) -9025. (4) -2601. (5) -0625. (6) -000729. (7) 17242-3161. (8) 1-002001. (9) 44415-5625. (10) 18947-5225. 125. In finding the square root of a decimal fraction we must be careful to make the decimal such, that the index of its order is an even number. Thus, if we have to find the square root of -4, we change the decimal into an equivalent decimal of the second, fourth, sixth... order, thus, -40/4000, -400000,... This is done in order that the denominator of the equivalent fraction may be a perfect square, which is the case in the fractions 40_ 4000 400000 100 10000 1000000"" but not in the fractions jL 4QQ 40000 TO 1000 100000'" Also, since for every pair of figures in the square we have one figure in the root, we shall have to take a number of figures in the decimal part of the square double the number of decimal places we are to have in the root. SQUARE ROOT. 107 Suppose, for example, we have to find the square root of *144 to four places of decimals. We must have eight decimal places in the square, thus, -14400000, and we mark off these and proceed aa in the extraction of the root of whole numbers, the root being a decimal of the fourth order : thus, 14|40IOO!00 ( -3794 .... 67 540 749 7584 7100 6741 35900 30336 5564 J*OTE. The square root of a decimal of an odd order is a non- terminating decimal. Examples- Extract to four places of decimals the square roots of (1) 20. (2) 80. (8) -9. (4) -121. (5) -169. (6) -016. (7) -00064. (8) -00121. (9) 16-245. (10) -9. (11) -25. (12) 42-08. 126. If we have to find the square root of a vulgar fraction, we can always by multiplication make the denominator a perfect square, if it be not already so, multiplying the numerator by the same number. We then find the square root of the denominator, and find, exactly or approximately, the square root of the numerator, and make the results respectively the de- nominator and numerator of a fraction, which is the root required, exactly or approximately. 36 v 36 6 j2 * 2x3 ... fc ' < 2 >- \ 8 * Af 3X8 " 108 SQUARE ROOT. We can now extract the square root of 6 to, say, three places of decimals, thus : 6-100100100(2-449 4 44 484 4889 200 176 2400 1936 46400 44001 2399 /2 2-449... ' V = 3 = - 816 Or we might have reduced f to a decimal, thus : 666666..., and then have extracted the square root of this decimal. 8 -j ,OQ Ex. (4). To find the square root of y^r Here we can reduce the fraction to lower terms. Thus, ^,-y = _, = _ 12.5 v/6-25 2-5 127. An integer can always be changed into a per- fect square by multiplying by a number equal to or less than the proposed integer. For example, 7 is changed into a perfect square if multiplied by 7 18 is changed into a perfect sqflare if multiplied by 2. Examples, (Iviii). Find the square roots of (1) Jt^ (2) A 4 T (8) HI (A\ 1369 V/ /\ 15129 /fi\ W TsF9 1 D ; T23irsr Iw (7) m (8) 3^, (9) (10) 88# (11) 17i (12) OCBB BOO 109 and find to four places of decimals the square roots of (18) * (14) (15) 6f (16) 9i (17) XIV. Cube Root. 128. When a number is multiplied by itself twice, the result is called the CUBE of the number. Thus 27 is the cube of 8, and 216 is the cube of 6. 129. The CUBE ROOT of a given number is that number, whose cube is equal to the given number. Thus the cube root of 843 is 7, because the cube of 7 is 848 The symbol ty , placed before a number, denotes thai the cube root of that number is to be taken : thus /125 is read, "the cube root of 125." 130. A number, which has an integer for its Cube Boot, is called a PEBFECT CUBE. The numbers, less than 1000, which are perfect cubes, should be committed to memory : they are 1, 8, 27, 64, 125, 216, 843, 512, 729, and the cube roots of these numbers are respectively 1, 2, 3, 4, 5, 6, 7, 8, 9. 131. To find the Cube Eoot of a perfect cube, greater than 1000, we proceed by a rule, which we shall now explain. Ex. (1). To find the cube root of 91126, 4 91 I 125 64 4300 625 5425 27125 27125 First divide the number 91125 into two periods by drawing a line marking off three figures on the right. Then take the nearest perfect cube not greater than 91, which is 64, and set down its cube- root, which is 4, in a CUBE ROOt. line with 91125, and some way to the left. This is the first figure of the root. Then subtract 64 from 91, and to the remainder attach the second period, 125. Now place three times the first figure of the root, 12, to the extreme left, and three times the square of the first figure of the root, 48, with two zeros annexed to it, just on the left of the 27125. Divide 27125 by 4800, and set the quotient, 5, midway between 12 and 4800. Then read 12 5 as 125 ; multiply this by 5 ; put the result, 625, under the 4800 ; add it to the 4800; this gives 5425; multiply this by 5 ; put the result, which is 27125, under the first remainder; subtract, and as there is no remainder, the process is complete, and the root is 45. Examples- (Kx) Find the cube roots of (1) 4096. (2) 82768. (8) 74088. (4) 493089. (5) 614125. (6) 262144. (7) 89304. (8) 389017. (9) 614125. (10) 970299. (11) 59319. (12) 250047. Next, let us take the case in which the cube root has fchi-ee figures, and extract the cube root of 428661064. 7 428 | 661 | 064 843 81 6 14700 85661 1075V 16775 f 78876 2gJ $io 4 1687500 6786064 9016 1696516 6786064 We separate the number 428661064 into three periods, and take the nearest perfect cube not greater than 428, which is 848, and we set down its cube root, which is 7. We then subtract 348 from 428, and annex to the remainder 661, the second period. CUBE BOOT. Ill Then we set down three times 7, which is 21, and three times the square of 7, which is 147, and add two zeros to it. Then we divide 85661 by 14700, which gives the quotient 5, and this we put down midway between 21 and 14700. Then we multiply 215 by 5, which gives 1075 ; we add this to 14700 ; we multiply the result, 15775, by 5 ; and subtract the product, 78875, from 85661 ; and to the remainder we annex the third period, 064. We then set down three times 75, which is 225, and three times the square of 75, which is 16875. N.B. This last result can be obtained by setting the square of 5, the second figure of the root, under the second divisor, and adding the three numbers coupled by the bracket. We then annex two zeros, to 16875 and repeat the process explained above, to find 4, the third figure of the cube root, which is in this case 754. Next, take the case in which the root h&afour figures and find the cube root of 14832537993, 14 8 832 { 537 | 998 5824 72 785 172800 1008687 882126 176425T 25J 7 18007500 126412998 61499 18058999 126412998 Hence the root required is 2457. NOTE. In dividing 6832 by 1200 the quotient is 5, but if we took this for the second figure of the root we should find that the addition of 6 times 65, or 826, to 1200, would give 112 CUBE ROOT. 1525, and this multiplied by 5 would give 7625, a number too large to be subtracted from 6832. Examples, Find the cube roots of (1) 14706125. I) 300763000. (2) 149721291. (3) 28934443. (5) 2097152. (6) 5735339. (7) 99252847. (8) 1092727. (9) 16777216. (10) 194104539. (11) 84027672. (12) 130323843. (13) 322828856. (14) 354894912. (15) 400227072. (16) 134217728. (17) 122615327232. (18) 673373097125. 132. To extract the Cube Boot of a Decimal Fraction. In order that a Decimal Fraction may be a Perfect Cube, it must be of the 3rd, 6th, 9th order, the Index of the order being some multiple of 8. We then proceed in the following way : Ex. (2) To find the cube root of -343. Ex. 12) To find the cube root of -039304. 89304 _ 34 _ ~ 100 ~ * 34 ^039304 1000000 Ex. (3) To find the cube root of "012812904. 234 -012812904 = ^-066006660 = 1000 234. 133. To extract the cube root of an integer or deci- mal expression to . a particular place of decimals, w must take three times the number of decimal places in the expression. Thus, to find the cube root of 4*23 accurately to three places of decimals we extract the cube root of 4-230000000, making the given expression a decimal of the ninth order. In working this example we find the cnbe root CUBE BOOT. 118 of 4280000000, regarded as a whole number, and mark off three decimal places in the result. 134. The cube root of a Vulgar fraction may be found by taking the roots of the numerator and denominator, or by reducing the fraction to a decimal of the 8rd, 6th, 9th..... order, and proceeding as in Art. 183. Examples. (fri). Find the cube roots of ) -389017 (2) -048228544 (8) 27054-086008 5| (5) m (6) 5*& (7) 405^; and find to three places of decimals the cube roots O A (8) 5 (9) 576 (10) -121861281 (11) 16-926972504 (12) f (13) * (14) i (15) 7f (16) fy 135. The fourth root of a number is found by taking the square root of the square root of the number. (1 (4 Thus */4096 = ^64 = 8 The sixth root of a number is found by taking the cube root of the square root of the number. Thus f 64 = V& == 2. Examples. (Ixii) lina the fourth roots of (1) 681441 (2) 4100625 (3) 1576-2961; and the sixth roots of (4) 4826809 (6) 24794911296 (6) 282429-586481. COMMERCIAL ARITHMETIC. XV. On English, Canadian, and United States Currencies. 136* Having explained the principles and processes of Pure Arithmetic, we proceed to show how they are applied to Commercial affairs. MEASURES OP MONEY. 4 farthings are equivalent to 1 penny, 12 pence are equivalent to . 1 shilling 20 shillings are. equivalent to 1 pound. The symbol placed before or over a number denotes pounds, s after shillings, d after .pence. s. d. JChus 14. 5s. 7df., or 14. 5. 7 stands for fourteen pounds, five shillings and seven pence. Since 1 farthing is one-fourth of a penny, 2 farthings are one-half of a penny, 8 farthings are three-fourths of a penny. Hence the symbol %d. is placed for 1 farthing, \d 2 farthings or a halfpenny. fcZ 3 farthings. The symbol q., placed after a number, is sometimes used to denote farthings: thus, Sq. stands for three farthings. 137. "We call 14 a simple quantity, and 14. 6s. Id. a compound quantity, because the former is expressed with reference to a single unit, while the latter is ex- pressed with reference to three diff&r&ni units. CANADIAN AND UNITED STATES COINS. 116 138. The unit in Canadian and United States cur- rencies is called a dollar. The tenth part of this unit is called a dime ; the tenth part of the dime is called a cent ; and the tenth of the cent is called a mill. We may conceive the unit, then, to be divided into ten equal parts, each of these parts into ten other equal parts, and so on. Hence Canadian and United States currencies are based on the Decimal System of Notation, and, therefore, all operations in these currencies are performed by means of the rules in Decimal Fractions. It is to this circumstance that they owe their great simplicity. TABLE OF CANADIAN AND UNITED STATES COINS. CANADIAN COINS. UNITED STATES COINS. Gold. Gold. British Sovereign, worth Double Eagle, or... $20 $4.86$. Eagle, or $10 British Half-Sovereign. Half Eagle, or $5 Three Dollar Piece. Quarter Eagle, or... $2 Dollar. Silix*. Silver. Dollar. 60-oent piece answers to Half dollar. 25-cent piece answers to Quarter dollar. 20- cent piece (no longer coined). 10-cent piece answers to Dime. NickeL 6-oent piece answers to 6-cent piece. 8-cent piece. Bronze. Bronze. 1 cent. 1 cent. Mill, not coined. Mill, not coined. Ex. (1). $251, 7 cents, 8 mills. = 1(261 + A + Tiv + T*o) = $(251 + T^ + y^ + = $(251 TT $ 251-073. REDUCTION OP MONEY. Ex. (2). $55-923 = $ 55 + 92 cents -f 3 mills = ft 55, 92 cts. 3 mills. The English gold coinage consists of |i pure metal and of lV alloy. The gold and silver coinage of the United States consists of -fa pure metal and -fa alloy. The silver coin in Canada and England is # pure metal and vV copper. Gold and silver thus alloyed are called standard. The gold or silver before it is coined is called bullion. The 1-cent. piece is made of bronze, and is 1 inch in diameter, and 100 of them weigh 1 Ib. avoirdupois. The term carat is employed to denote the fineness of gold, Perfectly pure gold is said to be 24 carats fine ; a mixture of 18 parts pure gold and 6 parts of some other metal, is said to be 18 carats fine. This latter is termed jellewer's gold. REDUCTION OF MONEY. 139. The expression 5s. Id. stands for a sum of money, which is made up of five shillings and seven pence. Now, since one shilling is equivalent to twelve pence, five shillings are equivalent to sixty pence ; and therefore five shillings and seven pence are equivalent to sixty-seven pence. The process by which we change the compound ex- pression 5s. Id. into the equivalent simple expression 67 d., is arranged thus : s. d. 5 . 7 12 Q7d. and we describe the process thus : We change the 5 shillings into pence by multiplying by 12, and add to the product the 7 pence. UCTION OF MONET. 117 i, to change the compound expression 4. 7s. into an equivalent number of farthings, we proceed thus : 0. A. 4 . 7 . lOfc 87. 12 10544. 4 f'irst we change 4 to shillings and add 7*., making 87. ; then .............. 87*. to pence ..... ...... 10d., making 1054d. ; then .......... 1054d. to farthings ........ 2g,, making 4218g, Examples- (luii) Bednce to farthings 1) 8^.; 7|.; 9d.; llfd. 2) 2a. 3d.; 5s. 7$d.; 12*. 9|d.; 17*. 3) ^3. 12.; 5 ; 2. 17 Beduce to pence (4) 6.t.; 4*. 10d.; 7*. 10^.; 8*. 9d.; 13. Id. (5) ^4 ; 5. 2. 4d.; ^17. 14*. 6d.; ^58. 18s. lid (6) 174. 10.; ^432. 15a. 10d.; ^1274. 17. 9d. 140. The converse operation, by which we express a simple quantity in terms of an equivalent compound quantity, will be best explained by the following Ex- amples. Ex. (1). Nine farthings will be expressed as pence and farthings, if we divide 9 by 4 (since 4 farthings = 1 penny), set down the quotient as pence, and the remainder as farthings, thus : 9 farthings = f d. = 2d. Ex. (2). Again, 33 pence will be expressed as shillings and pence, if we divide 33 by 12 (since 12 pence = 1 shilling), set down the quotient as shillings, and the remainder as pence, thus : 33 pence = f f shillings = 2s. 9d. Ex. (8), Also, 75 shillings = ft pounds = B. 15s. 118 COMPOUND ADDITION. Ex. (4). To express 4275639 farthings in terms of s. d. farthings. 4 12 20 4275639 1068909(2. and 3 farthings over. 8907,5*. and 9 pence over. 4453 and 15%. over. .'. 4275639 farthings = 4453 15*. These methods of expressing a given sum of money in another, hut equivalent, form are included in the word Reduction. Examples* (l*iv) Reduce to pence and farthings the following numbers of farthings : (1) 57. (2) 173. (3) 197. Reduce to shillings, pence and farthings the following numbers of farthings : (4) 857. (5) 479. (6) 747. Reduce to & s. d. the following numbers of farthings : (7) 4288. (S) 876289. (9) 642880. 141. The copper coins in use in Great Britain are the Farthing, the Halfpenny, and the Penny. The silver coins in use are the Crown, (5s.), the Half- crown (2s. 6d.), the Florin (2*.), the Shilling, the Six- pence, the Fourpenny piece (or Groat), and the Three- penny piece. The gold coins in use are the Sovereign or Pound, and the H*lf-sovereign. The Guinea (21*.) and the Half- guinea (10s. 6d.) are not in use, but reference is fre- quently made to them. COMPOUND ADDITION. 142. In adding compound expressions together, we follow the principles which regulate the process of Addition in the case of pure numbers. COMPOUND ADDITION. 119 Thus, in adding sums of money we- arrange them so that the pounds stand under pounds in vertical col- umns, shillings under shillings, pence under pence, and farthings under farthings. For example, if we have to add together 4s. 3d., 3*. 3m 94 12 . 75 . 9 d, 1 6 tal ie 58 47 9. 9 8 d. 2. 8. (3) 58 . 13 4 ' 49 14 5 (4) 276 . 17 5\ 37 19 7i (5) 1247 . 5 10; 1246 11 8^ (6) 8000 . 10 7i 2998 13 11: (7) 199 . 0. 198 19 10 (8) 80609 . 5 2; 79089 12 5: (9) 44005 . 7 94 7896 10 2, (10) 80704 . 5 29484 61 COMPOUND MULTIPLICATION. 144. To multiply a compound expression, as 4 8s. 9fd., by a number, as 9, is equivalent to taking the sum of nine expressions, each equal to 4. Ss. 9d, Instead of writing these expressions one under the other, and finding their sum by tlie process of addition, we obtain the required result by multiplying each of the four quantities, composing the expression, sepa- rately by 9, calculating the value of each result as in addition, setting down part of those results under the several columns, and carrying on part, as in addition, thus: 9 89 . 19 . 8| The process may be more fully explained thus : 9 times 3 farthings =27 farthings = 6| inches. (3) Reduce 74325 yd. to poles ; 2423694 in. to furlongs. (4) Reduce 728964 ft. to miles ; 82976432 in. to miles. (5) Addition. yd. ft. in. mi. fur. po. fur. po. yd. 4 2 7 (6) 18 . 4 . 20 (7) 2 19 2 19 1 9 43 . 3 . 9 25 21 5 2 10 66 . 2 . 13 6 11 31 23 2 8 4 . 7 . 32 6 23 4 85 17 1 2 6 4 16 . 8 . 15 19 . 5 . 11 8 1 21 if Subtraction. yd ft. in. mi. fox. po. far. po. yd. (8) 134 . 2 . 7 (9) 235 . . 19 (10) 6 . 28 . 1* 69 . 1 . 11 184 . 6 . 24 4 , 27 . 4 (11) Multiply 7 yd. 2 ft. 9 in t by 11 ; 16 mi 6 fur. 7 po. 56. (12) Multiply 32 po. 8 yd. 1 ft. by 67 ; 36 mi. 8 fur. 6 pa. 8* yd. by 49. (12) Divide 26 yd. 1 ft. 8 in. by 4 ; 17 mi. 8 fur. 7 po. by . (14) Divide 14 po. 2 jd. 1 ft 8 in. by 82; 11 mL 7 fur. 7 po. by 56 ON MEASURES. l33 153. MEASURES OF SURFACE. 144 square inches make 1 square foot, written 1 sq. ft. 9 square feet 1 square yard, 1 sq. yd. 30 square yards 1 square pole, 1 sq. po. 40 square poles 1 rood, 1 ro. 4 roods 1 acre, 1 ac. Hence 1 acre = 4840 square yards. 640 acres = 1 square mile. Land surveyors make use of a Chain 22 yards in length, divided into 100 equal parts, called Links. The square of 22 is 484, and therefore 10 Square Chains make an Acre. NOTE. The Square Inch is a square whose side is an inch in length. Ex. (1). How many square inches are there in 3 ac. 2 ro. 27 po. 27 sq. yd.' 7 sq. ft. 23 sq. in. ? ac. u. po. sq.-yd. aq. t'c. sq. hi. 3 . 2 . 27 . 27 . 7 , 23 4 14 ro. 40 587 po. 146 17637 17783| sq. yd. 9 146| the result of the division of 587 by 4,. r637 160054 6 the result of multiplying f by 9. 160060| sq. ft. 144 640263 640240 160060 108 the result of multiplying by 144. 23047771 sq. in 184 ON MEASUKES* Ex. (2). Reduce 74237 sq. yards to poles. 74237 sq. yd. = (74237 -5- 80-J) poles -= (74237 -f i|i) poleg *i (74237 X jjj) poleg We may proceed thus : 121 I" 111 74237 yards 4 296948 quarter-yards 26995 and 3 quarter-yards over 2454 po. and 1 parcel of 11 quartei yards over. The remainder is (11 + 8) quarter-yards, or 14 yards, or 8$ yd. /. 74^37 sq. yd. = 2454 po. 3* sq. yd. Examples- Reduction. (1) Reduce 5 ao. 8 ro. 17 po. 13 sq. yd. 6 sq. ft. 16 sq. in. to square inches. (2) Reduce 7 ac. 15 po. 5 sq. yd. 3 sq. ft. to square inches. (8) Reduce 250 acres to square ysxds, and 78 sq. yd. to square inches. (4) Reduce 5239 sq. in, to sq. yd., and 15376 sq. yd. to acres. (5) Reduce 84729 sq. yd. to poles, and 562984 sq. in. to square poles. Addition, (6) c. 47 ro. . 2 8 (7) 19 r fiC.ln. 42 (8) 46 re . 2 . R *fi 72 . 1 24 27 5 52 17 . 8 . 14 13 80 . 2 82 82 8 124 7 . 1 . 39 14 4 2 28 5 2 72 24 2 . 15 19 27 . 3 8 21 6 98 12 . . 17 22 42 . S 5 56 8 185 4 . 1 . 9 16 ON MBA8UBK8. 185 Subtraction. M. to. pc. sq. yd. nq.it. q. to- ae.ra.fo. (9) 57 . 2 . 80 (10) 42 . 8 . 124 (11) 16 . 2 . 29 . 3 . 34 86 . 8 . 139 14 . 3 . 24 ac. ro. po. sq. yd. sq. ft. sq. in. ac. ro. pa (12) 247 . 1 . 14 (13) 39 . 7 . 12 (14) 245 . 3 . 19 243 . 3 . 24 32 . 8 . 134 178 . 3 . 23 (15) Multiply 5 ac. 3 ro. 24 po. by 15 ; 17 ac. 2 ro. 13 po. by 53. (16) Divide 7 ac. 2 ro. 18 po. by 21 ; 29 ac. 2 ro. 87 po. by 71. 154. MEASURES OF SOLIDITY 1728 cubic inches make 1 cubic foot, written 1 cub . ft. 27 cubic feet make 1 cubic yard, written 1 cub. yd. A Cube is a solid figure contained by six equal squares. Hence a cubic inch is a six-sided figure, each of whose sides is a square inch. The lines that form the bound- aries of the sides are called the Edges 01 the Cube. Examples. (Ixxvi) Reduction. (1) Reduce 7 cub. yd. 18 cub. ft. to cubic feet ; 25 cub. yd. 5 cub. ft. 143 cub. in. to cubic inches; 14 cub. yd. 1374 cub. in. to cubic inches. (2) Reduce 74825 cub. in. to cubic feet ; 439284 cub. in. to cubic yards. (3) Reduce 5 \ cub. yd. to cubic inches ; 3 cub. yd. 6$ cub. ft. to cubic inches. Addition. cnb.yd. cnbJt. cnb.ln, cub.yd. cubJt. cnb.ln. (4) 57 . 32 . 46 . 76 . 4 . 52 . 13 . 25 . 19 . 8 . 26 . 14 . 572 (5) 493 374 587 1249 1324 43 . 26 . 19 . 45 . 26 . 88 . 7 . 22 . 16 . 13 . 5 . 18 . 1638 472 1384 427 1286 276 (6) 528 237 764 446 729 852 WM . 16 . . 19 . . 10 . . . . 11 . . 5 . vutuu. 432 683 1359 1275 846 1478 136 ON MEASURES. Subtraction. eab.yd. c*>.ft. Cuban. eab.yd.cnb.ft. cob. In. unb. yd. <*-*Jt rotv.n (7) 47 . 17 . 543 (8) 247 . 19 . 1274 (9) 527 . 38 . 28 . 726 239 . 18 . 1868 499 . U . 265 (10) Multiply 26 cub. yd. 5 cub. ft. 49 cub. in. by '1 ; 472 cub. yd. 17 cub. ft. 238 cub. in. by 53. (11) Divide 78 cub. yd. 13 cub. ft. 252 cub. in. by 12; 472 cub. yd. cub. ft. 1416 cub. in. by 59. 15 b MEASURES OF CAPACITY. 2 pints make 1 quart, written 1 qt., 4 quarts 1 gallon, 1 gall., 2 gallons .... 1 peck, 1 pk., 4 pecks 1 bushel, 1 bus., 8 bushels ... 1 quarter, > 1 qr., Examples, (Ixxvii) Reduction. (1) Reduce 3 pk. 1 gall. 8 pt. to pints, and 214 qr. 8 bus. to pint*. (2) Reduce 4234 pt. to quarters, and 3047 gall, to quarters. Addition. gall. qt. pt. bus. pk. galL qr. bus. pk. (3) 4.3.1 (4) 4 . 8 . 1 (5) 42 . 5 . 3 8 . 2 . l\ 6,2.1* 27 . 7 . 2 1*2. 8.0 1.8.1 64. 3.1 14 . . 1$ 4 . 2 . 1$ 49 . 6 . 2 6.2.1 8.1.0 12. 4.0 Subtraction. gait qt. pt. bus. pk. gall. qr. bus. pk. (6) 5.2.0 (7) 6 . 8 . (8) 36 . 7 . 2 4.8.1 5.8.1 29. 7. 8 (9) Multiply 6 qr. 8 bus. 2 pk. by 63, and 16 qr. 2 1 pk. by 73. (10) Divide 13 gall. 1 pt. by 15, and 348 qr. bus. 1 pk. by 43 155. TBOY WEIGHT. 24 grains make 1 pennyweight, written 1 dwt. 20 pennyweights make 1 ounce, written 1 oz. 12 ounces make 1 pound, written 1 Ib, Oluefly used for weighing gold, silver, and jewel*. ON MEASURES. 187 Examples- (Ixxviii Reduction. In 27 ounces of go' I how many Tains are there ? Reduce 7 Ib.; 14 Ib. 8 oz.; 25 Ib. 9 oz. 6 dwt. to pen- nyweights. (3) Reduce 8 Ib. 10 oz. 7 dwt. 5 gr.; 7 Ib. 4 OE. )7 dwt. 15 grains to grains. (4) Reduce 3145 gr. to ounces ; 42672 gr. .o Ib. (5) Reduce 72469 gr. to Ib. ; 3246 dwt. to ib. Addition. (1) (2) (6) Ib. 21 27 3 14 oz. . 2 . 9 . 8 . 3 dwt . 12 . 4 . 17 . 19 oz. (7)7 6 2 dwt . 13 . 6 . 17 . 9 ."a . 19 . 23 . 6 (8) Ib. 15 . 12 . 5 . 42 . oz. 8 . 4 . 10 . 7 . dwt. 6 . 17 . 18 . 15 . B 8 21 7.6.8 8 . 16 . 18 12 . 11 . 19 . 23 Subtraction. oz. dwt. gr. Ib. oz. dwt. Ib. oz. dwt. gr. (9) 6 . 19 . 13 (10) 37 . 8 . 6 (11) 35 . 9 . 8 . 22 8 . 14 . 16 29 . 10 . 13 84 . 11 . 15 . 23 (12) Multiply 7 Ib. 5 oz. 9 dwt. by 12; 6 Ib. 8 oz. 19 dwt. 57 21. (13; Multiply 10 oz. 16 dwt. 23 gr. by 37 ; 8 Ib. 7 oz. 10 dwt. 21 gr. by 41. (14) Divide 16 Ib. 4 oz. 16 dwt. by 8 ; 7 Ib. 10 oz. 17 dwt. 7 gr. by 15. (15) Divide 9 oz. 17 dwt. 8 gr. by 37 ; 15 Ib. 8 oz. 9 dwt. 12 gr.' by 63. 157. AVOTRDUPOIS WEIGHT ^6 drachms make 1 ounce, written 1 oz. 16 ounces 1 pound, lib. 14 pounds 1 stone, 1 st. 25 pounds 1 quarter, 1 qr. 4 quarters 1 hundredweight or cental 1 cwt. 20 hundredweight 1 ton. The pound Avoirdupois contains 7000 grains Troy. The pound Troy contains 5760 grains Troy. NOT&. in Great Britain 28 pounds make 1 quarter. 188 ON MEASUBBS. Examples- (Ixxix) Reduction. .deduce 11 cwt. to oz. ; 17 Ib. to dr. ; 5 tons to Ib. Reduce 6 tons 7 cwt. to oz. ; 15 tons 2 qr. to Ib. Eeduce 8 cwt. 6)b. 5 oz. to dr. ; 3 tons 15 cwt. 71b. tolb. Reduce 4763 oz. to cwt. ; 3749 Ib. to tons, Reduce 7432 oz. to cwt. ; 247294 dr. to cwt. Addition. Ib. oz. dr. qr. Ib. OB. (6) 3.3.9 (7) 3 . 16 . 8 19 . 8 . 6 4 . 7 . 12 7 . 10 . 13 16 . 19 . 5 14 . 5 . 7 8 . 20 . 13 8 . 15 . 14 12 . 6 . 9 cwt. qr. Ib, 13 . 2 . 24 11 . 3 . 5 29 . 1 . 19 16 . 2 . 9 17 . . 7 Subtraction. Ib. oz. dr. qr. Ib. oz. owl. qr. Ib. (9) 16 . 13 . 6 (10) 17 . 13 . 3 (11) 19 . 1 . 4 14 , 11 .12 14 . 15 .11 17 . 3 . 18 tons. cwt. qr. cwt. qr. Ib. tons. cwt. qr. Ib. (12) 37 . 19 . 2 (13) 16 . . 3 (14) 74 . 15 . 1 . 13 29 . 19 . 3 15 . 3 . 25 89 . 16 . 3 . 25 (15) Multiply 17 owt. 23 Ib. 14 oz. by 7; 4 cwt. 17 Ib. by 45. (16) Multiply 6 cwt. 8 qr. 5 Ib. by 23; 10 oz. 9 dr. by 37. (17) Divide JL4 cwt. 2 qr. 8 Ib. by 12 ; 32 tons 15 cwt. 1 qr. by 40. (18) Divide 16 cwt. 3 qr. 9 Ib. by 65 ; 37 tons 4 owt. 3 qr. 7 Ib. by 17. 158. APOTHECARIES' WEIGHT. 1. Measures of Weight. 437i grains make 1 ounce, 16 ounces make 1 pound. The grain is the same as the grain Troy The ounce is the same as the ounce Avoirdupois. This is the table given in the British Pharmacopooia, The Avoirdupoib ounce and pound are taken, in prefer- ON MEASURES. tt ence to the ounce and pound Troy of the old table, be- cause the former are used by wholesale dealers in drugs and medicines. In prescribing, many physicians still employ the scruple 3, of 20 grains, and the drachm 5, of 60 grains. 159. 2. Measures of Capacity. 60 Minims make 1 fluid drachm, written fl dr., 8 Fluid Drachms 1 fluid ounce, fl oz., 20 Fluid Ounces 1 pint, O, 8 Pints 1 gallon, C. is a contraction for Octavus or eight, and C for Congius, a Roman liquid measure. The relation of the measures of capacity to those of weight in these tables is given by the definition that 1 Minim is the measure of -91 Grain of Water. The connection may be better remembered by the old rhyme. A Pint- of Water Weighs a Pound and a Quarter. 160. Multiplication of Compound Quantities when the multiplier contains a fraction. (See page 128). Examples, (Ixxx). Multiply (1) 3. cwt. 2 qr. 12 Ib. by 3f (2) 6 Ib. 5 oz. 4 dr. by 2| (3) 4 mi. 3 fur. 10 po. by 181 (4) 15 yd. 2 ft. 3 in. by 43* (5) 37 ac. 3 ro. 8po. by 4-^- (6) 25 ac. 2 ro. 15 po. by 29 g (7) 27 sq. yd. 7 sq. ft. 36 sq. in. by 2f . 161. Division of Compound Quantities when the divisor contains a fraction. (See page 128). Examples. (Ixxxi). Divide (1) 5 cwt. 2 qr. 11 Ib. by 2f (2) 7 Ib. 4 oz. 14 dr. by 11^ (8) 7 mi. 2 fur. 12 po. by 4-&- (4) 17 yd. 1 ft. 3 in. by 5 25 ac. 2 ro. 12 po. by 41 (6) 14 ac. 3 ro. 8 po. by 8 107 sq. yd. 4 sq. ft. 132 sq. in. by 18|. (5) (7) 140 FRACTIONAL MEASURES. 162. XVII. Fractional Measures. Ex. (1). How many shillings and pence are there in f of a pound ? ft D a pound = of 20 shillings. -^p 5 - shillings. =*'*i*B. = 12s. 6cL Ex. (2). Find the value off of 15 5s. 8eL 2- of 15 5s. Sd. = 3 times | of .15 5s. 8cZ. = B times 2 3s. Sd. = ,6 1 Is. Or thus : 15 . 5.8 3 7 I 45 . 17 . Q . 11 Ex. (3). Fiad the value of 2f of 7 3 T of 5 acres. 2| of ->- of 5 acres = Y- of -,- f 5 acres. = - 11 ^ of 5 acres. 4x22 = of 5 ac. = 1 ac. 3 ro. 20 po. Examples, (Ixxxii). Find the value of the following ; (1) | of 1 ; | of 2 10s. ; | of 5 18s. 5d. (2) f of a mile ; -fy of an acre ; f of a cwt. (3) 2| of 54 9s. Sd. ; 3/ 4 - of half-a-guinea ; f of 3| of a mile. (4) | of | of 1-&- of Ig of 2470 guineas ; f of ^ of 4^ guineas. , (5) f of 1 + f of Is. + $ of 16s. 4d. (6) fa of 1 + i of 2s. 6d. + 7 of a guinea. (7) f of 5 ac. 3 ro. + | of 7 ae. 2 ro. 2 J po. + f of 8 ro. 15 po. (8) Jj of a year + - 5 V of a week + -fa of an hour. (9) !* of a mile + | of a furlong + | of a yard. FRACTIONAL MEASURES. i-ii (10) I of 2 cwt. 3 qr. + j of 5 cwt. 3 qr. 14 Ib. -r g- of 7 Ib. 163. The following are examples of an operation which is the converse of that just explained. Ex. (1). Express 14s. Id. as the fraction of 5. Us. 7d. = 175d., and ^5 .-. I75d. is -iVtfV- of 1200. Hence the fraction required is -^AV* or ~2 3 4 5 (T> or iV- Ex. (2). Express 6 Ib. 5 oz. avoird. as the fraction of 3 Ib. 12 oz. 6 Ib. 5 oz. = 101 oz., and 3 Ib. 12 oz. = 60 oz. ; . the fraction required is Ytf- Ex. (3) Express f of 5s. 9<1 as the fraction of 4s. Id. 5s. 9d.=69d. and 4*. 7. =7-5*. and -Sofia. =('5 X .. '875 of 1 = 7*. Qd. The operation is performed more briefly thus : 875 20 . 7-500 12 d. 6-000 Ex. (2). Find the value of 8-16876 of Jgl. J68-16875 *. 8-87500 12 f. 2-00000 ^8-16876 = ^8. 8*. 4K OIMAL MEASURES. 148 Ex. (8). Find the value of -4256 of 125. Sd. 4256 of 12*. Sd. = '4256 of 152eJ. = (-4256 X 152)d. 4256 162 8512 21280 4256 64-6912 .-. Valne required is 64*6912d. Ex. (4). Multiply 27 ac. 8 ro. 14 po. by -285. ac. ro. po. 27 . 8 . 14 4 111 ro. 40 4454 po. 235 22270 18362 8908 40 1046-690 26 . 6-69 po. ac. 6 . 2 ro. 6*69 po. Ex. (5). Find the value of -25 of 1. 25 of 1 = *- of 1 = if of 1 = yj. = 6>. Or thus : J6-2555 20 d. 1-8888 Value required is 5a. 144 DECIMAL MKASURB8. Examples- (Ixxxiv) Find the value of (1) (8) (5; (7) (9) (11) (13) (15) (16) (17) 625 of 1. -009765. 046875 of 1 Ib. avoir. 425 of 8*. 4d. 83 of 5s. 35 of 2qr. 14 Ib. 2-1372 of 2 tons 5 cwt. (2) 15-275. (4) -9375 of a cwt. (6) 2-003125 of 8. (8) 2-46875 of 1 3s. (10) 4-13 of 12s 3ri (12) 2-125 of 8$ -guineas. (14) 5-247 of 5. 2*, 6& 45 of 3. 10s.+-75 of 4s. 8d.+3-245 of 3. 4d. 7 of l+-8 of 7s. Qd. - 2-45 of Is. Sd. 285714 of 3. 3s. -|- -142857 of 2. 17s.-f-34 df 16s. 6a. 165. The following examples illustrate the operation, which is the converse of that already explained. Ex. (1). Express 5s. 6d. as the decimal of 1. 5s. Qd. = 66d., and 1 = 240 , of 10s. 78 . 9 . 82 . 18 . 9 2 each. 10* 2* 2072 . 7 . 9 = cost at 2. 12*. lOd. each, 167. ^"h fractions of a Unit, which have for theii numerator Unity, are called ALIQUOT PARTS of the unit. Thus 5s., being of 1, is an aliquot part of a pound ; and 5 Ib. being ? of 1 qr., is an aliquot part of a quarter. Examples- (Ixxxvi) Find the cost of the following articles : 4321 at 1. 17s. 3fd, (2) 2175 at 2. 15*. 4$d. 8768fc at 1. 7*. 4^. 4) 4276 at 12. 11s. &%d. 5783 at 14. 9s. 6d (6) 8689f at 16. 12s. 9eJ. 7483 at 22. 13s. tyd. What is the dividend on 4234. 10s. at 5*. Qd. in the ? What is the dividend on 4975 at 3*. tyd. in the ? What is the dividend on 3729. 18*. Qd. at 7*. 9$d. in the? II. COMPOUND PRACTICE. Ex. (1). When we have to find the cost of a quantity of goods of mixed denomination (as 14 cwt. 3 qr. 17 Ib.), the cost of a single unit of one of the denominations being given (as 3. Is. 6d. per cwt. of 112 Ibs.) we proceed thus : . *. d. 14 cwt. is 14 x 1 cwt. 8 . 7 . 6 cost of 1 cw' = cost of 14 cwt. 2 qr. , 1 qr. 14 Ib. 21b. 7ii= lib. (1) (3) (5) (7) (8) (9) (10) 2 qr. is of 1 cwt. 1 qr. is I of 2 qr. 14 Ib. is i of 1 qr. 21b.is}ofl41b. lib. iB*of21b. 6. 15 . 7 47. 1 . 5 . 13. 16 8. 1 . to c* o o o 50 . 5 14cwt.3qr.17lb. COMPOUND PRACTICE. 151 '. *. 5 . d = the rent of 1 acre. 39 12 ao. 1 12 6 2 ro 16 3 = 1 ro. 8 20 po. 2 0-375 .. 5 po. 4-875 1 po. Ex. 2. What is the rent of 12 ac. 8 ro. 26 po. a. 8. 5*. an acre ? 12 ac. is 12 X 1 ac. 2 ro. is of 1 ac. 1 ro. is of 2 ro. 20 po. is | of 1 ro. 6 po. is { of 20 po. 1 po. is of 5 po. 41. 19. 8-75 = therentofl2a.3r.26p. NOTB. When the divisor is any number less than 12 (except 7) it is desirable to employ decimals, instead of vulgar fractions, to express the result of the division after the line of pence. Examples- (ixxxvii) (1) 5 ac. 8ro. 4 po. 4 yd. at 10 per rood. (2) 12 cwt. 3 qr. 22 Ib. 12 oz. at 3. 18*. 2d. per cwt, (8) 10 ac. 3 ro. 26 po. at 2. 18s. 101 d. per acre. (4) 6 tons 12 cwt. 3 qr. 10lb. at 3. 14*. Sd. per cwt. (5) 63 cwt. 8 qr. 17$ Ib. at 12 guineas per cwt. (6) 29 ac. 8 ro. 5 po. at 100 guineas per acre. (7) 16 oz. 6 dwt. 20 gr. at 3. 17*. 6d. per oz. (8) 25 ac. 1 ro. 10 po. at 42. 2*. 4d. per acre. (9) 13. cwt. 3 qr. 17 Ib. at 22 8*. per cwt. (10) 319 cwt. 3 qr. 16 Ib. at 2. 12*. Qd. per cwt Invoices and Accounts. 168. An INVOICB is a statement in detail, sent by a Seller to the Buyer at the time the goods are delivered to the Buyer, of the quantity, description, and price of the goods. An ACCOUNT is a statement sent by the Seller to the Buyer at the end of a term of credit, shewing the totals and dates of each Invoice and the sum total of the whole. 152 COMPOUND PRACTICE. Each separate article or amount in an Invoice or an Account is called an ITEM. A DETAILED ACCOUNT is a full statement, sent by the Seller to the Buyer at the end of a term of credit, shew- ing the dates of delivery, the quantities, description, prices, and sum total of the goods delivered by the Seller to the Buyer during that term of credit. When an account has been made out it is rendered, i.e., sent in to the Buyer. Specimen of an Invoice, Toronto, June 20, 1877. John Smith, Esq., Bought of 0. Jones & Co., 21 Front-st. 6 Ibs. of Tea at 75 cts. . 8 Ibs. of Loaf Sugar... at 12^ cts... 2 Ibs. of Butter at 30 cts. ., $ jets. 3 75 1 I 00 i 75 5 ! 50 Specimen of an Account Toronto, July 21, 1877. John Smith, Esq., To J. Jones & Co., 21 Front-st. 1877 June 20 June 28 July 3.. Julvl2. To To To To $ 6 7 3 2 cts. 50 80 CO 27 do do do .. 19 17 COMPOUND PRACTICE, 153 Specimen of a Detailed Account. Toronto, July 21, 1877. John Smith, Esq., To J. Jones & Co., 21 Front-st. 1877 June 20 5 Ibs. of Tea at 75 cts $ cts. 3 11 " 20 8 Ibs. of Loaf Sugar. ..at 12 cts 1 00 " 2J 2J- Ibs. of Butter .... ..at SO cts 75 June 23 1 bbl. of Flour . at $6 6 00 " 23 18 Ibs. of Cheese.... ..at 10 cts . . .. 1 80 July 3 12 Ibs. of Biscuit.... ..at 15 cts 1 80 " 3 6 jars of Pickles .... ..at bO cts 1 80 July 12 41 12 " 12 1 gal. of Coal Oil.... 8 Ibs. of Sugar 8^ Ibs. Raisins ..37 cts ..11 cts ..12 cts 37 88 1 02 19 1 17 Examples (Ixxxviii) Make out invoices of the following sales, supplying names and dates of your own selection : (1) 100yds. of broadcloth at $3.25 per yard; 2500 yards of sheeting at 12 cts. per yard ; 3000 yards of prints at 18 cts. per yard ; 300 yds. of French silk at $1.75 per yard. (2) 5 Ibs. of black tea at 70 cts. ; 2} Ibs. of green tea at 90 cts. ; 15 Ib. of lump sugar at 12 cts. ; 17 Ib. of brown sugar at 9 cts. ; 74- Ib. of raisins at 20 .4 Ib. currants at at 13 cts. Make out accounts of the llowing sales, supplying names and dates of your own selection: (3) 39^ yd. of Brussels carpet at $1.50; 62| yd. of Kid- derminster carpet at $1.10; 27yd. of avitting at 23 cts.; 34 \ yd. of drugget at 65 cts. ; 43^- yd. of India matting at 18"cts. (4) 23 yd. of black silk at $2.15; 17 yd. of ribbon at 23 cts. ; 13$ yd. of silk velvet at 25 cts. \\\ doz. pairs of stock- ings at 45 cts. a pair ; 5 pairs of gloves at $1.25 ; 18 yd. of muslin at 17 cts. (5) 6 pairs of blankets at $5.50 ; 12J yd. of merino at 45 cts. ; 15| yd. of cloth at $3.25 ; 5 J yd. of flannel at 30 cts.; 2 counterpanes at $4.ia cadi ; 2.34 yd. of calico at 15 cts. 154 PROBLEMS. XX. Problems. 169. The Unitary Method, which is rapidly displac- ing the Rule of Three, will be gradually explained in this and the succeeding Sections. Ex. (1). If 23 bullocks cost $483, what is the cost of 1 bullock e f Since 23 bullocks cost $483, 1 bullock will cost $<& 3 or $ 21 - Ex. (2). If 7 men do a piece of work in 12 days, how long will it take 1 man to do it ? Since 7 men can do the work in 12 days, 1 man can do the work in (7 X 12) days, or 84 days. Ex. (8). If 28 men do a piece of work in 42 days, in how many days can 21 men do it ? Time for 28 men to do the work = 42 days. 1 man " " = 28 < 42 days. 21 men = ?*2<-l? days. = 56 days. Ex. (4). If 7o men finish a piece of work in 12 days, how many men will finish it in 20 days ? In 12 days the work is done by 75 men, In 1 day the work is done by (12 X 75) men, In 20 days the work is done by men, or 45 men. Ex. (5). A bankrupt's debts are $2520, and his assets (that is the value of his property) are $1890 ; what can he pay in the dollar. In the place of $2520, he can pay $1890, In the place of $1, he can pay $$ffg or $ , or 75 cts. ; .*. he pays 75 cents, in the dollar. Ex. (6). A bankrupt's debts are 4264, and he pays 12s. 6d. in the pound ; what are his assets ? That which he has to meet a debt of 1 is 12. That which he has to meet a debt of ^4264 is (4264 X 12>- ' .-. his assets are " 4 **'j t or 2665. PROBLEMS 155 Ex. (7). If 27 men can do apiece of work in 14 days, working 10 hours a day, how, many hours a day must 12 men work to do the same in 45 days ? Since 27 men can do the work in (14x10) hours, or 140 hours, 1 man can do the work in (27 X 140) hr. .-. 12 men can do the work in ^~ nr - or 315 Dr - Now 815 hours have to be distributed equally over 45 days; /. the number of hours they work each day ^ or 7. Ex. (8). If 7 Ibs. of tea cost $5.60, what will be the cost of 12 Ibs. ? Since 7 Ib. of tea cost $5.60, 1 Ib. of tea costs *M, or 80 cts., .*. 12 Ib. of tea cost 12 x SOcts. or $9.60 Ex. (9). If 9 horses can plough 46 acres in a certain time, how many acres can 12 horses plough in the same time ? Since 9 horses can in the given time plough 46 ac., 1 horse can in the given time plough 4 ^ ac. .*. 12 horses can in the given tune plough IgX4g ac., or 61 ac. Ex. (10). If 15 horses can plough a certain quantity of land in five days, how many horses will be required to plough it in three days ? In 5 days the land can be ploughed by 15 horses ; In 1 day the land can be ploughed by (5x15) horses ; In 3 days the laud can be ploughed by , or 25 horses. 3 NOTE I. In simple questions of this kind we ha\*e a supposition and a demand. Each contains two kinds of things ; in the supposition the magnitudes of both kinds are given ; in the demand a magnitude of one kind is given, and the appropriate corresponding magnitude of the other kind has to be found. The first line of the solution contains the magnitudes of the supposition so 166 arranged, that at the end of the line we have that kind oj thing, of which the magnitude is required in the demand. Thus in Ex. (10) the order of the supposition is changed, and the magnitude, 15 horses, put at the end of the line, because we have to find how many horses will be required in the demand. Examples- (l xxxix ) (1) If a man walk 62 miles in 4 days, in how many days will he walk 93 miles ? (2) If 12 men reap a field in 4 days, in what time will 82 men reap it ? (3) If 850 acres of land cost $61250, what will 273 acres cost ? (4) How many men can perfgrm in 12 days a" piece of work which 15 men can perform in 20 days ? (5) The rent of 17 acres is* $297, what is the rent of 86 acres ? (6) If a man walk 116 miles in 8 days, how far will he walk in 14 days ? (7) A farmer sells a flock of 270 sheep at $240 a score, what does he get for them ? (8) A servant's wages being $108 per annum, how much ought she to receive for 7 weeks ? (9) A clerk's salary is 191. 12s. 6d. per annum; what ought he to receive for 60 days service ? (10) A ship performs a voyage in 63 days, sailing at the rate of 6 knots an hour ; how long would it take her, if she sailed c the rate of 7 knots an hour ? (11) A bankrupt's effects are worth $860, and his debts are $ 1300 ; what does he pay in the dollar ? NOTE II. To one of the magnitudes in a supposi- tion there is a corresponding magnitude of the same kind in the demand, and these magnitudes must be ex- pressed in units of the same denomination. Ex. A man walks 1 m. 1 fur. 7 po. in 20 minutes; how loni* will he take to walk 41 m. 2 fur. 12 po. ? Here 1 m. 1 fur. 7 po. = 367 poles, and 41 m. 2 fur. 12 po. = 13212 poles. Then he walks 367 poles in 20 minutes ; he walks 1 pole in ^/V min.; he waiks 13212 poles in > V 6 2 7 - 2 -? min., or 720 mm. PROBLEMS tNVCWaVING FRACTIONS 157 Examples (Ixxxix) continued. (12) If 3 bushels of wheat be worth $3.50 what fa the worth of 43 qr. 6 bus. ? (13) If 15 yards of silk cost $6.75 how much will 20 yd, 1 ft. cost ? (14) If 3 cwt. 3 qr. cost $27, what will be the cost of 2 cwt. (15) If 2 cwt. 8 qr. 7 Ib. cost 5 17s. 8%d. what is the cost of 9 cwt. ? 170. Problems involving Fractions. Ex. Iff of an estate be worth $1500, what is the value of -J of the estate ? Since f of the estate is worth $1500, j of the estate is worth $ 1 V UL - . . the estate is worth $ 7xl a * or $3500, Hence of the estate is worth $ 4x35 2 or $2800. o Examples (xc). (1) If f of an estate be- worth $7520 ; what is the value of 4 of the estate ? (2) A persons owns f of a ship and sells f of his share for $1260 ; what is the value of the ship ? (3) If 3t Ib. of tea cost 15s. 3d. how much can I buy for 3s, 10U. (4) If -nr of a piece of work be done in 25 days, how much will be done in llf days ? (5) A man walks 18 ra. 2 fur. 26 po. 3| yd. in 5J hours. How long does he take to walk a mUe and a half? (6) A gentleman possessing -ft- of an estate sold T of-Trr of his- share for $603.12^- ; what would i of iV of the estate sell for at the same rate ? (7) If the carriage of 15 '5 cwt. of goods for 69 miles cost $3.10, how far ought 3.25 cwt. to be earned for the same money ? (8) What is the value of -^V of -^V of a vessel, if a person who owns -jV of it sell of -J of his "share for $1400. (9) When the ounce of gold is worth <3'89, what is the cost of .04 Ib. ? 168 COMPLEX PROBLEMS. ^10) If the price of candles 8 inches long be 9d. per half-dozen, and that of candles of the same thickness and quality 10J inches long be lid. per half-dozen, which kind do you advise a person to buy ? (11) If the carriage of 60 cwt. for 2.0 miles cost 14|, what weight can be carried the same distance for 5^ ? COMPLEX PROBLEMS. 171. We now proceed to cases in which the suppo- sition, expressed in the simplest form, contains more than two magnitudes, the demand containing the same number of magnitudes, all of which are given, except one, which has to be found. Ex. (1). If 12 horses can plough 96 acres in 6 days, how many horses will plough 64 acres in 8 days ? In 6 days 96 acres can be ploughed by 12 horses. In 1 day 96 acres can be ploughed by 6x12 horses. In 1 day 1 acre can be ploughed by 5-2-JL2. horses. 9 8 In 8 days 1 acre can be ploughed by ^ x Ia horses. 8X96 In 8 days 64 acres can be ploughed by *A X JS x 1 * horses 8X96 .*. the number of horses required is 6. Ex. (2). If 35 bushels of oats last 7 horses for 20 days, how many days will 96 bushels last 18 horses ? 85 bushels last 7 horses for 20 days. 1 bushel lasts 7 horses for f$ days. 1 bushel lasts 1 horse for T x 8o days. 96 bushels last 1 horse for or in an hour ; .'. A can do in an hour ; .. A can do the work in 6 hours. NOTE II. If a tap can 11 a vessel in 5 hours, the part filled by it in 1 hour will be represented by ^. Ex. (1). A vessel can be filled by three taps, run- ning separately, in 20, 30, and 40 minutes respectively. In what time will they fill it when they all run at the same time ? They fill & + ^ + -& of the vessel in 1 minute ; .-. they fill 6 *** 3 , or ^ in 1 minute ; .*. they fill T$T in iS of a minute ; .*. they fill the vessel in Va 4 or 9 & minutes. Ex. (2). A bath is filled by a pipe in 40 minutes. It is emptied by a waste pipe in an hour. In what time will the bath be full if both pipes be opened at once ? One pipe fills -fa of the bath in a minute. % The other empties $ of the bath in a minute. .'. when both are running, ,V TO* or -j-^j- 01 the bath is filled in a minute ; .*. the bath is filled in 120 minutes. Examples, (xcii). (1) A can do a piece of work in 6 hours; B can do it in 9 hours. TT ^hat tune will they do it if they work together ? 162 PROBLEMS RELATING TO CLOCKS. (2) A can do a piece of work in 85 days ; B can do it in 40 days ; C can do it in 45 days. In what time will they do it, all working together? (3) A and B can reap a field of wheat in 8 days ; A and C in 8 days ; B and C in four days. In what time could they reap it, all working together ? (4) If three pipes fill a vessel in 6, 8, and 12 minutes respectively, in what time will the vessel be filled when all three are opened at once ? (5) A does ^ of a piece of work in 14 days. He then calls in B, and they finish the work in 2 days. How long would B take to do the whole work by himself? (6) A does a piece of work in 8 hours, which is twice the time B and C together take to do it ; A and C could together do it in 1 hours. How long would B alone take to do it ? (7) A can do a piece of work in 27 days, and B in 15 days ; A works at it alone for 12 days, B then works alone 5 days, and then C finishes the work in 4 days. In what time could C have done the work by himself ? (8) A cistern is filled by two pipes in 18 and 20 minutes respectively, and emptied by a tap in 40 minutes ; what part of it will be filled in 10 minutes when all are opened at the same instant ? 173. Problems relating to Clocks. The minute-hand moves 12 times as fast as the hour- hand, and therefore in 12 minutes the minute-hand gains 11 minute-divisions on the hour-hand. Ex. (1). Find the time between 3 and 4 o'clock when the hands of of a watch are together. At 3 o'clock there are 15 minute-divisions between the hands ; we have therefore to find how long it will take the minute-hand to gain 15 minute-divisions on the hour-hand. The minute- hand gains 11 minute-divisions in 12 minutes; 3 minute-division in |f minutes ; 15 minute -divisions in ~ min. ; .'. the time required is Ig ] x 1 min., or l^pj- min. past 8. Ex. (2). At what time between 2 and 8 are the hands of a clock at' right angles to each other ? When the hands are at right, angles there is a space of 16 minute -divisions between them. PROBLEMS RELATING TO CLOCKS. Hence, since at 2 o'clock there are 10 minute-divis- ions between the hands, we have to find how long it will take the minute-hand to gain 10 + 15, or 25 min- ute-divisions on the hour-hand. The minute-hand gains 11 minute -divisions in 12 minutes ; 1 minute-division in |f minutes ; 25 minute-divisions in ^-^ min. ; .. the time required is - x ~ min., or 27 T 3 T min. past 2. Ex. (8). At what times between 6 and 7 are the hands of a clock at right angles to each other ? Twice between 6 and 7 this will occur : first, before the minute-hand has overtaken the hour-hand ; second- ly, after the minute-hand has passed the hour-hand. Now, since at 6 o'clock there are 30 minute-divisions between the hands, we have to find : First, how long it will take the minute-hand to gain 80 15, or 15 minute-divisions on the hour-hand. Secondly, how long it will take the minute-hand to gain 30 + 15, or 45 minute-divisions on the hour-hand. The process in each case will be similar to that in the preceding examples, and the results are 16^ min. and 49^ min. past 6. Ex. (4). Find the time between 7 and 8 o'clock when the hands of a watch are opposite to each other. When the hands are opposite there is a space of 80 minutes between them, and at 7 o'clock there is a space of 35 minutes between the hands. Hence in this case we have to find how long it will take the minute-hand to gain a space of 8580, or 5 minutes on the hour-hand. The process will be similar to that in the preceding examples, and the result is 5^ min. past 7. Examples* (xoiii) At what time are the hands of a watch together be- tween the hours of 164 EXAMINATION PAPERS. vl) 4 and 5. (2) 6 and 7. (3) 9 and 10? At what time are the hands of a watch at right angles to each other between (4) 4 and 5. (5) 7 and 8. (6) il and 12 ? At what time are the hands of a, watch opposite to each other between (7) 1 and 2. (8) 4 and 5. (9) 8 and 9 ? EXAMINATION PAPERS. I. (1) If for a given sum I can have 1200 Ibs. earned 86 miles, how many pounds can I have canied 24 miles for the same sum ? (2) If i- of a ship be worth $18056, wnat is the value of of the ship ? (3) A 1 silver tankara weighs JL ib. 10 oz. ; what is its value, when a dozen spoons, weighing of oz. each, are worth $54? (4) A man spends $61.60 every 85 days, and saves $400 a year. What is his annual income ? (5) When the income-tax is 6d. in the & a man pays 15 7s. &d. ; what is his income ? IL (1) A man s income is reduced from $2720 to $2640.66 when he has paid his income tax. What is his tax on the dollar ? (2) If 10 horses and 132 sheep can be kept 8 days for $202, what sum will keep 15 horses and 148 sheep for the same time, supposing 5 horses to eat as much as 84 sheep ? (3) A man receives 75 cents in the dollar of what was due to him and thereby loses $602.10. What was due to him ? (4) If 15 men can perform a piece of work in 22 days, how many men will finish another piece of work 4 times a? large, in a fifth part of the time ? (5) If 72 men dig a trench in 63 days, in how many days will 42 men^dig another trench three times as great ? III. (1) The wages of A and B together for 7 days amount to the same sum as the wages of A alone for 12f days. For how many days will the sum pay the wages of J alone ? EXAMINATION PAPK&S. 165 (2) If 100 men can perform a piece of work in 80 days, how many men can perform another piece of work thrice as large in one-fourth of the ^ime ?" (3) If 5 men or 7 women can do a piece of work in 87 days, how long will a piece of work twice as great occupy 7 men and 5 women ? (4) Two persons, A and B, finish a work hi 20 days, which B by. himself could do in 50 days. In what time could A finish it by himself? How much more of the work is done by A than B ? (5) If a cistern when full of water can be emptied in 15 minutes by a pipe, an'd when empty can be filled by another hi 20 minutes ; if the cistern be full, in what time can it be emptied by both pipes being opened at the same time ? IV. (1) A and B can do a piece of work alone in . 15 and 18 days respectively ; they work together at it for 3 days, when B leaves, but A continues, ana after 3 days is joined by 0, and they finish it together in 4 days. In what time would C do the work by himself? (2) If a man can do treble, and a woman double the work of a boy in the same time, how long would 9 men, 15 women, and 18 boys take to do double the work which 7 men, 12 women, and 9 boys complete hi 250 days ? (3) A and B walk to meet each other from two places 100 miles distant. A walks 6 miles an hour and B four miles an hour. At what point on the road do they meet, and at what two times are they fifty miles apart from each other ? (4) A watch which is 10 minutes too fast at noon on Mon- day loses 8 min. 10 sec. daily. What will be the time indi- cated by the watch at a quarter past 10 on the morning of the following Saturday ? (5) A watch set accurately at 12 o'clock indicates 10 min. to 5 at 5 o'clock. What is the exact time when the watch indicates 5 o'clock ? If it indicated 10 minutes past 5 at 5 o'clock, what would be the exact time when the hands indi- cated 5 o'clock ? V. (1) A laborer agreed to work for 60 days on this condition : (hat every day he worked he should receive $2, and for every day he was idle, he should pay $1.50 for hie board. At the eipiration of the time, he received $92. How many days did he work ? 166 SIMPLE INTEREST. (2) A piece of work can be done in a day of 11 hours by 2 men, or 5 women, or 12 boys ; in what time could it be done by 1 man, 2 women, and 3 boys together ? (3) A cistern has two supplying pipes A and B, and a tap (7. When the cistern is empty, A and B are turned on, and it is filled in 4 hours ; then B is shut and C turned on, and the cistern is quite emptied in 40 hours ; when, lastly, A is shut and B turned on, and in 60 hours afterwards- the cistern is again filled. In what time could the cistern be filled by jach of the pipes A and B, singly ? (4) A clock is set at 12 o'clock on Saturday night, and at noon on Tuesday it is 3 minutes too fast. Supposing its rate regular, what will be the true time when the clock strikes four on Thursday afternoon ? (5) A contractor engages what he considers a sufficient number of men to execute a piece of work in 84 days ; but he ascertains that three of his men do, respectively, , \ , and less than an average day's work, and two others ^ and vo more, and in order to complete the work in the 14 weeks, he procures the help of 17 -additional men for the 84th day. How much less or more than an average day's work on the part of these 17 men is required ? XXI. Simple Interest. 174. INTEREST is that which is paid by one, who bor- rows money, for the use of the money. The money lent is called the PRINCIPAL. The Borrower agrees to pay at what is called a certain BATE of interest, which is usually reckoned by the sum paid for the use of $100 for 1 year. Thus, if I borrow $500 for 1 year, and agree to pay $25 for the use 6f the money, I am said to borrow at the Rate of 5 per cent, per annum, that is, I agree to pay $5 for the use of every $100 in the loan at the end of the year. The sum made up of the Principal and Interest added together, is called the AMOUNT at the end of the time for which the money is borrowed. 175. The solution of questions relating to Interest depends on precisely the same principles as those ex- plained in the last Section, and it is only because of the SIMPLE T NTEREST. 1*57 necessity of explaining technical terms that tnsre is any occasion to separate this or the succeeding Sections from Section XX. For, just as we reason about the question What must I pay for the hire of 4 horses for 5 months, if I pay $18 for the hire of 8 horses for a month ? so do we reason about the question What must I pay for the use of $550 for 3 years, if I pay $5 for the use of $100 for a year 9 Ex. (1). To find the Simple Interest on $2675 for 8 years, at 8 per cent., we reason thus : Interest on $100 for 1 year is $8 ; , on $1 for 1 year is $ T ? )ff ; 011 $2675 for 1 year is $ 2675x ; on $2675 for 3 years is $ 3x2G75x8 . 100 ' .'. the interest is $642. Hence we derive the following Rule : Multiply the Principal by the Rate per Cent., and the result by the Time expressed in years, and divide the pro- duct by 100. The process stands thus : 2675 8 21400 3 $642.00 .-. the interest is $642. Ex. (2). Find the interest on $3200 for 2 years and 7 months at 7 per cent. MS8 SIMPLE Since 7 months is T 7 j of a year the time is 2/3- years. Interest on $100 for 1 year is $7.53. $1 for 1 year is $ TV. o $3200 for 1 year is $ 3S .- * o 3 2 $3200 for 2 T V yean is 2 T 7 y X $ ~ ~- , 3 1_x ,T ? * 7 .SO . lux 100 = $020; .-. the interest is $620. Ex. (3). Find the interest on $101178 from January 28th, 1876, to Sept. 15th, 1876, at 6 p^r cent. The number of days between January 28th and Sept. 15th is 231, and 231 days is -5^ of a year. Interest = ^JJJ-^ x ||J $3841-992. NOTE I In calculating the number of days between two given days of the year, the rule is to imimh one of them only in the calculation. Thus from Jan. 4 to Jan. 9 will be 5 days. In the preceding example if we multiply numerator and denominator by & we have 101178 x 2 x 231 730oO Hence, in computing the interest for any number of days, we have the following rule : Multiply the Principal by twice the rate, and the remit by ilie number of days and divide the product by 73000. When the Principal is not very large the division is most readily effected by dividing the product by 3, the quotient by 10, and the new quotient bv TQ, and adding these quotients and the product tog aid pointing off five places of decimals. ESTEKKS'A. Thus: Find the interest O A &1000 for. 121 davs at 8 per cent. 16 160,0 121 8 I 1033000 1') I 645303 10 i Since 73000 increased by \ of itself, J 9 cf itself, and ^y of iU*uf becomes 1UOUIO. 7300') Thus, 2433 of 100010 and considering this as 100000 the reason for tne at>ove process is evident. NOTE II. In actual practice the time, when not an exacfc number of years, is always expressed in days, or in years and days. Examples, (xciv). Find the simple interest (1) On 62750 for 6 years at 5 per cent, per annum. (2) On $o625 for 4 years at 8 per cent, per annum. (3) On 127 for 6 years at 7\ per cent, per annum. (4) On $8825 for 6^' years at 8 per cent, per annum. (5) On $1160 for 11 months at 9 per cent, per annum. (6) On S9125 for 78 days at 8 per cent, per annum. (7) On $5913 from Nov. 23, 1876, to April 7, 1877, at 1\ per cent, per annum. (8) On 204 17s. Id. from Aug. 3 to Jan. 9 at 5 per cent. 176. We have explained how to find the Interest (and Amount) when the Principal, Rate and Tjme are given. We shall now explain how to find the Eate, or 170 SIMPLE INTEREST. Time, or Principal, -vhen the other two and also the Interest (or Amount) are given. Ex. (1). At what Rate percent, will $520 amount ^o $800-80 iii 9 years ? As the rate is the interest on $100 for 1 year, to find the rate we must find the interest on $100 for 1 year. Here interest = $800.80 -$520=$280. 80, Thus, the interest on $520 for 9 years is $280.80 ; .-. the interest on $520 for 1 year is $?-*~>- 8 - on $1 for 1 year is $ff^ on $100 for 1 year is $ 1 -.-^i|^ = $ 6 . .'. Bate required is 6 per cent. Ex. (2). Iii what Time will the Interest on $360 amount to 126 at 5 per cent. ? , Interest on $360 for 1 year is $ 3 ~y~p S or $18 Then, since $18 is the interest for 1 year, $1 is the interest for T ' g ^eai, $126 is the interest for Vs y ear > or 7 years. .'. Time required is 7 years. Ex. (3). What Principal will amount to $980 in 3 years at 7 per cent. ? Interest on $100 ior 3 years at 7| per cent, is $22'50, .'. $122.50 is the amount which has for its Principal $100; $1 is the amount which has for its Principal $p*, f $980 is the amount which has for its Principal ft- 9 -- 80 x A or $800. .*. Principal required is $800. Ex. (4). At what rate will any sum triple itself in 20 years at simple interest ? Here the interest is twice the Principal. Thus the interest on the Principal for 20 years is 2 x Prin- cipal ; 2 x Principal .*. interest on the Principal for 1 yeans ^n 2 X Principal interest on $1 for 1 year is - 100 X 2 X Principal Principal X 20" Bate required is 10 per cent. SIMPLE INTEREST. 173 Examples- (XCY) (1) At what rate will the interest on $-326 for 16 years be $220.05? (2) In what time will $700 amount to $920.50 at 6 per cent. ? (3) What sum will amount to $1325 in 8 months at 9 per oent. ? (4) The interest on a sum of money for 12 years at 4| per cent, is $202.50 ; what is the sum ? (5) In what time will any sum double itself at 5 per cent, simple interest ? (6) What must be the rate per cent, that the interest at the end of 16 years 8 months may be equal to seven-eighths of the sum lent ? (7) A sum of money amounts in ten years at 7 per cent, to $1275 ; in how many years will it amount to $1406.25 ? (8) The sum of $500 is borrowed at the beginning of the year at a certain rate per cent., and after 9 m*nths $400 more is borrowed at double the previous rate. At the end of the year the interest on both loans is $35 ; what is the rate at which the first sum was borrowed ? (9) In how many days will the interest on .243. Qa. 8 Ex. (I). $4000. TOBONTO, June 1, 1872. Two years after date I promise to pay William Smith, or order, four thousand dollars, for value received, with interest at 7 per cent. RICHARD PAYWELL, On this note were the following endorsements : Sept. 15, 1872, Four hundred and fifty Dollars, Dec. 15, 1872, Fifty Dollars. Mar. 1, 1878, Five hundred Dollars, JP:I, 1, 1874, One thousand Dollars What remained due June 4, 1874 ? ABT1AL PAYMENt. 178 Principal on interest June 1, 1872 $4000 00 Interest to Sept. 15, 1872 ; 80 89 Amount $4080 89 Less 1st payment 450 00 Remainder for a new principal '. $363089 t Interest from Sept. 15 to Dec. 15, 1872, is ) / $63.44, which exceeeds the payment. J Interest from Sept. 15, 1872 to March 1, 1873 117 20 Amount $3748 09 Less the sum of the 2nd and 3rd payments 550 00 Kemainder for a new principal $319809 Interest from March 1, 1873 to Jan. 1, 1874 18647 Amount $3384 56 Less payment Jan. 1, 1874 1000 00 Kemainder for a new principal $238456 Interest from Jan. 1 to June 4, 1874 70 94 Balance doe June 4, 1874 .................. $2455 50 Examples* (1) $1500. Hamilton, Jan. 1, 1877. Orie year after date, we promise to pay S. White, or order, fifteen hundred dollars, with interest. Value received. GEORGE BROWN & Co. The following payments were made on this note : March 16, 1877, $100 ; June 18, 1877, $400 ; Sept. 1, 1877, $200. What was due Jan. 1, 1878, interest at 6 per cent ? (2) $3500. BeHeviEe, March 15, 1876. For value received, we jointly and severally promise 174 COMPOUND INTKBE8T, to pay Wm. Smith, or order, three thousand five hun- dred dollars, with interest. JAMES JONES & Go. Endorse.! as follows : June 1, 1876, $800; Sept. 1, 1876, $100; Jan. 1, 1877, $1560; March 1, 1877, $300. What was due May 16, 1877, interest at 6 per cent. ? (3) $1200. Toronto, Oct. 15, 1859. One year from date we promise to pay James Smith, > or order, twelve hundred dollars, for value received, with interest. WILDER & SON. Endorsed as follows : Oct. 15, 1860, $1000; April 15, 1861, $200. How much remained due Oct. 15, 1861, interest at 6 per cent. ? XXII. Compound Interest. 178. Compound Interest is that which is paid, not only for the use of the original sum lent, but also for use of the interest as it becomes due. The interest on $500 for 1 year at 4 per cent, is $20. If then $500 be -lent at Compound Interest for 2 years at 4 per cent., the Interest for the first year is $20. Now, as the borrower has to pay for the use of this $20, the Interest for the second year must be calculated on $520. Hence Interest for second year = $^^ 4 = $20.80. To put the matter in a more simple way, we have supposed the borrower to retain the interest due at the end of the first year, but the reasoning will be the same if we suppose the lender to receive the interest at the end of the first year, and to put it out immediately at the same rate of interest. 179. We may calculate Compound Interest by the following rule : COMPOUND INTEREST. 176 Find the interest for the first year : add it to the, original principal : call the result the Second Principal : find the interest on this for the second year : add it to the second principal: call the result the Third Principal', find the interest on this for the third year, and so on. Ex. (1). Find the Compound Interest on $7500 for 8 years at 4 per cent. $7500 is the Principal for the first year, 4 $300.00 The interest for the first year is 1300. Add this to the Original Principal, $7500. Then $7800 is the Principal for the second year. 4 $312.00 The interest tor the second year is $312. Add this to the Second year's principal, $7800. Then $8112 is the Principal for the third year. 4 $324.48 The interest for the third year is $324.48. .*. Compound interest required is $300+$312+$324.48 = $936.48. If the A mount at Compound Interest be required, add the original Principal, $7500, to the Compound Inter- est, $936.48. Then Amount required =$8436.48. Ex. (2). What is the compound interest of $250 foi 2 years, at 7 per cent. ? $250 Principal for 1st year. $250X0.07 = 17.50 Interest for the 1st year. 267.50 Principal for 2nd year. $267.50x0.07 = 18.725 Interest for the 2d year. 286.225 AmountatCom.Int.for2yrs. First Principal 250.00 $36.225 Com. Int. for 2 years. 176 COMPOUND Examples- Find the Compound Interest 03 (1) 0875 for 3 years at 5 per cent. (2) $564 for 4 years at 7 per cent. (3) $1154.37 for 4 years at 5 per cent (4) $740 for 5 years at 7 per cent. NOTE I. When the Compound Interest is required for 8 years, it is usual to find the compound interest for the whole of the fourth year, and take hall' the result as the compound interest for the half year. This really implies that the interest is paid half-yearly, but the approximation does not differ much from the exact truth. 180. The process for finding the Amount of a sum at Compound Interest may be presented in a very brief and neat form as follows : If the rate of interest be 4 per cent., Amount of $100 at the end of 1 year is $104, of $1 at the end of 1 year is | j of $1. Hence it follows that Amount of any sum at 4 per cent, in 1 year = |g$ of that sum. Again, Amount for second year = -Jgj of amount for the first year; .'. Amount of any^sum at 4 per cent, in 2 years = i8 J of f of that sum. Suppose, then, we have to find the amount of $540 in 8 years at 4 per cent, compound interest. The amount is }gj of fg$ of jgj of $540 = $540 x (1.04) 3 = $607.426. From the above example it will be noticed that the amount of $1 for a year at 4 per cent, is raised to the power indicated by the number of years for which com- pound interest is to be calculated. Hence we have the following rule: To find the sum to which any^principal mil amount if put out to Compound Interest at a given rate COMPOUND INTEREST. 177 number of years, find the amount of $1 for a year at the given rate, raise that sum to th'e power which is denoted by the given numlwr of years, and multiply the result by the number of dollars in the given principal. Ex. (1). Find the amount of $850 in three years at 6 per cent, compound interest. The Amount = $850 X (1'06) = $850 X 1 -191016 = $1012-363. The Compound Interest = $1012-3G-$850 = $162.36. NOTE IH. When the number of years is large the student is recommended to employ the contracted method of multiplication, explained in Art. 111. Interest may be payable either yearly, half-yearly, or quarterly, or at some other stated period. In finding the Compound Interest on $2000 in two years, when the interest is payable half-yearly, at 6 per cent., we reason thus : 5 per cent, for a year = 2 per cent, half-yearly, 2 years = 4 half-years. Hence we have to find the Compound Interest on $2000, for four times of payment, at 2 per cent. The Amount = $2000 x (1-025)* = $2000 X 1-1038127 = $2207-625. The Interest = $2207-625 -$2000 = 1207-625. Ex. (2). What Principal will amount to $ 1012-8C8 in 8 years at 6 per cent ? Principal X (1'06) 8 = $1012-863 ,. Principal =$^^ = $850. Examples (xcviii) (1) What is the Compound Interest on $1000 for 2 years, at 6 per cent., payable half-yearly ? (2) What is the amount of $200 for 8 years, at 6 per cent., payable half-yearly. (3) Find the Compound interest on $675.75 for 8 years, at 6 per cent, per annum. 178 PRESENT WORTH AND DISCOUNT. (4) A money dealer borrowed $1000 for 2 years, at 6 per cent, interest; and loaned the same in such a manner as to Compound the Interest every 6 mouths. What profit did he make in 2 years by this proceeding ? (6) Find the difference in Compound Interest on J65000 for 2 years at 4 per cent., according as it is reckoned yearly or half-yearly. (6) What is the difference between the Compound Interest on 140000 for 4 years, and on $80000 for 2 years, the rate in both cases being 5 per cent. ? (7) A and B lend each 0248 for 3 years at 3 per cent, one at Simple, the other at Compound Interest ; find the difference of the amount of interest which they respectively receive. (8) What sum at four per cent. Compound Interest wilS amount in 2$ years to $16989'7728. (9) What sum will amount to $27783 in 3 years at 5 per cent. Compound Interest. XXIII. Present Worth and Discount. 181. Suppose A owes B $105, to be paid at the end of a year. If A be disposed to pay off the debt at once the sum which he ought to pay should be such tha^, if put out at interest by JB, it will amount at the end of a year to $105. Suppose further that B can put out his money at 5 per cent, interest : then if he put out $100 at interest, this is the sum which will amount at the end of a year to $105. Hence $100 is the sum, which A ought to pay at once, and this is called the PRESENT WORTH of the debt, and is evidently such a sum as would, if put out to ir- terest for the given time and rate, amount to the debt. The difference between the Debt and the Present Wortb, which is in the case under consideration $5, is called the Discount. DISCOUNT is therefore the abatement made when a sum of money is paid before it is due ar.d is equal to the interest on the Present Worth of the Debt. Ex. (1). Thus, to find the Proser.t Worth of $1781.40, due 4 years hence, reo*juiug interest at 5 per cent. PRESENT WORTH AND DISCOUNT. 179 The interest on $100 for 4 years at 5 per cent, is $20. .-. $120 has for its Present Worth $100 ; .-. $1 lias for its present Worth $|- ; .-. $1781.40 has for its present worth $ v - 8 - 1 -'*~ LL ^- -$1484.50. ,'. Present Worth required is $1484.50. Ex. (2). Find the Discount on $1781.40, due 4 years oence, reckoning interest at 5 per cent The Present Worth is $1484.50, as we have just shown ; the discount == $1781.40 -$1484.50 = $296,90 When the Discount alone is required to ^e found the following is the solution : The Interest on $100 for 4 years at 5 per cent, is $20. .-. $120 has for its Discount $20 ; ,-. $1 has for its Discount pfo ; .-. $1781.40 has for its Discount $- l78 \'^ xS ^ = $296.90 Ex. (3). What was the debt of which the discount for 8 months at 9 per cent, was $44.46 ? The interest on $100 for 8 months at 9 per cent, is $6. .-. $6 is the discount on $106 .'. $1 is the discount on ,: $44.46 is the discount on $785.46. Ex. (4). The interest on a certain sum bf money for two years is $50, and the discount for the sa^ne time and rate is $45. Find the sum and the rate per cent, per annum. Since $50 is the interest on a sum of money wnich sun> = (its Present Worth + its Discount ' = (its Present Worth + $45) . and $45 is the interest on its Present Wort) .*. 85 is the interest on 845 .-. $1 is tne interest on $* ; 10 PRESENT WORTH AND J3B30O&NT. .-. $50 is the interest on $?.* 4 ,* or $450, .*. $450 is the sum required. Again, the interest on $45 for 2 years is $5. .*. the interest on $45 for 1 year is \ ; .*. the interest on $1 for 1 year is $ .- the interast on $100 for 1 year $ l Ji* = w. .*. the rate is 5f per cent. NOTE I. From the above it -will be seen that the Dis- count oil any sura is the Present Worth of the interest of that sum for the same time and rate : thus $45 is the Present Worth of $50 for two years at a certain rate per cent. Ex. (5). If $20 be allowed off a bill of $420 due in 6 months, how much shall b* allowed off the same bi)' due in 12 months ? $20 is the discount off $420 for 6 months ; .'. $20 is the interest on $400 for 6 months; .-. $40 is the interest on $400 for 12 months; .-. 840 is the discount off $440 for 12 months; * $4VV is the discount off $1 for 12 months ; ... $1E X J^ is the discount off $420 for 12 months. /. the Discount required is NOTE II. The student will observe that the Discount is not proportional to either the time or the rate. Ex. (6). If $15 be the Interest on $115 for a given time, what should be the Discount off $115 for the same time : $ : 5 is the interest on $115 ; .-. $15 is the discount off U 30 ; the discount off SI ; is the discount off $115. ,'. the Discount required PRESENT WOBTH AND DISCOUNT. 181 i Ex. (7). If $10 be allowed off a bill of $110 due 8 months hence, what should be the bill from which the same sum is allowed as 4 months discount: $10 is the discount off $110 for 8 months ; /. ft 10 is the interest on $100 for 8 months ; .'. $10 is the interest on $200 for 4 months ; .-. $10 is the discount off $210 for 4 months ; .*. the sum required is $210. Ex. (8). Find the present worth of $842.70 for two years at 6 per cent. Compound Interest. The compound interest on $100 for 2 years at 6 per cent. U $12.86. .-. $112.36 has for its present worth .\ $1 has for its present worth .-. $842.70 has for its present worth $^ a ~?!^ 1I 113 * 3 6 = $750. .*, Present worth required = $750. Examples- (xclx) Find the Present Worth of (1) $5520, due 4 years hence, at 6 per cent. (2) $84.70, due 2^ years hence, at 9 per cent. (3) $615, due 1 year 4 months hence, at 7 per cent. * (4) $1120, due 16 months hence, at 5 per cent. (5) 618. 2*. 6d., due 8| years hence, at 4 per cent. Find the Discount on ^ ~(6) $636, due in 9 months, at 8 per cent. <*(7) $1884.30, due in 3$ years, at 10 per cent. (8) $637.50, due in 6 years, at 5 per cent. (9) 1165. 16*, 3d., due in 2| years, at 6 per cent. (10) 'Jf>2. 19s. 3d., due in 9 months, at 4^ per cent. (11) Find the present worth of $0945.75, due 8 years heuce, reckoning compound interest at 5 per cent. *(12) Find the discount on $245.25, due 1| years hence, at 5 per cent, compound interest, payable quarterly. v (13) A tradesman accepts $19-3125 in payment of a debt of $20g^ due in 12 months, in consideration of being paid *t once. What rate of discount does be allow ? 182 PBESKNT WORTH AND DISCOUNT. } (14) Find the present worth of a bill for $1127.10 drawn Jan. 1 at 4 months, and discounted Feb. 20 at 10 per cent, per annum. \ (15) The discount on $275 for a certain time is $25 ; what is the discount on the same sum (1) for twice that time, and (2) for half the time ? (16) A tradesman marks his goods with 2 price*, one for cash and the other for credit of 6 months ; what relation should the two prices bear to each other, allowing interest at 7| por cent.? If the credit price of an article be $33.20, what is the cash price ? %(17) If $98 be accepted in present payment of $128, due some time hence, what should be a proper discount off a bill of $128 which has only half the time to run ? (18) A certain sum ought to have $20.80 allowed as 8 months interest on it ; but a bill for the same sum due in 8 months at the same rate, should have $20 only allowed oft as discount in consideration of present payment. What is the sum and the rate per cent. ? 182. The Discount, of which we have been treating, is called Mathematical Discount or True Discount, to distinguish it from Practical Discount, of which there are two kinds: (1) The deduction made by a trader, when an account is paid to him before the time when he proposes to demand payment. It is then calculated as interest on the account. Thus if a trader gives notice on his bill that he will allow 10 per cent, discount for immediate payment, and if the amount of the bill be $25.50, he deducts $2.55, and the customer pays him $22.95. (2) The deduction made by a lender of money from the sum which lie proposes to lend. Thus if a borrower binds himself by a bill to pay $100 a year hence, and a discounter advances money on the security of this bill, at the rate of 5 per cent., he gives to the holder of the bill $95, and takes the bill. True Discount is the Interest on the Present Worth of a debt. Practical Discount is the Interest on the Debt itself. Hence Practical Discount is greater than True Discount. PRESENT WORTH AND DISCOUNT. 188 183. Three days, called Days of Grace, are always allowed, after a bill of exchange, or a promissory note is nominally due before it is Ugally due. Thus a bill drawn on July 5, for 8 months would be nominally due on Oct. 5, but legally on Oct. 8. Calendar months are always reckoned so that a bill of 3 months whether drawn on the 28th, 29th, or 80th of Nov. 1876, would be due on the 8rd of March 1877. The banker or money lender who discounts a note always charges interest on the note from the time it is discounted till it is legally due ; hence in computing Practical Discount of this nature interest must be calculated for 8 days more than the time the note has to run. Ex. (1). What would a banker gain by discounting on Sept. 21 a bill of $318.15, dated July 81, at 4 months at 5 per cent. ? The bill is legally due on Dec. 3. The number of days from Sept. 21 to Dec. 3 is 73. The interest on $318.15 for 73 days at 5 per cent, is $3.1815. The Mathematical discount is $3.15. . . the banker's gain is $.0315. Ex. (2). A merchant wishes to borrow $96.91 on a bill made on July 5 for 8 months. What must be the face of the bill, interest being reckoned at & per cent. ? Time between July 5 and Oct. 8 is 95 days. Interest on $100 for 95 days at 8 per cent, is $2. .-. a note for $100 would produce $97f ; 100 .*. a note for $ ----- would produce $1 ; 97f .'. a note for $ 96 ' 9 ^ Xl00 would produce $96.91. .-. the face of the note is $99. Examples, (c) (1) What is the difference between the trne and the bani discount of $950 for 3 mos. at 7 per cent. ? 184 EXAMINATION PAPERS. (2) A bill is drawn for $722.70 on July 17' at 2 months, and discounted on Aug. 11 at 7 per cent.; how much did the holder receive ? (3) Find the discount charged in discounting a bill for $7850 drawn &pril 9 at 7 months and discounted June 19th at 10 per cent. (4) For what sum must a note be drawn on July 3, at 3 months, so that discounted immediately it may produce $501.69, money being worth 7 per cent. ? (5) Find the difference between the true and bank dis- eounts on $5555 at 6 per cent, for 1 year. EXAMINATION PAPERS. I. (1) Explain the difference between Simple and Compound Interest. Find the Interest on $25000 for three years at 4 per cent, supposing Interest to make Capital at the end of each year. (2) The difference between the Compound and Simple Interest of a certain sum of money for 3 years at 4 per cent, is $3.80. Find the sum. (3) Find at what rate Simple Interest in two years a sum of money would amount to the same sum as at 4 per cent. Compound Interest. (4) Find the Compound Interest on $1000 at 8 per cent, per annum for 2 years .and 195 days. (5) A person puts out to interest $8000 at 4 per cent. ; he spends annually $300, and adds the remainder of his divi- dend to his stock. What is he worth at the end of 6 years ? II. (1) Explain the distinction between true discount and bank discount. Does the creditor or the debtor gain by computing the interest intead of the discount ? (2) Find the discount on $400, due one year hence, if money bear interest at 6 per cent, per annum. Calculate the interest on this discount for the same time, and show that it is equal to the difference between the interest and the discount of $400. (8) If 10 be the interest on 110 for a given time, what should be the discount of 110 for the same time ? EXAMINATION PAPEfcS. 186 (4) What must be the rate of interest in order that the discount on $10292 payable at the end of 1 year 73 days may be $372 ? (5) A tradesman who is ready to allow 5 per cent, per annum Compound Interest, for ready money, is asked to give credit for two years. If he charge $110.25 in his bill, what ought the ready money price to have been ? III. (1) A speculator borrowed $5000, which he immediately invested in land. Six months afterwards he sold the land for $7500, on a credit of 12 months, with interest. Money being at 6 per cent., what is the speculator's profit at the end of 12 months, at which time he pays $5000 ? (2) A merchant bought 43 cwt. 3 qr. of sugar at $5.25 per cwt., which he immediately sold at $7 per cwt., on a credit of 90 days, and then had the purchaser's note for the amount discounted in the bank, at 6 per cent. What profit did the merchant make? (3) Find the present worth of $1000 due 2 years hence at 5 per cent, per annum ; and show that the discount of the -given sum is equal to the interest of the present wor^b for the same time and at the same rate of interest ? (4) A man having lent $10000 at 5 per cent, interest, pay- able half-yearly, wishes to receive his interest in equal por- tions monthly, and in advance ; how much ought he to receive every month ? (5) Show that the interest on 266. 13s. 4d. for three months, at 4 per cent, per annum, is equal to the discount of J083 for 15 mos. at 8 per cent, per annum. IV. (1) How much may be gained by hiring money at 6 % to pay a debt of $6400, due in 8 months, allowing the present worth of this debt to be reckoned by deducting 5 % per annum discount? (2) The difference between the pimple and compound interests of a sum of money for 8 years at 8 per cent, is 1985.60. What is the sum? (3) The interest on a certain sum of money for two years is 71 16?. 7d., and the discount on the same sum, for the ame time, is .63 17*., simple interest being reckoned in both oases. Find the rate per cent, per annum, and the stun. 186 JBQtJATION OF PAYMENTS. (4) A offers $8000 for a farm ; B offers $9500. to be paid at the end of 4 years. Which is now the better offer, and by how much, allowing 5 per cent, compound interest ? (5) A person borrows money at 6 per cent, per annum, and pays the interest at the end of the year ; he lends it out at 8 per cent, per annum, payable quarterly, and receives the interest at the end of the year ; by this means he gains $269-18592 a year. How much did he borrow ? 'XXIV. Equation of Payments- 184. When several sums of money are due from A to B, payable at different times, it is often required to find the time, called the EQUATED TIME, at which all may be paid together, without injustice to A or B. When great exactness is demanded, interest must be added to the sums paid after they are due, and discount subtracted from the sums paid before they are due. But in practice the following rule is sufficiently accu- rate: Multiply each debt by the number of days [or months] after which it is due : add the results togetfar: divide this sum by the sum of the debts : the quotient will be the num- ber of days [or montJis] in the equated time. Take the following Examples : Ex. (1). If $300 be due from A to B at the end of 6 months, and $700 at the end of 9 months, when may both sums be paid in a single payment without unfair- ness to A or to B ? Number of months in equated time = -^^ -tti = w .*. the whole amount of the debt should be paid at the end of 7f months. The principle on which this solution depends is, that the interest of the money, the "payment of which is de- layed beyond the time at which it is due, is equal to the interest of that which is to be paid before it becomes dne. EQUATION OP PAYMENTS. 187 In the above example $800 is kept 2 months after it is due, and the interest on it for that time is the same as the interest on $840, $(300 x 2$), for one month. But $700 is paid 1| months before it is due, and the interest on it for that time is the same as the interest on $840, $(700 x 1|) for one month. Ex. (2). A is indebted to B in the following amounts : $500 due in 6 months ; $600 due in 7 months ; and $800 due in 10 months. Find the time when all these payments should be made together. 600 x 6 - 3000 600 x 7 - 4200 80D x 10 * 8000 1900 1900)15200 8 *. the equated time is 8 months. NOTE. This method is but a rough approximation, and can only be taken as eq litable when the various times of payment are not widely apart. It will, in short, be applicable only to cases which occur in the ordinary course of trade, and is therefore all that we require in the present work. It is also to be observed that the error involved in this method is slightly in favour of the payer, because interest is calculated on the payments made before they are due, instead of discount, in the algebraical process, from which the method is derived. See Appendix. Examples- (ci) What is the equated time of (1) $250 due 4 months hence, and $350 due 10 months hence. Find the equated time of (2) $300 due 3 months hence, $400 due 4 months hence, and $500 due 6 months hence. (3) Of a debt of $1400, $100 is due immediately, $600 at the end of 1 month, $400 at the end of 7 months, and the 188 EQUATION OF ACCOUNTS. remainder at the end of a year. At what time might th whole debt fairly be paid in one sum ? (4) A grocer ought to receive from a customer $50 at the end of 2 months, $30 at the end of 4 months, and $20 at the end of 6 months. What would be the proper time for re- ceiving the whole sum together ? (5) A debt is to be paid as follows : One-sixth now, and one-sixth every 8 months until the whole is paid. When might the whole debt be paid at once ? (6) If $450 be due in 16 months, and $250 be due in 13 months ; find the sum which if paid now would be equiva- lent to the whole debt at the equated time, interest at 4 per cent. (7) There is due to a merchant $800, one-sixth of which is to be paid in 2 months, one-third in 3 months, and the remainder in 6 months ; but the debtor agrees to pay one- half down. How long may he retain the other half so that neither party may sustain loss ? (8) A sold goods to B at sundry times, and on different terms of credit as follows: Sept. 30, 1868, $80.75, on 4 months credit; Nov. 8, 1868, $150, on 5 months credit ; Jan. 1, 1869, $30.80, on 6 months credit; March 10, 1869, $40.50, on 5 months credit; April 25, 1869, $60.30, on 4 months credit ; how much will balance the account June 2, 1869 ? (9) A owes B on the 1st of March the following sums : 140 due on 20th of April, .120 due on 14th of May, .380 due on 15th of June. On what day may B pay these debts together ? (10) M buys goods of j^, and has 6 months' credit from the date of invoice. The goods are delivered on 6 different days, to the following amount : 101. 14s. Wd. on Aug. 8, 144. 2s. Wd. on Sept. 5, 303. 18s. Wd. on Sept. 18, 757. Os. Sd. on Nov. 13, 123. 11s. 6d. on Nov. 28, 123. 11s. 6d. on Dec. 5. On the 13th January, N, who desires to receive all the debts in one payment, reckons that this payment should be made in 100 days. Show that this is approximately correct. EQUATION OF ACCOUNTS 185. EQUATION OF ACCOUNTS (also called "Averaging of Accounts " and " Compound Equation of Payments ") is the process of finding at what time the balance of an account can be paid without gain or loss to either party. EQUATION OF ACCOUNTS. 189 The BALANCE OF AN ACCOUNT is the difference between the two sides of it and is what one owes the other. Ex. Dr. A in Account with B. Or. 1877. Jan. 1 Feb. 4 Mar. 10 To Mdse < $500.00 600.00 800.00 1877. Feb. 10 Mar. 4 By Cash ft *< $1000.00 600.00 Jan. 1, 500X 0= Feb. 4, 600x34=20400 Mar.10, 800x68=54400 1900) 74800(89fj 5700 17800 17100 700 89 days from Jan. 1 is Feb. 9. Due Feb. 9... Feb. 10, 1000X00- Mar. 4, 600x22-18200 1600) 18200(8* 12800 400 8 days from Feb. 10 is Feb. 18. .$1900 Due Feb. 18 $1600 If the account be settled on Feb. 9 it is evident the credits would have been paid 9 days, or the time from Feb. 9 to Feb. 18, before they are due. This would have been a loss of interest to the credit side and a cor- responding gain to the debit side. Now as the settle- ment should be one of equity we find how long it will take the balance, $800, to gain the same interest that $1900 would in 9 days. If $1900 gain a certain interest in 9 days, $1 will gain the same interest in 1900X9 days, and $300 will gain the same interest in 1 -^^, or 57 days. Hence the balance became due 67 days before Feb. 19, or on Dec. 24. NOTE. Fractional parts of a day are not counted, unless the fraction amounts to half a day or upwards ; it then counts another day. 190 EQUATION OF ACCOUNTS. Hence we have the following rule : First find the equated time for each .aide of the account separately. Then multiply the amount due on that ride which Jills due FIRST, by the number of days between the dates of the equated times, and divide the product by the balance of the account. The quotient will be the number of days to be counted FORWARD from the LATEST DATE when the smaller side of the account falls due FIRST ; and BACKWARD when the larger side falls due FEBST. Examples- (cii) (1) Average the following account : DR. J. Hughes in account with S. Adams. CR. 1875. 1 1875. July 4 To Balance $375.90 Aug. 10 By Cash $316.00 Aug. 20 " Mdse. 815.58 Sept. 1 a 675.00 Aug. 29 44 ( 178.25 Sept 25 " Mdse. 512.25 Sept 25 387.20 Nov. 20 " Cash 161.75 Dec. 5 _ ; That which I eold for $7.20 I bought for $li^ or $6. PROFIT AND LOSS. 201 Ex. (4). If by selling coffee at Is. Id. per Ib. I lose 5 percent., what must I sell it,at to gain 5 per cent. ? That, which I sell at 95d., I bought for that, which I sell at Id., I bought for that, which I sell at 19d. 1 bought for 9 x 10 d. or 20d. Having thus found the cost price, we proceed thus : To gain 5 per cent, that, for which I gave lOOd., I wust sell for 105d. that, for which 1 gave Id., I must sell for $y%d. ; that, for which I gave 20d. , I must sell for 20x 105 d., or Is. 9d. .100 Or thus : In the first case, that which costs lOOd. sells for 95d. In the second case, that which costs lOOd., sells for 105d. ; .-. that which sells for 95d. must bring 105d. ; " M Id. must bring *rd. ; *' " 19d. must bring liL^l^d , 95 or Is. 9d..as before Ex. (5). A quantity of tea is sold for 83 cents, per pound, the gain is 10 per cent., and the total gain is What is the quantity of the tea ? That which sells for SI 10 cost $100 ; .*. the cost price per Ib. = |^ o .*. the gain on 1 Ib. = - But, the gain per Ib. X No. of Ibs. sold = total gain, or T 1 T - of $0-8of X No. of Ibs. sold = $48 ; No. of Ibs. sold = = 633f . 633f Ibs. is, therefore, quantity sold. 202 PROFIT AND LOSS. 194. When tea, spirits, wine and such commodities are mixed it must be observed that quantity of ingredients = quantity of mixture, cost of ingredients = cost of mixture. Thus, if a mixture is made of 1 gallon of ale at 8 cts. a gallon, 8 at 15 cts., 4 at 20 cts., aiid 12 at 7 cts. quantity of ingredients = (1-j- 34- 4 + 12) galls, or 20 galls.; cost of ingredients = (8-f-45-f-80-f-b4) c t s ., or $2.17. If I want to know what gain per cent. I shall make by selling this mixture at 26 cts. a gallon, I reason thus : 20 gall, at 26 cts. will sell for $5.20 ; .-. that for which I gave $2.17, I sell for $5.20 ; .-. $2.17 gains, ($5. 20$ 2. 17), $3.03; .-. $1 gains $3,03 . .'. $100 gains $IoPJ[ld>, or $139.68. .*. I gain $139.63 per cent. 195. In solving questions on Profit and Lose the student must be very careful to notice whether the gain is calculated on the selling price or cost price. Thus, it is sometimes said that a retailer's profit is 25 per cent, meaning that he gave 75 cents for an article which he sells for $1. His profit, in this case, is 88 per cent, on his outlay. Care must, therefore, be taken to express distinctly which is meant. The profit on a single transaction or set of transactions by no means represents a net profit, as it is not charged with a variety of expenses which belong to the business in general rather than to the set of transactions in question. Ex. If 100 articles of a given kind can be made in a week out of $40 worth of raw materials, cost of labour &c. being $10, fixed charges for rent &c. being $250 a year, find (1) the cost price of each article,- (2) the invoice price in order that a profit of 80 per cent, on the cost price may be realized, the following allowances be- ing necessary, viz., 10 per cent, commission to agents on money received for sales, and 12 per cent, for bad , and (8) the amount of profit in a year. pfcorrr AND LOSS. 808 (1) The fixed charges must be referred to the same unit of time as the rest of the estimate, viz. : 1 week Cost of 100 articles = $50 + $ VV 8 = $54-8077 ; .-. cost of 1 article = $0-548077. (2) The profit on capital may be regarded as part of the cost of production. It would be so, in fact, if the money were borrowed at 30 per cent, interest. 80 per cent, added to $-548077 gives |>J2*. Again, the commission is paid on the money actually received ; to provide for it, we must take the l s of glSOX'g48077 t or ftlOXl30X-348077. To 9X100 Next : 12 per cent, on bad debts means that 12 do not pay lor 88 who do. To provide for it, we take l ff of the selling price. The invoice price will, therefore, be $1 OOX IOX 1 30X'48077 (8) To find the profit we must take ^& of the cost price, and multiply by 100 x 52. Annual profit = |aoxiooxsax4>oTT Examples (1) If I buy an article for $3.20 and sell it for $4, what is my gain gain per cent. ? (2) If I sell goods for $2240 and gam 12 per cent., what was the cost price ? (3) If 375 yards of silk be sold for $1960, and 20 per cent. profit be made, what did it cost per yard ? (4) If, by selling wine at 17s. 5d. a gallon, I lose 6 per cent., at what price must I sell it to gain 15 per cent. ? (5) If, by selling goods for $544, I lose 16 per cent. , how much per cent, should I have lost or gained, if I had sold them for $672. (6) The manufacturer will supply a certain article at lid.; if a tradesman charge 2rf., what profit per cent, will" he make? (7) A tradesman's prices are 20 per cent, above cost price. If he allow a customer 10 per cent, on nig bill, what profit does he make ? (*) A tradesman's prices are 26 por cent, above ooft price. 204 SfOCKS AND SHARES. If he allow a customer 12 per cent, on his bill, what profit does he make ? (9) A man bnys goods at 29. 5*. 5d., and sells them at J622. 2. l$d. How much does he lose per cent. ? (10) A man buys goods at 15. 6s. 3d., and sells them again at jll. 15*. 9%d. How much does he lose percent. ? (11) A man buys goods at the rate of $96 per cwt., and sells 2 tons 14 cwt. 8 qr. 12 Ib. for $6000. How much has he gained or lost per cent, on his outlay ? (12) If 8 per cent, be gained by selling a piece of ground for $4125.60, what would be gained per cent, by selling it for $4202 ? (13) If 3 per cent, more be gained by selling a horse for than by selling him for $324, what must his original price have been ? (14) A grocer mixes 12 Ib. of tea at 2*. 6d. per Ib. with 4 Ib. at. 8*. 2Wg- and sVYfc. .'. Income for $1 in the 6 per cents is (-sWs VsVs ) of a $1 better than in the 5 per cents. .-. Income for $100 in the 6 per cents is 100 X (^Ws aWo of a SI better than in the 5 per cents. Now 100 X ('AW-sV&) = -91... per cent, required. : Ex. (11). A person transfers 5000 stock from a 3 per cent, stock at 72, and invests the proceeds in a 4 per cent stock at 90, Find the difference in his in- come. STOCKS AND 9HABEB 811 First, he aella 5000 stock at 72, and gets (72x50) or 3600. Then he invests 8600 in .the 4 per cent, stock at 90, and buys ^8.800X100 stock, or 4000 stock. Now his fint income on the 6000 stock was *M*- S , or 150. And his tecond income on the 4000 stock IB iMp, or 160 ; .. he increases his income by 10. Ex. (12). A person invests 1075. 10*. in Consols when they are at 89, and sells out when they are at 93| ; what is his gain, brokerage at ^ per cent, on each transaction. Here an annuity which costs (89-H-) is sold for i); .*. on 89$ the gain is 8$ ; 8$ .*. on 1 the gain is ggl, or ^- ; .% on 1075 10*. the gain is 1075. 5 x ^, or 43. 10, Ex. (18). A person invested in Bank, stock at 89$ and sold out at 108 J, and cleared $897.50 ; how much did he invest, brokerage being i per cent, on each tran- saction ? Here what cost $90 is sold for $103} ; .'. he gained $13.25 by investing $90 ; .. he gained $1 by investing $ff;f~j ; .-. he gamed $397.50 by investing $*^^ 9 -, or $2700. Ex. (14). A person having to pay $36()6 7 s j two years hence, invested a certain sum in the Toronto 6 per cent, city bonds to accumulate interest until the debt be paid, and also an equal sum next year ; supposing the invest- ments to be made when the stock was at 99, and the first year's interest also invested in stock, and the price to remain the same, what must be the sum invested on each occasion that there may be just sumcient to pay the debt at the proper time ? 212 STOCKS AND SHARE*, Every $99 invested will give $6 interest; .*. every $1 invested will give $^- interest. .-. $ sum invested will give $ sum X g % interest. Now $ sum X -fy invested will give $ sum x / ff X ^ interest. Hence at the end of the Becond year there were on hand the two sums invested. Two years' interest on the first investment = 2 x sum X re> One year's interest on the second investment = sum X A And the interest on the first year's interest = sum x A x & Or 2 sums + 3 x sum X / f + sum x X to meet .'. (2 + H + v8i) snm = $3606^ ; 3606 A ..aum = ^ 1A o a = S1650. Examples (ex). Find the value of (1) $7645 stock in the 6 per cents, at 95. (2) $9800 stock in the 5 per cents, at 80. (3) $7650 stock in the 7 per cents, at 118^ (4) .3850 stock in the 8 per cents., at 92. (5) .572 1O*. stock in the 3 per cents, at 91. How ranch stock will (6) $8400 buy in the 4 per cents, at 75 ? (7) $3757.50 buy in the 8 per cents, at 125 ? (8) $994.50 buy in the 7 per cents, at 117 ? (9) J2199 buy in the 3 per cents, at 91| ? (10) ^5527 10*. buy in the 3 per cents, at 92J ? What income is got from investing (11) $934.25 in the 6 per cents, at 101 ? {12) $4147 in 4 per cent, stock at 72 ? (13) $6720 in 5per cent, stock at 96.? (H) $3725 in 3 per cent, stock at 74*? (15) ,8475 10*. in 3 per cent, stock at 92 ? STOCKS AND SHAKES. 218 What amount of stock must be sold (16) In the S per cents, at 125 to 'produce $750 ? (17) In the Dominion 5's at 92 to produce $629 ? (18) In the 6 per cents, at 101 to produce $959.50 (19) In the 7 per cents, at 128 to produce $4096? What per cent, is made by investing in the (20) 8 per cents, at 120? (21) 5 per cents, at 95 ? (22) 6 per cents, at 104 ? (23) 3 per cents, at 75 ? When Greenbacks are at (24) 90, what is the price of gold 1 (25) 92, what is the price of gold ? (26) 84, what is the price of gold ? When gold is at a premium of (27) 10 per cent., what are " Greenbacks " quoted at ? (28) 25 per cent., what are "Greenbacks " quoted at? (29) 14 per cent., what is $5700 of American Currency worth ? What sum must be invested in the (30) 8 per cents., at 120 so as to produce an income oi $640. (31) 5 per cents., at 90 so as to produce an income of $3750 ? (32) 4| jper cents., at 67 so as to produce an income of $2790. What is the selling price of stock when (33) $550 stock in the 6 per cents, produce $558.25 ? (34) $7840 stock in the 4 per cents, produce $6664? (35) 840 stock in the 8 per cents, produce 773. 17*. ? - (36) What must I pay for U. S. 10-40's ( Interest at 6 %) that my investment may yield 6 per cent ? (37) Which is the better investment, the buying of 9 per cent, stocks at 25 per cent, advance, or 6 per cent, stocks al 25 per cent, discount, and how much per cent, better ? (38) The difference between the i ncomes derived from in- a certain sum in 6 per ce nt. stock at 126, and in 9 214 STOCKS AND SHAEES. per cent, stock at 210, is ^22. 10*. What is the amount in- vested? - (89) I sell ont of the 8 per cents, at 96, and invest tke proceeds in Railway 6 per cent, stock at par ; find by how much per cent, my income is increased. _ (40) If a 3 per cent, stock be at 91, how much must I invest in it, so as to have a yearly income of J805B, after pay- ing Id in the pound income-tax ? 93 (41) By selling out 4500 in the India 5 per cent, stock at 112, and investing the proceeds in Egyptian 7 per cent, stock, a person finds his income increased by 168. 15*. What is the price of the latter stock ? v (42) Find the alteration in income occasioned by shifting 3200 stock from the 3 per cents, at 86$, to 4 per cent, stock at 114& : the brokerage being per cent. (43) A owns a farm which rents for $411.45 per annum. If he sell the farm for $8229, and invest the proceeds in U. S. 6's, 6-20's of 84, at 105, paying per cent, brokerage, will his yearly income be increased or diminished, and how much ? (44) Through a broker I invested a certain sum of money in U.S. 6's, 5-20 at 107, and twice as much in U.S. 5'e 10-40 at 98, brokerage in each case per cent. My in- come from both investments was $1674. How much did I invest in each kind of stock ? ' (45) A purchased goods for which he was to pay $7000 hi currency, or $6500 in gold. Will he gain or lose by accept- ing the latter proposal, gold being at 125, aud how much ? - (46) I invest in the 3 per cents, at 92. They fall to 85, and I sell out and obtain a safe investment paying 5 per cent., but not subject to fluctuation of value. How long must I hold it before I shall make a profit by the change, in case 8 per cents rose to their former value ? - (47) I own $4000 Montreal. Bank stock paying an annual dividend of 14 per cent. I sell at 180 and. invest in Toronto Gaw Company stock at 125 and receive an annual dividend of 9 per cent. What change is made hi my income, broker- age being f % and f % on the respective transactions? c(48) A person bought stock at 95, and after receiving the half yearly dividend at the rate of 7 per cent, per annum, sold out at 92f and made a profit of $37.50. How much stock did he buy ? (49) Whether is it better to invest in the 6 per cents at 084, or in the 6 per cents at 85, brokerage being \ per cent.? BtAMINAlION PAPERS. (60) What sum must a man invest in the Dominion 6's at 101 in order to have a clear income of $1775.50, after pay- ing an income tax of If cents on the dollar on all over $400? (51) A gentleman has been receiving 12 per cent, on hie capital in Canada. He goes to England to reside, and in- vests it in the 3 per cents, at 94f , and his iacome in England is 2400. What was his income in Canada, being equal -to$4.86f ? (52) By selling out ^4500 in the India Five per Cent. Stock at 112f , and investing the proceeds in Egyptian Seven per Qent. Stock, A finds his income increased by 168. 15a. What was the price of the latter stock, brokerage on each transaction being ^ per cent. ? (53) The 6 per cents, are at 91 and the 7 per cents, at 102. A person has a sum of money to invest which will give him $3500 more of the former stock than of the latter. Find the difference of income he could obtain by investing in the two stocks. (54) One company guarantees to pay 5 per cent, on shares of $100 each ; another guarantees at the rate of 4| per cent, on shares of $80 each ; the price of the former is 124^, and of the latter $84 each ; compare the rates of interest which the shares return to the purchasers. (55) The present income of a railway company would justify a dividend of 3f per cent., if there were no preference shares ; but as $1200000 of the stock consists of such shares, wlr'eh are guaranteed 5 per cent, per annum, the ordinary shareholders receive only 3 per cent. What is the whole amount of stock ? (56) Received from my correspondent in New York $6150 U.S. currency, with instructions to deduct my com- mission at 2 per cent., and invest the remainder in Cana- dian Tweeds worth $1.03 per yard. How many yards should I send him, gold being quoted at 115 ? Examination Papers. I. (1) In a sale of goods for $728 there is a loss of 9 per cent. ; for what must 8 times the quantity be sold in order to gain 7 per cent. ? (2) If 20 per cent, be gained by selling an article for $2.10; what is the gain or loss per cent, when it is sold for $1.60 ? 216 EXAMINATION FAPEB8. (8) A grocer had 150 Ibs. of tea, of which he sold 50 lb&. at $1.80 per pound, and found he wa8 gaurng only 7^ per cent., but he wished to gain 10 per cent, on the whole. At what rate must the remaining 100 Ibs. be sold that he may attain his wishes ? (4) A tradesman adds 85 per cent, to the cost price of his goods, and gives his customers a reduction of 10 per cent, on their bills ; what profit does he make ? (5) A bill of $2520 due a year hence can be taken up now at 5 per cent, discount. Supposing a tradesman can employ his capital so as to obtain interest at the end of every quarter at the rate of 4 per cent, per annum, had he better BO employ it or take up the bill ; and what will be the difference to him? IL (1) A tradesman marks his goods with two prices, one for ready money, and the other for one year's credit, allow- ing discount at 5 per cent. If the credit price be marked $2.45, what ought the cash price to be ? (2) If goods be sold on condition to allow 10 per cent, discount, if payment be made at the end of six months, what discount ought to be allowed if payment be actually made (1) three months before, and (2) three months after the stated time, if money bear interest at 5 per cent, per annum ? (3) A person purchases goods at $1.20 per pound Troy weight and sells them again by Avoirdupois weight ; at what rate per ounce must he sell so as exactly to reimburse his outlay ? (4) What is meant when it is said that Consols are at 88i ? .What are they at when 9000 is paid for ,10000 Consols ? (5) A person sells $1200 stock in the 3 per cents, at 86, in order to invest in bank stock paying 8 per cent.; what price must he pay for it to be neither a gainer nor loser ? III. (1) I send $3060 to my agent in Montreal to invest in tea at 75c per Ib. He deducts his commission of 2 per cent, and purchases the tea. How many pounds do I receive and at what must I sell per Ib. so as to make a profit of 40 % after paying freightage $30 and insurance at the rate of ^ per cent.? (2) Bought land at $50 an acre ; how much must I ask an acre that I mav take off 25 per cent, from my asking EXAMINATION PAPERS. 217 price, and still make 20 per cent, profit on the purchase money ? (3) A buys silks at $2.25 per yard on a credit of 6 months. B buys the "same quality of silks for $2.15 per yard, cash. Which makes the best purchase, money being worth 10 per cent., and what must the goods be marked at to insure a gain of 25 per cent.? Or, if the silks be sold at $3 per yard, what profit per cent, does each make ? (4) A person buys an article and sells it so as to gain 5 per cent. If he had bought it at 6 per cent, less, and sold it for 5 cents less, he would have gained 10 per cent. Find the cost price. (5) A person buys 6 per cent, city of Toronto bonds, the interest on which is paid yearly, and which are to be paid off at par, 8 years after the time of purchase ; if money be worth 6 per cent., what price should he give for the bonds ? IV. (1) Bought cloth at $8 in gold, and sold at $4 in currency. Did I gain or lose by the transaction, and how much per cent, in currency, gold being at 118 ? (2) A merchant sold 24 cheese at $30 each. On one half he gained 30 per cent., and on the remainder he lost 80 per cent. ; did he gain or lose on the whole, and how much ? (8) A man wishing to sell his farm asked 86 per cent, more than it cost him, but .he finally sold it for 20 per cent, less than his asking price. He gained $528 by the transaction. How much did the farm cost, what was his asking price, and for how much did he sell it ? (4) A person having to pay $1085 at the end of 2 years invested a certain sum in 3 per cent, stock, allowing the dividends to accumulate until the. payment of the debt, and also an equal sum next year, and also the previous year's interest. If the investment is made and the debt paid when stock was at 73, what must be the sum invested on each occasion that there may be just sufficient to pay the debt at the proper time ? () A merchant's stock-in-trade is valued on Jan. 1, 1875, at $40000, he has $1750 in cash and owes $9350 ; during the year his personal expenses, $1500 are paid out of the pro- ceeds of the business, and on Jan. 1, 1876, his stock is valued at $39750, he has $2850 in cash and owes $7550. What is the whole profit of the year's transactions after deducting 6 per cent interest on the capital with which he began the year? 218 DIVISION INTO PROPORTIONAL V. (1) I received an 8 per cent, dividend on railway stock, and invested the money in the same stock at 80. My stock having increased to $13750, what was the amount of my dividend ? (2) How many shares of $50 each must be bought at 25 per cent, discount, brokerage If per cent., and sold at 16 per cent, discount, brokerage 1 per cent., to gain $121. 66f ? (3) What sum must be invested in United States 10-40'a bearing interest at 5 per cent., payable in gold purchased at par, to produce a semi-annual income of $400 U. S. currency, when gold is quoted at 175 per cent. ? (4) The charter of a new railroad company limits the stock to $1500000, of which 3 instalments of 10 per cent., 20 per cent., and 40 per cent, respectively having been paid in ; the cost of construction has reached $850000, and the estimated cost of completion is $850000. If the company call in the final instalment of its stock, and assess the stock- holders for the remaining outlay, what will be the rate, per cent. ? (5) A person invests $16380 in the 8 per cents, at 91 ; he sells out S 12000 stock when they have risen to 93, and the remainder when they have fallen to 85. How much does he gain or lose by the transaction. If he invests the pro- duce in 4 per cent, stock at 102, what is the difference in his income ? XXVIII. Division into Propoi*tional Parts. 202. Suppose 3 persons, A, J3, and C, to be in part- nership, and an arrangement made that the profits of the business, in which they are engaged, are to be di- vided into 6 equal parts, of which A is to take 3 parts, B 2 parts, and Q I part. The shares of A, B, and C are then said to be in the proportion of 8, 2, and 1. ET. (1). Divide $1275 among 8 persons, whose sha. es are to be in the proportion of 3, 5, and 7. This may be regarded as a case in which one holds 8 shares, one 5, and one 7, and they hold 15 shares in alL Hence, if we divide $1275 by 15, and we got the amount of one share, that is, amount of one share = $ --- = $86. Then one of the persons receives 3 X $85, or $255 ; the second receives 6 x $35, or $425; the third receives 7 X $85, or $595. DIVISION INTO PK P.;i:i IOXAL PA-.T3. Ex. (2), Divide- $837 shares are to be in propoi The common denominator of L '., nr\,1 $ is 30, .-, the shares are to bo in the proportion of . J : j, J". an 1 -/V; that is the proportion of 15, 10, and G; Now 15 + 10 4- 6 = 31. .*. amount of one share out of 31 shares == (y-^\ l ----- .$27. Then one of the partners receives 15 X $27, or $405 ; the second receives 10 x S27, or $270 ; the third receives 6 X $27, or $162. Ex. (3). A. rate of $4212 is to be paid by three town- ships, and the property on which it is levied is $2-1700 in the first, $37250 in the second, and $43350 in the third. What sum is paid by each ? Amount of property on which the rate is levied is $105300. Then $105300 has to pay a rate $4212. .\ $1 has to pay o. rate $TO I a oir ; *. $24700 has to pay a rate $^- J * ^_i_s ^ or ^^ . $37250 has to pay a rate $' 7 J J J J^ 2 l 9 , or $1490 ; $43350 has to pay a rate $*- 3 -J * 4 Q 2 ' 2 , or $1734. Ex. (4). Divide $1000 among A, B, and C, so that ^4 may have half as much again as /?, and B a third as much again as C. Eepresenting C's part by 1, B'spart will be 1$, and A 's part will be 1 ^ + ^ of 1 ^ =2 ; Mid, therefore, the parts are to bo as the numbers 2, 1J, 1, .*. All the shares ^=2 + 1^ + 1 = 4^ times C Y 's fihai'e. 4| times C"s = ^lOCO, $1000 Cs = 41 $230-769, Fs f of C's = $307-692. -4's = 2 times C's = $461-538. Ex. (5). Divide the number 237 into three parts, such that three times the first may be equal to 5 times the second and to 8 times the third. Take the first part as the unit; then by the question 220 DIVISION INTO PROPORTIONAL PARTS. the second part will be f of the first, and the third will te of the first. Sura of the parts = 1 + f + $ == ] f times the first. Hence, H times the 1st = 237, the 1st = 2:57 -4- i i = 120, the 2nd = f of 1st = f- of 120 = 72. the 3rd == f of 1st ~ f of 120 = 45. Ex. (6). Divide $3400 among A, B, and C, so that A may have $800 more than J of -#'s share, and YJ $000 less than ol C"s share. Representing C's share by 1, then L"s sliare = f of C's share $600 A 's share == f of B's sliare -f $800 = I ( J of C's - $600) + $800 = i of C"s -f 400 Sum of all the shares = C's + $ C's $600 J- C's + $400 = ! C's $200. .'. !'>; *300 -as. $8400 f C's = $3400 4- S200 = ^3600 C's ^ $1600. jB's = | of $1600 $600 = $600. A's = i of 11600 -1- $400 = $1200, Examples, (cxi). (1) Divide $60 into two parts proportional to 11 and 9. (2) Divide $2500 into parts proportional to 2, 3, 7, 8. (3) Divide $8470 into parts proportional to ," , and . (4) Gunpowder is made of saltpetre, sulphur, and charcoal, in parts proportional to 75, 10 and 15 ; how many pounds of each are contained in 12 cwt. of gunpowder ? (5) The sides of a triangle are as 3, 4, 5, and the sum of the lengths of the sides is 480 yards ; find the sides. (6) Divide $640 among A, B and C, so that A may have three times as much as B, and C as much as A and B to- gether. (7) Divide 100 apples among three boys, so that the first may receive 7 as often as the seeond receives 8, and the third may receive 5 as often as the second receives 4. (8) A bankrupt owns 272 10s. to A, ,354 5s. to B, and 490 10s. to C: his assets are .418 19s. 4^. What wiU of the creditors receive? DIVISION INTO PROPORTIONAL, PARTS. 221 (9) A force of police 1921 strong is to be distributed among 4 towns in proportion to the number of inhabitants in each ; the population being 4150, 12450, 24900, and 29050, respec- tively. Determine the number of men sent to each. (10) Divide 29 into an equal number of half-sovereigns, crowns, half-crowns, shillings, sixpences, and fourpences. (11) A piece of land of 200 acres is to be divided among four persons, in proportion to their rentals from surrounding property : supposing these rents to be ,500, 350, 800, and 90, how many acres must be allotted to each ? (12) Divide 2. 5. among A, B, and (7, so that for each threepenny piece received by A, B may receive a four- penny piece, and that there are as many shillings in the Bum received by O as there are sixpences in th sum received by B (13) Divide $10.40 among 6 men, 7 women, and 14 boys, so that each woman may have \ of each man's share, and each boy f of each woman's share. (14) *A number of men, women, and children, are in the proportions 2, 3, 6 ; divide $517.65 among them, so that the shares of a man, a woman, and a child may be proportional to 3, 2, 1, there being 9 women. (15) A man left his property to be divided among his 8 eons in proportion to their ages which are ^0. 18, and 12 years. The share of the youngest is $1440. What was the value of the property ? (16) Divide $5000 among A, B, and C, so that A may get $300 less than f of C's share, and C, $800 more than I of B's share. What are the shares ol each ? (17) Divide $5000 among A, B, C, and D, so that A may get f of B's share and $ii50 ; B, $200 more than $ of C's share, C, $100 less than A of D's share. What are the shares of each? (18) The sum of three fractions is ^||- ; and 22 times the first, 23 times the second, and 24 times the third give equal products. Find the fraction. (19) Divide the simple interest on $1171 for 13 years at 6 per cent, in parts which shall have the same relation as 908 TT TT TF (20) Of the boys in a school one-third are over 15 years of age, one third between 10 and 15. A legacy of $400 can be exactly divided amongst them by giving $ to each boy over 15, $} to each between 10 and 15, and $} to each of the rest. How many boys are there in the schooL 222 PARTNERSHIP. PARTNERSHIP. 203. When persons unite to carry on any particular branch of business the connection so formed is called a PARTNERSHIP. The method of working questions in partnership is the same as that explained in the preced ing article. Ex. (1). A, B, and C entered into partnership to carry on a mercantile business for two years. A puts in $9000, B $6000, and C $3000. They gained $4500. Wbat is each one's share of the gain ? The whole capital invested is $18000. Then $18000 gain $4500. .-. $1 gains iffffif or $*. $9000 gains $^- ? = $2250. $6000 gains $^- = $1600. $3000 gains $^p = $750. B>nce J. 'a share of the gain is $2250; IT*, $1500; and CTa, $750. Ex. (2). A, B, and C entered into partnership for trading. A put in $600 for 4 months ; B $400 for 5 months, and C $200 for 6 months. They gained $980 ; what was each man's share of the gain ? $600 for 4 months = $2400 for 1 month. 0400 5 " = $2000 " $200 " 6 =$1200 " The whole capital is equivalent to $5600 for 1 month. Then $5600 gain $980 ; /. $1 gains $7nnnj=TV A2400 gains $H?-JJiI = $420. $2000 gains 8M2LI = $350. $1200 gains $ 12 ^ = $210. .-. A's share is $420, B's $350, and (7s $210. Examples (cxii) (1) Two men jointly purchased a house for $2592, the firet contributing $864 towards the purchase and the second PARTNERSHIP. 228 $1728. They afterwards rented the house for $182.75 annu- ally. What share of the rent ought each to have ? (2) A t B, and C jointly rented a pasture for 8 months, agreeing to pay $22.50 for the use of the same. A put in 6 horses, B put in 18 cows, and C 90 sheep. Considering each horse as equivalent to two cows, and each cow as equal to 3 sheep, what part of the rent ought each to pay ? (3) A, B, and Centered into partnership for- speculating in cotton, their joint capital being $25780, of which A fur- nished f, B contributed $ of the remainder, and G the bal- ance. Their clear profit was 20 per cent, of the original in- vestment. How should it be divided ? (4) A starts a business with a capital of $2400 on the 19th of March, and on the 17th of July admits a partner B with a capital of $1800. The profits amount to $943 by the 31st oi December. What is -each person's share ? (5) D and E enter into partnership ; D puts in $40 for 8 months, and E $75 for 4 months. They gain $70. What is each man's share in the gain ? (6) A, B, C are partners ; A pute in $500 for 7 months, B $600 for 8 months, and C $900 for 9 months. The profit is $410. What is the share of each ? (7) Three, graziers hire a pasture for their common use, for which they pay 8106. One puts in 10 oxen tor 3 months, another 12 oxen for 4 months, and the third 14 oxen for 2 months. How much of the rent should each pay ? (8) Two m^n complete in a fortnight a piece of work for which they are paid $29.55. One of them works alternately 9 hours and 8 hours a day. The other works 9^ hours for 5 days in the week, and does nothing on the remaining day. What part of the sum should each receive ? (9) A and B begin to trade in partnership. A puts in $400 at first, and $500 at the end of two months. B puts in $300 at first, and $000 at the end of three months. The profit at the end of the year is $470. How should this bf divided ? (10) Johnston and Wilson formed a copartnership in business for 2 years. Johnston at first contributed $3000 to joint capital, and at the end of 12 months put in $1500 more. Wilson at first put in $3500, but at the e'nd of 15 months from the beginning withdrew $1000. At tho end of the firs; year they admitted Miller into the firm, he contributing $2250. Their joint profits were $1248. How ought this t< be apportioned ? 224 PABTNEHSHIP SETTLEMENTS. (11) A and B rent a field for $88.20. A puts in 10 horses for 1 months, 30 oxen for 2 months, and 100 sheep for 8 months ; B, 40 horses for 2 months, 50 oxen for 1 months and 115 seeep for 8 months. If the food consumed in the same time by a horse, an ox and a sheep, be as the numbers 3, 2, 1, what proportion of the rent must each pay ? (12) A person in his will directed that his property should be given to A, % to B, to C, and J to D ; shew that this disposition cannot be fulfilled. If his property amount to $1886.50, dispose of it so that their shares may have to one another the relation he intended. (13) A, B, and Chad each a cask of rum containing re- spectively 86, 54, and 78 gallons. They blended their rum and then refilled their casks from the mixture ; how much of the rums of A and B are contained in C'B cask ? (14) A rents a house for $187.20, at the end of 4 months he takes in B as a co-tenant, and they admit C in like man- ner for the last 2 months ; what portion of the rent must each of them pay ? PARTNERSHIP SETTLEMENTS. 204. When a partnership is dissolved, either by mu- tual consent or by limitation of contract, the adjustment of the proceeds between the members is called a Partner- ship Settlement. If the RESOURCES are found to exceed the LIABILITIES, the difference is termed NET CAPITAL ; if the Liabilities exceed the Resources, tlie difference is NET INSOLVENCY. The investment of the partners is the Net Capital at commencement. If the net capital at closing exceeds the net capital at commencement, the difference is the NET GAIN ; if the opposite, NET Loss. This net gain, or net loss, is then shared between the partners in accordance with the original agreement between them. This division is frequently not made in exact proportion to the amount invested ; sometimes the skill of one partner is considered equal to the capital of another ; sometimes a stated salary is allowed each partner according to his ability or reputa- tion; and sometimes, where unequal amounts are in- vested, interest is allowed each partner on his invest- ment ; but whatever allowance is made suck allowance must be classed as a liability and go to reduce the aain PARTNERSHIP SKTTLEMENTS. 225 Ex. (1). A and B are partners. The following is a state- ment of their property and debts :, they have cash, $3240 ; merchandise, % 2575 ; Bills Receivable, $860; J. Brown owes 011 account, $375. TJiey owe on Bills Payable, $1250; and /J. Jones on account, $370. .4 invested at commencing, , $2500, and drew out, during business, $560. B invested 2500, *nd drew out, during business, $280. They agreed to ', share equally in gains and looses. What was the net gain ? and what was the net capital of each at closing ? RESOURCES AND LIABILITIKK. i OWNERSHIP. DR. CR. DR. OR. $3:i40 $1'250 560 A withdrew. $2500 L'575 370 j liftO/f 2500 660 840 Total investment 5000 " witudvawu fc*0 Firm's net investment 4160 375 1<320 70.50 Resources at Closing. 11/20 Liabilities 5430 Present Worth of Firm. 4160 Credit excess of Ownership. 1270 Net Gain. 025 A's share of net gain. 635 '* ' \ Hence 4's present net capital $2500 $560 4- $635 = $2575, and 's present net capital = $2500 $280 -f $635 = $2855. Examples, (cxiii). (1) A and B having conducted business 1 year as partners, close with the following resources and liabilities: They have cash, $3456; Mdse., $2120; Bills Keceivable, $1874; E. Corby owes $630. They owe on Bills Payable, $3250; W. Smith on account, $346. A invested $150 and withdrew $175. B invested fcloOO and drew out $3 15. What is the net gain, and net capital of each at closing'? (2) A and B close business as follows: They have cash, $1424; Mdse., $1562; Fixtures, $383; Mortgages Receiv- able, $3485 ; Bills Receivable, $826, . They owe on Bills Payable, $2450 ; on accounts, $124 >. A invented $6000, and a debt for $1000 was assumed by the firm, and paid during business. He drew out $685; and is allowed interest on capital invested, $420. B invested $4i)00, and drew out $1860, and is allowed interest on capital, $280. A is to share f- and B f of gains and losses. What is the net loss ? What is the net capital of each ? (3) A and B close business, and wish to know the finan- cial standing of each. They have cash, $2263, ami Jleal ALLIGATIoN. Estate worth $5000. They owe on Mortgages, $3846 : on Notes, $4462 ; on Personal Accounts, $G7o. A invested $GJi!0 and drew out $2800*. B invested $4000, drew out, $5560, and is allowed for extra services $-'f>0. A shares | and B I of the gains and losses. "What is the net loss ? What is the financial standing of each ? XXIX. Alligation. 205. Alligation is the process by which we find the ni'-aa or average price of a compound when we mix or unite two or more articles of different values. Ex. (1). A merchant has brown sugar worth 8 cents per pound. New Orleans worth 9 cents, and refined sugar worth 14 cents ; how many pounds of each kind must he use in order to form a mixture worth 12 cents per pound ? By selling the mixture at 12 cents per pound, we see that 8 cents (brown) gains 4 cents on 1 Ib. ; /. 1 cent, is gained on \ Ib. 9 cents (New Orleans) gains 3 cents on 1 Ib. ; /.I cent, is gained on \ Ib. 14 cents (refined) loses 2 cents on 1 Ib. ; .*. 1 cent, is lost on \ Ib. Now with every cent, gain he must combine a cent, loss, hence lie must have | Ib. at 8 ets.\ (3 Ibs. at 8 cts. | Ib. " 14 cts. [ J6 Ibs. " 14 cts. | Ib. " 9 cts. (" ~]4 Ibs. " 9 cts. | Ib. " 14 cts.) (6 Ibs. " 14 cts. He must, therefore, have 3 Ibs. brown sugar, 4 Ibs. New Orleans, and 12 Ibs. refined. "\Yo may show that these quantities will make the mixture required, as follows: 8 Ibs. at 8 cts. per Ib. = 24 cts. 4 Ibs. ' " 9 cts. " = 30 cts. 12 Ibs. " 14 cts. " = 168 cts. 1 9 cts. = whole mixture. 228 cts. = value of mixture. Hence if 19 Ibs. bo woitli 228 cents, rib. is worth -i 2 o a = 12 cts. Or we may reason thus : The 1 ct. gained on the | Ib. of brown exactly balances the 1 ct. lout on the 5 Ib. of the ALLIGATION. 227 refined. Hence he must take \ Ib. of the brown and i Ib. of the refined, or 2 Ibs. of the one and 4 Ibs. of the other. Similarly, for every 2 Ibs. of New Orleans, there must be 8 Ibs. of refined. As 4 Ibs. of refined were required to bal- ance the brown, and 8 Ibs. of the refined to balance the New Orleans, there must be 7 Ibs. of the refined in the compound. Therefore the respective quantities are 2 Ibs. brown, 2 Ibs. New Orleans, and 7 Ibs. refined. From the above, we see that in examples of this kind a variety of answers may frequently be obtained, and all of them may be correct. To ascertain their correctness we resort to the method of proof given in this example. 206. From the above analysis we derive an easy practical method of solving such Question*, Ex. (2). How much sugar at 10, 18, 15, 17, and 18 cents per pound must be taken to make a mixture worth 16 cents? We proceed as follows : Differences.! 16 | Write down the prices in a vertical column, and place the 10 | 1 Differences between these prices 13 1 and the mean in a second verti- 15 | 1 cal column to the left. Now ... ! ... take 1 @ 10, 1 @13, and 1 @15, 17 2, 4, 6, 8 (the lowest that can be taken) ; 18 4, 3, 2, 1 this would represent a loss of 10 as compared with the mean; and this loss must be balanced by taking the necessary multiples of the differences 1 and 2, which represent gain as compared with the mean. It is seen that this loss ol 10 can be made up in four ways, :>y 2@17, 4@18, 4@17, U @ 18, 6@17, 2@18, 8@ 17, and Other combinations may be made, as e.g. : 8 10 I 1 Here 1 @ 10, 1 @ 13, and 2 @ 15,- 3 13 j 1 give loss of 11, which can be made up 1 15 j 2 by multiples of the. differences 1 and ... j ... ! ... 2 (opposite 17 and 18) in jive ways, 1 I 17 ' 1, P, 6, 7, 9 as indicated. 2 i 18 6, 4, 8, 2, 1 Also, 228 ALLIGATION. 10 1 18 2 15 1 17 18 Again, 1, 3, 5, 7, 9, 11 6,5,4,3,2, 1 6 I 10 2 3 13 1 1 15 1 1 17 'a, 2 18 7, 2, 4, 6, 8, 10, 12, 14 7,6,5,4, 8, 2, 1 Where 1 @ 10, 2 @ 13 and 1 @ 15 give 13 loss; which may be made up in nx different ways. Where 2 @ 10, 1 @ 13 and 1 @ 15 give loss of 16, which may be made up in seven ways. Also, 6 10 1 8 13 1 1 15 3 T 17 2* 2 18 5, Where 1 @ 10, 1 @ 13 and 3 @ 15 give loss of 12, which may be made up in five ways ; and thus an indefi- nate number of combinations may be 2, 4, 6, 8, 10 formed. 5,4,3,2, 1 It should be observed that if the differences opposite the prices less than the mean are together greater than the sum of the other differences (as in the example) we assign num- bers (the lowest possible) to the prices less than the mean FIRST, and vice versa ; e.g. of the latter case : How much coffee at 25, 24, 23, 22, 21, 19, 18 and 17 cents per pound must be taken to make a mixture worth 20 cents per pound ? Here the sum of the dif- ferences in excess of the mean is greater than that of the differences below the mean; we therefore assign first numbers to the prices which are greater than the mean, viz., 1 @ 21, 1 @ 22, 1 @ 23, 1 @ 24, and 1 @ 25 ; this gives a gain of 15, which may be balanced as 19, 1 @ 18, and 4 @ 17 ; or by 2 @ 19, 2 @ 18 and 8 @ 17, &c., &c. Ex. (8). A grocer has 12 Ibs. of brown sngar, worth 10 cents per pound, which he wishes to mix with clari- fied dugar worth 16 cents per pound, so that the mix- tt'8. 8 uu 17 4, 3, 2, 1, 1, 2 2 18 1, 2, 3, 5, 4, 2 &c. 1 19 1, 2, 3, 2, 4, 3 1 21 i" 2 22 i 3 23 i 4 24 i 5 25 i 1 above by 1 ALLIGATION. 229 fare may be worth 14 cents per pound; how many pounds of clarified sugar must he take ? Proceeding as in the previous examples, without refer- ence to the quantity of the brown sugar, we find that there must be 1 Ib. brown sugar to 2 Ibs. clarified sugar. But as 12 Ibs. of brown sugar are required, we must multiply each of these quantities by 12 in order that the gain and loss maybe equal. We shall therefore have 12 x 2=24 Ibs. of clarified sugar. Ex. (4). A grocer wishes to mix 20 Ibs. of sugar, worth 9 cents per pound, and 10 Ibs. worth 12 cents per pound, with clarified sugar, worth 15 cents, so that the compound may sell for 13 cents ; how much of the clarified must he fake ? 20 Ibs. at 9 cents = $1.80 10 Ibs. at 12 cents = $1.20 80 $3.00 Then, if 80 Ibs. is worth $8, lib. " $& = 10 cents. The value of 1 Ib. of the mixture is, therefore, worth 10 cents. The question may then be read as follows : How many pounds of clarified sugar, worth 15 cents per pound, must be mixed with 80 Ibs. of another kind of sugar, worth 10 cents per pound, so that the mixture may be sold for 13 cents per pound ? The question in this form has already been fully ex- plained. Ex. (5). A merchant has West India sugar worth 8 cents per pound, and New Orleans sugar worth 18 cents. He wishes to combine these so as to make a barrel, containing 175 Ibs., which he may sell at 11 cents per pound. How many pounds of each kind must he take ? Solving the question without reference to the 175 Ibs., we find that 2 Ibs. of West India sugar, and 8 Ibs. of New Orleans sugar will form a mixture worth 11 cents per pound. Adding these quantities, we find that they form a mixture of 5 Ibs, But *bp ^nircd mixture i& 280 EXCHANGE. to contain 176 Ibs., or 85 times 6. We shall therefor* have 85 x 2 Ibs. * 70 Ibs. West India sugar. 85 X 8 Ibs. = 105 Ibs. New Orleans sugar. Examples, (cxiv) (1) What quantities of coffee, worth 23 and 86 cents respectively per pound, must be mixed together so that the compound may be sold for 80 cents a pound ? (2) What quantity of oats at 35 cents per bushel, rye at 60 cents per bushel, and barley at 80 cents, must be taken u form a mixture worth 55 cents per bushel ? (8) How much tea, worth respectively 55 cents and 75 cents per pound, must be mixed with 30 Ibs., worth 90 cents per pound, in order that the compound may be sold for 70 cents per pound ? (4) How much water will it require to dilute 60 gallons of alcohol, worth $1.50 per gallon, so that the mixture may be worth only $1.20 per gallon ? (5) How many gallons of kerosene oil, worth 60 cents per gallon, must be mixed with 12 gallons of coal oil, worth 86 cents, and 8 gallons of Aurora oil, worth 66 cents, so that the compound may be sold for 50 cents per gallon ? (6) A farmer has 16 bushels of corn, worth 48 cents per bushel, and 12 bushels of oats, at 34 cents per bushel, which he wishes to mix with rye, at 60 cents, and barley, at 80 cents, in order to sell the compound at 66 cents per bushel. How many bushels of rye and barley will be required ? (7) A confectioner mixes three different qualities of candy worth respectively 14 cents, 18 cents, and 30 cents per pound, so as to make a box of 84 Ibs.; how many pounds of each sort must he take so as to sell the compound at an average price of 24 cents per pound ? (8) A farmer has three different qualities of wool, worth respectively 83 cents, 87 cents, and 45 cents per pound. He wishes to make up a package amounting to 120 Ibs., which he can afford to sell at 89 cents per pound, flow many pounds of each kind must he take ? XXX. Exchange. 207, l"he term Exchange is here used for giving or receiving in the money of one country a sum equal in value to a sum of money of another country. For ex- EXCHANGE. 281 ample, if an English merchant pays to a Frmch mer- chant 100 sovereigns and receives in return 25UO francs, it is a case of Exchange. In countries which carry on considerable trade with each other, the debts reciprocally due from the one to the other are generally nearly equal. In England there is always a large number of persons indebted to others in America, and likewise a large number in America owing money in England. Now if coin, or specie, as it is called, were sent from Euglaiiu to pay the debts in America, and from America to England, the specie would have to be transmitted twice, and would neces- sarily involve risk, loss of interest, and expense of tran- sportation. To avoid this risk, &c., Bn^s OF EXCHANGE are used to liquidate debts reciprocally due between two places without any actual transmission of money. 208. A BILL OF EXCHANGE is a written order, ad- dressed to a person in a distant place, directing him to pay a certain sum of money, at a specified time, to another, or to his order. The person who signs the bill is called the DRAWER, or MAKER. The person to whom it is addressed is the DRAWEE, and after the Drawee agrees to pay it, and writes "accepted" with his signature and the date, across the face of it, he be- comes the ACCEPTOR. The person to whom the money is to be paid is the PAYEE ; if he transfers payment to another he ENDORSES it, i.e., he writes his name across the back of it and becomes responsible for its payment in case the Drawee fails to make payment. 209. The Par of Exchange between two countries denotes the nominal value of a unit of coinage in one country, as estimated in terms of a unit of coinage in the other country. As we supposed the exports from England and Amer- ica to be equal, creditors hi England will be as anxious to sell bills on America as debtors to buy them, and the exchange will deviate but slightly from the par of Ex- change. But if the exports from America are in excess uf those from England, or the Balance of Trade is in 232 KXCHANGH. favor of America, the claims of America in England will exceed its liabilities, and the English will give more than the par value of such bills to avoid the cost of transmitting specie ; and on the other hand, the export- ers in America not finding sufficient purchasers for all their bills on England, will sell them at less than their par value. Now the real rate of exchange depending on the balance of trade is called the COURSE OF EX- CHANGE ; and it is at a premium or discount according as it is above or below the par of exchange. Of course no one would give a premium greater than the cost of transmitting specie. But if the balance of trade is against England as regards America, but in favor of England as against France, the English merchant may find it advantageous to remit to France, and then for France to remit'to America, and this mode is adopted when the co,urse of exchange by this circuitous route is less than the direct course of exchange. The finding the course of exchange between two places, by com- paring the courses of exchange between them and one or more intervening places is called ARBITRATION OP EXCHANGE. The arbitration is Simple when only one place intervenes, and Compound when more than one. Bills of Exchange are usually drawn in sets, three bills constituting a set These are distinguished from one another by being called the first, second, and third of exchange. These are forwarded by different routes so as to guard against delay or their being lost. The first that arrives is paid, and the other two become void. 210. By Act of Parliament the value of the pound sterling was fixed at $4. This was much below its intrinsic value, which is now fixed at $4.86f. The rates of exchange which are quoted in commercial papers are still calculated at a certain per cent, on tfie old par of exchange. Exchange is at par between Great Britain and Canada when it is at a premium of 9 per 0ent., for $4 increased by 9J per cent., equals $>4.86f. [CHANG*. FORM OF DRAFT OB INLAND BILL OF EXCHANGE. $1000. Toronto, July 12, 1877. At ten days' sight, pay to the order of Adam Miller & Co., One Thousand Dollars, value received, and charge Stamp to account of W. E. JONEB. To. J. Smith & Co., Montreal. FORM OF, A FOREIGN BILL OF EXCHANGE. Exchange for 200. Toronto, July 12, 1877 Three days after sight of this first of exchange (second and third of same date and tenor unpaid), pay to Stamp Adam Miller & Co., or order, Two Hundred Pounds Sterling, value received, and charge tne same to the account of W. B. TAYLOR. To Geo. H. Simpson, Banker, London. FOREIGN MONEYS OF ACCOUNT, With the. par value of the unit, as fixed by commercial iwige, expressed in dollars and cents. AUSTRIA. 60 kreutzers = 1 florin (silver) = $-485 BELGIUM. 100 cents = 1 guilder or florin ; 1 guilder (silver) = -40 BRAZIL. 1000 rees = l milree = -828 BRITISH INDIA. 12 pice = 1 anna ; 16 annas = 1 Company's rupee = -445 BUENOS AYRES. 8 rials = 1 dollar currency, mean value = '93 CANTON. 10 cash = 1 candarines; 10 cand. = 1 mace ; 10 mace = 1 tael = 1-48 CUBA, COLUMBIA, AND CHILI. 8 rials = 1 dollar = ... 1.00 DENMARK. 12 pfenning = 3 skilling ; 16 skilling = 1 marc ; 6 marcs = 1 rix-dollar -52 FRANCE. 10 centimes = 1 decime ; 10 decimes = 1 franc = -186 GREECE. 100 lepta = 1 drachme ; 1 drachme (sil- ver) = -166 HOLLAND. 100 cents = 1 florin or guilder; 1 florin (silver) = -4p SiS* EXOHANGB. HAMBUBO. 12 pfenning = 1 schilling ; 16 achil. = 1 niarc ; 8 marcs = 1 rix-dollar = 6/ MEXICO. 8 rials = 1 dollar = 1.0 POKTUGAL. 400 rees 1 cruzado ; 1000 rees = 1 milree or crown = ..... 1.12 PBUSSIA. 12 pfennings = 1 grosch (silver) ; 30 gros- . chen = 1 thaler or dollar = -69 RUSSIA. 100 copecks = 1 ruble (silver) = '78 SWEDEN. 48 skillings = 1 rix-dollar specie = 1.06 SPAIN. 84 maravedis = 1 real of old plate * = -10 8 reals = 1 piastre ; 4 piastres = 1 pistole of ex- change ; 20 reals vellon =' 1 Spanish dollar = ... l.(K TURKEY. 8 aspers = 1 para ; 40 paras = 1 piastre (variable) about '096 VENICE. 100 centesimi = 1 lira = .... *186 VALUE OF FOREIGN COINS. Sovereign of Great Britain. 4.863 Milree of Azores .83} Crown of England 1.216 Roal-Vellou of Spain ... .05 Half Crown of I-'.ngland .608 Real-Plate of Spain .10 Shilling of England . .. . .24i Pistole of Spain . 39? Franc of France isj Rial of Spain 12 Five Franc Piece of France .98 Pista^een 18 Livre Touruois of France ... .18* Cross Pistareen .16 Forty Franc Piece of France Crown of France 7.66 106 Ruble (silver) of Russia ... Imperial of Russia .75 783 Louis-d'Or of France Florin of the Netherlands... Guilder of the Netherlands. Florin of South Germany ... 4.56 .40 .40 .40 I)o;il>loou of Mexico Half-Joe of Portugal Lira of Tuscany and Lom- bard v 15.60 8.53 .16 Thaler of Rix-Dolla r of Prus- sia atrtl North Germany Lira of Sardinia Ounce of Sicily .186 240 Rix- Dollar of Bremen .78? Ducat of NII pies 80 Florin of Prussia Marc-Banco of Hamburg .. .2-4 .35 Crown of Tuscany Florence Livre . ... 1.05 .15 Genoa Li'vre Florin of Saxony, Bohemia * Geneva Livre .21 and Trieste .. .. .48 Leghorn Dollar "... .90 Florin of Nuremburg and Swiss Livre .27 Frankfort .. .40 Scudo of Malta .40 Kix-Dollar of Denmark .... 1.00 Turkish Piastre .05 Specie-Dollar of Denmark.. 1.C5 Pagoda of India 184 1 06 Rupee of India 44S Milraa of Portugal.... 1.12 Taelof China.... 1.48 Ex. (1). A broker in Toronto sold a bill of exchange on London, the face of which was for 750. 8s. ; what did he receive for the bill, exchange being quoted at ^The old plate real is not a coin, but is th denomination ia which exchangee are usually made. SXOHANGB. 285 Since 1 = $4$ Xl.lOJ, i.e., $4f increased by 10$ per cent .*. 750. 4 = $ 750.4 X4fxl.l0i. =- $3676.96. .-. he got = $3676.96 for the bill. Ex. (2). What is the value in English money of 4528'7 francs, when the course of exchange between Paris and London is at 25-3 francs per pound sterling ? ^ince 25-3 francs = 1, 1 franc = ^ & .'. 4528-7 Jrancs = *%?-> or 179., Ex. (3). A merchant pays a debt of 4379 milrees in Portugal with 971. Us. 9|d ; what is the course of exchange in pence per milree ? 971. Us. 9fd. = 932727 farthings Then since 4379 inilrees = 982727 farthings, 1 milree ? "y" farthings, or 23 j iar- .*. the course of exchange is 58| pence per milree. Ex. (4). If 11-65 Dutch florins are given for 24-42 francs, 352 florins for 407 marks of Hamburg, and 58 marks for 32 silver rubles of St. Petersburg ; how many francs should be given for 932 silver rubles ? Here 1 silver ruble =- 5 4^|- in; 1 mark = ^f- florins, 1 florin = J ; - francs: 1165 .-. 1 silver ruble- -4^5- x-fj? X f^f francs, or 8-8 ..ancs ; .-. 932 silver rub!es = 932x3 % 3 francs, or 3075*6 francs. Ex. (5). A New York merchant remits 27940 florins to Amsterdam by way of London and Paris, at a time when the exchange of New York on London is $4'885 for 1, of London on Paris is 25-4 francs for 1, and of Pans on Amsterdam is 212 francs for 100 florins ; per cent, brokerage being paid in London and in Paris, how many dollars will purchase the bill of exchange ? 236 EXCHANGE. ct*. fa cf Since 100 florins = 212 francs, )1\ '- l florin =fuo- francs. IT** But to buy a bill of 100 fr. requires a bill of 100 fr. .'.to buy a bill of 1 fr. requires a bill of f^ fr- gain 25.40 fr. = 1 ; ' but to buy a bill of 100 requires . , '** . rofits is to be divided in proportion to the capital employed. t?ind the net receipts of A and B. (3) Bills on Amsterdam, bought in London at 12 florins 15 cents per 1 sterling, are sold in Paris at 57 florins for 120 francs ; what is the course of exchange between London .;ud Paris ? (4) On the 1st Jan., A brought into a business $1400, and on 1st April $'2000 more ; on the 1st June he took out $1600, and 3 months after this he brought in $2400. B brought into the business $2000 ; 4 months after this he took out $000, and on the 1st Nov. brought in $2600. l^heir clear profit for the year is $4032. How much ought each to receive ? (5) A cask contains 12 gals, of wine and 18 gals, of water ; another cask contains 9 gals, of wine and 3 gals, of water ; how many gallons must be drawn from each cask so as to produce by their mixture 7 gals, of wine and 7 gala, of water ? III. on his capital, of which he spends .1200 per annum in house and other ex- penses. At the end of 4 years he finds himself in possession 4 times as large as what he had at commencing business ; what was his original capital ? (5) There are two mixtures of wine and water, the quan- tities of wine in which are respectively '34 and '46 of the whole. If a gallon of the first is mixed with two gallons of the second, what decimal part will the wine be in the com- pound, and how much per cent, will the first mixture be strengthened ? XXXI. Ratio and Proportion. 211. If A and B be quantities of the same kind, the relative greatness of A with respect to B is called the RATIO of A to B. 212- The ratio of one quantity to another quantity is represented in Arithmetic by the fraction, which ex- presses the measures of the first when the second is taken as the unit of measurement. Thus if 5 shillings be the unit, the measure of 8 shillings is $, and the ratio of 3 shillings to 5 shillings is represented by the fraction f . The words " the ratio of 8 shillings to 6 shilling^ " are abbreviated thus : 8 shillings : 5 shillings. 213. Katies may be compared with each other by comparing the fractions by which they are represented. Thus 2 pence : 5 pence is represented by J and 3 pence : 7 pence is represented by f Now | = it, and I = J| .'. ?is greater than | and .-. 8 pence : 7 pence is greater than 2 pence : 5 pence. When we thus compare the ratios existing between two pairs of quantities, it is not necessary that all fowr quantities should be of the same kind ; it is only neces- sary that each pair should be of the same kind. 242 RATIO AND PROPORTION. For example, we can compare the ratio of 4 shillings to 7 shillings with the ratio of 7 days to 12 days, and finding that | is less than T 7 y, we may say that the ratio of 4 shillings to 7 shillings is less than the ratio of 7 Jays to 12 days. 214. When the ratio symbol ( : ) is placed between two numbers we may substitute for it the fraction symbol. Thus if we have to compare the ratios 2 : 8 and 5:7, we effect it by comparing the fractions f and \ , 215. Ratios are compounded by multiplying together the fractions by which they are represented, and ex- pressing the resulting fraction as a ratio. Thus the ratio compounded of 2 : 3 and 5 : 7 is 10 : 21. 2 and 8 are called the TERMS of. the ratio 2:8. 2 is called the ANTECEDENT and 3 the CONSEQUENT of the ratio. 216. Ratios are either direct or inverse. A direct ratio is the quotient of the antecedent divi- ded by the consequent. An inverse ratio, or reciprocal ratio, is the quotient of the consequent divided by the antecedent. Examples- (cxvi) (1) Compare the ratios 2 : 5 and 4 : 9, (2) Compare the ratios 17 : 39 and 19 : 41. (3) Compare the ratios 4 : 7, 8 : 15 and 13 : 24. (4) Compound the ratios 5 : 7, 13 : 15, 21 : 91, and 45 : 52. (5) Compound the ratios 3| : 4, 3 : 7, l\: 3|, 2 : If (6) If the ratio be 25 and the consequent $1.25, what is the antecedent ? (7) How much does the ratio 36x4x3:12x16x2 exceed that of 60 -r- (3x5) : 20x2 ' 8 ? (8) What is the ricprocal ratio of \ : ; of 2fc : 7'9 ? (9) A owns a farm of 180 acres. There are 36 sq. miles in the township in which it is situated. What is the rela- tion of the latter to the former ? PROPORTION. 243 (10) The ratio 63 : 52 results from compounding four ratios together; three of these are 7:8, 1'2 : 15, and 4- : -* ; express the fourth ratio in its simplest form. (11) What effect has adding the same quantity to both terms of a ratio. PROPORTION. 6l7. PBCPORTION consists in the equality of two ratios. The Arithmetical test of Proportion is therefore that the two fractions representing the ratios must be equal. Thus the ratio 6 : 12 is equal to the ratio 4 : 8, be- cause the fraction T 6 y = the fraction . The four numbers 6, 12, 4, 8, written in the order in which they stand in the ratios, are said to be in propor- tion, or proportional*, and this relation is thus ex- pressed 6 : 12 s= 4 : 8. The two terms 6 and 8 are called the EXTREMES. 12 and 4 the MEANS. .The sign of equality is usually expressed thus, :: and then the ratios read 6 is 12 as 4 is to 8. 218. When four numbers are in proportion, the product of the extremes = the product of the means. For example, if 6 : 1 2 :: 4:8 6 X 8 -=: i2 X4 For, since r e .> = $, by hypothesis, C X 9 __ 4X12 m ' ' Y 2 x 8 8x12* JTow the denominators of these fractions are equal, and therefore the numerators must also be equal, that is 6 x 8 =- 4 X 12 From this it is evident that if three out of the four numbers that form a proportion are given, we can find the fourth, 244 * Ex. (1). Find a fourth proportional to 8, 15, 7. 8 : 15 = 7 : number required, .*. 3 X number required = 15 x 7, .*. number required = * g * y = 85. Ex. (2). What number has the same ratio to 9 that 8 has to 5? 8:5 = number required : 9, ;. o X number required = 3x9, .'. number required = 2 ^ 7 = 5J. 219. Three numbers are said to be in CONTINUED PROPORTION when the ratio of the first to the second is equal to the ratio of the second to the third. ' Thus 3, 6, 12 are continued proportion, for | = A- The second number is called a MEAN PROPORTIONAL between the first and the third. Ex. Find a mean proportional between 6 and 24. 6 : required number = required number : 24 ; .-. required number X required number = 6 X 2 ; .*. square of required number = 144, . . required number is 12. 220. When two quantities are connected in such a way, that when one is increased 2, 8, .... times, the other is also increased 2, 8, times, they are in direct proportion. For example, if 1 Ib. of sugar cost 9 cents, 2 Ibs. will cost 2x9 cents, 8 Ibs. " 3x9 cents; hence 7 Ibs. " 7x9 cents, And 25 Ibs. " 25 x 9 cents, .-. 7 Ibs. : 25 Ibs. : : 7 x 9 cents : 25 x 9 cents. That is, the cost of sugar is directly proportional to its weight. 221. When two quantities are connected in such a way, fiat when one is increased 2, 8, times, the utiier is diminished 2, 8, times, they are inversely SIMPLE PROPORTION. 245 proportional ; thus, if one man can mow a field in 12 days, 2 men can mow it in half the time, or in y* days ; 8 men in a third of the time, or in \g. days, &c. hence four men can mow it in -^ days ; and 12 " " f| days; 4 men : 12 men : : -}-| days : ~ 4ays ; that is, the number of men required to do a certain work is \nversely proportional to the number of days, or vice versa. Examples- (cxvii) (1) Arrange 4, 3, 9 and 12 so that they may be in propor- tion. (2) ind the second term when 18, 2*6 and 1-8 are the other 3 terms of a proportion. (3) Find a mean proportional to -038 and '00152. (4) If A = 3$ of B, and 0=5 of 5, find the ratio of A to C. (5) Find a fourth proportional to 6, 7, and 15. (6) Find a fourth proportional to f , ^, and f . (7) Find a fourth proportional to '8, '16, arid -09. (8) Fiud a mean proportidnal to 14 and 56. (9) Find a mean proportional to ^ rand f f . (JO) Divide $1587 among A, B, C, D, so that A's shari. to J5's share =6:5, IPs share : (7* share = 4:3, and C\, : D's share = 3:2. SIMPLE PROPORTION OR RULE OF THREE. 222. When Three terras of a proportion are given to nd ike fourth it is a SIMPLE PROPORTION. In a simple proportion we have two ratios given; one of these has both terms, the other is incomplete, having only one term. Two of the given terms must be of one kind and the third and the answer of another kind. Ex. (1). If 5 horses eat 20 bushels of oats in a given time, how many bushels will 8 horses eat in the same time ? Here the number of bushels consumed is directly proportional to the number of horses, hence 5 : 8 :: 20 bu. : bu. required; ;. bu. required = 8 - =82. 246 SIMPLE PROPORTION. Ex (2). If 6 men can do a piece of work in 5 days, in what time can 9 men do the same work ? Here the time is inversely proportioned to the number of men, hence 9 : 6 :: 5 days : days required ; ;. days required = - 8fc. Ex. (8). If 8 cwt. 1 qr of hay cost $2.21, what should 8 t. 5 cwt. cost ? Here the cost is directly proportional to the quantity. Hence 8 cwt. 1 qr. : 8t. 5 cwt. :: $2.21 : dollars required ; Here we reduce the 1st and 2nd terms to the common denomination, quarters, and the proportion be- comes 13 : 260 :: $2.21 : dollars required; .-. Dollars required= ***!. =$44.20. From these examples we deduce the following rule : Write the given number that is of the same kind as the required fourth term, for the third term of tlie proportion. Tlien consider from the nature of the question whether the answer is to be greater or less tlw,n the third term. Ij greater, place the larger of the two remaining numbers in the second place; if less, in the first. Then having re- duced the first and second terms to the same denomination, multiply the second and third terms together, and divide the product by the first term. The quotient will be the answer required. NOTE. After the third term has been written down the order of the other two may be ascertained by a question. Thus, in Ex. (1) : " If 6 horses eat 20 bu., will 8 horses eat more or less than 20 bu. ?" More ; hence 5 : 8. In Ex. (2) : " If 6 men do a piece of work in 5 days, will it take 9 men a longer or shorter period than 6 days ?" Shorter ; hence 9 : 6. COMPOUND PROPORTION. 247 Examples, (oxviii) (1) A person after paying an income tax of 7d. in the .1 Las a net income of 1247. 10s. 5d. ; 'what was his gross income ? (2) A watch which is 10 minutes too fast at 12 o'clock uoon on Monday, gains 3in. 10s. a day ; what will be the time by the watch at a quarter past 10 a.m. on the following Saturday ? (3) In running a 3 mile race on a course ^ of a mile round, A overlaps B at the middle of the 7th round. By what distance will A win at the same rate of running ? (4) A watch was 6/ T min. slow at noon ; it loses 12 min. in 20 hours ; find the true time when its hands are together for the fourth time after noon. (5) If 4 men or 6 women or 9 boys can perform a piece oi work in 27 days, in what time can (1) 5 men and 9 women perform it ? and (2) 5 men and 8 boys perform it ? (6) If 14 1 shares of a property are worth $116.15, what are 5| shares worth ? (7 ) A floor can be covered by 32 yards of carpet 7 quar- ters wide ; how many yards of Brussels carpet 26 in width will cover the same room ? (8) Two clocks, of which one gains 4m. 15s. and the other loses 3m. 15s. in 24 hours, were both within 2 min of the true time, the former fast and the latter slow, at noon on Monday ; they now differ from one another by half an hour ; find tlie day of the week and the hour of the day. (9) If 6336 stones 3| ft. long complete a certain quantity of wall, how many similar stones of 2| ft. long will raise a like quantity ? (10) A besieged town, containing 22400 inhabitants, has provisions to last 3 weeks ; how many must be sent away that they may be able to hold out 7 weeks ? COMPOUND PROPORTION. 223. Where Jiv.*, seven, nine, &c.,- terms of a propor- tion are given to find a aixth, eighth, tenth, &c., term it is called, COMPOUND PROPORTION OH THE DOUBLE RULE OF THREE. In Compound Proportion there are three or more ratios given, all being complete but one. 248 COMPOUND PROPORTION. A Compound Proportion ie produced by multiplying together the corresponding terms of two or more simple proportions. Thus, 12 : 6 : : 4 : 2 9:8:: 6:2 5 :. 4 : : 10 : 8 multiplied together produce the proportion 540 : 72 : : 240 : 32. Ex. If 6 men in 8 days, working 10 hours a day, can reap 24 acres of wheat, how many acres cu.Jd 10 men reap in 15 days of 12 hours each ? 6 : 10 : : 24 : acres required 8 : 15 10 : 12 480 : 1800 : : 24 : arces required. /. acres required = ^ 8 ^ = 90. 24 the term of the imperfect raito is put in the 8rd place ; the other ratios are then considered separately and treated as in Simple Proportion. After all the ratios have been stated, all the first terms are multi- plied together for a new first term and similarly with the second terms. The answer is then got as in Simple Proportion. NOTE. I. Before compounding the complete ratios it is convenient to cancel all the factors common to the first terms, and to the 2nd or 3rd terms. When any of the 1st and 2nd terms are not of the same denomina- tion they must be reduced to a common denomination before proceeding with the solution. NOTE II. Before stating the question it is conveni- ent to write down the terms of the supposition under one another and opposite these to place the correspond- ing terms of the demand with an x opposite the term of the same name as the answer required. Thus, in l&e above example 6 men 10, 8 days 16, 10 hours 12, 24 acres x. COMPOUND PROPORTION. 249 Examples (1) If 18 men in 12 days build a wall 40 feet long, 3 feet thick, and 16 feet high, how many men must be employed to build a wall 120 yards long, 8 feet thick, and 10 feet high, in 60 days ? (2) An engineer engages to complete a tunnel 3f miles long in 2 years 10 months ; for a year and a half he em- ploys 1200 meni and then finds he has completed only three- eighths of his work ; how many additional men must he employ to complete it in the required time ? (3) Two sets of men perform the same amount of work. Each man in the first set is stronger than each one in the second in the ratio of 7 to 6 ; the first set works 6 days a week for 10 weeks, and the second set 5 days a week for 7 weeks. If there are 9 men in the first set, how many are there in the second ? (4) If 20 men can excavate 185 cubic yards of earth in 9 hours, how many men could do half the work in a fifth of the time ? (5) At the siege of Sebastopol it was found that a certain length of trench could be dug by the soldiers and navvies in 4 days, but that when only half the navvies were present it required 7 days to dig the same length of trench. Compare the amount of work done by the navvies with that done by the soldiers. (6) Two elephants which are ten in length, 9 in breadth, 36 in girt, and 7 in height, consume one drona of grain ; how much will be the rations of 10 other elephants, which are a quarter more in length and other dimensions ? (7) How many revolutions will be made by a wheel which revolves at the rate of 360 revolutions in 7 minutes, while another wheel, which revolves at the rate of 470 in 8 min., makes 658 revolutions ? (8) A piece of work is to be done in 36 days ; 15 men work at it 15 hours a day, but after 24 days only f of it is done ; if three more men are put on, how many hours a day must all work to finish it in the given time ? (9) If 248 men, in 5^ days of 12 hours each, dig a ditch of 7 degrees of hardness, 232 yds. long, 3| yds. wide, and 2 yds. deep ; in how many days of 9 hours each, will 24 men dig a ditch of 4 degrees of hardness, 387 yds. long, 5 yds. wide, and 3 yds. deep? (10) If 5 compositors in 16 days of 11 hours each, can compose 25 sheets of 24 pages in a sheet, 44 lines in a page, 250 THE METRIC SYSTKM. and 40 letters in a line, in how many days of 10 hours each, can 9 compositors compose a volume (to be printed in the same kind of type), consisting of 36 sheets, 16 pages to a sheet, 50 lines to a page, and 45 letters to a line ? XXXII. The Metric System. 224- The Metric System of Weights and Measures is now in use in many countries of Europe. The follow- ing is an acount of the system as it is established in France, where it originated at the end of the last century. The basis of all measurement is the METRE, a measure of length equal to the ten-millionth part of the distance from the North Pole to the Equator. The length of the Metre in English Measure is 89.37 inches, nearly. Units of Metric Measures. 1. LENGTH. The METRE. 2. SURFACE. The ARE = 100 square metres. 8. SOLIDITY. The STERE = 1 cubic metre. 4. CAPACITY. The LITRE = the cube of the tenth part of a metre. 5. WEIGHT. The GRAMME, which is the weight of a quantity of distilled water which fills the cube of the hundredth part of a metre. The Tables of Weights and Measures under the Metric System are constructed upon one uniform principle. Prefixes derived from Greek and Latin are attached to each of the units. Greek Prefixes. \ Deca stands for 10 times Hecto stands for 100 times i ,, Kilo stands for 1000 times f 11 Myria stands for 10000 times Latin Prefixes. Deci stands for the 10th part \ Centi stands for the 100th part J- of the unit. Milfi stands for the 1000th part) THE METRIC SYSTEM, 25J Tims, A decametre = 10 metres. A hectolitre = j.00 litres. A kilogramme = 1000 grammes. A myriametre = 10000 metres. Uso, A decilitre = -1 litre. A centimetre = -01 metre. A milligramme = '001 gramme. NOTE. In English measures the following are rough ap- proximations of some of the French measures : The Kilogramme is about 2J Ib. Avoird. The Litre is about If pints. The Kilometre is about 5 furlongs. The Hectare is about 2 acres. MEASURES OF LENGTH. 10 decimetres (dcm.) 1 metre (m.). 100 centimetres (cm.) " 1000 millimetres (mm.) * 1000 metres 1 kilometre. 1 inch = 2.539954 centimetres. t l foot = 3*047945 decimetres. 1 yard = 0-914383 metres. 1 mile = 1-609315 kilometres. NOTE. A rough rule for converting French metres into English yards is to add 10 per cent, to them. Thus 40 metres are nearly equal to 44 yards. MEASURES OF SURFACE. 100 square decimetres (sq. dcm.) = 1 square metre or centiare (sq. m.) 10000 ' centimetres (sq. cm.) = " 1000000 " millimetres (sq.mrn.) = 100 square metres 1 are. 10000 " 1 hectare. 1 square inch = 6-4518669 sq. cm. 1 " foot == 9-2899683 sq. dcm. I " yard = 0-83609715 sq. m. 1 acre = 0-40467101 hectare. 252 THE MBTKIC 8YSTGEM. MEASURES OF CAPACITY. 1000 cubic decimetres (cb. dcm.)...l cubic metre or ster^ 1000000 cubic centimetres (cb. cm.) " 1000000000 cubic milltmetres (cb. mrn.) " 1 cubic decimetre 1 litre. 1 " inch == 16-386176 cb. cm. 1 foot = 28-815312 dcm. 1 gallon == 4-54345797 litres. MEASURES OF WEIGHT. 1 cubic centimetre of distilled water at 4C. at the sea's level in the latitude of Paris is 1 gram (grm.) 1000 cubic centimetres of distilled water weighed under :he same conditions 1 kilogram (kilo.) 1000 grams (grms.) 1 kilogram. 10000 decigrams .... " 100000 centigrams ... " 1000000 milligrams .... 1 grain = 0*06479895 gram. Itroyoz. = 31-108496 grams. 1 Ib. avd . = 0-45859265 kilo. 1 cwt. = 50-80237^89 kilos. Examples- (cxx) (1) What is the fundamental unit IB this system ? Whence and why was it chosen ? (2) Name the units of weight and capacity, and show how larger and smaller meatnires are attained. (3) Give the English equivalents of a kilometre and kilo- gram. (4) How many millimetres are contained in 5 metres ? (5) How many decimetres are equivalent to 106725 millimetres ? (6) Required the number of milligrams in 15 cb. cm. of water measured at 4C. ? (7) How many millimetres and centimetres are repective- ly contained in 0.437 of a decimetre. (8) How many square centimetres are there in 15.5 square metres ? MEASUREMENT OF AREA. 25* (9) How many square decimetres are contained in 108642 square centimetres ? (10) Define the gram and litee. How many grams are contained in 1.725 kilograms ? (11) How many milligrams are there in a decigram ? How many decigrams in a kilogram ? (12) How many centigrams are contained in 2.567 kilo- grams ? (13) Required the number of milligrams contained in 6 culic centimetres of water measured at 4C. (14) In an English inch are contained 25.3995 millimetres. How many kilometres are there in a mile ? (15) A gallon is equal to 4 .543 litres. How many cubic centimetres are contained in one pint ? (16) Three pipes furnish respectively 30 litres, 45 litres, and 80 litres an hour. What quantity of water do they sup- ply together in 24 hours ? XXXIII. Measurement of Area. 225. The Unit of Measurement, by which we measure Area or Surface, is derived from the unit of Length. Thus, if we take an inch as the unit of length, and con- struct a square whose side is an inch, this Square Inch may be taken as the Unit of Area, and the measure of any given area will be the number of times it contains this unit, in accordance with the remarks in Art. 58. Let ABDC be a rectangle, and let the side AB be 4 inches in length, and the side AC 8 inches in length, A - - B 254 MEASUREMENT OP AREA. Then, if the Unit of Length be an inch, the measure of AB is 4, and the measwre of AC is 8. Divide AB, AC into four and three equal parts respec- tively, and draw lines through the points of division parallel to AC, AB respectively. Then the rectangle ABDC is divided into a number of equal squares, each of which is a square inch. If one of these squares be taken as the Unit of Area, the measure of the area of ABDC will be the number of these squares. Now this number is the same as that obtained by multiplying the measure of AB by the measure of AC : that is, measure of ABDC = 3x4 = 12 ; .*. area of ABDC is 12 square inches. Hence to find the area of a rectangle we multiply the measure of the length by the measure of the breadth, and the product will be the measure of the area. Ex. (1). A rectangular garden is 48 feet long and 25 feet broad, what is its area ? Taking a foot as the unit of length, and therefore a square foot as the unit of area, measure of the area = 48x25 = 1200 ; .'. the area is 1200 square feet. Ex. (2). A rectangular board is 2 ft. 7 in. long and 1 ft. 4 in. broad, what is the area ot its surface ? Taking 1 inch as the unit of length, and therefore 1 square inch as the unit of area, measure of the area = 31 X 16 == 496 ; .. the area is 496 square inches. Or we might take 1 foot as the unit of length, and then measure of area=2^ x H=f |-* * = V = 8* : . '. the area is 3f square feet. Ex. (8). The length of the side of a square croquet- ground is 49 yards, what is its area ? Taking 1 yard as a unit of length, area = (49 X 49) sq. yds. = 2401 eq. yds. atflASUfcEMENT OF AREA. 266 NOTE, Observe the difference between the expres- sions 49 yards square and 49 square yards. The former refers to a square whose side is 49 yards, and whose area is 2401 square yards, the latter to a surface whose area is 49 square yards. Ex. (4). A rectangular bowling-green is 56 yards long and 42 yards broad. Find the distance from corner to corner. By Euclid I. 47, we know that in a right-angled tri- a^agle the square on the side opposite the right angle is equal to the sum of the squares on the sides containing the right angle. Hence the square of the measure of the side opposite the right angle is equal to the sum of the squares of the measures of the sides containing the right angle. Thus in our present example, square of measure of distance from corner to corner = (56 X 5.6) -h (42 X 42) = 4900 ; .". distance is 70 yards. Examples- (cxxi) Find the area of the rectangles having the following dimensions : (1) 7 ft. by 5 ft. (2) 13 ft. by 10 ft. (3) 22 ft. by 13 ft. (4) 5 ft. 4 in. by 2 ft. 3 in. (5) 17 ft. 5 in. by 8 yd. 2 ft. (6) 5 yd. 1 ft. by 4 yd. 2 ft. (7) 12 yd. 2 ft. by 5 yd. 1 ft. (8) 6 yd. 2 ft. 3 in. by 2 yd. 1 ft. 5 in, (9) 7 yd. 2 ft. by 5 yd. 2 ft. 6 in. Find the area of the squares whose sides have the following lengths : (10) 5^yd. (11) 37yd. (12) 17f ft. (13) 29$ ft. (14) 9ft. 7 in. (15) 3 ft. 4 in. (16) 7 yd. 1 ft. 5 in. (17) 15 yd. 2 ft. 3 in. Find the breadth of the following rectangles, having given the area and length : (18) Area 176 sq. ft., length 11 ft. (19) Area 71 sq. ft. 100 sq. in., length 9 ft. 8 in. 256 MEASUREMENT OF AREA. (20) Area 854 sq. ft. 84 sq. in., length 97 it. 8 m. (21) Area 1 acre, length 440 yd. (22) Area 5 acres, length 275 yd. (23) Area 6 ac. 1 ro. 36 po., length 267 yd. 2 ft. What are the sides of the squares whose areas are (24) 81 sq. ft. (25) 256 sq. ft. (26) 1178 sq. yd. 7 sq. ft. (27) 33 ac. 4305 sq. yd. ? (28) A rectangular field is 225 yards in length and 120 yards in breadth ; what will be the length of a straight path from corner to corner ? (29) A rectangular field is 300 yards long and 200 yards broad ; find the distance from corner to corner. (30) A rectangular plantation, whose width is 88 yards, contains 2 acres ; find the distance from corner to corner. (31 ) What is the length of the diagonal of a square, whose aide is 5 inches ? (82) The area of a square is 390625 square feet ; what is the length of the diagonal ? CARPETING ROOMS. 226. If we know the area of the floor of a room, we know how many square inches of carpet will be required to cover it. Carpets are sold in strips, and when the width of a strip is known, we shall know how much length of carpet will be required to cover a given sur- i'ace. For instance, if the surface be 162 square feet, and the carpet selected be 27 inches wide, we reason thus : 162 sq. ft. = 162 X 144 sq. inches ; .. length of carpet required = J- 62 ^ 144 in. == 864 in. = 24 yards. Then we find the cost of 24 yards at $1.20 per yard to be $28.80. Examples, (cxxii) How many yards of carpet, 27 inches wide, wiU be required for rooms whose dimensions are ; (1) 15 ft. by 13 ft. (2) 25 ft. by 12 ft. 6 in. (8) 22 ft. 4 in. by 20 ft. 3 in. (4) 27 ft. by 14$ ft. (5) 85 ft. 4 in. by 27 feet 8 in. BIEAGTTRE^IEXT OF . AK1CA. 257 Find the expense of carpeting i^oms wnose dimen- sions are : (6) 13 ft. by 14 ft. with carpet 30 i.ichfes wide, at $1 a yd. .(7) 22 ft. by leU ft,, with carpet 27 inches wide, at $1.80 a yard. (3) 29 ft. 9 in. by 23 ft. 6 in., with carpet a yard wide, at $1.03 a yard. (9) 34 ft. 8 in. by 13 ft. 3 in., with carpet f yard wide, at 3s. 4.W. a yard. PAPERING THE WALLS OF A ROOM. 227. To find the quantity of paper required to cover one wall of a room, we find the area of the surface of the wall by taking the product of the measures of the length and breadth of that wall, the latter being the same as the height of the room. Hence we shall find the area of the four walls of the room if we take the measure of the compass of the room a'nd multiply it by the measure of the height. By the compass of a room we mean the length of a string stretched tight on the floor, going all round the room- Deductions for doors and windows and fireplace must be made in practice. Suppose then we have to find how much paper is re- quired for the walls of a room, whose length is 22 ft. 3 in., breadth 17 ft. 4 in. and height 9 ft. 6 in. We first find the compass of the room thus : ft. in. f f ur sides. . compass of the room. To get the area of paper required, we multiply the measure of the compass of the room by the measure of the height, thus : area =(9J X 79fc) sq. ft. = 19 ** 75 sq. ft. = 752,\ sq. ft. V * 258 MEASUREMENT OF A.R3A. NOTE. Papers, like carpets, are sold in strips, and if we know the width of a strip we shall know how many feet in length will be required to cover a given surface. Thus, in the room under consideration, if the paper be 20 inches wide, length of paper required =(762^-^) ft.= ?-* ft. =451 ft. Examples, (cxxiii) How many square feet of paper will be required for rooms whose dimensions are : (1) Length, 19 ft. ; breadth, 16 ft. ; height, 9 ft.? (2) Length, 24 i ft. ; breadth, 18 ft.; height, 10 ft. ? (3) Length, 25 ft. 7 in. ; breadth, 19 ft. 4 in. ; height, 9 ft. 9 in, ? <4) Length, 23 ft. 5 in.; breadth, 18 ft. 7 in.; height, 9 ft. 6 in. ? Find the expense of papering rooms whose dimen- sions are : (5) Length, 18 ft. ; breadth, 14 ft. ; height, 8 ft. ; with paper 16 inches wide, at 20 cents a yard. (6) Length, 20 ft. 6 in. ; breadth, 17 ft. 4 in.; height, 9ft.; with paper 20 inches wide, at 10 cents a yard. (7) Length, 30 ft. 8 in.; breadth, 26 ft. 5 in.; height, 10 ft. 6 in.; with paper 2 ft. wide, at Sd. a yard. (8) Length, 26 ft. ; breadth, 21 ft. ; height, 10 ft. ; with paper 20 inches wide at 9ught A and B to pay C to settle their accounts ? (21) Find the value of * , 1 ,JL X5 Hi 6 14X3 ~~15 and reduce to its lowest terms |f. (22) Express as vulgar fractions in the lowest terms 24-0025 and '0008125 ; and divide 1-1214 by 5'34 and 1121-4 by -534. ^(23) What fraction is 7 cwt. 4 Ib, of 3 tons 1 qr. (long ton) ? How often must one go round a square field of 10 acres to run 1 mile ? (24) A gunboat's crew, consisting of a lieutenant, a gun- ner, and 15 seamen, captured a prize worth .399. 7s. ; the lieutenant's share is 10 times and the gunner's share 3 timee as much as that of each seaman. What is the value of each person's share ? . (25) Extract the square root of 167'9616, and of 5 \% T . "* (26) A clock which loses 4 minutes in 12 hours is 10 min utes fast at midnight on Sunday. What o'clock will it indi- cate at 6 o'clock on Wednesday evening ? Y(27) The distance between two wickets was marked out for 22 yards, but the yard measure was ^ of an inch too short ; what was the actual distance ? (28) What is the difference between simple interest, com pound interest, and discount ? Find the difference between the simple interest and the true discount on $1900 for 1^ years at 8 per cent. -* (29) What is the present worth of a bill of $170 due in 4 niG_., reckoning money at &% per annum ? EXAMINATION PAPEKS. 265 Find the interest on $880 for 1 years at 4fper cent., and the discount on $929.50 for 2 years, at 2 per cent. 7 H of - .(31) Simplify __ 1/ 14 1-01015 Y-(32) Find the vulgar fraction equivalent to *gg ^foSj Which is the better investment, the 3 per cents, at (H, or the 4 per cents, at 103? How much must a man invest in the former that he may have a yearly income of $4851, after paying an income tax jf 2 cents in the dollar? -^ (34) Two ships get under weigh at tlie same time for the came port, distant 1200 miles ; the faster vessel averages 10 knots an hour, and arrives at tlie port a day and a half before the other ; what will the latter vessel average an hour ? yL(35) Divide $87.50 between two men, so that one may receive half as much again as the other. (3.6) A man has $3430 stock in the 3 per cents, at 83^; when the stock rises il per ccut. he transfers his capital to tho 4 per cents, at 98 ; find the alteration in his income. f^37) The weight of the water contained in a rectangular cistern 8 ft. long, 7 ft. wide, is 93| cwt. If a cubic foot ofoTtfto- water weigh 10uO oz., find the depth of water in the cistern. ^ :S) If $3 is the discount off $333 for 2 mos., what was the rate per cent. ? What should be the discount off $333 for 1 year ? (39) The height of a tower on a river's bank is 55 feet, the length of a line from the top to the opposite bank is 78 feet ; what is the breadth of the river ? -(40) How many yards of matting^4 feet broad will cover a floor that is 27'3 feet long and 20'16 feet broad ? (41) Simplify the fraction Ijofll-f oflj _2_ il ' : 17 2 + 1U 266 EXAMINATION PAPERS. (42) If | of 1 of an estate be worth $300, what will be 2 2 the value "of ~JT of the estate ? 14 43. Of an electric cable -j- rests on the bottom of the sea, iV hangs in the water, and 234f yards are employed on land : what is the length of the cable ? * (44) Extract the cube root of 1 677721G. (45) At what price must an article, which cost 15s., be sold so as to gain ID per cent. ? (46) The njimber of -disposable seamen at Portsmouth is ' 800, at Plymouth 756, and at Sheerness 404. A ship is com- missioned, whose complement is 490 seamen. How many must be drafted from each place so as to taks an equal pro- portion ? (47) (a) Find the difference between the simple and com" pound interest of $416. 66| for 2 years at 8 per cent. (Z>) Find the rate of interest, when the discount on $211.60 due "at the end of'l years is $:.7.60. (48) What sum will amount to $3213 in ten years at 8 per cent, simple interest -? (49) The length of a rectangular field which contains 4 ac, 3 ro. 14 po. 26- sq. yd. is 260 yd. 1 ft. 4 in., what is its breadth ? (50) A room is 14 ft. 3 in. high, 20 ft. wide, 24 ft. long, ' what will it cost to paper it with paper 2^ ft. wide, whose price 11 %d. per yard; allowing 8 ft. by 5 ft. 3 in. for each oi four doors, 10 ft. by 6ft. 8 in. for each' of two windows, and 8 ft. 6 in. by 5 ft. for a fireplace ? (51) Simplify the fraction qv o JL d ~~3 ftf nj a + 1 * 2 + i s + r^+i 8=1 f 2 i (52) Find tht, ^e of 003 ot 1 5s. + -069 of 5 - -8 of 2s. 3d. (;'3) If f of the cargo of a ship be worth $16000, what will be the value of f- of $ of the remainder ? -V (54) A can mow 5 acres of grass in 3 days, B 7 acres in 9 days, (711 acres in 12 days: in how many days can they jbinilT mow 121 acres ? , EXAMINATION PAPERS* 267 * (55) A watch, which is 5 m. 40 s. fast on Monday at noon, , is 2 m. 51 s. fast at midnight on the lollowing Sunday : what * did it lose in a day ? ";G) The rent of "a farm is $720, and the taxes are 142 per cent, on the rent: find the amount of rent and taxes together ? ?. (57) Three persons divide the cost of an entertainment amongst them in such a manner that the first pays ^ of the whole, and the second | of what the first pays, and the third pays the remainder, which is 82. 50: what is the amount of the bill? - (58) If an income of $1200 pays $18 for income tax, how much must be po.id on an income of $750 wlien the tax is half as much again ? (59) A invests $552 in the 3.^ per cents, when they are at 92 ; B invests $079 in the 3 per cents, when they are at 97. Find the difference of their incomes. ^ (60) What is the cost of the carpet for a room, the dimen- sions of which are 21 feet long, 15 1 feet wide, a* 2f cents per square yard ? (61) Simplify: x ' 281 2* ' 8ft/ 1-405 *-~(62) A regiment marching 3j miles an hour makes 110 steps a minute : what is the length of the step ? (63) How long would a column of men, extending 3420 feet in length, take to march through a street a mile long at the rate of 58 paces in a minute, each pace being 21? feet? (64) A street being 850 feet long, and the width of the pavement on each side being 5 feet 3 in. find the cost of pav- ing it at 37.1 cents a square foot? (65) Two pipes together fill a cistern in 1 hour: one of them alone fills it in 1| hour. How long will it take the other to fill it ? (66) How many hours a day must 42 boys work, to do in 45 days what 27 men can do in 28 days of 10 hours long ; the work of a boy being half that of a man^? (67) At what rate will the simple interest on $125 amount to $13.12.| in H years ? (68) What principal will give $616 simple interest in 5 years at 6| per cent. ? 268 EXAMINATION PAPERS. (69) A log of timber is 18 feet long, 1 foot 4 inches wido, and 15 inches thick. If a piece containing 2 solid feet, bo cut off the end of it, what length will be left ? " (70) If 8 guineas be expended in purchasing Brussels carpet, f yd. wide, at 3s. 6d. a yd., for a room '20 ft. long and 16 ft. 9 in. broad ; how much of the floor will remain uncovered ? (71) Simplify ||| + -ty- 'f (72) Find the value of 02 of 1 + -03 of 7s. 6d. + 014 of 2s (73) Extract the square root of 30712-5625 of -000000133225. (74) .A bankrupt owes $7850, and pays 37 ^ cents in the dollar ; how much did his creditors jointly lose ? "(75) If 14 men can mow 35 acres of grass in 6 days of 10 hours each, in how many days of 12 hours each can 3 men mow 24 acres ? >(76) If 9 men or 16 women could do a piece of work in 144 days, in what time would 7 men and 9 women do it, working together ? "-(77) Divide $2849 among A, J5, and C, in the .proportion of -7, '28, and '056. (78) The mathematical discount on a sum of money fof 2 years is $360; the interest on the same sum for the same time is $400 : find the sum and the rate percent. (79) Find the gain or loss per cent., in buying oranges at $2.50 per hundred and selling them at 8 for .\2 cents. (80) What will be the cost of papering a room 21 ft. long by 15 ft. broad and 11 ft. high, which has two windows each 9 ft. high and 3 ft. wide, a door 7 ft. high and 3 ft. 6 in. wide, and a fire place 4 ft. high by 4 ft. 6 in. wide, with paper, 2 ft. 3 in. wide at 9s. a piece ; the price of putting it on being Qd. per piece, and each piece containing 12 yards ? (81) Simplify (1) 2f - H + 9 T 1 T (2) (3-71 1-M) X 7)3 EXAMINATION PAJ"E&S. (82) A man owns ^g of a mine, and sells -1351 of his share ; what fraction of the mine has he left ? (83) A and B can do a piece of work in 8 days, B and C can do it in 12 days, and J., B, and C can do it in 6 days. In how many days can A and C do it ? (84) A clock which gains 7% minutes in 24 hours is 12 minutes fast at midnight on Sunday. What o'clock will it indicate at 4 o'clock on Wednesday afternoon ? (85) Gunpowder being composed of 33 parts of nitre, 7 of charcoal, and 5 of sulphur ; find how many pounds of each will be required to make 30 Ibs. of powder. (86) What is the difference between Interest and Dis- count ? Which of the two is greater ? Find the difference between the Interest and Discount on $1639 for 43 mos. at 6A per cent. (87) Find the difference between the true and bank dis- counts on a note of $10400 due in six months, (days of grace included) at 8% per annum. (88) $ of A's stock was destroyed by fire, of the re- mainder was injured by water and smoke ; he sold the uninjured goods at cost price, and the injured goods at a third of cost price. He realized $1155. What did he lose by the fire. (89) Having given that the weight of a cubic foot of water is 1000 oz., and that the imperial gallon contains 277*274 cubic inches, find the weight of a pint of water. (90) A room is 22 ft. 6 in. long, 20 ft. 3 in. wide, and 10 ft. 9 in. high. Find the cost of carpeting the room at $1.20 a square yard, and of papering the walls at 20 cents a square yard. (91) Simplify : 004 -r- -0005 2-423 -f 3-576 -f 2-0001911 (92) The quotient in a division question equals six times the divisor, and the divisor equals six times the remainder ; the three amount together to 516 ; find the dividend. (93) Add together -60625 of 1 -f -142857 of 14*. 10K, and f f of A of J-3. 5*. Id., and express the result as the decimal of 27 shillings. (94) A clock gains 8$ minutes a day; how must the 270 EXAMINATION PAPESg. hands be placed at noon so as to point to true time at 7 h. 30 m. P.M. ? (95) A person invests $750 at simple interest, and at the end of 3 years and 8 months he finds that he possesses $956.25 ; at what rate per cent, per an. was his profit ? (96) A person's half-yearly income is derived from the proceeds of $4550 at a certain rate per cent., and $5420 at 1 per cent, more than the former. His whole income is $453. Determine the rates. \ (97) What will be the cost of enclosing a rectangular gar- den, 90 yd. long and 30 yd. 2 ft. 3 in. broad, with a wall 8 ft. 4 in. high, at the rate of $1.20 per superficial square yard $ (98) A person invests .10000 in 3 per cents, at 75, and when they rise to 78 he sells out and invests the produce in bank shares at 208 each, which pay a dividend of 8 per share. Show that his income is not altered. (99) What must be the least number of soldiers in a regi- aient to admit of its being drawn up 2, 3, 4, 5 or 6 deep, and also of its being formed into a solid square ? (100) If $40 is a proper discount off $360 for 8 months, what should be the 12 months' interest on $360 ? (101) Multiply 57875 by 729819, with three lines of multi- plication, and divide 123456 by 63, using short division. (102) A French metre = 1-0936 of a yard, and a centi- metre is the hundred part of a metre. Find a centimetre in decimals of an inch to 4 places. (103) A and B can do a piece of work in 4 days, B and C in 5f days, and A and G in 4f days. In what time can each do the work separately ? (104) M starts from and travels towards D at a rate of 6 miles per hour; two hours afterwards N starts from C, and going 10 miles per hour reaches D 4 hours before M. Find the distance from C to D. (105) Find the simple interest on $2733$ at 4 percent, for 3 years and 9 months ; and determine what sum will amount to $926.10 in 8 years at 5 per cent, compound interest. (106) Find the difference between the discount on $1622.50 for 14 months at 7 per cent, per ann. and the interest on $1760 for 16 months at 6 per cent, per ann. (107) A women buys a certain nnmber of p.ppl^R at 3 * PAPERS. 271 penny, and the same numjber at 2 a penny ; she then mixes them and sells them at 5 for twopence. How much does she gain or lose per cent. (108) A person, by disposing of goods for $182, loses 9 per cent. Wbat ought tLey to have been sold at to realise a profit of 7 per cent. ? (109) Find the cost of papering a room 14 ft. 6 in. long, 18 ft. 7 in broad, and 12 ft. 3 in. high, with paper at 14f cents per square yard. In the room are 4 windows 4 ft. by 3 ft., 2 doors 6 ft. 6 in. by 2 ft. 5 in., and a fireplace 5 ft. by 4 ft. (110) The external dimensions of a box without a lid are, length 4 feet, breadth 3 feet, depth 2 feet, and the thickness of the sides and bottom is the same, namely 1 inch ; if the cost of a cubic yard of the material is 9., and the cost o* making the box = f\ of the cost of the material, what will the box cost ? (111) Eight bells begin tolling together at the same in- stant, and they toll at intervals of 1, 2, 3, 4, 5, 6, 7, 8 seconds respectively : after what time will they be again tolling at the same instant ? (112) Simplify Ltt ( -6i" "*"*"" ( . , I of iH-6i """" ( 7 10* ~ 18 01 7 (113) A, B, and G are partners. A receives two-fifths of the profits, B and C dividing the remainder equally. ,4's income is increased by $220 when the rate of profit rises from 8 to 10 per cent. Find the capital of B and C. (114) A railway train having left a terminus at noon is overtaken at 6 P.M. by another train, which left ,the same terminus at 1 P.M. If the former train had been 10 miles farther on the road when the latter started, it would not have been overtaken till 8 P.M. Find the rates of the trains. (115) A person invests 5000 in Turkish 6 per cent, stock at 80 ; find the rate of interest he gets for his money. When his stock has risen to 104 he sells out, and buys .20 railway shares at 18, which pay dividend at the rate of 4* per cent. Find the alteration in his income. (116) If 6 men and 2 boys can reap 13 acres in 2 days, and 7 men and 5 boys can reap 33 acres in 4 days, how long will it take 2 men and 2 boys to reap 10 acres ? 272 EXAMINATION PAPERS. (117) The cost price of a book is $4.75, expense of th sale 6 ^o, profit 24 % ; what is the retail price 1 (118) Show that the simple interest on $025 for 8 months at 7 % is equal to that on $1093. 75 at 8 i for 4 months. (119) One clock gains 4 minutes in 12 hours, and anotner loses 4 minutes m 24 hours. They are set right at noon on Monday. Determine the time indicated by each clock, when the one appears to have gained 16 minutes on the other. (120) A rectangular court is 50 yards long and 30 yards broad. It has paths joining the middle points of the oppo- site sides, of 6 ft. in breadth, and also paths of the same breadth running all round it. The remainder is covered with grass. If the cost of the pavement be 12| cents per square foot, and of the grass 70 cents per square yard, find the whole cost of laying out the court. (121) How many times does the 29th day of the month occur in 400 consecutive years ? (122) A creditor, agreeing to receive $281.25 for a debt, finds that he has been paid at the rate of 62 cents in the dollar ; how much was the debt ? (123) A, B, and C rent a meadow for $43. A puts in 10 horses for 1 month, B 12 oxen for 2 months, and C 20 sheep for 3 months. How should the expense be divided, if the quantities eaten by a horse, an ox, and a sheep, during the same time, be in the ratio of 4, 3, and 1 ? (124) If the price of 9760 bricks, of which the length, breadth, and thickness are 20 inches, 10 inches, and 12^ inches respectively, be $213.50, what will be the price of 100 bricks, which are one-filth smaller in every dimension ? (125) How many years' purchase should I give for an estate, so as to get 3 per cent, interest for my money ? (126) How often between 11 and 12 are the hands of a clock an integral number of minute spaces apart ? (127) A and B walk a race of 25 miles ; A gives B 45 minutes' start ; A walks uniformly a mile in 11 minutes, and catches B at the 20th milestone : find B'B rate, and by how much he lost in time and space. (128) A debt is due at the end of 4 months ; is paid immediately, and at the end of 8 months ; when ought the remainder to be paid ? EXAMINATION PAPERS. 278 (129) A man, by selling out of a 8 per cent, stock at 99 gains 10 per cent, on his investment. At" what price did he buy, and what was his income, supposing that he realized $15345 ? (130) A tack is 8 ft. long, 5 ft. 4 in. wide, 4 ft. 6 in. deep. Find the number of gallons it contains, having given that 1 cub. ft. of water weighs 1000 oz., and that a pint of water weighs a pound and a quarter. (131) Simplify 2 .8 7T 01 7T /_8^ 2\ /JL8 1\ \ 13 ~ 9 / " \ 3 + 6 / * _^ 7J of 3A + 3 A \ 13 ~ 9 / " \ 3 6 / 3 8 of 63. (132) In a dormitory *- of the boys are in the upper school, f of the remainder in the middle, and the rest, 8 in number, in the lower ; find the number in the dormitory. (133) The circumference of the fore-wheel of a carriage is 8 feet, and that of the hind-wheel is 10 feet ; in what dis- tance will the fore-wheel make 100 revolutions more than the hind-wheel ? (134) A and B receive $1.37 for digging a garden. They work at it together for 4\ hours ; B then left, and A finished the work in 3 hours. How should the pay be divided ? (135) What are the two exact times when the hands of a watch are equally distant from fig. III. ? (136) In how many years will $320 double itself, at 7| per cent, simple interest ? (137) A person invests the present value of 2358 due two years hence at 4 per cent, in gas-shares, which pay at the rate of nine per cent. ; he gives 144 for each share of 100 ; wfcat is his annual income, and what rate per cent, does he make of his money invested in the gas-shares ? (138) At billiards A can give B 5 points in a game of 50, and C 10 points in 50 ; how many points can B give in a game of 90 ? (139) How much money must one invest in 3 per cent- Consols, when they are at 10 per cent, below par, in order to have an income of .2000 a year ? (140) A level reach in a canal, 14 miles 6 fhrlongs long, and 48 feet broad, is kept up by a lock 80 feet long, 12 feet broad, and having a fall of 8 ft. 6 in. ; how many barges 274 EXAMINATION PAPERS. might pass through the lock before the water in the uppei canal was lowered one inch ? (141) Find the value of 7 r~ X jq r| of $5.67. (142) ^1 can do a piece of work in 6 days, which r> can y in 4. A lias worked for 10 clays, during the last 5 ot which U has been destroying ; how many days must A now work alone, in order to complete his task ? (143) Two cisterns of equal dimensions are filled with water, and the taps for both are opened at the same time. If the water in one will run out in 5 hours, and that in the other in 4 hours, find when one cistern will have twice as much water in it as the other has. (144) If 3 men, 4 women, 5 boys, or 6 girls, can perform a piece of work in 60 days, how long will it take 1 man, 2 women, 3 boys, and 4 girls, all working together? (145) Two trains start at the same time from London and Edinburgh, and proceed towards each other at the rates of 30 and 50 miles per hour respectively. When they meet, it is found that one train has run 100 miles farther than the other. Find the distance between London and Edinburgh. (146) Two persons buy respectively with the same sums into the 3 and 3i per cents., and get the same amount of interest. The 3 per cents, are at 75: at what price are the 3 per cents. ? "(i47) Divide 81986.50 among A, B, and 0, in the proper' tion of 2-3, T15, and '524 respectively. (148) If for a sovereign one can buy 11 gulden 12 kreut- zers or 25*5 francs, and for one 20- franc piece 9 gulden 20 kreutzers, how much per cent, is gained by buying French gold witli English gold before buying German money ? (149) Express 69 miles in metres, 32 metres being taken to be equivalent to 35 yards. (150) Find the cost of painting the walls ot a square room 14 ft. high and 18 ft. long, with two doors 8 ft. by 4, and three windows 10 ft. by 5, the amount saved by each window being ,2 16s. ocL What additional height would increase the cost by nine shillings. (151) Simplify + 4- + *of A- EXAMINATION PAPEBS. 275 (152) Two lines are 41*06328 inches and '0488 of an inch long respectively. How many lines as long as the latter can be cut off from the former, and what will be the length of the remaining line ? (153) A and B start to run a race ; their speeds are as 17 to 18. A runs 2 miles in 16 min. 48 sec. ; B finishes the course in 34 minutes : determine the length of the course. (154) A boat's crew row over a course of a mile and a quarter against a stream which flows at the rate of 2 miles an hour, in 10 minutes. The usual rafe of the stream is half a mile an hour. Find the time which the boat would take in the usual state of the river. (155) A person pays one tax of Wd. in the , and another of 5 per cent, on his income. His remaining income is .545. What was his original income ? (156) A man invested $14350 in the U. S. 6's at 107 the brokerage being % ; what will be his clear income after an income-tax of 5% is deducted? (157) A soldier has 5 hours' leave of absence : how far may he ride on a coach which travels 10 miles an hour, so as to retur._ to the camp in time, walking at the rate of 5 miles an hour? (158) Two trains start at the same time, the one from London to Norwich, the other from Norwich to London. If they arrive in Norwich and London respectively 1 hour and 4 hours after they passed each other, show that one travels twice as fast as the other. (15S) When 170 will purchase 4233 francs, what is the course of exchange between London and Paris ? And if 603 gold pieces of 20 francs contain as much pure gold as 400 sovereigns, what is the par of exchange between London and Paris? (160) A hollow cubical box, made of material which is 1/8 inches in thicknes's, has an interior capacity of 50*653 cubic feet : determine the length of the outside edge of the box. (161) Simplify ( 6| of (162) Gold of the value of 423267 arrives from Australia; what is its weight in Ib. avoirdupois, the price being 3. 18 t. per oz. troy ? 276 EXAMINATION PAPERS. (163) A can do one-half of a piece of work in 1 hour, B can do three-fourths of the remainder in one hour, and C can finish it m 20 minutes : how long would J., B, and fl together take to do it ? (164) If I pay $750 now for a debt of $771.09| not yet payable, and money be considered worth 7 per cent, per annum, when will the debt be due ? (165) Two equal wine-glasses are filled with mixtures of spirit and water in the ratios of 1 of spirit to 3 of water and 1 of spirit to 4 of water : when the contents are mixed in a tumbler, find the strength of the mixture. (166) At what per cent, in advance of cost must a mer- chant mark his goods so that after throwing off 20 % of the marked price he may make a profit of 25 per cent. (167) A man receiving a legacy of $34510 invested one- half in Dominion 6 per cents, at 101, and the other half in U. S, 5 per cents, at 84^, paying brokerage at % ; what annual income did he secure from his legacy ? (168) A piece of work must be finished in 36 days, and 16 men are set to do it, working 9 hours a day ; but after 24 dajs it is found that only three-fifths of the work is done. If 3 additional men be then put on, how many hours a day will they all have to labour, in order to finish the work iu time ? (169) Of two stalactites hanging from the flat roof of a cavern, one is 1*02 inches longer than the other, and the shorter one increases in length at the rate of 3*014 inches in a century. Find the rate of increase of the other, in order that they may bo of the same length at the end of 125 years. (170) Two men A and J5, start from Cambridge, at 4 and 5 o'clock, A.M. respectively, to walk to London, a distance of 50 miles ; B passes A at the twentieth milestone, and reach- es London at 5 P.M. When will A arrive there ? (171) Find the square root of T0747'4689, and the cube root of 189119220. (172) A person can read a book containing 220 pages, each of which contains 28 lines, and each line on an average 12 words, in 5^ hours ; how long will it takp him to read a book containing 400 pages, each of whicL c< ntahiF **6 lines, and each line on an average 14 words ? - (173) The whole time occupied by a train 120 yarc,., long, BXAM1NATION PAPERS. 277 travelling at the rate of 20 miles an hour, in crossing a bridge is 18 seconds ; find the length of the bridge. (174) If 20 men, 40 women, and 50 children receive $4200 among them for seven weeks' work, and 2 men receive as much as 3 women or 5 children, what sum does a woman receive per week ? (175) Two clocks begin to strike 12 together ; one strikes in 35 seconds, the other in 25 ; what fraction of a minute ia there between their seventh strokes ? (176) A speculator bought 43 shares in a mine at 35, and kept them till they dropped to 11, when he sold out and bought with the proceeds 6 per cent, railway stock at 28 pre- mium ; find his annual income from the latter investment. (177) Two clocks strike 9 together on Tuesday morning. On Wednesday morning one wants 10 minutes to 11 when the other strikes 11. How much must the faster be put back, that they may strike 9 together on Wednesday even- ing ? (178) How much ore must one raise, that on losing $1 in roasting and T 8 8 of the residue in smelting, there may result 506 tons of pure metal ? (179) It' a population is now ten millions, and the births are 1 in 20 and the deaths 1 in 30 annually, what will the population become in 5 years ? (180) There are two rectangular fields equal in area ; the sides of one are 945 yards and 1344 yards in length, and the longer side of the second is 1134 yards ; what is the length of its shorter side, and how many acres are there in each field? . (181) The masters of a school are "0416 of its whole num- ber, but after 40 new boys have been added the masters be- came *0375 of the whole. How many boys and masters were there before the new boys came ? (182) Divide $350 among 4 persons, so that B may have three times as much as A, C half as much again as A jind B together, and D as much as A, B, and C together. (183) By selling a house for $3700 I lost 7 per cent.; what must I have sold it for to have gained 12J per cent. ? (184) Find the difference between the interest and dis- count on $1265 for 73 days at 6% ? ft 85) A merchant sells tea to a tradesman at a profit of 6U ^or c-ut., but the tra^esnj.i;; becoming a bankrupt pays 278 MXAMINATION PAPERS. 87 cts. in the dollar. How much does the merchant gain or lose by the sale ? (1 86) What sum must a man invest in the 6 per cent. County bonds at 101 in order to have a clear income of 81424 40 after paying an income tax of 1% on all over $400? (187) A baker's outlay for flour is 70 per cent, of his gross receipts, and other trade expenses 20 per cent. The price of flour falls 50 per cent, and other trade expenses are thereby reduced 25 per cent. What reduction should he make in the price of a 15c. loaf, allowing him still to realize the same amount of profit ? (188) What is the average annual profit of a business, when a partner, entitled to $ of the profits, receives as hie share for 2 years aud 4 months the sum of $7890.50. (189) If a tradesman adds to the cost price of his goods a profit of 12 per cent., what is the cost price of an article which he sells for $3. 82 ? (190) A rectangular piecte of ground 72 yards by 45 yards is to be laid out in 4 plots of grass, each 27 ft. by 13 ft., and a pond in the centre 6 yards square, to contain 252 cubic yards of water ; find the expense of gravelling the remainder at 2| cte. per square yard, and the depth of the pond. (191) Find the value of 51 of | of 2*-l (192) If 12 men*OTTl8 boys can do \ of a piece of work in 6 hours, in what time will 11 men and 9 boys do the rest ? (193) Find the principal sum on which the. simple interest in 2 years at 6% per annum is $1068.75. (194) The compound interest on a certain sum at 4 per cent, for 2 years exceeds the simple interest for the same time at the same rate by $6 ; what is the sum ? (195) Two ships are built. Twice as many ship-carpen- ters are employed about the first as about the second ; the first is built in 9 months, the second in 8 months ; the wages of each man of the first set are 25 cents per hour, and they work 12 hours a day ; the wages of each of the second set are 1 } cents per hour, and they work 10 hours a day. T]K cost of the first in carpenters' wages was $30000 ; what was that of the second ? EXAMINATION PAPERS. 279 (196) A person leaves $12670 to be divided among his five children and three brothers, so that, after the legacy duty has been paid, each child's share shall.be twice as great as each brother's. The legacy duty on a child's share being one per cent., and on a brother's three per cent., find what each will receive ? (197) Two persons, A and B, meet to settle their accounts. A has 3^ years previously lent B $500 ; and B has a bill of $360'50 against A, for which he is to allow nine months' dis- count ; if the interest in each case is 4 per cent, per annum, what has B to pay A ? (198) A grocer buys 4 cwt. B qr. 14 lb. of sugar at 1 16*. Sd. per cwt. (long cwt.), and sells it at A^d. per lb. ; how much does he gain or lose per cent. ? (199) If when 25 per cent; is lost in grinding wheat, a country has to import 10000000 quarters, but can maintain itself on its own produce if only 5 per cent, be lost, find the quantity of wheat grown in the country. (200) A man rows down a river 18 miles in 4 hours with the stream, and returns in 12 hours; find the rate at which he rows, and the rate at which the stream flows. (201) Show that 1 ^202) A and B can do a piece of work in 6 days, B and G in 8 days, A, B, and C in 4 days. How long would A and C take to do it ? (203) If, by selling an article for $38.25, 8 percent, is lost, what per cent, is gained or lost by selling it for $57 ? (204) A French metre contains 39-371 English inches : express to three decimal places an English mile in metres, (205) A tradesman marks his goods with two prices : the one for ready money, the other for 6 months' credit, the rate of interest being o per cent, per annum ; if the credit price of an article be $2.05, what ought its ready-money price to be? 280 EXAMINATION (206) Compound interest reckoned quarterly at 2% is equal to what interest reckoned yearly ? (207) A person having $9790 in the Toronto city 6 per cent, bonds sells out at 98^, and invests the proceeds in Bank of Montreal stork at 177, which pays a dividend of 12 per cent, per annnm. Find the change in his income, broker- age in each transaction being %. (208) I buy wheat at 39s. a quarter, and some of a supe- rior quality at 6s. per bushel : in what proportions must I mix them, so as to gain 25 per cent, by selling the mixture at 57s. 6c?. per quarter ? (209) The weight of a cubic foot of water being 1000 oz., find the weight of a rectangular block of gold, 8 inches in length, 2 inches in thickness, and 3 in breadth ; the weight of a mass of gold being 19*26 times the weight ot an equal bulk of water. (210) The content of a cistern is the sum of two cubes whose edges are 10 inches and 2 inches, and the area of its base is the difference between two squares whose sides are 1J and Ig feet. Find its depth. (211) Find the value of '857142807 of 10. 14s. Id. accu- rately ; and show that the error committed by neglecting all decimals of an order higher than the fifth is less than-j-^-y of a penny. (212) The sum of $327 is borrowed at the beginning of a year at interest, and after 9 months have passed $400 more is borrowed at a rate of interest double that which the former sum bears. At the end of the year the interest on both loans is $26.35. What is the rate of interest in each case ? (213) A dealer purchases a liquid at 4s. the gallon, and dilutes it w~.th so much water that, when he sells the com- pound at 3s. a gallon, he gains 20 per cent, on his outlay. How much water is there in every gallon of the compound sold? (214) The discount on $566.50 for 9 months is $16.50: find the rate of interest. (215) A merchant lost a cargo at sea which he had in- rared ; the broker offered him a sum of money for his loss, which the merchant refused as being 10 per cent, below the estimated value of his loss; the broker then offered $379.75 more than he offered at first, and the whole amount of the 1XAMINATION PAPERS. 281 Second offer was 6 per cent, in excess of the flstimated value. What was that value ? (216) A man -wishing to sell a horse, asked 25 % more than it cost ; he finally sold it for 15 % less than his asking price, and gamed $5.75. How much did the horse cost and what was the asking price ? (217) If 15 masons, working 10 hours a day, can build a wall 6 ft. high and 100 yd. long, in 6 days, how long will it take 7 masons, working 9 hours a day, to build a wall 9 ft. high and 140 yd. long ? (218) A bankrupt's assets are $2700, out of which he pays 75 cents in the dollar on half his debts, and 60 cents on the other half. What is the amount of his debts? (219) If a ship containing 150 hhd. of wine pays for toll at the Suez Canal, the value of 2 hhd., wanting $30 ; and another, containing 240 hhd., pays at the same rate, the value of 2 hhd. and $90 besides ; what is the value of wine per hhd. ? (220) A picture-gallery consists of three large rooms ; the first is 20 yd. long, 20 yd. broad, and 6 yd. high ; the other two are 20 yd. long, 20 yd. broad and 5 yd. high. Sup- posing the walls to be covered with pictures, except the doors, which are 8 ft. high and 8 ft. wide, and of which each room has two, what will be the number of pictures, the average size being 8 feet by 3 feet ? 8-5 - 1-8*3 i 8-1 X -l6i (221) Simplify - f j^ x - + ~ ~ (222) Find the square roots of 15376-248001 and (223) A general, after losing a battle, found that he had only two-thirds of his army left fit for action ; one ninth of the army had been wounded, and the remainder, 2000 men, killed or missing ; of how many did the army consist before the battle ? (224) A contractor sends in a tender of $20,000 for a cer- tain work ; a second sends in a tender of $19000, but stipu- lates to be paid $2000 every three months ; find the differ- ence between tenders, supposing the work in both cases to be finished in two years, and money to be worth 7 per cent simple interest- 282 EXAMINATION PAPJCK8. (225) What snm of money must be left in order that, after a legacy duty of 10 % has been paid, the remainder being invested in the Dominion 5 per cents, at 91^ may give a yearly income of $450, brokerage at | %. (226) If two boys and one man can do a piece of work in 4 hours, and two men and one boy can do the same in 3 hours, find in. what times a man, a boy, and a man and a boy together, respectively, could do the same. (227) Show that the interest on $15840 for 8 months at 8 per cent, is equal to the discount on $3696 for 15 montha at 7| per cent. (228) A piece of work has to be finished in 36 days, and 15 men are set to do it, working 9 hours a day ; but after 24 days it is found that only three-fifths of the work is done ', if 3 additional men be then put on, how many hours a day will all have to work so as to finish the task in time ? (229) The interest on a certain sum at simple interest is $360, and the discount $340 for the same time and rate. What is the sum ? (230) The breadth of a room is twice its height and half its length, and the contents are 4096 cubic feet. Find the dimensions of the room. (231) If 1 Ib. of tea be worth 50 oranges, and 70 oranges be worth 84 lemons, what is the value of a pound of tea when a lemon is worth a penny ? (282) At a certain battle two-thirds of the defeated army ran away with their arms, five-sevenths of the remainder left their arms on the field, and of the rest seven-eights were missing, the remaining 500 being either killed or wounded. Find the whole number of the army. (288) If gold be at a premium of 20 per cent., and a person buy goods marked 135 dollars, and offer gold to the amount of 135 dollars, what change ought he to receive in notes, 5 per cent, being abated for ready payment ? (234) Show that the difference between the interest and the discount on the same sum for the same time is the interest of the discount. (235) I bought 20 Ibs. of opium by Avoir, wt. at 55 cts. per oz., ani sold by Troy wt. at 60 cts. per oz. Did I gain or lose and how much ? (236) By investing a certain sum of money in the 6 per cents at 91^ a man obtains an income of $320 ; what would EXAMINATION PAPEB8. 288 he obtain by investing an equal sum in the 5 per cents at 80. (237) A tradesman makes a deduction of 10 per cent, for ready money on a bill of $28 due in 12 'months, receiving $25.20. Find the difference between this sum and the pre- sent worth of the debt, reckoning interest at 10 per cent. (238) M invests one-third of his property in bank stock, one- sixth in Consols, and the remainder in railway shares. When he sells out he makes a profit of 6 per cent., 3 per cent., and 2 per cent, respectively on the investments, and realise ^6190. Required the amount of his property originally. (239) Mr. A sent $3681 to his agent with instructions to deduct his com. at 2^%, and invest the balance in fiour at $7.50 per bbl. If the cost of freightage and insurance amounts to $119, at what must the flour be sold per bbl. so as to make a profit of 20% ? (240) How many bricks, 9 inches long, 4| broad, and 4 thick, will be required for a wall 60 feet long, 20 ft. high, and 4 ft. thick, allowing 6 per cent, of the space for mortar ? i (241) What is the value of 25 of ,V of ."in. f (242) A work can be accomplished by A and B in 4 days ; By A and G in 6 days ; by B and G in 8 days. Find in what time it would be accomplished by all working together, i (243) A man hired a labourer to do a certain amount of work, on the agreement that for every day he worked he should have $1.50, but that for every day he absented him- self he should lose 60 cents. He worked twice as many days ;is he absented himself, and received on the whole $72. Find how long he was doing the work. , (244) A legacy of $146000 is left to three sons in the pro- portion of i, \ % and } respectively : how much will each receive ? . (245) If $10 is a proper discount off $210 for 8 months, what should be a proper discount off the same sum for 1 year? (246) The price of gold in this country is 3. 17. 10^, 'an ounce ; find the least number of ounces which can be- coined into an exact number of sovereigns, and the numbtj of sovereigns so coined 284 EXAMINATION PAPEBS. (247) A merchant in Toronto instructed his agent in Mon- treal to sell a consignment of flour at $7.50 per barrel and invest the proceeds in Montreal bank stock at 174^, which pays half-yearly dividends of 7 %. If the merchant's first dividend is $445.60, and commissions of 1 % and $ % be allowed on the transactions, respectively, how many barrels of flour were sold ? (248) State the conuection between Troy and Avoirdupois weights. A ring weighs 1 dwt. 4 gr., and is wortn 1. 2a. ; if 1050 of such rings be packed in a box weighing 8 lb., what would it cost to convey them 144 miles at the rate of 5s. per long ton per mile, insurance being demanded at thf rate of per cent. ? (249) How long will it be before $2500 put out to com- pound interest at 10 per cent, per annum will obtain $1727. 58 1 as interest ? -f (250) The breadth of a room is two-thirds of its length and three-halves of its height, and the contents are 6832 cubic feet. Find the dimensions of the room. (251) Multiply 82866 by 121711, using 8 linea of multi- plication only. 2-8 of 2-27 4-4-2-83 6'8 of 8 (252) Simplify -TJ- + r ;r-- of - (258) An agent received $21.70 for collecting a debt of $2480. What rate was his commission ? (254) A man sells out of the U. S. 6's 6-20 of 85 at 92| and realizes $25760. If he invest the proceeds in Erie R. B. stock at 45, which pays a yearly dividend of 8%, what alteration in his income has ensued, brokerage on each oi the two transactions being % ? (255) A farmer bought a horse for a bill of $292, due in 1 month, and sold him for a bill of $348, due in 4 months. What did he gam per cent., money being worth 4% ? (256) A man and a boy are to work on alternate day's at a piece of work, which would have occupied the boy alone 13 days. If the boy take the first day, the work will be finished half a day later than if the man commences. Find how long they would take to do it working together. (257) Two men invest $300 and $100 in a machine ; it works 5 months for each of them; determine what one must pay the oilier, if they would _ have made 30 per cent. PJJ the money by letting the machine. BXAHINATION PAPERS . 285 (268) A owes B $2725, and offers to pay him at a certain rate of discount instantly, instead of at the end of two years, when the debt will be due. B can place out the money, which he will receive, at 5 per cent, interest, and by that means gain $25 on the transaction. At what rate is the discount calculated ? (259) If 36 men, working 8 hours a day for 16 days, can dig a trench 72 yards long, 18 wide, and 12 deep, in how many days will 32 men, working 12 hours a day, dig a trench 64' yards long, 27 wide, and 18 deep? (260) A man discounts a bml of 180 drawn at 4. months at 60 per cent, per ann., and insists on giving in part pay- ment 5 dozen of wine, which he charges at 4 guineas a dozen, and a picture, which he charges at 19. How much ready money does he pay ? If the cost to the man of the wine and the picture be only one-fourth of the sum he has charged for them, what is the real interest the man has been charged ? (261) One-tenth of a rod is coloured red, one-twentieth orange, one-thirtieth yellow, one-fortieth green, one-fiftieth blue, one-sixtieth indigo, and the remainder, which is 302 inches long, violet, what is the length of the rod ? (262) The discount on a certain sum due 9 months hence is $20, and the interest on the same sum for the same time is $20.75. Find the sum and the rate of interest. (263) Two persons, walking at the rate of 8 and 4 miles per hour respectively, set off from the same place in opposite directions to walk around a park, and meet in 10 minutes. Find the length of the walk round the park. (264) In a hundred yards race A can give B four and C fwe yards start : if B were to race 0, giving him one yard in a hundred, which would win ? v (265) A man buys an article and sells it again so as to gain 6 per cent. If he had bought it at 5 per cent, less, and sold it for $1 less, he would have gained 10 per cent. Find the cost price. (266) If the difference between the simple and compound interest or a sum of money for two years at 5 per cent, be $3, find the sum. (267) If 7 per cent, be gained by selling goods for $09.55, what will be gained or lost by selling them for $61.75. (263) A draft on Dublin for .860 cost $l,7ii6.10; what 286 EXAMJ^ ATION PAPERS. was the course of exchange, com. charged at the rate of per cent. ? (269) A banker, in discounting a bill due in 3 months at 8 per cent., charges $16 more than the true discount. Find the amount of the bill. (270) A grocer mixes 18 pounds of coffee at 30 cents a pound with 12 pounds of chicory at 5 cents a pound ; at what piice must he sell the mixture to gain 25 per cent. ? (271) The following rule has been given to divide by 3-14159 : " Multiply by 7, divitle by 11, then by 2, and add l .th of Tifoirth of the result." Find the error made in obtain- ing 1 -r- 3'14159 by this process. 3 + 4 % (272) Prove that ^-^ is greater than f and less than f. - (273) The estate of a bankrupt (value $21000) is to be divided among four creditors, whose claims are, .4's to #'s as 2 to 3, 's to O's as 4 to 5, C"s to Z)'s as 6 to 7. What must each receive ? x (274) Which is the more profitable to buy flour at $6.50 on 6 months, or $6.30 cash, money being worth 8 % ? '(275) If $10.50 be a person's income tax at lj cents on the dollar, how much in the dollar is it when his income- tax is $12.25 ? '(276) If 9 tons 7 cwt. of iron be sold for $1260, and the gain on it be 20 per cent., what was the cost per cwt. ? (277) I send to my agent in Montreal $3060 to invest in tea at 75 cents per Ib. ; he deducts his commission of 2 per cent., and invests the balance. At what must I sell per Ib. so as to make a clear profit of 25 per cent, after paying freightage $30, and insurance at the rate of per cent. ? (278) A banker borrows money at 3 per cent., and pays the interest at the end of the year ; he lends it out at 5 per cent., but receives the interest half-yearly, arid by this means gains $200 a year ; how much does he borrow ? (279) A dealer sends out 250 Ibs. of tea at 80 cents per Ib. and allows 2 per cent, on the price for the expense o car- riage. Supposing the whole amount of carriage to come to $9.30, how 'much will the customer have to pay? (280) A plate of gold, 3 inches square and one-eighth of an inch thick, is extended by hammering so ns to cover a o" 7 vards squ;>.r< : find its; proper thickness. EXAMINATION PAPERS. 287 (281) A man having a flock of sheep sold 8 per cent, of them to A, 90 to B, 8 per cent, of the remainder to 6', and 29 to D. He then had 550 left. How many had he at first ? (282) The product of the 1st and 2nd of three numbers is 176382 ; of the 1st and 3rd is 279152 ; of the 2nd and 3rd is 215496 : what are the numbers? (283) Find the rate of 2 trains 150 ft. and 180 ft. long respectively which pass each other going the same way in 15 sees., and going in opposite directions in 3 sees. (284) By selling tea at 72 cents a pound a grocer clears % of his outlay. He then raised the price to 90 cents. What does he clear per cent, by the latter price ? (285) A grocer buys If cwt. of tea at 60 cents per Ib. and 2J cwt. of tea at 50 cents per Ib., and mixes them; he sells 2J cwt. at 55 cents per Ib. : at what rate must he sell the remainder to gain 20 per cent, on his outlay ? (286) In England gunpowder is made of 75 parts nitre, 10 sulphur, and i 5 charcoal ; in France of 77 parts nitre, 9 sulphur, and 14 charcoal: if half a ton of each be mixed, what weight of nitre, sulphur, and charcoal will there be in the compound ? (287) A ship 40 miles from the shore springs a leak, which admits 3| tons of water in 12 minutes. 60 tons would suffice to sink her ; but the ships pumps can throw out 12 tons of water in an hour. Find the average rate of sailing that she may reach the shore just as she begins to sink. (288) The receipts of a- railway company are apportioned in the following manner : 48 per cent, for the working ex- penses, 10 per cent, on one-fifth of the capital, and the remainder, $320,0, .for division among the holders of the rest of the stock, being a dividend at the rate of 4 per cent. ; find the capital and the receipts. (280) If the discount on a sum due at the end of 2 years is f 2- of the simple interest, at what rate i? that calculated ? (^90) If a crew, which can row from At B in 60 minutes, can row from B to A in 55 minutes, compare the rates of the stream and boat. (291) Simplify ( a ) 2 4- - 5 +6 288 EXAMINATION PAPERS. (292) If 3 men and 5 women do a piece of work in 8 days, which 2 men and 7 children can do in 12 ; find how long Ifc men, 14 children, and 15 women working together will take -j do it. (293) A person possessing 10000, 8 per cent, consols, sells out when they are at 93|, and invests the proceeds in 4 per cent, stock at 101 & : find the change in his income, allowing per cent, commission on each transaction. (294) Five men do -6006 of a piece of wo:k in 2*12 hours: how long will 6 boys take to finish it, it being known that 3 men and 7 boys have done a similiar piece of work in 8 hours ? (295) A watch set accurately at 12 o'clock indicates 10 minutes to 5 at 5 o'clock ; what is the exact time when the watch indicates 5 o'clock ? (296) A does | of a piece of work in 20 days, and then gets B to help him. They work together for 2 days, when B leaves and A finishes the work in half a day more. How long would B have taken to do the whole ? (297) The wages of 5 men, 3 women, and 1 child amount to $34, a man receiving twice as much as a woman, and a woman three times as much as a child. What will be th< wages of 6 men, 2 women, and 6 children ? (298) If 6 per cent, be gained by selling a horse for $132.50, how much per cent, is lost by selling him for $115 ? (299) A person invests $6825 in 6 per cent, stock at 91 ; he sells out $5000 stock when it has risen to 93, and the remainder when it has fallen to 85. How much does he gain or lose by the transaction ? If he invest the produce in M. B. S., which pays a dividend of 12 per cent., at 175, what is the difference of his income ? (300) The flooring of a room, 14 ft. 8 in. long by 18 ft. 4 in. broad, is composed of in. planks, each 8 in. wide and 10 ft. long. How many will be required ; and what will be the weight of the whole, if 1 cubic inch of wood weighs Lall an ounce ? ' 129*4947 (801) Find the square roots of 4957 '5681 and (302) At what rate will $167.50 amount to 1189 in 6 years ? (308) Two bills for $278.75 and $456.87^ are due on the 2nd and 22nd July respectively. What is their value on the EXAMINATION PAPERS. 289 12th July, interest being reckoned at the rate of 5 per cent, per annum ? (304) If a cask contain 3 parts wine and 1 part water, how much of the mixture must be drawn off and water sub- stituted for the mixture in the cask to become half and hall? (305) Three tramps meet together for a meal ; the first has 5 loaves, the second 3, and the third, who has his share of the bread, pays the other two 8 half-pence ; how ought they to divide the money ? (336) If the discount on a bill due 8 months hence at 7 per cent, per annum be , E, in the proportions of 4, 3, 2, 1 ; of the remainder, half vote for B, and divide their votes among (7, D, E, in the proportions of 3, 1, 1 ; two-thirds of the remainder vote for D and jk\ and 540 do not vote at all ; find the order on the poll, and the whole numb 3r of electors. (813) When the New York gold market |s at 104|, what would I get for $2304.50 currency. (314) A person invests $9450 in 5} per cent, stock, so as to receive an income of $787.50 ; what was the price of the stock ? (345) Two pipes A and B, would fill a cistern in 25 min- utes and 30 minutes respectively. Both are opened together, but at the end of 85- minutes the second is turned off. In how many minutes will the cistern be filled ? (346) A man for 5 years spends .40 a year more than his income. If he, at the end of that time, reduce his expendi- ture 10 per cent, in 4 years he will have paid off his debts and saved 120. Find his income. (347) The sum of ^177 is to,be divided among 15 men, 2 298 women, and 80 children, in such a manner that a man and a child may receive together as much as two women, and all the women may together receive .60 ; what will they each respectively receive ? (348) If 8000 metres he equal to 6 miles, and if a cuhic fathom of water weigh six tons, and a cubic metre of water 1000 kilogrammes, find the ratio ,of a kilogramme to a pound avoirdupois, (LongtonJ. (849) A mixture of soda ana potash, dissolved in 2540 grains of water, took up 980 grains of aqueous sulphuric acid, and the weight of the compound solution was 4285 grains. Find how much potash and how much soda the mixture contained, assuming that aqueous sulphuric acid anites with soda in the proportion of 49 grains to 82, and with potash in the proportion of 49 to 48. (350) A room is 21 ft. long, 15 ft. 6 in. wide, 10 ft. high ; it contains 8 windows, the recesses of which reach to the ceiling, and a^e 4 ft. 6 in. wide ; there are in it 4 doors, each 6 ft. 6 in. high and 3 ft. 8 in. wide ; the fire-place is 6 ft. wide and 4 ft. high; a skirting 1 .ft. 8 in. deep runs round the walls ; find the expense of papering the room at 5 cents, a square ANSWERS Ex. (i), p. 5. (1) geyen ; thirteen ; forty-five ; fifty-nine ; three hundred and twenty-six ; four thousand, five hundred and seventy- eight. (2) Ninety ; one hundred and ten ; two hundred and seven ; four thousand, three hundred ; four thousand and thirty-six ; four thousand, three hundred and six. (3) Seven hundred and eighty; six hundred and nine; five thousand, three hundred and sixty ; two 'thousand and twenty ; one thousand, one hundred and one. (4) Thirty-six thousand, four hundred and ninety-seven ; forty-nine thousand, five hundred and thirty-two ; six hundred and fifty-four thousand, three hundred and twenty- one; seven hundred and forty-three thousand, two hundred and sixty-nine. (5) Forty-five thousand; thirty-two thousand, six hundred; seventy-five thousand, two hundred and thirty ; five hundred thousand. (6) Eight millions, five hundred and seventy-two thousand, nine hundred and fourteen ; three millions, four hundred and sixty-nine thousand, two hundred and eighteen ; four millions, six hundred and twenty-nine thousand, eight hundred and seventeen. (7) Nine millions ; twenty-nine millions ; seven hundred and fifteen millions. (8) Nine hundred and ten millions, three hundred and seven thousand, two hundred and forty ; three hundred and seven millions, four thousand two hundred and five ; three hundred and eighty millions, five hundred and three thousand and forty. (9) Two hundred and forty-three billions, seven hundred and fifty-nine millions, two hundred and sixty-eight thou- sand,;three hundred and forty-two ; three hundred and seven billions, four hundred and five millions, six thousand, two hundred and seventy. ANSWKB8. 295 (1) (2) (3) . 8! (10) (11) (12) (13) (14) (15; (16) (17) (18) Ex. (ii), p. 6. 9; 12; 17; 19; 13; 16; 11. 23 ; 27 ; 35 ; 38 ; 44 ; 40 ; 26 ; 34. 67 ; 75 ; 62 ; 83 ; 74 ; 92 ; 68 ; 95. 76; 22; 50; 15, 28; 61; 49; 18; 90; 73. 107 ; 130 ; 246 ; 372 ; 608 ; 740 ; 990. 836 ; 747 ; 410 ; 913 ; 750 ; 384. 818 ; 808 ; 206 ; 430 ; 512 ; 787. 7845 ; 9637 ; 12000 ; 8400 ; 6003 ; 85040. 5470; 3650; 8780; 1247; 4808. 6004 ; 7022 ; 3500 ; 9047 ; 2017 ; 19402. 70007 ; 60060 ; 14014 ; 70017 ; 12303 ; 16005. 356728 ; 640842 ; 900000 ; 800040. 7000000 ; 4576865 ; 75806940. 315000000; 5040000; 8000700; 18000020; 70000000ii. 315674018003 ; 35600000520. 7000000000; 5800000600047; 8000000043007. 305005004006003 ; 53000053053. 9000000000009; 90000000000900; 1900000001900. 1000001001101. 1. Ex. (iii) p. 9. (1) (4) (6) (7) (8) (9) (10) (1) 4) 7) flO) (1) (4) (7) (10) Twenty-seven. (2) Forty-nine. (3) Sixty-eight Seventy-three. (5) Ninety-two. One hundred and forty-four. One hundred and sixty-three, One hundred and ninety-nine. Six hundred and sixty-four. One thousand eight hundred and seventy-two. 2. XXXVII. (2) LIX. (3) LXII. LXXXVIL (5) XCV. (6) CXXXIX. CXLV. (8) CLXXIX. (9) DCCCXLVI MDCCLXIII. Ex. (iv), p. 10. 11; 16; 20; 26. (2) 98. 60. 738. 74338. 1671. (22) 33633. 208. 12932. (5) 1409. (8) 4971. (11) 2008. (14) 880. (17) 28206. (20) 1163. (23) 106384. (3) 67. (6) 949. (9) 23406. (12) 3310. (15) 28493, (18) 18526. (21) 9289. (24) 69223. 2*6 31) 284271. 1843088. '8782272. 34) 50333150. 37) 112251. 40) 227056697. (26) 1450741. (27) 680077891, (29) 1979628. (30) 1184946. (32) 8476908. (33) 799819. (35) 20826857. (36) 14621293. (38) 764368. (39) 1825947. Ex. (v), p. 14. I 7. 29. 35. (2) (5) (8) 8 84. 29. (8) (6) (9) 19. 54. 45. (10) 66. (11) 509. (12) 82. (13) 8808. (14) 2112. (15) 4228. (16) 6222. (17) 61471. (18) 108. (19) 2779. (20) 28828. (21) 2761. (22) 381. (23) 46. (24) 32. (25 2042. (26) 6457. (27) 6780. (28) 51195. (29) 10999. (30) 1096. (31) 18467. (32) 60023. (33) 18378402. (34)' 1. (85) 999000. (36) 99900000. (37) 9999998999. (38) 8. (39) 26. (40) 610. (41) 598. (42) 150. (48) 619. Ex. (vi), p. 18. 0) 45. (2) 804. (8) 490. (4; 87ft. (5) 684. (6) 861. M 9273. (8) 11364, (9) 50; 75 : 175; 225 (10 635; 1016; 1270; 1397; 2540; 8890. 11 9868; 148i)2 ; 27137 ; 29604 ; 1233500 ; 17269000. (12) 836861 ; 411719 ; 449148 ; 1871450000 ; 2994320000000. 845. (4) 1820, (7) 8815. (10) 1646992. Ex. (vii), p. 19. (2) 1073. (5) 8000. (8) 80086. (11) 7417784. Ex. (viii), p. 20. 3) 1620. 6) 13734. '9) 93940. (12) 679826952. (1) 173482. (4) 872302. (7) 89842154. (10) 152847420. (12) 848087421500. (14) 8340400440. 128904. (3) 409354. (5) 2274048. (6) 2667640. (8) 61212122. (9) 819766614 (11) 58376823669. 18) 88871923744. 16) 296990965442. (16) 609485012763918. (18) 703(K)4503. (20) 3454309838. (22) 61110346167. (24) 24259354428. 13575555747. (26) (17) 18426705851000. (19) 3590386740. (2lj 1930038124. (23) 1407009621. (25) 248155914760. (27) 249493596792. Ex. (ix). p. 21. v l) 6840. (2) 1909680. (2) 1121111043841- \ Ex. (x) p. 21. (1) 225. (4) 3249. (7) 7569. (10) 56169. (18) 622521. (16) 16625. (19) 804357. (22) 45118016. (25) 961504803. (1)8. (5) 14. (9) 108. (13) 56285. (17) 2104. (21) 56169. (25) 317649. (29) 3469805. (32) 5642300741. (35) 300071. (2) (6) (10) (14) (18) (22) (26) 576. 4761. 10000. 390625. 1331. 103823. 1000000. 156590819. . (xi), p. 24. (3) 12. (7) 24. (11) 528. (3) 1600. (8) 5184. (9) 12996. (12) 804609. (15) 2197. (18) 314432. (21) 16974593. (24) 348913664 (4) 11. (8) 103. (12) 1082. (16) 7589523. (1) 3420. ,15) 458097 ,.., 17553. (19) 24000729. (20) 2019. 5678094. (23) 4348432. (24) 5072. 391525. (27) 39876548. (28) 30207. (30) 68274625. (31) 472304974. (33) 8462974231. (34) 90807. (86) 29970. Ex. (xii), p. 25. (2) 6487. (ff) 64008924. Ex. (xiii), p. 26. (1) 3826. (2) 241987. (4) 1749864, (5) 1243904. (7) 79267440. (8) 896547. (10) 444513674545. (11) 6947544611 (13) 2131962, 1421308, 1065981. (14) 310218774, 206812516, 155109887. (15) 13770459132, 9180306088, 6885229568. ,'16) 9085784, 5816115, 5169880. (3) 2162558. (6) 500603. (9) 659372. (12) 8007490200467. ANSWEBS. (17) 196350840, 122719275, 109083800. (18) 46918400, 29320850, 26063000. (19) 1188764, 724668, 664279. (20) 4224924, 2688588, 2464539. (21) 962341 116, 612398892, 661365651. (1) 3, 9. Ex (2) . M, p 2, 3, 4, . 28. 8,9, (8) 8 , 5,9, 11 (4) (7) 2, 2, 3, 4, 5, 8. 10. (5) (8) 2, 3, 4, 6. 8,9. (6) 3,5,9. (9) 2,3,4. (10) 2, 3, 4, 5, 8, 9. (11) 2, 11. Ex . (xvi), p. 29. (1) 2, 3, 3. (2) 2, 2, 2, 8. (8) 8, 8, 3. (4) 2, 2, 2, 2, 2. (5) 2, 2, 3, 3. (6) 3, 13 . (7) 2, 3, 7. (8) 3,17. (9) 2,3, 3,3; (10) 3, 19 (11) 2, 2, 3, 3, (12) 5,17 (13) 7, 18 (14) 8, 3, 11 . (15) 2,2, 5,5. (16) 3, 5, 7. (17) 2, 2, 3, 3,3. (18) 2,2, 2, 2, 7. (19) 2, 2, 3, 11 . (20) 2, 2, 2, 2, 11. (21) 2, 2, 2, 2, 2, 3, 8. (22) 2, 2, 2, 2, 3,8, 8. (23) 3, 5, 5, 7. (24) 6, 5, 5, 5. (25) 3, 3, 3, 8, 3, 3. (26) 8, 3, 3, 37. (27) 2, 2, 2, 2, 3, 3, 3, 8 (28) 2, 2, 2, 2, 2,5, 11 (29) 2, 2, 2, 2, 2, 2, 2, 3, 3, 6. Ex. (xvii), p . 29. (1) 4858. (2) 9306. .(3) 147474. (4) 805892. (5) 420077. (6) 1594432. (7) 8104199. (8) 11976096. (8) 32661000. (10) 4342356. in) 48482280. (12) 10138680000 Ex. (xviii), p. 30. d) 2472. (4) 42370218. (7) 74232657. (10) 486876. (18) 37296. (2) 452736. (5) 8642934. (8) 14287262. (11) 8781076. (3) 41798032. (6) 42765328. (9) 204074. (12) 784978. Ex. (rix), p. 31. 94, reui. 14. 18573, rem. 17. (fi) 105531, rem. 35. ) 849, rem. 20. (9) 11447, rem. 72. (11) 10005, rem. 8569. 11860, rem. 86. 878, rem. 22. 844380, rem. 85. 2392, rem. 134. ._.. 965316, rem. 718. (12) 10000821, rem. 1812, ANSWERS. Ex. (xx), p. 32. 278, rem. 18. (2) 36724, rem. 11. (3) 2378, rem. 9. (4) 20174, rem. 18. (6) 28998, rem. 22. (6) 21074998, rem. 25. (7) 85629* rem. 23. (8) 246925, rem. 21. (9) 7429, rejn. 7. (10) 129295, rem. 33. (11) 2987635, rem. 19. (12) 4236, rem. 57. (13) 423, rem. 72. (14) 504, rem. 128. (15) 6687, rem. 207. Ex. (xxi), p. 33. (1) 1. (2) 472369. (3) 624. (4) 8012. Examination Papers. (Page 40.) (I.) (1) Four millions, two hundred and thirty-seven thousand, four hundred and ninety- six ; 653812. (2) 196181. (3) 7829. (4) 4253111 ; 15362894. (5) 935977 ; 7429. (II.) (1) 25257630 ; four hundred and two millions, fifty thou- sand four hundred and seven. (2) 16992009. (8) 26488313; 99914800. (4) 338091, rem. 58. (5) 1175427; 130603. (in.) (1) Ten billions, ten millions, two hundred and one thou- sand, four hundred and one; 1023001; 10011224402; 2046002. (2) 1546478344; 1577918816. (3; 2237069, rem. 11. (4) 81405999. (5) 5226, rem. 83. (IV.) 1888. (2) 7482229, 2, 2, 2, 7 ; 2, 3, 13 ; 2, 8, 19. (4) 12000590. ) 65299476, rem. 6346. (2) 88652792964. ) 2,2,2,5; 2,3,3,5; * 3, 8, 7. 4( xxly ; xlTii; olxxviii. (o) 12000. BOO ANSWERS. (VI) i2; 8670344882024 (4S $14541. (1) 619161890. (8) 7283 (5. 7684 and 978 (VII.) (!) $269. (2; 72 days. (8) 252. (4) 296237 (6> $13300, $11900, $10500 (1) 7000 (8) 81239 (5) $30 ;) 9999000025 11796 steps (VIII. (IX.) (2) 450 Ibs. (4) $232. (3) $37569. (5) $21000; $5400. (2 ) 170680900742874252. (4) 786543. Quot , 432 (3) 2796219. (5) Bern, 12 ; Divisor, 72 ; Ex. (xxii.) p. 44. ;i) 2. (2) 6. (8) 20. (4) 18. (5) 48. [*) 7 fe 3 (8) 16. (9) 16. (10) 8. Ex, (xxiii.), p. 45. (1) 48. (2) 82. (3) 8. (4) 8. () 8453. (6) 36. (7) 936. (8) 855. (9) iiS. (10) 2345. (1) 4. (2) 2 Ex. (xxiv.), p. 46. (3) 73. (4) 29. Ex. (xxv.), p. 47. (5) 41. (6) 87. (1) 54 17000. ) 31759. (1) 860. (g) 86036. (9; 27720 (2) 2376. (6) 85800. (3) 2532. (7) 23400. Ex. (xxvi,), p. 49. (2) 1320. (6) 27324 (10) 228150. (3 288. (7) 3570. (4) 9555. (8) 16128. (4) 5040. [8) 2340. ANSWEfcS. 301 Examination Papers. (Page 49). (I.) (1) 3327, (2) 35 times. (8) 44496 rails. (4) 7. (5) 84, 36 and 182. (II.) (2) Bags of 1, 2, or 3 bu. each ; bins of 300, 150, or 200 bu. (3) $1650. (4) 60min. (5) 982882. (1) 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 25, 30, 36, 40, 45, 50, 60, 72, 75, 90, 100, 120, 150, 180, 200, 225, 300, 360, 450, 600, 9uO, 1800. (2) 29. (3) 8391 and 2699 are prime; 14787 and 1477 are com- posite. (4) 60 hours ; A, 800 mi ; B, 240 mi. ; C, 180 mi. (5) 40 grains. (IV.) (2) '203. (3) 9fmi. (4) 70560. (5) 24 firkins. 00- (2) 60. (3) 3 and 6. (4) 44 times ; 9284 trees. (5) 8366000. Ex. (xxvii.), p. 54. (1) TV (2) & (3) *. (4) i. (5) f. (6) (7) H- (8) fa. (9) fif. (10) M. Ex. (xxviii.), p. 55. (i) ft, It (2) f, M, if. (3) *ft, g, ftj. (6) 18, M, M, ?f (7) Ex. (xxix.), p. 56. The fractions are arranged in descending order. (1) *, rV- f (2) f, i, ^ 2 y . (3) , , 9 T, W A. li. A- (5) ^, &, 4^. (6) ^, ^., 802 ANSWERS. Ex. (xxx), p 56, (1) . (2) If (3) >f (4) ||. (5) j*|. (6) M- (7) mi-' (8) itni- (9) MM- Ex. (xxxi), p. 57. (1) A- (2) T \V (3) T i F . (4) (6) T'A- (7) rVrV ( 8 ) *** () Ex. (xxxii), p. 59. (1) A- (2) *. (3) f. (4) (6) iY*V (7) if|^. (8) AV (9) Ex. (xxxiii), p. 60. (1) f (2) |. (8) A. (4) if (5) f. (6) |f (7) $f (8) f (9) f. Ex. (xxxiv.), p. 61. W-r- (2) 4f- () 4F ( 4 ) (5) 42 T V (6) SiVfo- (7) 81,Vr. (8) Ex. (xxxv.), p. 62. (1) 1 T \V (2) If. (3) If^f. (4) 66|. (5) 43i. (6) 178f (7) 5 T V (8) (9) 38 A. (10) 2^. (11) 8^J. (12) Ex. (xxxvi.), p. 63. (1) 27|. (2) 744. (3) 718f. (4) i|. (5) li. (6) 4. (7) 2|. (8) 26. Ex. (xxxvii.), p. 64. (i) A- (2) T^- ( 3 ) A- < 4 ) (5) 8J. (6) 142i. ( 7) 8 350j. ( 8 ) 66. Ex. (xxxviiii), p. 67. (1) 1A- (2) ^- (3) if- (4) (5) 1A- ( 6 ) (7) M- (8) (9) 2f c (10) H- ANSWERS. 303 Ex. (xxxix.), p. 68. (1) ft- (2) T V ( 3 ) lyV (4) f (5) 5Sf (6) f. (7) fff. (8) 20113. (9) 1^. (10) Iff (ID 7 V (12) |, Ex. (xl.), p. 68. (1) l|f. (2) 1&, ft- (3) I, 6f. (4) tf, W> 3 - W 4 t- ( 6 ) n H> 20. (7) * 1 8 V i- (8) 3 (9) 7f (10) 3. (11) 10|ff. (12) li. If (13) |, T V (14) -f, If. (15) 1, j. (16) 5, 18. (17) 66. (18) 14 T %V (19) |f. (20) 1. (21) f (22) Iff. (23) 2. Examination Papers. (Page 71.) (I.) (2) $49i. (8) SrW- (4) $13860. (5) 3f^ and M- (H.) (2) H, H, A, & (3) Wo- (5) Ship, $24000; Cargo, $36000. (in.) (2) $18|. (4) yftihr. (5) A, 20 ; B. 48 ; C, 84. (IV.) (2) H- (3) 34 A. (4) Horse, $120; Carriage, $105; Harness, $25. (5) A, $4334 ; B, $1474; C, $3080. (V.) (2) 3|f. (3) $2015; 465 sheep; 390 calves; 806 pigs. (4) $180. (5) 18 ft. (VI.) (2) A, ft, iV (3) 1. (4) $40. (5) 3 * B ft. (VH4 (2) $1333i, aV' (3) 1000000. (4) 30 min. : A, 6 times ; B, 5 times ; C, 4 times ; (5) A ; A ; B, & ; C, M ; D, #. ANSWEK8U (vni.) (2) J;A- ( 3 ) 86. (4) 80 min.;4500 rods; 8600 rods ; 3000 rods. (5) 252. Ex. (xli), p. 76. (1) i (2) i (8) f (5) rtfoJW (6) TuSSim (7) V (9) 128001 no) looooi rm -9 -37 V ' 8500 V ' 1000 V ' l ' (18) -4579. (14) -003. (15) 172-95. (16) -0000059. (17) -025679. (18) 8'25793. (19) -0019. (4) I (8) 2 AI V ' 200 Ex. (xlii), p. 78. (1) 7. (2) -2464. (8) -0012. (4 758-279832. (5) 885-260863. (6) 8741-2062. (7) G964-72672. (8) 970-17047. Ez. (xliii), p. 79. a) 61-211. (2) 1-543. (3) 48-2293. (4) -001 () 0607. (6) 579-1274. (7) -0000014. (8) 004385. (9) 9-9998. (10) -00101. Ex. (xliv), p. 81. (1) 85-25. (2) 18-9326. (S) -100345. (4) 00041588. (5) 12-08980432. (6) -9. (7) 14977-92625425. (8) -0000465131. (9) 057746898828045. (10) 203 175662750726562. (11 00984126. (12) 1-01. (13) -00031304. (14 15205806. (15) -1009981674. (16) 20-570824 (17) 150-0625. Ex. (xlv), p, 85. (1) 12. (2) 14400. (8) -0018. 4 12700. (5) 43-078. (6) 10000. (7 430. (S) 147. (9) -0000002004. (10) 98-476. (11) -0065889. (12) 876540000. (13) 0000771039. (14) 299846000. (15) -20162. 16) 2469300000 . (17) 8596. (18) -00000029. 19) 1290. (20) 8-59. (21) 457-61. N 76.371. (23) 905741000. Ex. (xlvi),p.87. (i) 28.28125. (2) 1-119296875. (8) 8-4608. (4) 83U35-448.. (6) -00192. (6) -0001736. 305 Ex. (xlvii.), p. 87. (1) 26-654875. (2) -0010902475. (4) -00001614. (6) 375-03J99875, (7) '154468-75. (8) 25000000. (10) -0000005005005. (3) 14498-8. (6) -0000926. (9) -00001. Ex. (xlviii.), p. 88. (1) 18478-260, (2) -249. (3) -092. (4) 8658146-964. (5) -095. (6J 82714-285. Ex. (xlix.), p. 92. (1) -35. (2) 44. (3) -857142. (4) -6i. (5) -OOi. (6) 02439. (7) -523809. (8) -216. (9) -01236. (1) -09484. (4) 235-104. (7) '0374. (10) -928. (1) (5) (1) (10) 2-345. Ex. (1.), p. 95. (2) -00252J . (5) 26-38702-. (8) 426-104. (3) 165-6995. (6) 1-811. (9) 170-8367. <2) T 3 i (6) Ex. (1L), p. 96. (7) Ex. (lii.), p. 97. (8) TTiTT- Ex, (liii.), p. 99. (2) 20-51662025. (3) 1-7780052. (5) 20|. 6 (1) 15-8430, (4) -02067249. (7) fifr . (81 Examination Papers. (Page 99.) (I.) (3) 1st. (4) 263 times; |. (5) -0000006 and -0000009. (H) (i; ^ Tffo, , fifa. (2) $14.90. (3) 8.715. (4) U. (5) -7142. 306 ANSWERS. (in.) (3) 13; $3000. (4) $232. (IV.) (5) (2) $21.60. (3) 425, (4) 4000004-00000010000090007 ; Seventy-four millions, three hundred and six, and sixty millions and seven trillionths. (5) 82 yd. (V.) (2) T| fHfor- (3) 10-7608 miles. (4) 13||. (5) ^,$192.23^; B, $145.53^; C, 11.0.94 J. (VI.) (3) $34^. (4) 3-141592. (5) -00097061. Ex. (lv.), p. 105. (1) 14. (5) 297. (9) 440. (13) 23456. (17) 5678. (2) 23.- (6) 345. (10) 835. (14) 72500. (18) 437962. (3) 32. (7) 327. (11) 6031. (15) 2031. (4) 75. (8) 867. (12) 4698. (16) 739000. Ex. (Ivi.), p. 106. (1) 4-1. (5) -25. (9) 210-75. (1) 4-4721. (5) -4110. (9) 4-0305. (2) 16-79. (6) -027. (10) 137-65. (3) -95. (7) 131-31 Ex. (Ivii.), p. 107. (2) 5 4772. (3) 9486. (6) -1264. (7) -0252. (10) -9999. (11) -5025. (4) -61. (8) 1-Onl. (4) -3478. (8) -0347. (12) 6-483*. (1) (5) (9) (13) (17) f,. iff s|. 7905. 8-7649. (2) (6) (10) (14) A. 2i. 4 6454. (3) (7) (11) (15) II 2i 4* 2-5298. (4) (8) (12) (16) 11- Hi. 8f 3-0822. Ex. (lix.), p. 110. (1) 16. (2) 32. (3) 42. (4) 79. (5) 85. (6) 64. (7) 34. (8) 73. (9) 85. (10) 99. (11) 39- (12) 63. 128. (9-j 256. (13) 686. (17) 4968. (1) .78. (5) I (9) 8-320. (13) -908. (1) 27. (5) 54. ANSWERS. Ex (Ix), p. 112. (2) 531. (6) 179. (10) 579. (14) 708. (18) 8765. (8) '307. (7) 463. (11 j 458. (15) 888. Ex. (Ixi), p. 113. (2) '364. (6) l^ (10) -495. (14) -693. (3) 30-02. (7) 7| (11) 2-516. (15) 1-966. 807 (4) 670. (8) 103. (12) 507. (16) 512. (4) (8) (12) (16) H 1-709. 822. 1-473. Ex. (Ixii), p. 113. (2) 45. (6) 8-1. (3) 6-3. (4) 13. Ex. (Ixiii), p. 117. (1) 13 ; 80 ; 86 ; 47. (2) 108 ; 270 ; 615 ; 845. (3) 3456 ; 4800 ; 2762 ; 16535. (4) 72 ; 58 ; 94 ; 105 ; 163. (5) 960 ; 1228 ; 4253 ; 14087. (6) 41880 ; 103870 ; 305973. Ex. (Ixiv), p. 118. (2) 43^d. (3) 49d. (5) 9s. llfd. (6) 15*. 6fd. . 3^d. (8} 391. 19s. 4$d. (9) 564. 19s. Id. Ex. (Ixv), p. 120. (1) (4) 7*. (7) 4. (2) 31. 8. Od. (3) 32. 15*. 2d. (5) (1) 21. 15*. Id. (4) 31. 12*. 9d. (5) 23. 13*. 8d. (6) 33. 18s. 4d. (7) 32. 9*. 3fd. (8) 32. 6*. 5d. (9) 169. 5*. Id. (10) 181. 18*. 6d. (11) 240. 19s. 7d. (12) 168. 11*. (13) 200. 17*. Hid. (14) 220. 6*. 9|d. a 5) 3602. 17*. 6d. Ex. (Ixvi), p. 122. \" I? 36. 8*. 5d. 8. 18*. lid. 14*. IK (9) 86108. 17. (2) (4) (6) (8) (10) 28. 0. lOd. 238. 17*. 10|d. 1. 16*. 7|d. 1519. 12*. 9d. 1219. 19. 4N8W1KB*. Ex. (ixvii), p. laa, r 1. 9s. (2) 5s. lOd. (8) 8s. 9d. ' 3. 6s. 6<2. (5) 18s. Sd. (6) 1. 2s Id. (7) 22. 14s. 8d. (8) 14. 11s. (9) 12. 5s. ' (10) 5. 18s. l$d. (11) 21. 12s. (12) 15. 15. (13) 111. 6.. 8d. (14) 5. 18s. l\d. (15) 122. 9s. 4d. (16) 104. 12s. (17) 5. 12s. 6d. (18) 8. Os. (19) 3. 14s. Sd. (20) 48. (21) 36. 6s. (22) 39 75. Qd. Ex. (Ixviii), p. 125. (1) 6. 10s. 6d. (2) 24. 9s. (3) 3. 12s. 5H. (4) 5. 18s. 4d. 29. 13s. 4d. (6) 34. Is. 7R (7) 167. 19s. U. (8) 15212. 1 (9) 6189. 6. 7id. (10) 6022. Os. (5) (7) 167. 19s. U. (8) 15212. 12s. 6d. (9) (11) 8615. 8s. Ex. (Ixix), p. 127. I. (1) 7s. 10$d. (2) 5. 12s. Qd. (3) 18s. (4) 8. 19s. 4cZ. (5) 12s. Sfcd. (6) 1. 17s. II. (1) 1. 16s. 6id. ' (2) 4s. 8d. (3) 5s. 6d. (4) 2s. 4d. (5) Is. 6id. (6) 19s. lOd. I. (1) 1. 3s. 2d. (2) 8s. 4Jd. (3) 1. 4s. (4) 4. 4s. 3*d. (5) 6s. 4|d. (6) 1. 8s. 9ifd. Ex. (Ixx), p. 127. (1) 100. (2) 22. (3) 42. (4) 79. (6) 231. (6) 10. Ex. (Ixxi), p, 128. a) 3*. fffd. (2) 4s. 62rf. (3) 6s, (4) Is. (5) Is. 9d. (6) 24. 16s. Sd (7) 10s. 6ci. (8) 14s. 8d. (9) 13. 6s. Gd. (10) 48. ls.4fd. (11) 77. 5s. (12) 1. 15s. (13) 8. 8s 3id (14) 8. 12s. Id. Ex. (Ixxii), p. 129. (1) 1412. 11s. Sd. (2) 3226. 0. 6d. (8) 28299. Is. lOd. (4) 31282. 8s. 5d. (6) ^18873. Is. 6d (6) 27877. 13s. W. 309 Ex. (Ixxiii), p. ISO. (1) 22645 sec. ; 61243 sec. (2) 107020800 sec. ; 544324 mm. (3) 88 da. 17 hr. 27 min. ; 6 hr. 32 min. 56 sec. (4) 8 da. 14 hr. 13 min. 12 sec. ; wk. 2 da. hr. 24 min. 56 sec. (5) 118; 151; 286; 120; 151. (6) 76 hr. 34 min 36 sec. (7) 136 da. 1 hr. 42 min. (8) 26 wk. 2 da. 2 hr. (9) 22 yr. 293 da. 1 hr.. (10) 77 hr. 8 min. 41 sec. (11) 250 da. 23 hr. 1 inin. 18 sec. (12) 2 br. 54 min. 48 sec. (13) 83 da. 17 hr. 47 min. (14) 6 da. 22 hr. (15) 298 da. 21 hr. (16) 1 yr. 331 da. 21 hr. (17) 5 da. 9 hr. 36 min. 46 sec. (18) 463 hr. 35 min. 5 sec. ; 740 hr. 46 min. 57 sec. (19) 2 da. 6 hr. 14 min. ; 12 min. 17 sec. Ex. (Ixxiv), p. 132. (1) 182 in. ; 23166 ft. (2) 446418 in. ; 6499 in. (8) 18513 po. 3 yd. ; 306 fur. po. 4 yd. 2 ft. 6 in. (4) 137 mi. 86 po. 8 yd. 1 ft. ; 1309 mi. 4 fur. 32 po. 4 yd. 2 ft. 8 in. (5) 107 yd. 1 ft. 8 in. (6) 154 mi. 2 fur. 20 po. (7) 23 fur. 21 po. 4 yd. (8) 75 yd. 8 in. (9) 50 mi. 2 fur. 85 po. (10) 85 p. 8 yd. (11) 87 yd. 3 in. ; 932 mi. 1 fur. 31 po. (12) 1858 po. 3 yd. ; 1783 mi. 3 fur. 5 po. 1 yd. (13) 6 yd. 1 ft. 2 in. ; 5 fur. 6^y po. (14) 2 yd 1 ft. 5i in. ; 1 fur. 291! po. Ex. (Ixxv), p. 134. (1) 36751875 sq. in. (2) 44425044 sq. in. (3) 1210000 sq. yd. ; 94608 sq. in. 4 sq. yd. 55 sq. in. ; 3 ac. 28 po. 9 sq. yd. 1148 po. 2 sq. yd. ; 14 po. 10 sq. yd. 7 sq, ft. 110 sq. in, f6) 284 ac. 2 ro. 25 p. 7) 163 sq. yd. 7 sq. ft. 91 sq. in. (8 (9 (10 (11) 1 ac. 2 ro. 16 po. (12) 8 ac. 1 ro. 80 po. (13) 6 sq. yd. 7 sq. ft. 22 sq. in. (14) 66 ac. 8 ro. 36 po. (15) 88 ac. 2 ro. ; 931 ac. 8 ro. 9 po, (16) 1 ro. 18 po. ; 1 ro. 27 po. i v t 4) 5) 1- y 112 ac. 8 ro. 38 po. 15 sq. yd. 27 ac. 2 ro. 36 po. 6 sq. yd. 8 sq. ft. 129 sq. in. 310 ANSWERS. Ex. (Ixxvi.), p. 135. (1) 202 cub. ft. ; 1175183 cub. in. ; 654558 cub. in. (2) 43 cub. ft. 21 cub. in. ; 9 cub yd. 11 cub. ft. 372 cub. in. (3) 244944 cub in.; 149904 cub. in. (4) 270 cub. yd. 26 cub, ft. 1143 cub. in. (5) 195 cub. yd. 3 cub. ft. 298 cub. in. (6) 3558 cub.' yd. 10 cub. ft 284 cub. in. (7) 8 cub. yd. 20 cub. ft. 1545 cub. in. (8) 8 cub. yd. 1634 cub. in. (9) 27 cub. yd. 7 cub. ft. 1472 cub. in. (10) 707 cub. yd. 1323 cub. in. ; 2312 cub. yd. 17 cub. ft. '518 cub. in. (11) 6 cub. yd. 14 cub. ft. 1029 cub. in.; 8 cub. yd. 24 cub. in. Ex. (Ixxvii.), p. 136. (1) 59 pts. ; 109792 pts. (2) 8 qr. 2 bu. 1 gall. 2 pt. ; 47 qr. 4 bus. 3 pk. 1 gall. (3) 41 gaU. 1 pt. (4) 20 bus. 1 pk. 1 gall. (5) 197 qr. 3 bus. (6) 2 qt. 1 pt. (7) 3 pk. 1 gall. (8) 6 qr, 7 bus. 3 pk. (9) 342 qr. 4 bus. 2 pk. ; 1115 qr. 4 bus. 1 pk. (10; 3 qt. 1 pt. 8 qr. 3 pk. Ex, (bLXviii.). p 137. (1) 12960 gr. (2) 1680 dwt.: 3420 dwt. 6185 dwt. (3) 22253 gr.; 42663 gr., (4) 6 oz. 11 dwt. 1 gr ; 7 Ib 4oz. 18 dm. (5) 12 Ib. 6 oz. 19 dwt 13 gr. - 13 Ib. 6 oz. 6 dwt. (6) 74 Ib. 7 oz. 7) 30 oz. 4 dwt. 9 gr. (8) 87 Ib. 7 oz. 12 dwt. 18 gr (9) 3 oz. 4 dwt. 21 gr. (10) 7 Ib. 9 oz. 13 dwt. (11) 9 oz. 12 dwt. 23 gr. (12) 89 Ib. 5 oz. 8 dwt. ; 141 Ib. 7 oz. 19 dwt. (13) 401 oz. 7 dwt. 11 gr ; 148 Ib. 9 oz, 5 dwt. 21 gr. (14) 2 Ib. 12 dwt. ; 6 oz. 6 dwt. llf gr. (15) 5 dwt. 8 gr. - 2 oz. 19 dwt. 20 gr. Ex. (Ixxix.) p 136 (1) 17600 oz. r 4352 dr. ; 10000. (2) 203200 oz. ; 30050 Ibs. (3) 78416 dr. ; 7507 Ibs. (4) 2 cwt. 3 qrs. 22 Ibs. 11 oz. : X ton 17 owt. 1 qr. 24 Ibs ANSWERS. 811 (5) 4 cwt. 2 qrs. 14 Ibs. 8 oz. ; 9 cwt. 2. qrs. 15 Ibs. 15 oz. 14 dis. (6) 53 Ib. 12 oz. 1 dr. (7) 45 qr. 19 Ibs. 15 oz. (8) 88 cwt. 2 qr. 14 Ibs. (9) 2 Ib. 1 oz 9 dr. (10) 2 qr. 22 Ib. 8 oz. (11) 1 cwt. 1 qr. 11 Ibs. (12) 7t. 19 cwt. 3 q. (13) 3 Ibs. (14) 34t. 18 cwt. 1 qr. 13 Ibs. (15) 120 cwt. 67 Ibs. 2 oz. ; 187 cwt. 65 Ibs. (16) 156 cwt. 1 qr. 15 Ibs. ; 390 oz. 13 dr. )17) 1 cwt. 21 Ibs. 8 oz. ; 16 cwt 1 qr. 13 Ibs. 2 oz. (18) 1 qr. 14J| oz. ; 2t. 3 cwt. 3 qr. 6 y 5 T Ibs. Ex. (Ixxx.), p. 139 (1) 13 cwt. 1 qr. 2* Ib. (2) 13 Ib. 14 oz. 12 dr. (3) 80 mi. 1 fur. 22 po. (4) 679 yd. 1 ft. 6 in. (5) 166 ac. 3 ro. 32 po. (6) 757 ac. 2 ro. 12 po. (7) 78 sq. yd. 7 sq. ft. 6 sq. in. Ex. (Ixxxi.), p. 139. (1) 2 cwt. 4 Ib. (2) 10 oz. 5 dr. (3) 1 mi. 5 fur. 8 po. (4) 3 yd. 6 in. (5) 5 ac. 3 ro. 4 po. (6) 1 ac. 3 ro. 8 po. (7) 5 sq. yd. 7 sq. A. 87 sq, in. Ex. (Ixxxii.), p. 140. (1) 13s. 4d. ; 1 ils. 3d. ; 2 10s. 9d. (2) 6 fur. 16 po. ; 30 po. ; 3 qr. 8 Ib. (3) 152 11s. 0d.; 1 13s. 9d.; 2 mi. 2 fur. (4) 514 16s. 15s. 9d. (5) 1 2s. W%d. (6) 13s. 6d. (7) 9 ac. 2 ro. 13 po. (8) 16 da. 3 hr. 35 min. (9) 2 for. 37 yd. 1* in. (10) 4 cwt. 2 qrs. 11 Ibs. lOf oz. Ex. (Ixxxiii.), p. 141. (1) A- ( 2 ) Mf () 44- ( 4 ) iff- (*) JWV- (6) f. (7) f . (8) (9) T 3 y . (10) I (11) iff*. (12) Ex. (Ixxxiv.), p. 144 (1) 12s. 6d. (2) 15 5s. 6d. (3) 2-3436rf. (4) 3 qr. 18 Ib. 12 oz. (5) 12 dr. (6) 16. Os. 6rf. (7) Is. 5d. (8) 2 16s. 9'375d. (9) 4* 2d. (10) 2 10s. 7-6d. 812 (11) 22 Ib. 6|o. (12; 7.16*. (13) 4 tons 16 cwt. 17.4 Ib. (14) 26, 17*. (15) 2, 6*. 9'Sd. (16) 16*. Id. (17) 1. 14*. &*. Ex. (Ixxxy), p. 145. (1) -3285 (2) -002088. (8) -1876. (4) -43. (5) 14-49. (6) '24. (7) 2-64. (8) 1-382890626. (9) -0027.' (10) 1-4318. (11) -3. (12) -00091876. (18) 2-445916. (14) -6581.... (16) -1406 Examination Papers. (Page 146.) I. (1) 866 ft V (2) 8413 d. 9 hr. (3) 8 d. 2 hrs. 20 min. (5) 191. II. 12) ISiilJ. (8) 49 min. past 1 P.M. ; 149$ mi. (4) 19 mi. 1464H yds. (5) 28160. III. C2) 8600 ; 7. 10*. (3) A 8 a. 1 r. 20 p. 21 yd. 77} in.; 6 a. 8 r. 1 p. 11 yd. 7 ft. 118? in, ; C 7 a. 2r. 16 p. 17 yd. 1 ft. 29* in. (4) 29 yds. (6) 16 1. 4 cwt.; 10 cwt. 8 qrs.6 Ibs. IV. (1) 111.87$. (2) 259 bus. 2 pk. 1 gal. liih pt. (8) 41 bus. 3 pks. 21 qts. (4) 47 bags. (5) $96.98 .... V. (1) Loses $2. (2) 13|?| cents. (3) 20 grs. is largest waight (4) 12 t. 7 cwt. 3 qrs. 16" 67 Ibs. (5) 250 Ibs. Ex* (Ixxxvi), p. 150. (1) 8061. 7*. 8$ A 6 3; /A- (19) $175.50; $218.40; $252.72; $117.00; $149.76. (20) 1200 boys. Ex. (cxii.), p. 222. (1) First, $44.25 ; Second, $88.50. (2) ^,$4.50; 5, $6.75; C, $11.25. (3) A, $2062.40; B, $2320.^0; G, $773.40. (4) 'A, *656rWr: 5, $286 A%. (5) D, $20; E, $50. 6) A, $87.50; B, $120; C, $202.50. ANSWKKb. 819 (7) $80; $48; $28. (8) $15.30; $14.25. (9) A, $245; B, $225. 10) Johnston, $585 ; Wilson, $487.50 ; Miller, $175.50. 11) ^,$34.30; B, $53.90. 12) A, $735; B, $490 ; C, $367.50 ; D, $294. (13) 16 gall, and 25^ gall. (14) A, $118.30; B, $55.90; 0, $13. Ex. (cxiii), p. 225. (1) Net gain, $1974 ; A'B, $2312 ; B\ $2172. (2) Net loss, $3165 ; 4's, $2836 ; 's, $1154. (3) Net loss, $3550 ; A'*, $1010 ; jB's, net insolvency, $2730. Ex. (cxiv), p. 230. (1) 5 Ibs. of first, 7 Ibs. of second. (2) 30 bu. oats ; 20 bu. rye ; 20 bu. barley. (3) 60 Ibs. at 55 cts. ; 30 Ibs. at 75 cts. (4) 1 5 gall, water. (5) 12 gall, kerosene. (6) 14 bu. rye ; 14 bu. barley. (7) 18 Ib. at 14 cts. ; 18 Ib. at 18 cts. ; 48 Ib. at 30 cts. (8) 86 Ib. at 33 cts. ; 36 Ib. at 37 ots. ; 48 Ib. at 45 cts. (1) 109* (4) 1760 cope ks. (7) 576.12*. Qd. 25 fr. 45c. for 1. (11) $4.86. 1. mikee. (nearly). Ex. (oxv), p. 837. (2) 44693.20 (8) 2 fr. 18 cent. (5) 9 fl. 20 kr. (6) 3345.44. (8) 1 = $4-8665. (9) London gives (10) 1 = I3j> marcs Banco. (12)2602 $fl. (13) 58H per (14) ()1020i5 oz : 25.17 francs. Examination Papers. (I.) ) 1-2372. (2) A $6075; B$5400; C$6000. '8)5774.43. (4) Direct $14224.91 ; Cir, $14476.72; gain $251.81 (5) 2*341'% discount. (II.) (1) 78J cents and 66$ cents. (2) A $4912 ; B $6168. (3) 1 = 25. 86;|. (4) A, $2324; B $1708. (5) 10 and 4. (HI.) (1) 83$ Ibs. of 8, 10 and 12 cents and 100 Ibs of 20 cts. C8i $1212. (3) $1257 . (4) ,V. (5) 2. 8*. 2*d. (nearly). 320 ANSWERS. (IV.) (1) $2211 ft. (2) $43.63. (8) $5. (4) Paris, $14285.71f ; London, $14600; Amsterdam, $14640. (5) 1 Ib. at 8 ; 8 Ib. at 13 ; 8 Ib. at 14. (V.) (1) 59, 17, and 106. (3) 8, 10 and 12 months. (5) -42 ; 23 1 3 T per cent. Ex. (cxvi.), p. 242. \ is least. (6) $31.25. (9) 128 1 Ex. (cxvfi.) p 245. (1) 4 : 3 : : 12 : 9. (2) 12$. (4) A : C : : 25 : 39. (5) 21. (7) -048. (8) 28. (1) $ is greater. (3) -J-f is greatest (5) Il2: 405. (8) i : 3i. (2) 118-f per cent. (4; 9176-A. (2) -H- is greater. (4) 45 : 364. (?) H- (10) 9 : 13. (3) -0076 (6) if (9) 2 V (10) A $552 ; B $460 ; C $345 , Z> $230. Ex. (oxviiii.) p 247. (1) 1285. (2) 10 h, 40 m. 36 A sec. (3) T A, mi. (4) 3 h. 25 min. P.M. (5) 10 d. ; 12f d, (6) $47.13. (7) 7|. (8) 8 P.M, Thursday. (9) 7722 stones. (10) 12800 Ex, (cxix. /p p, 249. (1) 54 men. (2; 105G men, (3) 18. (4) 50 men. (5) Navvies did 6 times as much as soldiers (6) 12M dronas. (7) 576 (8) 16|. (9) 155 (10 12 days. Ex. (cxx).. p, 252. (4) 5000 mm. (6) 15 milligrams, (8) 155000 sq. cm. (10) 1725 grams. (11) (12) 256-7 centigrams. (14) 1-60931 kilometres. (16) 3720Utres. (5) 1067.25 dcm. (7) 43-7 mm, ; 4 37 cm. (9) 1086-42 sq. dcm. 100 milligrams ; 10000 decigrams. (13) 5000 milligrams. (15) 567*875 cu. cm. ANSWERS, 821 (1) 35 sq. ft. (4) 12 sq. ft. (7) 608 sq ft. 10) 30^ sq. yd. (13) 870i sq. ft- (15) 11' sq. ft. (17) 2232 sq. ft. 81 (19) 7 ft. 5 in. (22) 88 yd. (25) 16 ft. (28) 255yd. (31) 5^ 2 in. Ex. (cxxi), p. 265. ' (2) 135 sq. ft. (3) 800| sq. ft. (5) 452| sq. ft. (6) 224 sq. ft. (8) 150 T 3 9 sq. ft. (9) 402 sq. ft. (11) 1387 A sq. yd. (12) 315 T V eq- ft. (14) 91 sq. ft. 121 in. (16) 502 sq. ft. 73 sq. in. sq. in. (18) 16 ft. (20) 8 ft 9 in. (21) 11 yd. (23) 99 yd. (26) 103ft. (32) 360-5 yd. (32) 625^ 2 ft. Ex.(cxxii), p. 256. (24) 9 ft. (27) 405 yd. (30) 163-25 yd nearly. (1) 28|. (4)" 58. (7) (2) 46JV. (3) 67. (5) 142ff. (6) $33.60. (8) $83.89*. (9) 11. y*. Ex. (cxxiii), p. 258. (1) 630. (5) 125.60. (8) 6. 6*. (2) 855. (6) $13.62. (3) 875*. (7) 6. 13s. (4) 79a Ex. (cxxiv), p. 258. (1) (4 CD (10) (13) (16) (18) (20) (23) $4.93^9. 135 ft. $1.20. 22^ a. $12. $13.50. 10511* sq. $69.20. 2ft. (2) $520.96*. (5) 13. 10s. (8) 12 ft. (11) 17i ft. (14) 5952 stones. (17) 429 yds. ; 715 yd. ; 2955fr sq. yd. (21) $2955 fr (24) 300. (3) (6) (9) (12) (15) yds. '19) (22) 210 ft. $5.95. 62 yd. 1 ft. 1ft. 9 1 in. $12.96. 26. $112.64. Ex. (cxxv), p. 262. (1) 836 cub. ft. (2) 548| cub. ft. (3) 83 J| cub. ft. (4) 850 i-ff cub. ft. (5) 1058 y T cub. ft. (6) 9600. (7) 16335 tons. (8) 500 men, (9) 1 ft. 7 in. (10) 203U lb. (11) It owt. (12) 5* ft. ,W) 100. (14) 31ft. <16)38 19s Zd 322 ANSWERS, Examination Papers. (1) 118. (4) 30 inches. (2) $75. (5) 6 of each. 1000010001 (3) 81; H; (6) -02 ; 2000 ; -000002 : 2000-020002; 500000 (7) -432. (8) 7899 mi. 1 fur. 25 po. 3 ft. 6 in. (9) 45 miles. (10) $210. (12) $5670, $7560. (14) |; -75. (15) 14. (17) 12 days. (18) 108. (30) A, $2.49; B, $15.81. W; WtfW; -21; 2100. (22) (24) (11) 9405 steps. (13) -rtfa; '0189. (16) 17695260 in. ; (19) $12000. (21) 1; if. (23) flft : twice. 142 12s. Qd. ; 42 15s. 9d. ; 14 5s. U. . 1^-96; j|. (27) 21 yd. 2 ft. 2f in. (29) $166-66|. (31) 8*. (33) 4 per cents.; $128700. (35) $35.00 and $52.50. (37) 3 ft. (39) 55-3 ft. (42) $4200. (45) 16s. Qd. (47) $2; 10 %. (48) $1785. (50) 5 15s. 0< (53) $37331. (G6) $823.68. (59) $1.50. (62) 33-6 in. (65) 3 hr. (40) 75 J yds. (43) (54) 36 days (57) $9.37. (60) $15f. (63) 1 hr. (66) 8 hr. (68) $1680. (69) 16i ft. (71) ij|8. (72) 7-976 d. (74) $4906-25. (75) 16 day. (77) A gets $1925; B $770; C $154. (79) Loss of 40%. (26) 5 h. 48 min. (28) $32-66f. (30) $49-50 and $49.50. (32) f *H. (34) 7& knots. (36) 35 cents less. (38) 5A%; 17if (41) 1. (44) 256. (46) 200, 189, 101. (49) 270ft. (52) 5s. Id. (55) 26 sec. loss, (58) $16*. (61) a $3346.87^ 7 %. 11 sq. ft. ; -000365. (81) 6.33403. (82) &. (84) 4 hr. 32 min. charcoal, 3^ Ib. of sulphur. (87) $16. (88) $12705. (90) $60.75 ; $20.42*. (02) 31116. (94) 1^ min. to 12. (96) 4% & 5%. (97) $805, (99) 900. ;64) 67) (70) (73) (76) (78) $3600 ;5| %. (80) 4 Is. 6f4d. (83) 8 days. (85) 22 Ib. of nitre, 4|lb. of (86) 95 T V cents. (89) 1-2535 Ib. (91) 1. (93) 1 ; -740. (100) $67,50. ANSWEBS. (101) 42238274625. (102> -8937 in. (103) A in 6|$ da.; B in 9-f| da.; C in 14^ da. (104) 90 miles. (105) $410; $800; (106) $9.50. (107) Loses 4/o. (108) $214. (109) $9.38^- (110) Id Ifd. (Ill) 14min. (112) -J. (113) $8250. (114) 30 mi.; 25 mi. per hour. (115) 7$ ; 50 less. (116) 4 days. (117) $6.17*. (119) On Tuesday p. m. when one clock marks 9h. llm., and the other 8JL 54m. 30sec. (120) $1238.70 (121) 4497 times. (122) $450. (123) $10, $18, $15. (124) $1.12. (125) 80. (126) 4 times. (127) B walks a mile in 13 min ; he loses by 11 J mir. and by ff mi (128) 7* months. (129) 90 ; $465. (130) 1200 gal. (131) 4 a *r. (132) 46. (133) 4000ft. 134) A t gets 88*cents ; B, 49* cents. (135) 18 j^ min. and 16^ min. past 3. 136) 13* years. (137) 136. 9s. 2d. ; 6 % 138) 10. (139) 60000. (140) 88. 141) $1.76. (142) 8* days. (143) 8* hrs. .144) 28^ days. (145) 400 miles. (146) 87*. (147) A gets $1155 ; B, $572 ; C, $259.50. (148) 6 J. (149) 111885$- metres. (150) 44. 13s. 3d. ; -J- ft. (151) 2. (152) 987 ; -02268 of an inch. (153) 7* miles. (154) 8k (155) 600. (156) $760. (157) 16f miles. (159) S&-9 fr. = 1. ; 25-15 fir. = 1. (160) 8ft. 11^, in. (161) 1. (162) 7442.fr (163) 48 min. (164) 4i months. (165) 9 of spirit to 31 of water. (166) 56%. 167) $2035. (168) 10. (169) 2'198in. in a century 170) 6.30P.M. (171) 103-67; 574. (172) 16 hr. 173) 56 yd. (174) $6. (175) ^ min. (176) $23 17f *. (177) ISff min. .(178; 1520 tons. (179) 10861578 nearly. (180) 1120 yd. ; 262^ ac, (181) 15 masters, 845 boys. (182) A gets $17.50; J5, $52.50 ; C, $105 ; D, $175. 83) $4500. (184) 18 cents. (185) 40 % of loss. 186) $24360. (187) 6 cents. (188) $11835.75, 189) $3.40. (190) $81.12; 7 yds. (191) 2. 92) 1A hr. (193) $7500. 94) $3750. (195) $8400. 96) Each child gets $1920.60 ; each brother, $960.30. 824 ANSWERS. (197) $215. (198) He gains 14ft f. (199) 50000000 quarters. (200) 8 miles an hour ; 1 miles an hour. (202) 4J. (203) 87 5 T . (204) 1609-306 metres. (205) $2.00. (206) 8-243 %. (207) $5 40 of increase. (208) 2:7. (209) 8b lb. 7 oz. Avoir. (210) 2 ft. 8 in. (211) 9. 3s. Gd. (212) 5 fi and 10 %. (213) 3 pints (214 4 %. (215) $2450. (216) $92 ; $115. (217) 80 days. (218) $4000. (219) $115. (220) 474. (221) -05. (222) 124.001 and f. (223) 9000 men. (224) Second greater by $50. (225) $9125. (226) 71 hr. ; 18 hr. ; 5} hr. (228 10 hours. (229) $6120. (230) Length 32 ft. ; breath 16 ft. ; height 8 ft. (231) 5s. (232) 42000. (233) $33.75. (235) Lost$l. (236) $305. (287) $if. (238) 6000. (239) $9.50. (240) 48000. (241) 8s. (242) 3 T \ days. (243) 90 days. (244) $56000, $48000, $42000. (245) $35. (246) 160 ; 623. (247) 1500 bbls. (248) 1. 11s. 4 \d. (249) 5$ years. (250) Length, 27 ft. ; breadth, 18 ft. ; height, 12ft. (251) 3998936616. (252) 9. (253) |%. (254) Increase $160. (255) 17$%. (256) 4 days. (257) $30. (258) 4 per cent. (259) 24 days. (260) 110; 150 per cent (261) 400 in. (262) $553 ; 5%. (263) llmi. (264) B. (265) $200. (266) $1200. (267) Loses 5%. (268) $4.81 nearly. (269) $40800. (270) 25 cents. (271) Between -0001 and -0002. (273) A $3200 ; J5$4800; C$6000; D $7000. (274) $6.50. (275) If cents. (276) $5.60. (277) 96&cenis. (278) $12800. (279) $4.80. (280)^^. in. (281) 750 (282) 478, 369, and 584. (283) 45 mi. and 80 mi. per hr. (284) 50 per cent. (285) 82 cents. (286) 15 \ cwt. of nitre ; 1 T - W cwt. of sulphur; 2^ cwt. oi charcoal. (287) 4 miles per hour. (288) Capital $1000000 ; Receipts $100000. (289) 3 %. (290) 1 ; 23. (291) 86fl J 10 T Y*. (292) \\\. (293) 66. 13. 4d. (294) 2-8523809 hr. ANSW3EB9. (295) 5hr. lOig min. (298) 32 days. (297) $39.95. (298) 8 per cent. (299) $16?.^, G JfeU (300) 28i;427^1b. . (301) 70-41; 1-46. (302) 4 per cent. (303) $274.12* ; $456.1' 5 (304) . (305) 8and. (806) $1023.75. (807) $20000. (308) $80 ; $133^ ; Loss 8l3i (309) $5100. (810) $12.31 (311) l|f3; i*. (312) 6f per cent. (313) 16 years. 314) $3000. (315) $7800. 316) 73f cents. (317) 1 months. 318) 5, by 16 yds. (319) per cent, loss 320) 26 yd. (321) ^ ; 6^ ; 7. (322) G2. 5s. (323) 14 min. 43| sec, 824) 9 days. (325) 1 ; 2. 326) $1440. (327) $10000 ; $4000. 328) 70. (329) 2s. Qd. (330) 8. 2d. 331) $;$;!. (332) 5 hr. (333) 5 miles. 334) $234; $266.40 ; $306; $345.60. (335) 3 hours. (336) 70 rents. (337) First is $50. (338) $14600. fl&9) 25 oxen. (340) A, 3240 ; JB, 2916; D, 2052; (7, 1944; E, 1728, in all, 6480. (341) 12 J|. (842) 4||. (843) $2200. (344) 66. (345) 18 min. (346) 1160. (347) Man gets 4. 4*. ; woman gets 3 ; ohild 1 16f. (848) 27951: 12500. (349) 875 grains of potash : 890 grains of soda. (350) $20.95. APPENDIX -I. INTEREST, ANNUITIES, #c. 1. To find the amount of a given sum, in any given time, at Simple Interest. If P be the principal in dollars, n the length of time in years, r the interest of $1 for 1 year; then the interest of $P for 1 year will be Pr, and for n years will be Prn ,- where- fore, if I be the interest, then I = Prn. If M be the amount, we have M = P + P = P(l+m). 2. To find the amount of a given sum, in <*ny given time, ut Compound Intercut. Let P = the principle in dollars ; " r = the interest of $1 for 1 year ; n = the number of years ; " R = the amount of $1 forl year = 1 -f- r\ then PR will be the amount of $P for 1 year, and this becomes the Principal for the 2nd year-, .-. PR-R = PR a will be the amount of $P for 2 years, and this becomes the Principal for 3rd year ; /. PR 2 R =- PR 8 will be the amount of $P for 3 years, etc. ; hence, M = PR" = P(l+r), will b the amount of $P for n years. Interest = PR" P = P(R 1). 3. To show that the formula, M= PR" is true when n is fractional. If n is fractional we can always find a whole ncimber such that na is a whole number = ., suppose. Divide the APPENDIX. 327 year into a equal intervals, and let m be, the amount of $1 in one of these intervals, then the amount of $1 in a in- tervals is m a , and is equal to R; also the amount of $1 in n years, that is na intervals, is equal to m, and therefore equal to R n ; hence the amount of $P = PR" , therefore the formula is true for fractional values of n. Thus, if r' is the nominal yearly rate of interest of $1 pay- able q times a year, meaning that is the interest payablo at the end of each qtla part of a year, then the amount of SI / r '\ q in a year is J$ II + I , and the true yearly rate of interest is + i'!-i Ex. (1). Find the amount of $100 in 2 years at 8 per cent. Compound Interest. = 100 (1 4- '02 + -012 + -000155 + ...) --= $121-215... Ex. (2). Find the advantage when Compound Inter- est is reckoned, of having the interest paid half-yearly, quarterly, &c., instead of yearly. The advantage per $1 for a year, when the interest is paid half-yearly, arid the half-yearly payment is half the yearly = l + r +-- + ...-(1 + r ) r 2 = ~r~ nearly, since r is a small fraction. Similarly, when the interest is paid quarterly, the advan- 3r 2 tage = -- nearly. 3^8 APPENDIX. And generally, when the interest is paid n times a year the advantage Ex. (3). Find the amount of a given sum at com- pound interest, the interest being supposed due every instant. If the interest were paid q times per annum, then ( = P ( 1 + wr + rJu - ) r 2 + ... J Now, if q fee indefinitely great, that is. the intervals be- tween the payments indefinitely small, then, neglecting - and its powers, we heve = Pe 111 , where e = 2-7182818. Todhunter's Algebra, Art. 54". Ex. (4). If P represents the population of any place at a certain time, and every year the number of deaths is ~th, and the number of births ^th, of the whole population at the beginning of that year; required the amount of population at the end of n years from that time. At the end ol one year rom the time the population was P, P P .-. the increase = - APPENDIX 329 /. population at end of 1st year Similarly population at end of second year ( p a] ( p a . - = p, ]l + - ~[= Pjl + - ( pq. } i p f i ' id so on as in Compound Interest. f p q} n Hence, population at end of nth yeni P 1 4- I 4. 'J-'o deduce the formula for simple Interest from th formula for Compound Interest. M = PK M P 1 + nr + -r . r 3 4- &c. Now Compound Interest may be regarded as consisting of two parts : (1) Interest on principal, and (2) Interest on interest. If from the value of M, given above, we take away the part that represents interest on interest, there remains the interest on the principal or the Simple Interest. Now the third term contains r' 2 or r x r, that is interest on interest. Similarly for succeeding terms. Therefore for Simple Interest we have M = P (1 + nr), as before. Hence, any formula for Simple Interest may be deduced from the corresponding one for Com pound, by neglecting r 2 and all higher powers Therefore, in general, we shall find the icrmula ior Com pound Interest, and deduce the corresponding formula for Simple Interest. Indeed this is the only rational method of treating the subject. There is but cnc kind of interest, viz., Compound Interest. Simple Interest is incorrect in principle, and of course may lead to very incorrect results. When any sum of money is due, it matters not whether it is called principal or interest, it is of value to the owner, and should bear interest. The results obtained by the principle of Simple Interest are merely approximations to the correct results obtained by the principle of Compound Interest, 380 APPENDIX. Exercise I. (1) A sum of $P is put out at Simple Interest for n years ; find an expression for its amount at the end of that time. (2) If B be the amount of $1 in one year at any rate of interest, the amount of P dollars in years will be PB", . whether n be integral or fractional. (3) If $P at Compound Interest*amount to $M in * years ; what sum must be paid down to receive $P at the end of t years ? (4) If SP at Compound Interest, rate r, double itself in n years, and at rate 2r, in m years ; show that > A. n * (5) In what time will a sum of money treble itself, at 5 per cent., Compound Interest ? log 3 = -4771212, log 1-05= -0211893. (6) A sum of money, $P, is left among A, B, C, in sach ft manner that at the end of a, 6, c years, when they respective- ly come of age, they are to possess equal sums ; find the share of each at compound interest. (7) Two men invest sums of $4410 and $4400 respectively, at the same rate of interest, the former at simple, the latter at compound interest ; at the end of two years their pro- perties amount to equal sums; find the rate'of interest. (8) In a certain county the births in a year amount to an mth of the whole population, and the deaths to an nth ; in how many years will the population be doubled? (9) A person spends in the first year m times the interest of his property ; in the second year, 2m times that of the remainder ; in the third year, 3m times that at the end of the second, and so on ; and at the end of 2p years he has nothing left ; shew that in the pth year he spends as much as he has left at the end of that year. (] 0) If interest be payable at every instant, in how many years would $1 amount to $6, reckoning interest at 5 per cent. ? (11) A person starts with a certain capital which produces him 4 per cent, per annum compound interest. He spem 1 . every year a sum equal to twice the original interest on his capital. Find in how many years he will be ruined, having given log. 2 = -3010300, log. 13 = 1-1139434. (12) The population of a county is 35748. There is no emigration or immigration. The annual deaths are 27 in AWENDIX. 381 the 1000, and the births 62 in 1000. What will be the in- crease of Ihe population in live years ? (13) If the population of a country be P, and every year the number of deaths s -^-tk and the number of births - 4 Vth, of the whole population at the beginning of the year ; find in what time the population will be doubled. ,og 181 = ?,-25768, ,og 3 = '4771213. log 2 == -30103. (14) On a sum of money borrowed, interest is paid at the rate of 5 per cent. After a time $600 of the loan is paid off, and the interest on the remainder reduced to 4 percent., and the yearly interest is now lessened one-third What was the sum borrowed? \lo, If a debt a at compound interest is discharged in n d yearsjay annual payments of , show that DISCOUNT. 5. To find the Present Worth and Discount on any sum for a given time. (1) Compound Interest. (2) at Simple Interest. The principal difference between Amount and Present Worth is that the former is reckoned forwards from a given date while the latter is ckoned backwards from the same date. Hence it is ev t that if V represents the Present Worth, then, V = P(l + r)r* P = Vf . r \ n for Compound Interest ; expanding and neglecting r* and higher powers we have P ~. fox Simple Interest APPENDIX. If D be the Discount, tlien D = P V p = P TJ - 7^ Compound Interest. p = p -rr^ r > approximately, Pwr = f~T~~. Simple Interest. 6. If we expand P (1 + r)~", and neglect r 2 and higher- powers, we get P (1 nr} which may be called the common present worth. The true present worth is P ; bvdivision = P (1 -nr + n?r 2 n 3 r 3 + &>'. Subtracting the common from the true present worth, we get nr + n 2 r 2 - it is a negative quantity. That is, the common present worth of a bill for 1 100 due 20 years hence at 5 per cent, is nothing, and for any period beyond 20 years the holder of the bill would require to pay a certain sum to get quit of it, which is absurd. The true present worth of $100 due in 20 P years, as given by the formula j i $50. APPENDIX. 833 7. The interest is greater than the discount. Pnr D = 1+^ 1 + TMr 8. Since V = and D D Pnr j_ + i ' Pnr P .-. I > D. 1 + nr __ 1 -f- nr we see that ^/ie Discount is the Present Worth of the Interest. 9. The Discount is half the haniwnic mean between the Principal and the Interest. 1 + nr _PI_ P + Pro- Pi P+I . JTPI * P + I = half harmonic mean between principal and interest Ex. (1). The Simple Interest on A certain sum of money for a certain time is $28, and the discount for the same time at the same rate of simple interest is 884 APPENDIX. $24. What is the sum of money? If the time be years, what is the rate per cent. ?" From the above formula we have 24 - - P+ 28 24P-f 24 X28 = 4 P = 24 x P = $168 ; the sum required is 1168. Again, D 24 = 1-fnr, 28 7 ; 1 ~ ion .-. rate per cent.= 100r= ^- = Ex. (2) If the Simple Interest on a sum of for a given time and rate is ith of that sum itself, the True Discount will be - ,-of the sum. but, in this case, I = -i P ; Similarly, if the interest b'e-r-of the principal, the discount is rof the principal. APPENDIX. 885 Ex. (8). Bank Discount at 5 per cent, being $180.90, find the true discount on the same amount. _=_-, where n = ^=^, 20 .D = X $130.90. = $124- 66f. 10. Sank Discount exceeds True Discount by the Simple Interest on the True Discount. Bank Discount True Discount = ID Pnr 1 4-nr = Dnr = Simple Interest on the True Discount. Ex. (4) The True Discount on a bill due in 1 year, and discounted at 8 per cent, being $500, what would have been the Bank Discount thereon ? Bank Discount = True Discount -f Dnr = $500 -f $500 X == $540. - Exercise II. (1) Bank discount being 5 per cent., a person receives $37.10 less than the nominal value of bis bill. What should he receive for his bill if true discount only were deducted ? (2) A person possesses a sum of money, the simple inter- est of which at 4 per cent, is $536.25. "With this sum he purchases an estate, for which he pays by a note payable in 4 month's time, and which being discounted at 4 per cent., is worth at present exactly the money he possesses. For how mnch is the bill drawn. 886 APPENDIX. (8) True discount, at 4 per cent., on a sum of money being $15, find simple interest on the same sum at 5 per cent. (4) The interest on a certain sum of money is $180, and the discount on the same sum for the same time, and the same rate of interest is $150 ; find the sum. (5) If the interest on $A for a year be equal to the dis- count on $B for the same time, find the rate of interest. (6) If the three per cents are at 90 one month before the payment of the half-yearly dividend, what is the rate of interest ? (7) A gives B a bill for $a, due at the end of m years, in discharge of a bill for $6, due at the end of n years : for what sum should B give A a bill due at the end of p years, to balance the account at Compound Interest ? (8) Given A my income, a the premium for assuring $100, r the rate of Interest per cent, per annum ; find what sum I must lay out in assuring my life, so that my execu- tors may receive a sum, whose interest shall equal my reduced income. (9) A sells goods to B and allows him 10 per cent, dis- count, if he pay in six mouths ; what discount ought he to allow if payment be made in two months, at 5 per cent, per annum, simple interest *? (10) The discount on a promissory note of $100 amounted to $7.50, and the interest made by the banker was $5-405 per cent. ; find the interval at the end of which the note was payable. EQUATION OF PAYMENTS. 11. To find the equated time of payment of two sums dijLe at di/erent times at a given rate of interest, Pi (Pi + Pi) P* N H! n HZ Let Pj, P 2 , be the suras due at the end of the time n,, HI ; r the rate of interest ; take time N greater than H Then it is manifest that the amounts of P,. P,, at APPENDIX. 837 the time N, should in equity be together equal to the amount of their sum, (P l -f P 3 ), in the same time. Whence, / \N n, I \ N w 2 PI [l+>| + P 2 _ P. +P a (1 +)! " (1 + rj a : ~ (1 +r) n '' and expand, neglecting /* a , and higher powers, we have -?i ^P 2 PI +P 2 .1 + n,ij 1 + n.->r 1 + nr which is the A orm of the equation for Simple Interest. 330 Solving for n we get P, ; + P 2 +r(P~ 1 n~ J + P 9 ~P which is the correct value of the equated time. If r be a very small quantity, as in practice it usually is, and P 15 P 2 , not very large, we shall have ?,, + P ra n - is p " -, as before. -ti T -t 2 14. The term Annuity is understood to signify any interest of money, rent, or pension, payable from time to time, at particular periods ; and these payments may take place yearly, half-yearly, quarterly, &c. 15. To find the Amount of an annuity, to be paid for a given number of years, at Compound Interest. Let A be the annuity, n the number of years, E the amount of one dollar in one year, M the required amount. We have Amount due at the end of 1 year = A 2 " = A + AB 8 " = A -r AK + AE^ AC. " &c. n " - A + AE + AR2 T . ^B 1 -.'- -* ^Aj ^ ,o. J'or Simple Interest, expanding and neglecting ana higher powers, we get APPENDIX. 889 17- To find the Present Value of an annuity, to be paid for a given number of years, at Compound Interest. I. The amount of the annuity at the end of the first year is A, while the present value is AR -1 ; similarly, the amount at the end of the nth year is AB*" 1 , and the present value is AH"" 1 . Hence in order to obtain the present value from the amount we must first multiply the formula for the amount by B, and then change the sign of the index of B. Multiplying by B we get f A B"+i - B B-l Changing sign of index we have n. We may obtain the same result by proceeding on the principle that if the present value P be put out to compound interest for n years, it ought to amount to the same as the annuity for that time. B n 1 Hence PR" = A 340 APPENDIX. III. We will now proceed on the principle that the present value P is the sum of the present values of the respective annual payments. Present value of A due 1 year hence = AR~~* ** " 2 " = AR""* &c. = o + a-Q)r i + r Q + r o> A Hence, the limit of P, when n = oc = g = x . This result shows that an infinite aum of money is re- quired to be left, in order to insure an equal annual pay- ment for ever, which is absurd. It indicates, therefore, that the only correct method of computing annuities is on the compound interest principle. 20. To find the Present Value of an annuity, to com- mence at the end of p years, and then to continue q years. The present values of the first, second, &c. , qtli payments, due at the end of p + 1, &c., p + q years respectively, will evidently be AR- (P+1) AR- (p + 2) &o. A whence the present value P = AB-te- 1 - 1 '- ll + R-i + R- 5 = AR-< p ^ fl - E " q R" If the annuity is payable for ever after p years have ex- pired, by summing the above series ad infinitum, we have P= A These formula enable us to compute the values of Rever- sions, or Annuities in Reversion ; and the latter determines the value of the Fee Simple of the freehold estate, which is to fall in at the expiration of p years. 842 APPENDIX. Ex. 1. A sum of $a is borrowed for a period of m years, to be repaid by equal annual instalments, the first payment to be made after one year. Find the amount of the annual instalment. Let A be the annual instalment. Tken the amount of this annual payment in m years. r \ ~~ } Again, if the sum a be allowed to accumulate for m years at compound interest, its amount = a R . Now, these two amounts ought to be equaL Hence, we have Ex. 2. The present value of an annuity of $1, to continue x years, is $10 ; and the present value of an annuity of $1, to continue- r years, is $16; find the rate of interest. Here, 10 = 7 (1-B-), art. 17, and 16 = (1-B- 2 *); 2 - S43 Subtituting in the first equation, we get or 100 r = 4. The rate is, therefore, 4 per cent. Ex. 3. A mortgage of $5,000, interest at 6 per cent. per annum, has 7 years and 10 months to run ; find its present value, interest at 10 per cent, per annum, pay- able half-yearly. The first payment of interest is $300, and will be due in 10 months; its amount for seven years, at 10 per cent., pay- able half-yearly, will be 300(1 '05) 1 4 . Similarly, the amount of the second payment of interest at the end of the 7 .years, will be 800(1 '05) la ; and so on. The amount of the last payment will be $300. Hence, the whole amount of the mortgage and interest will be 5000 + 300 (1 05) u + 300(1 -05) 12 + ......... + 300. -P5UOO + 300 (l-04 w + (1.05) 12 + ............ + 1 = 8462-OG. Now, if the present value, P, be put out to compound interest at 10 per cent, per annum for 7 years and 10 mos., it ought to amount to the same as the mortgage for that .-. P (1-05) 15 * = 8462-06 .-. P = 3940.13. The Present Value of the Mortgage is, therefore, $3940.13. The value of (1-05) 158 may be found (1) By means of a table of logarithms. (2) By raising 1'05 to the 16th power, dividing this by 1'05 we obtain the 15th power, taking of the difference and adding to the 15th power, we get approximately the 15| power of 1 05. 344 4PPKNDIX, (8) By the Binomial Theorem, as follows : (1.05)* = (1 -}- Jd* = l + jy T ^ = 1.0163 ; then (1.06)" -f- 1.0163 will give a close approximation to (1.05) 1 * For additional information on this subject consult Loan Tables by Professors Cherriman and London. Exercise III. (1) A person's dividend from his Bank Stock is $530 a year. What is the present value of this income for five years to come, computing by simple, and also by compound interest, at 7 per cent. (2) What annuity, to continue 20 years, can be purchased for $10000, allowing compound interest, at 5 per cent. (3) For what sum might the Government of a country undertake to pay an annuity of $1000 a year, for ever, on the supposition that money may always be invested at 6 per cent. (4) For what sum might an annuity of $400 a year, for 10 years, to commence in 5 years, be purchased, allowing com- pound interest at 6 per cent ? (5) A person who enjoyed a perpetuity of $1000 per annum, provided in his will that, after his decease, it should descend to his only son for 10 years, to his only daughter for the next 20 years, and to a benevolent Instjitu- tion for ever afterwards. What was the value oi each bequest at the time of his decease, allowing compound inter- est at 6 per cent. ? (6) A person at the age of 22 put $100 at interest, at 6 per cent., and $100 each year afterwards, until he was 40 years old. He also collected the interest annually, and con- verted the same into principal ; what amount vvas, by these means, accumulated ? (6) A corporation borrows 3769 at 4 per cent., to be paid in 30 years by equal annual instalments. What will be the annual payment ? (7) A property is let out on lease for a years at an annual rental of $6, and after c years the lease is renewed on pay- ing a fine of $d. What is the additional rent equivalent to thig fine ? (8) A farm is let for n years at a fixed rent and a fine of $p. When p years of the lease remain, what fine must be paid to extend these p years to #, at compound interest ? APPENDIX. 345 (9) If two joint proprietors have an eqnal interest in a freehold estate worth $a per annum, but one of them pur- chased the whole to himself by allowing the other an equiva- lent annuity of $6 for n years, find the relation between a and 6. (10) Find the present value of an annuity of $1, paid n tunes per annum, and continuing for m years, allowing compound interest at the rate of r per cent, per annum ; and prove that, as n is indefinitely increased, this present value continually approaches the limit ^ . (11) A monthly instalment of $10 has 2 years one month to run, what sums must be paid at once to reduce the period iix months, money being worth one-half per cent, per month ? (12) A mortgage of $4000 interest at 6 per cent, per an- num, payable half-yearly, has 17 years and 8 months to run. Find its present value, interest 10 per cent, per an- num, payable half-yearly. (13) If two sums, $ lt 5 2 , due at times t r t v be paid to- gether at an intermediate tune t, t being determined from the equation. Show that whichever mode of payment be adopted (1) At any antecedent period, the present values are the same ; (2) At any subsequent period, the amounts are the same ; (3) At the intermediate time of payment, the interest of the sum overdue is the discount of that not due. OF THE ( UNIVERSITY ) OF APPENDIX II. CUBE ROOT. The extraction of the cube root, by the ordinary rale, is a troublesome process, seldom used and easily forgotten. The following process is much simpler and more easily remem- bered. Let a be an approximate value of the cube root of N, so that ] and, therefore, -j^N = a + sc _ 2N + as. N + 2ci3 ' Suppose we wanted to find the cube root of any number N , In the first place we find some number a whose cube is somewhere near the given number. Then the fraction, 2N + qg. N + 2a3 will be a nearer approximation to the cube root than a itself was. When we have found this value, we can take this as a and repeat the process. Thus, to find the cube root of 241 '804367, we observe that 216, the cube root of 6, is nearest to 241. Hence the first value of a is 6. APPENDIX. Therefore, --. a N + 2a _ 699-608734 ~ 673-804367 4197-636404 = 6'23 very nearly. On trying 6.28 we find it is correct. Ex. (1). Find the cube root 47. The newest cube to 47 is that of 4. 2 N + a Hence, N + 2 a 9 94 + 64 47 + 128 x 4 _ ~ 175 _ 2528 r 700 = 3.61 nearly. Next, take 3.61 for a, and substitute in the formula, and tre get 3.6088261, which is correct to seven places of deci- nals. Ex. (2). Find the cube root of 10. 2N + a 8 In this case. - . i N 4- 2 a 9 _20_+J 10 + 18 = 2.153. Next, substitute 2.15, instead of 2, and we get 20+9-938870 9<15 10 + 19-8707.50 > = 2-1544346, which is correct as far as six places of decimals. This method has also the practical advantage that an error o vrorV gets corrocUd at the next $3600. Exercise II. (1) $706.66*. (2) $13585. (3) $19.50. (4) $900. (5) 100 -^percent A. (6) m. (7) a~BP- m 6B?-. (8) j~-. (9) 11-463 per cent. (10) l^yean. Exercise III. (1) A.t simple interest, $2237.77 ; at compound interest $2173.10. (2) $802.42. 3) $16666. 66f. (4) $2199.95. 5) $7360.08; $6404.74; $2901.83. (6) $3000.56 150-76 drB (8) fZT (B--B-) - 10) - ! 1 Q , r \mn [ r is interest of $1 for 1 year- 11) $58.68. (12) $2422.85. SB. J. dage & 0.0' |Uto (Ebucational Gage's Practical Speller. A new Manual of Spelling and Dictation. Price, 3O Cents PROMINENT FEATURES The book is divided into five parts as follows : PART I. Contains the words in common use in daily life together with -abbrevia- tions, forms, etc. If a boy has to leave school early, he should at least know how to spell the words of common occurrence in connection with his business. PART II. Gives words liable to be spelled incorrectly because the same sounds are spelled in various ways in them. PART m. Contains words pronounced alike but spelled differently with different meanings. PART rv. Contains a large collection of the most difficult words in common use, and is intended to supply material for a general review, and for spelling matches and tests. PART V. Contains literary selections which are intended to be memorized and re- cited as well as used for dictation lessons and lessons in morals. DICTATION LESSONS. All the lessons are suitable for dictation lessons on the slate or In dicta- tion book. REVIEWS. These will be found throughout the book. An excellent compendium. Alex. McRae,Prin. Acafy,Digby,N.S. I regard it as a necessity and an excellent compendium of the subject of which it treats. Its natural and judicious arrangement well accords with its title. Pupils instructed in its principles, under the care of diligent teachers, cannot fail to become correct spellers. It great value will, doubt- less, secure for it a wide circulation. I have seen no book on the subject which I can more cordially recommend than " The Practical Speller." Supply a want long felt. John Johnston, I.P.S., Belleville. The hints for teaching spelling are excellent. I have shown it to a num- ber of experienced teachers, and they all think it is the best and most prac- tical work on spelling and dictation ever presented to the public. It will supply a want long felt by teachers. Admfrably adapted. Colin W. Howe, I.P.S., Wolfville, jr. S. The arrangement and grading of the different classes of words I regard ! as excellent. Much benefit must arise from committing to memory the "Literary Selections." The work is admirably adapted to our public schools, and I shall recommend it as the best I have seen. m. J . (iage ODote. Jfcto (Ebttmtional Work*. 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Containing Books I. to VI., and portions of Books XI. and XII.,of Euclid, with Exercises and Notes, by J. HAMBLIN SMITH, M. A., &c., and Examina- tion Papers, from the Toronto and McGill Universities, and Normal School, Toronto. Price, 9O Cents. Hamblin Smith's Geometry Books, i and 2. Price, SO Cents. Hamblin Smith's Statics. By J. HAMBLIN SMITH, MA, with Appendix by Thomas Kirkland, M. A., Science Master, Normal School, Toronto. Price, 9O Cuats. Hamblin Smith's Hydrostatics. 75 Cents. KEY. Statics and Hydrostatics, in one volume. $2.OO. Hamblin Smith's Trigonometry. $1.25. KEY. To the above. $2.5O. . J. - t C~ .._M -4 ^x ^t, CE c O z z - 3 FORM NO. DD6, UNIVERSITY OF CALIFORNIA, BERKELF BERKELEY, CA 94720 b .184004