A THEORETICAL AND PRACTICAL R I TH ET IC; DESIG TED FOR COMMON SCHOOLS AND ACADEMIES. BY DANIEL LEACH AND WILLIAM D. SWAN. PHILADELPHIA: THOMAS, COWPERTHWAIT, & CO.'1 851. Entered according to act of Congress, in the year 1850, BY DANIEL LEACH AND -WILLIAM D. SWAN, in the Clerk's,office of the District Court for the District of Mlassa husetts, STEREOTYPED AT THE BOSTON STEREOTYPE FOUNDRY. P RE FACE. IT has been the aim of the authors, in preparing this work, to make it eminently both a practical and a theoretical treatise on the science of numbers. They have.therefore adopted that arrangement which has appeared the most philosophical, and, at the same time, the best suited to the comprehension of the learner. They have bestowed great labor on the rules and definitions, in order to make them lucid, concise, and accurate; and they have carefully avoided introducing any illustrations or remarks not necessary to a clear understanding of the subject. The examples have been prepared with much discrimination. Many of them are questions which have actually occurred in ordinary business transactions. They would call attention first to the examples in addition, some of which have been so arranged as to bring together the same combination of figures 4 PREFACE. throughout the same line. The long leger columns are designed for those who wish to acquire a facility in adding long columnns. This is one of the most usefuld exercises in arithmetic to which pupils can be accustomed. They would also call particular attention to the rule for finding the least common multiple, the rule of alligation, and the rule for extracting the cube root. These rules are clear and concise, and can be most rigidly demonstrated; while in the processes indicated by them there is a saving of more than one half of the figures, as compared with the processes in similar works now in common use. The section on fractions, they think, will also commend itself to every experienced teacher. Although the preface is not the proper place for discussing the best method of teaching arithmetic, yet the authors cannot refrain from urging upon all teachers not to allow their pupils to attempt to solve a question till they fully understand all its conditions, and always to require them to state the principles upon which each solution is founded. Pupils should be accustomed to write questions of their own under each rule. This is a very important exercise. They would also suggest that, in every question in which there are both multiplication and division, the pupil should at first indicate the processes by their PREFACE. 5 appropriate signs, and then cancel the factors common to the dividend and divisor. In the preparation of this work, a large number of the recent English and American treatises on arithmetic have been consulted, and the most valuable mathematical works of the French. Aw CONTENTS. INTRODUCTION.. 4 o............ 1 Numeration........................................... 12 Numeration Table................................ 13 Addition....................................... 15 Multiplication................................. 22 Subtraction.............................. 28 Division..................................... 31 General Principles in Divisiion.................. 36 Practical Questions............................... 37 The'Divisibility of Numbers....................40 Prime and Composite Numbers............. 42 Table of Prime and Composite Numbers...................... 44 Cancellation.....4..................... 46 The Greatest Common Measure....................7..... 47 The Least Common Multiple................................ 49 Fractions.................................................. 52 Addition of Fractions..................... 0.......oe... 60 Multiplication of Fractions................................ 61 Subtraction of Fractions.................................... 62 Division of Fractions................................... 63 Practical Questions in Fractions......................... 66 Decimal Fractions.................................. e 70 8 CONTENTS. Reduction of Decimals..................................... 73 Addition of Decimals....................................... 75 Multiplication of Decimals.................................. 77 Subtraction of Decimals.................................... 78 Division of Decimals....................................... 79 Practical Questions in Decimals............................. 81 Contractions in Multiplication and Division................... 82 Federal Money............................................8.. 84 Table of Federal Money;................................... 84 Reduction of Federal Money................................ 85o Addition of Federal Money................................. 85 Multiplication of Federal Money............................ 86 Subtraction of Federal Money................................ 87 Division of Federal Money................................. 87 Practical Questions in Federal Money......................... 88 Bills of Parcels............................................. 89 Dlenominate Numbers....................................... 90 Tables of Money, Weights, and Measures..............:90 Reduction of Denominate Numbers........................... 95 Addition of Denominate Numbers........................... 99 Multiplication of Denominate Numbers........................ 102 Subtraction of Denominate Numbers........................... 104 Division of Denominate Numbers.............................. 107 Practical Questions in Denominate Numbers............ 109 Denominate Fractions.................................... 116 Addition and Subtraction of Denominate Fractions.............121 Practical Questions in Denominate Fractions.....................22 Percentage....................................... 124 Simple Interest........................................127 Partial Payments............................................. 1 36 Problems in Interest.............................. 142 Compound Interest........................................ 1-45 Table in Compound Interest................................. 147 CONTENTS. -9 Discount.................................................. 148 Bank Discount.............................................. 150 Commission................................................ 152 Stocks.................................................... 154 Insurance................................................. 155 Profit and Loss..................................... 157 Practical Questions in Profit and Loss....... 162 Ratio..................................................... 163 Proportion................................................ 165 Analysis........... 170 Compound Proportion...................................... 171 Partnership............................................. 174 Equation of Payments....................................... 179 Taxes.................................................... 182 Duties.................................................. 184 Alligation Medial.......................................... 186 Alligation Alternate........................................ 187 Duodecimals.............................................. 190 Multiplication of Duodecimals............................... 191 Involution..............................................194 Table of Roots and Powers................................. 195 Evolution................................................. 196 Square Root.............................................. 197 Practical Questions in Square Root........................ 203 Cube Root................................................ 203 Root of Higher Powers................................... 211 Arithmetical Progression................................... 212 Geometrical Progression.................................... 216 Annuities............................................ 220 Tables of Annuities......................... 221, 222 Exchange of Currencies.................................... 226 Exchange..... 229 Value of Foreign Coins...................................... 230 Tables of Exchange...... 231 10 CONTENTS. Forms of Bills of Exchange............................... 233 Arbitration of Exchange............................. 234 Permutation.............................................. 236 Mensuration of Surfaces................................... 237 Mensuration of Solids..................,................... 242 Similar Surfaces........................................ 246 Mensuration of Boards and Timber......................... 248 Guaging.................................................. 249 Tonnage of Vessels........................................ 250 Mechanical Powers....................................... 251 General Analysis.......................................... 254 Problems................................................. 258 Practical Questions........................................ 265 Forms of Promissory Notes, Drafts, &c................. 273 Form of Account Current....................... 76 THEORETICAL AND PRACTICAL ARITHMETIC. INTRODUCTION. SECTION I. 1. ARITHMETIC is the science of numbers. 2. Numbers express how many units, or parts of a unit, there are in any quantity. 3. Quantity is any thing that can be increased, diminished, or measured, 4. The least whole number employed to express or measure quantity of the same kind is called a unit. 5. A number expressing a particular kind of a unit is called a concrete number; as, i dollar, 2 books, &c. 6. When a number does not express any particular kind' of a unit, it is called an abstract number; as, 1, 4, 7, &c. OBs. 1. Concrete numbers are also called denomingzate or comnpotcd numbers, OBs. 2. Whole numbers are sometimes called integers. 7. In the computation of numbers, ten characters are employed, called figures; thus: 1, one; 2, two; 3, What is arithmetic? WVhat are numbers? What is quantity? W'That is a unit? WYhat is a concrete number? WVhat is an abstract number 12 NUMIElRATION. three; 4, four; 5, five; 6, six; 7, seven; 8, eight; 9, nine; 0, cipher. The first nine figures are called sgn.ficantj because they have a given value assigned them. The cipher has no representative value, but is used where no number is to be expressed. OBs. 1. By the aid of these ten characters any possible or conceivable quantity may be expressed. OBs. 2. The first nine figures are sometimes called digits, from the Latin word digitzs, signifying a finger. 8. The various operations of arithmetic are performed by NUMERATION, ADDITION, MULTIPLICATION, SUBTRACTION, and DIVISION. Addition and multiplication are employed to show how numbers may be increased; subtraction and division how they may be diminished. NUM{ ERATION. SECTION II. 9. NUMERATION is the art of expressing any number whatever by figures. Figures are arranged in different orders or places, and have different values assigned them, according to the place they occupy. The first place, which is always at the right, represents uniits; the second, tens; the third, hundreds; the fourth, thousands;. the fifth, tens of thousands; the sixth, hundreds of thousands; the seventh, millions, &c. Thus the figure 1 represents a unit, a ten, a hundred, a thousand, &c., according to the place it occupies. In all places in which no nlumber is to be expressed, ciphers must be written. Which are significant figures? How are the various operations of arithmetic performed'? What is numeration? How are figures arranged? WVhat- does the first place represent? The second? The third? The fourth?: The fifth? &c. When must ciphers be used? NUMEIRATION. 13 Thus, if any figure be written in the fourth place, or the place of thousands, a cipher must be written in t' place of units, and in the place of tens, and in place of hundreds; as, 7000. It is evident that the same figure represents in the second place a value ten times greater than in the first place, and in the third place ten times greater than itl the second, and a hundred times greater than in the first; and by each removal of any figure to the next place on the left its representative value is increased ten times. 10. The different places in which figures are arranged may be divided into periods of three figures each. NUMERATION TABLE. 0'~ ~ ~.,.~ t = = =' S: ~ = V498 5 1 51C) = e0 0 0 d) 0. O 1.) c. thus' six hundred and forty-nine quitillions; eight Wh effet rc t he representative value of a figre by rchan its place? The number indicated in the above table is read changing its place? 14 NUMIERATION. hundred and seventy-six quadrillions; five hundred and forty-eight trillions; seven hundred and nineteen billions; eight hundred and seventy-six mnillions; five hundred and forty-three thousand; six hundred and one. 1 1. Read the following numbers: — 1. 12 7. 1010101010 2. 21 8. 2020202020 3. 123 9. 6006006006 4. 101( 10. 2222222222 5. 10209 11. 54467841234 6. 202807 12. 123456789000 I. Express in figures the following numbers: — i. Twenty-four. 2. Two hundred and four. 3. Two hundred and forty. 4. Two thousand arid four. 5. Two thousand and forty. 6. Forty-six thousand, five hundred and twenty. 7. Four hundred and six thousand, five hundred and two. 8. Eight hundred thousand, one hundred and one. 9. One million, one thousand, one hundred and one. 10. Ten millions, ten thousand, and ten. 11. One hundred millions, one hundred thousand, and one hundred. 12. Two millions, six hundred and ten thousand, four hundred and forty-six. 13. Sixty-four millions, nine hundred and ten. 14. Two hundred and forty millions, three thousand. 15. Five hundred and sixty-seven billions, three huindred and forty-eight millions, seven hundred and twenty thousand, six hundred and forty. 16. Fourteen trillions, six billions, three hundred and forty millions, and twenty-two. ADDITION. 15 AD D IT IO N. SECTION III. 13. ADDITION is the process of finding the sum of two or more numbers of the same kind. OBS. The sum expresses the total value of the several numbers, or as many units as there are in all of them. 14. RULE. Write the numbers utnder each other, units under units, tens under tens, hundreds under hundreds, ~Sc. First, add the column of units, and write under this colutmn the right hand figure of the sum, and add the remaining figure or figures to the next column. Add all the colum~ns in the same manner, and under the last write the whole sum contained in it. PRoor. Beginning at the top of the column of units, add each column downwards; and if the result be the samze as the first, the work is supposed to be right. 15. Two signs are often employed in addition; the one, +-, called plus, which signifies added to, or and, and the other, -, called the sign of equality, which signifies equal to, or are: thus, 4 +- 6 = 10 is read, four and six are ten. 1. Add together the following numbers:2472 51856 27692 70347 81850'234217 The sum of the first column is 17; the 7 is written What is addition? What does the sum express? What signs are ased in addition? Recite the rule. 16 ADDITION. under the columln of units, and the 1 is added to the next column, whose sum is 31; the 1 is written under the column of tens, and the 3 is added to the next column, whose sum is 32; the 2 is written under the column of hundreds, and the 3 is added to the next column, whose sum is 14; the 4 is written under the colualn of thousands, and the I is added to the next column, whose sum is 23, which is written under the last column. 16. It is evident that adding the left hand figure to the next column is the same as adding the tens in the column of units, the hundreds in the column of tens, the thousands in the column of hundreds,. &c., which may be illustrated as follows:27 465 847 963 15 units. 16 tens. 21 hundreds. 2275 sum. The: sum of the first, or the column of units, is 15, which is 1 ten and 5 units; the 5 is written under the column of units, the 1 under the tens. The sum of the second, or the column of tens, is 16, which is 1 hundred and 6 tens; the 6 is written under the column of tens, and the 1 under the column of hundreds. The sum of the third, or the column of hundreds, is 21, which is 2 thousand and 1 hundred; the 1 is written under the column of hundreds, and the 2 is written in the next place on the left, which is the place of thousands. These added together give the sum total 2275. What is the effect of adding the left hand figure to the next columnii? ADDITION. 17 1 7. This rule depends upon the principle illustrated in Article 9th, that ten in the column of units is equal to one in the column of tens, and ten in the column of tens is equal to one in the column of hundreds, &c. EXAMPLES. 3. 4. 5. 6. 7. 2222 2222 2222 2222 2222 3333 3333 3333 3333 3333 4445 5556 6667 7778 8889 8. 9. 10. 11. 12. 3333 4444 5555 6666 7777 4444 5555 4444 3333 4444 7778 6667 7778 8889 7779 13. 14. 15. 16. 17. 3333 6666 5555 88S8 3333 4444 5555 4444 2222 9999 5555 4444 7777 6666 4444 6666 3333.4444 3333 7777 1113 2224 1113 3335. 0002 18. 19. 20. i21. 22. 3333 4444 5555 6666 9999 8765 8765 9876 9876 8765 3333 4444 5555 6666 9999 8765 8765 9876 9876 8765 3333 4444 5555 6666 9999 6545 5445 6546 5546 4315 Upon what principle does the rule' of addition depend? AB 2 18 ADDITION. 23. Add together 12345, 54321, 678, and 876. 24. Add together 6789, 3211, 7456, and 2546. 25. Add together 40396, 43745, 675, and 96. 26. Add together 70964, 84345, 4327, and 75. 27. Add together 90, 4360, 10345, and 467634. 28. Add together 3, 7506, 42, 90704, and 736. 29. Add together 78960, 45, 960, 301, and 84. 30: Add together 5360, 4263, 2, 7503, and 801. 31. Add together 934, 8375, 2013, 46, and 7640. 32. Add together 7560, 473, 421, 76, and 96341. 33. Add together 7345, 8403, 642, 24, and 8460. 34. Add together 9345, 6704, 6340, 54, and 760. 35. Add together 2317, 5742, 96043, and 8483. 36. Add together 2319, 64893, 45407, and 98. 37. Add together 4296, 3715, 39624, and 7434. 38. Add together 968732, 201, 28346, and 291. 39. Add together 234604, 8764, 2346, and 734. 40. Add together 424603, 26934, 80973, and 73. 41. 964564 + 843453 + 372131 + 234560?-? 42. 843735 + 345841 + 673450 + 343466 -? 43. 432846 + 648345 + 873459 + 673489 -? 44. 849321 -+734463 +- 734649 + 763921? 45. 734561 + 234764 + 245321 + 289641 --? 46. 346734- +824375 + 654342 +- 734931 =? 47. 729329 + 324643 + 320103 + 345603 -? 48. 678456 + 324321 + 735920 + 275463? 49. 372635 + 424851 + 734639 +- 273486 -? 50. 934275 ]- 604922 + 234769 + 832834 —? 51. 634296 - 234283 + 864234 + 234675 =? 52. 376263 + 456934 + 963264 + 239340-? 53. 763412 + 123456 + 789654 +- 236960? 54. 656789 + 756789 + 639936 + 7355353? 55. 324423 + 459954 + 863638 + 450207 =? 56. 875634+ 845673 + 437734 + 960784 -? 57. 735640 + 937456 + 784503 + 875406? 58. 345634 + 783453 + 456278 + 673840 -? ADDITION. 19 59. What is the sum of the following numbers? Six hundred and five; thirty-seven; four thousand five hundred and twenty; thirty-seven millions; two hundred and one; ninety-nine thousand and nine. 60. Find the sum of three thousand seven hundred and forty-four; nine million fourteen thousand and eleven; five hundred and eight millions two hundred and, three thousand and twenty-five. 61. A merchant, commencing business, had in cash 6330 dollars, goods valued at 9875 dollars, bank stock valued at 4320 dollars, railroad stock valued at 2700 dollars: during the year, he gained above his expenses 231.6 dollars. What was he worth at the end of the year? 62. A merchant sold five bales of cloth. For the first bale he received 735'dollars, for the second 637 dollars, for the third 573 dollars, for the fourth 391G dollars, for the fifth 721 dollars. How much did he receive? 63. A farmer received for the ptroduce of his farml in one year as follows: for hay 276 dollars, for pTotatoes 391 dollars, for oats and corn 234 dollars, for fruit 567 dollars. How much did he receive? 64. A man paid 3234 dollars for his farm, 5640 dollars for his house, 1500 dollars for his furnitulre, and 539 dollars for his stock and tools. What did he pay for the whole? 65. There are two numbers; the less is 93078, the di ference is 4796. What is the greater? 66. A man owvns three farms; the first is valtued at 5697 dollars, the second is valued at' 96230 dollars, the' third at 1639 dollars. How much are the three worth? 67. A gentleman left in his will to his three sons 1.930 dollars each, to his two dauli'hters 1737 dollars each, to his wife 930 dollars more than all his coildren. What was his wife's portion, and what was the value of the whole estate? Ao20 ADDITION. 68. 69. 70. 71. 76543 98765 76586 92345 76543 98765 34524 18765 76543 98765 76586 92345 76543 98765 340524 18765 76543 98765 76586 92345 76543 98765 34524 18765 76543 98765 76586 92345 76543 98765 34524 18765 76543 98765 76586 92345'76543 98765 34524 18765 76543 98765 76586 92345 76543 98765 34524 18765 76543 98765 76586 9234.5 76543:98765 34524 18765 54943 65555 47858 76425 72. 73. 74. 75. 33146 29956 45353 32142 40354 97194 88868 88459 37644 11613 22242 26976 72212 82453 84587 32785 31254 27644 86523 72329 27623 93216 48868 86788 53433 19259 82242 21863 37654 81621 24587 33369 12315 27434 83523 76785 61141 91956 13868 84956 37'654 12159 1.4242 26479 31234 97695 84587 31787 42222 23416 22523 73324 37654 87999 88867 86789 4A454 21984 16818 67809 ADDITION. 2 76. 77. 78. 79. 244658 275634 135790 123456 492327 386731 246824 789123 635425 987654 135790 456789 321465 321456 864212 123456 732849 989123 579246 788123 376731 456789 835792 456789 935746 1.23456 468357 123456 847963 789123 924689 789123 745-143 456789 753246 456789 23456 1 123456 835792 1234.56 746874 789123 468357 7891 23,934746 456789 924683 456789 872345 123456 579246 123456 934756 789123 835798 789123 842345 456789 642,875 456 789 873456 123456 324683 123456 864580 789123 579864 789t22 234672 456789 29 031 456789 325871 246842 135795 871178 479234 357931 246834 936639 845645 642248 824248 2488S42 823456 706139 357964 5'95255 24573.4 246842 872278 736376 8S72475 657931 375946 875578 896731 642248 624862 473468 456841 753139 375937 934579 314567 246842 -872459 894645 814563 357931 837645 123875'427831 642248 644875 767457 932768 753913 472963 875345 456345 375913 875847 874563 3- 4 5634 426428 8643 14 375534 734734 573931 734561 937565 734564 624824 273475 875734 834756 735913 845675 698945 MULTIPLICATION. M ULTIPLI CATION. SECTION IV. 18. MULTIPLICATION is the process of finding the sum of any number, when taken as many times as there are units in another number. OBs. This definition of multiplication is applicable-only to whole numbers. The number produced by multiplication is called the produxc t. The number multiplied is called the multiplicand. The number multiplied by is called the mniltiplier. The multiplicand and multiplier are called farctors. OBs. The termn jctlom is clerived from a Latin word, sinliying to?make, or produce. 19. The sign of nmultiplication is a cross, X, and is read, multiplied by, or times. Thtus, 9X8=72 is read, 9 times 8 are 72. 9 is the multiplicand, 8 the multiplier, and 72 the product..0. RULE. Write the nmultiplier,nder the multiplicand, units u tder units, tens uznder tens, Cc. W/hen there is butt one figure in the multiplier, vmultiply each figure inz the multiplicand, beginning with tnuits, and write the right hand figure of the produlct under its multiplier, and aadd the remaining fi gure or figures as in addition. l[Vhen there is,more than one figure in the multiplier, multiply each aigure in the multiplicand, beginning with units, by each figure of the multiplier in succession, and zwite the rilght hand fig-tt:e of the product ainder its multli~plier, and add the left hand ji7iure or figures as in, hatm is.mltip].;Caliozn?!A'Vhat is the nuember proTduced by mutl]p.iim;onu cnalled? flWhat is the m-ultidilicarcldi What is the multi]le'? What are fsacorl' e\Vat is the,igl of rauitiplicatio.1:eclite the rule. MULTIPLICATION. 23 addition. The sum of the several products will be the product sought. PROOF. Make the multiplicand the multiplier, and the multiplier the tmultiplicand, and, if their product be the same as before, the work is supposed to be right. It is the most convenient to take the larger number for the multiplicand, and the smaller for the multiplier. The result will be the same, whether the larger number be repeated as many times as there are units in the smaller, or the smaller as many times as there are units in the larger. Either the multiplicand or multiplier must always be considered as an abstract number, for it is absurd to suppose that one number canll be repeated as many times as there are pounds, yards, or dollars. 1. Multiply 4657 by 8. 4657 8 37256 The 7 units, multiplied by 8, are equal to 56: write the 6 under the 8, and reserve-the 5 to be added to the next product. 8 times 5 are 40, to which 5 being added, makes 45: write the 5 under the column of tens, and reserve the 4 to be added to the next product. 8 times 6 are 48, to which 4 being added, makes 52: write the 2 under the column of hundreds, and reserve the 5 to be added to the next product. 8 times 4 are 32, to which 5 being added, makes 37, which is to be written under the last column. 2. Multiply 65943 by 58. 65943 58 527544 329715 3824694 24 -MU LTIPLICATION. Multiply by the 8 units, as in the last example. Then multiply by the 5 tens. The first product is 15: write the 5 directly under its multiplier, reserve the I for the next product of 4 by 5, which is 20; the 1 being added, makes 21: write the I at the left of the 5, under the column of hundreds, and reserve the 2 for the next product of 9 by 5, which is 45, to which 2 being added, makes 47: write the 7 at the left of the 1, and reserve the 4 for the next product of 5 by 5, which is 25, to which 4 being added, makes 29: write the 9 at the left of the 7, and reserve the 2 for the next product of 6 by 5, which is 30, to which 2 being added, makes 32, which is to be written at the left of the 9. The sum of the several products is 3824694, the required product. EXAMPLES. 3. 4. 5. 6. 4294 3276 8752 9743 5 4 6 9 7. 8. 9. 10. 8349 2546 3767 5886 24 32 44 58 11. 12. 13. 14. 9654 4569 8756 6643 364 263 942 842 15. 16. 17. 18. 2756 3256 9463 3472 436 384 272 357 MULTIPLICATION. 25 19. 934567 X 13 =? 29. 456789 X 4361 -? 20. 845603 X 14-? 30. 865432 X 5624 =? 21. 945679 X 15? 31. 936745 X 6345 --? 22. 843475 X 16-? 32. 845346 X 7346? 23. 975349 17? 33. 934575 X 8431 =-? 24. 873945 X 18 -? 34. 654321 X 3214 =? 25. 943457 X 19 _? 35. 897654 X 4576 -? 26. 896321 X 21-? 36. 356891 X 5316=? 27. 456789 X 23 _ 5 37. 896453 X 6452-? 28. 876432 X'24-? 38. 943456 X 9345? 21. When there are ciphers at the right hand of either the multiplicand or multiplier, or both, - RULE. TVrite the significant figures under each other, and multiply only by the significant figures, and to the product annexe as mnany ciphers as there, are at the right hand of both of the factors. 39. 6456X20 -? 42. 45634X6200 —? 40. 3400 X 7300-? 43. 75000X32000? 41. 46000X 9300 -? 44. 8400X7300 -? OBS. Multiplying by 10 is the same as annexing one cipher to the multiplicand; multiplying by 100, the same as annexing two ciphers; by 1000, the same as annexing three ciphers, &c. 22. When there are ciphers between the significant figures of the multiplier, — RULE. Multiply by the significant figures only, and write the first product of eachfigure directly under its multiplier. 45. 7644X304 =? 47. 96964X3004 -? 46. 8456X40006? 48. 97564X20008 _? What is the rule for multiplication when there are ciphers at the right of the multiplicand, multiplier, or both? What is the rule when there are ciphers in the multiplier? c 26 MULTIPLICATION. 23. When the multiplier is a composite number, RULE. Separate the multiplier into its several factors, and multiply first by one factor, and that product by another, and so onr till all the factors have been used as multipliers. The last product will be the answer. OBS. A composite number is one which may be produced by multiplying together two or more numbers, both of which must be greater than a unit. The numbers producing the composite number, when multiplied together, are calledfactors. (ART. 17.) Thus, 36 is a composite number, and may be produced by multiplying together 3 and 12, 6 and 6, 4 and 9, 18 and 2, and 3 and 3 and 4. 49. {Multiply 6789 by 42 or 7X6 or 3X2X7. 50. Multiply 7845 by 56 or 8X7 or 4X2X 7. 51. Multiply 8643 by 54 or 9X6 or 3X3X3X2. 52. Multiply 9368 by 81 or 9X9 or 3 X 9X 3. 53. Multiply 8645 by 64 or 8X8 or 2X4X8. 24. When the multiplier consists wholly of 9's, - -RULE. Annex to the multiplicand as many ciphers as there are 9's in the nmultiplier, and front this numn — ber subtract the multiplicand. 25. This rule depends upon the principle, that annexing one cipher to the multiplicand multiplies it by 10; that annexing two ciphers, multiplies it by 100, &c. Now, it is evident that if the multiplicand be multiplied by 10, it is repeated once too many times for the product of the multiplicand by 9; if inultiplied by 100, it is repeated once too many times for the product of the multiplicand by 99, &c. (ART. 28.) 54. Multiply 67899 by 9999. 678990000 67899 678822101 What is the rule when the multiplier is a composite number? What is a composite number? What is the rule when the multiplier consists wholly of 9's? MULTIPLICATION. 27 55. What will 565 barrels of flour cost, at 7 dollars a barrel? 56. How many bushels of corn will 96 acres of land produce, if each acre produces 33 bushels? 57. In one bushel there are 32 quarts. How many quarts are there in 156 bushels? 58. In one year there are 8766 hours. How many hours will a boy have lived when he is 12 years old? 59. In one year there are 525960 minutes. How many minutes will a boy have lived when he is 15 years old? 60. In one acre of land there are 43560 feet. What would be the price of 1 acre, at 7 cents per foot? What at 8 cents? What at 12 cents? What at 25 cents? 61. Two men start from the same place, at the same time, and travel the same way; the one travels 36 miles a day, and the other 45 miles a day. How far apart will they be in 9 days? 62. Two men start from the same place, at the same time, and travel in opposite directions; the one travels 9 hours a day, at 7 miles an hour; the other travels 12 hours a day, at 8 miles an hour. How far apart will they be in 6 days? How far in 11 days? 63. If a railroad car moves 38 miles an hour, how far would it go in 30 days, of 24 hours each, allowing 2 hours each day for stopping? 64. If 9 men can do a piece of work in 13 days, how long would it take one man to do the same work? How many men would do it in one day? 65. In 1 mile there are 320 rods. How many rods are there in 42 miles? I n 336 miles? 66. A merchant bought 54 pieces of cloth, each piece containing 39 yards, and paid 5 dollars a yard? What did he give for the whole? 67. In one day there are 1440 minutes. How many ninutes in 365 days? 28 SUBTRACTION. SUBTR1ACTION. SECTION V. 26. SUBTRACTION is the process of finding the differetlce between two numbers of the same kind. The larger number is called the minuend; the smaller, the subtrahend. OBs. The term minuend is from a Latin word signifying to be diminisshed; su6btrahend from a Latin word signifying to be tak/en from. 27. The sign employed in subtraction is a short lhorizontal line, - and is read less. Thus 6 -4 -2 is read, 6 less 4 is equal to 2. 8 U....LE. WTrite the less nmber uzde7r the greater, uznits ulnder units, tens funder tens, (cc. Beginning with units, subtract each figure in the lower line from the one above it, and write underneath their difference. If the figutre in, the lower line be greater than the one above it, add 10 to the upper figure before subtracting, and add 1 to the next left hand figure in, the lower line. - PROOF. Add the remainder to the smaller number, and, if the work be right, it will be equal to the larger. 29. This rule depends upon the evident -principle, that the difference of two numbers remains the same when each of. them is increased by the addition of any given number; and the adding of 10 to any colimln, in the upper line, is the same as adding I to the next left hand column, in the lower line, (ART. 9.) 1. From 8434536 take 4530644. 8434536 4530644 3903892 — the difference. What is subtraction? What is the minuendi? What the subtra hend? From what are thev derived? WVhat is the sign of subtrae tiobn Recite the rtule. SUBTRACTION. 2t9 4 units from 6 units leaves 2 units, which is written underneath. As 4 cannot be taken from 3, 10 is added to the 3, making 13. 4 from 13 leaves 9, which is written Underneath, and 1 is added to the 6, the next figure on the left, in the lower line, making 7. As 7 cannot be taken from 5, 10 is added to the 5, making 15. 7 from 15 leaves 8, which is written underneath, and I is added to the cipher in the lower line, making 1, which taken from 4 leaves 3. 3 taken from 3 leaves 0, which is written underneath. As 5 cannot be taken from 4, 10 is added, which makes 14. 5 from 14 leaves 9, which is written underneath. I added to the 4 makes 5; 5 from 8 leaves 3. EXAMPLESo 2. 3. 4. 987654 404045 678932 123456 204024 456734 5. 6. 7, 456789 573456 875678 357901 345674 734567 8. 9. 10. 100000 1010101 9090909 1 10 90 11. 12. 13. 100000 100000 100000 33334 44445 55556 14. 15. 16. 4040404 6060606 8080808 404040 606060 808080 b _. -:.... we 30 SUBTRACTION. 17. 86401 -7356 -? 23. 960304- 730245 —? 18. 734756-340736 —? 24. 875734 — 805345? 19. 936475 - 463040=? 25. 730370- 370370(-? 20. 909909 - 98778 --? 26. 100000 - 99999 -? 21. 101010-90909? 27. 100000 88888 =? 22. 100000 -77777? 28. 666666-477777=? 29. From 67567 + 3456 take 9643+ 7345. 30. From 87567-+2678 take 6304+3456. 31. From 73456 +- 4345 take 9360+ 7561. 32. From 93464 Jr- 7560 take 4234 + 961. 33. From 8345 + 6734 take 9641 -- 1013. 34. From 99875 -. 2634 take 7342 + 206. 35. From 8756+ 937 take 7309 +561. 36. From 3456 + 9879 take 4-050 + 345. 37. From one thousand one hundred and one,subtract nine hundred and eleven. 38. From fifty thousand, subtract five thousand five hundred and five. 39. From one million,subtract one hundred and one. 40. From thirty millions, thirty thousand and thirty, subtract three millions three thousand and three. 41. What time elapsed from the flood, 2348 A. C., to the death of Abraham, 1821 A. C.? 42. What time elapsed from the death of Abraham to the revolt of the ten tribes of Israel, 957 A. C.? 43. How many years from the first settlement in Greece, 1850 A. C., to the founding of Rome by Romulus, 753 A. C.? 44. How many years from the destruction of Troy, 1184 A. C., to the founding of Rome? 45. How many years from the discovery of America by Christopher Columbus, in 1492, to the declaration of American independence, in 1776? 46. George Washington died in 1799, and was 67 years old. In what year was he borl)? DIVISION. 31 47. Benjamin Franklin was born in 1706, and died in 1790. How old was he when he died? 48. Cotton was first planted in the United States in 1769. How many years since? 49. Glass windows were first used in England in 1180. How many years since? 50. Newspapers were first published in 1630. How many years since? 51. Quills were used for writing in 636. How many years since? 52. The first permanent settlement in Virginia was made in 1607. How many years since? DIVISION. SECTION VI. 30. DIv-ISION is the process of finding how many times one number is contained in another. 31. The number divided is called the dividend. The number divided by is called the divisor. The result is called the quotient. When any thing remains after dividing, it is called the remainder, and is always of the same kind as the dividend. OBs. The term quotient is from the Latin word quoties, signifying how manzy times. 32. The sign of division is a horizontal line between two dots, -*-, and is read divided by. Thus 12 3 = 4 is read, 12 divided by 3 is equal to 4. What is division? What is the dividend? What is the divisor? What is the quotient? What is the remainder? What is the sign of division' 32 DIVISION. 33. When the divisor does not exceed 12, - RULE. Write the divisor at the left of the dividend. Find how many tines the divisor is contained in the first left hand figure or figures, and write underneath the result. If there be no remainder, divide the next figure or figures in the same manner. If there be a remainder, suppose it to be prefixed to the nextfigure of the dividend, and divide as before. Os. When the divisor is not contained in any figure of the dividend, excepting the first, a cipher must be written in the quotient. 1; Divide 8756 by 6. Divisor, 6) 8756, dividend. Quotient, 1459, and 2 remainder. 6 is contained in 8 once, and 2 over. Write the 1 underneath, in the quotient, and prefix the 2 to the next figure, 7, making 27. 6 in 27 4 times, and 3 over. Write the 4 in the quotient, and prefix the 3 to the 5, making 35. 6 in 35 5 times, and 5 over. Write the 5 in the quotient, and prefix the 5 to the next figure, 6, making 56. 6 in 56 9 times, and 2 over. Write the 9 in the quotient, and the 2 at the right for the remainder. EXAMPLES. 2. Divide 845678 by 4. 6. Divide 356742 by 9. 3. Divide 96783-4 by 2. 7. Divide 498756 by 12. 4. Divide 603406 by 3. S.. Divide 643275 by 7. 5. Divide 734842 by 8. 9. Divide 734562 by 5. 34. When the divisor exceeds 12 -- RULE. Write the divisor at the left of the dividend. Find how many times the divisor is contained in the smallest num,ber of figures that will contain it one or more times, and write the result in the quoRecite the rule for division when the divisor does not exceed 12. Wrhen the divisor exceeds 12, what is the rule? DIVISION. 33 tient at the right of the dividend. Multiply the divisor by this quotient figure, and subtract the product from the figures divided, and to the remainder annex the next figure of the dividend, and divide this number as before, and continue dividing in the same manner till all the figures are divided. PROOF. Multiply the divisor by the quotient, and to the product add the remainder, and if the sum be equal to the dividend, it is supposed to be right. OBs. 1. The dividend, divisor, and quotient must be separated by a line between them. OBS.. 2. If the remainder, after having one figure annexed, will not contain the divisor, write a cipher in the quotient, and annex another figure to the dividend. Ons. 3. If the product of the divisor by the quotient figure be larger than the dividend, the quotient figure is too large. OBs. 4. If the remainder, before a figure of the dividend has been annexed, be greater than the divisor, or equal to it, the last figure of the quotient is too small. Ons. 5. Annexing a figure is placing it at the right of another figure; prefixing a figure is placing it at the left of another.'10. Divide 8756424 by 324. 324 ) 8756424 ( 27026 648 2276 2268 842 648 1944 1944 The smallest number of figures that will contain the divisor is three, 875; in which the divisor is colitained 2 times and 227 remainder, to which annex 6, the next figure, making 2276, which contains the divisor 7 times and 8 remainder, to which annex 4, the next figure of the dividend, making 84, which is less than the divisor; write a cipher in the quotient, and annex 2, the next figure of the dividend, making 842, 3 34 DIVISION. which contains the divisor 2 times and 194 remainder to which annex 4, the next figure, making 1944, which contains the divisor 6 times, without a remainder. 11. 4875674 14-? 18. 9645045 804 -? 12. 5960345' 24? 19. 8756454 963 —? 13. 6876454 + 34? 20. 8340341 + 134 -? 14. 7936273 43? 21. 473276 ~ 912-? 15. 3732984- -61-? 22. 87345678 ~ 125? 16. 7345674 ~ 75? 23. 8930314. 635 —? 17. 96034567 - 83?- 24. 9181745 225 —? 35. When the divisor consists of two or more fig.."ures, products may be first formed of the divisor and the nine digits, which will enable the pupil to deternine at once how many times the divisor is contained in any partial dividend. Ons. The figures first selected to be divided, and those consisting of the remainders, with the several figures of the dividend annexed, are called the partial dividends. 25. 36) 96686964( 2685749 72 36X1 36 246 36X2- 72 216 36 X 3108 308 36X4_ 144 288 36X5 - 180 36 X 6 -216 206 36X7-252 180 36 X 8 - 288 269 36 X 9 — 324 252 176 144 324 324 26. Divide 7684564 by 324. 27. Divide 7675439 by 273. 28. Divide 8432564 by 346. DIVISION. 35 29. Divide 6543742 by 295. 30. Divide 4526437 by 425. By comparing the several products of the divisor and the partial dividends together, the pupil will discover how many times the divisor is contained in any partial dividend. Thus it will be seen that 36 is contained in 96, twice; in 246, 6 times; in 308, 8 times, &c. 36. When the divisor is a composite number, — RULE. Divide the dividend by one of the factors of the divisor, and the quotient thuis obtained by the other. To find the true remainder when there are two factors, - RULE. Multiply thefirst divisor by the last remainder, and to the product add the first remainder, which will be the true remainder. When there are more than two factors, - RULE. Multiply the product of the first and second divisor by the last remainder, and the first divisor by the second remainder; to the sum of their products add the first remainder, which will be the true remlainder. 37. This rule depends upon the principle stated in art. 31, that the remainder, after division, is of the same kind as the dividend. 31. Divide 96599 by 84 7X4X3. 7 )96599 7X4X2 =- 56 4 ) 13799, 6, 1st rem. 7X3=21 3) 3449, 3, 2d rem. 6= 6 1149, 2, 3d rern. 83, true rem. When the divisor is a composite number, what may be done? How dor you find the remainder when there are two factors? More than two factors? 36 DIVISION.The first remainder, 6, is obviously so many units. The second dividend being so many 7ths, the second remainder, 3, is so rnany 7ths, and must be multiplied by 7. The third remainder, 2, being so many 28ths, must be multiplied by 28, or 7 X 4, to reduce it to units. The saum dr all the remainders thus reduced wvill be the tirue remainder. 32. Divide 47516 by 35. 35. Divide 4275 by 32. 33. Divide 97234 by 27. 36. Divide 3654 by 42. 34. Divide 87544 by 20. 37. Divide 2743 by 18. 38. When there are ciphers at the right of the dividend. RULE. Cut off as many figures from the right of the dividend as there are ciphers in the divisor. Divide the re-maining figures of the dividend by the significant figures of the divisor, and to the renmainder annex the figures cut off from the dividend for the true remainder. Ois. 1. If there be no remainder, the figures cut off will be the true remainder. OBs. 2. If the divisor be 10, 100, 1000, &c., the figures cut off will be the remainder, and the remaining figures of the dividend will be the quotient. 38. Divide 4654 by 300. 41. Divide 6702 by 10. 39. Divide 3756 by 400. 42. Divide 4200 by 20. 40. Divide 2640 by 270. 43. Divide 1364 by 40. GENERAL PRINCIPLES IN DIVISION. 39. As the product and one factor inll division is given to find the other, it is evident that multiplying the dividend is the same in effect as multijplying the quotient, and dividing the dividend the same as dividing the quotient. Thus 48 divided by 6 gives 8 for When there are ciphers at the right of the dividend, what is the rule? What effect is produced on the quotient by multiplying or dividing the dividend? PRACTICAL QUESTIONS. 37 the quotient. Now, multiplying 48 by 2, 3, 4, &c., will make the quotient 2, 3, 4, &c., times larger. 40. It is also evidefit that multiplying the divisor is the same as dividing the dividend, and dividing the divisor the same as multiplying' the dividend. Thllus, having 48 for a dividend and 6 for a divisor, if the divisor be multiplied by 2, 3, &c., it will give the same quotient as if the dividend were divided by 2, 3, &c., and by dividing the divisor by 2, 3, &c., it will give the same quotient as if the dividend were multiplied' by the,.same numbers. Thus 48' 6X2 - 4, which is the same as 48. 2, and the quotient-. by 6. ART. 43. 41. If the dividend and divisor be both divided or multiplied by the same number, the quotient will notf be changed. Thus'48 divided by_ 6 will give the same quotient as 48X4 divided by 6X4; and 48' 3 divided by 6.~ 3. 42. When the sum and difference of two numbers are given, the smaller number may be found by subtracting the difference from the sum and dividing the remainder by 2. The larger number nay be Jbund by adding the difference to the smaller number. PRACTICAL QUESTIONS. 1. A merchant owes to one man 7361 dollars, to another 1969 dollars, to a third 2739 dollars; how much does he owe to all three? 2. A man bought a farm for 9375 dollars, but was obliged to sell it for 845 dollars less than he gave for it; what did he sell it for? 3. A farmer sells 354 cords of wood at 4 dollars a cord, and 37 tons of hay at 13 dollars a ton; ho w much does he receive for all? 4. A gentleman left in his will his estate, valued What effect is produced by multiplying or dividing the divisor?'What by multiplying both the dividend and divisor? When the sum and difference of two numbers are given, how are the two numbers found? D 38 PRACTICAL QUESTIONS. at 64331 dollars, as follows: 5630 dollars to his wife, 1245 dollars for charitable purposes, the remainder to be divided equally among his seven children; how much did each child receive? 5. The salary of the President of the United States is 25000 dollars a year; how much will he save in one year, if he spend 50 dollars a day? 6. A merchant bought a ship for 27342 dollars, and paid for cargo 37564 dollars; he sold the ship and cargo in California for 164000 dollars; how much did he gain, after deducting 2560 dollars for the expenses of the voyage? 7. If you were to count 2 every second for twelve hours a day, how long would it take to count a million? 8. What is the difference, between 644X46 and 615433 - 13? 9. A gentleman left his estate, valued at 16596 dollars, to be divided between his wife and three children in the following manner: his wife was to have one third of the whole estate; the oldest child was to receive one third of what was left; the remainder was to be equally divided between the two youngest; what did each receive? 10. The sum of two numbers is 1496, one of which is 984; what is the other number? 11. If a man's income be 1650 dollars a year, and he. expends 24 dollars a week, how much will he have left at the end of the year? 12. The difference between two numbers is 965, the greater is 1875; what is the less number? 13. A merchant owes 14560 dollars, which is less than what is due him by 9560 dollars; what is the sum due him? 14. The difference between two numbers is 564, and the less number is 896; what is the greater number? PRACTICAL QUESTIONS. 39 15. The product of two numbers is 1296, and one of the numbers is 9; what is the other? 16. If the quotient be 345, and the divisor 64, what must the dividend be? 17. If the remainder of a sum in division be 20, the quotient 423, and the divisor the sum of the quotient and remainder, plus 19, what will be the dividend? 18. There are two numbers, the greater of which is 37 times 45, their difference 4 times 19; what is their sum and product? 19. If the dividend be 1728, andthe quotient 24, what will be the divisor? 20. There are three numbers, whose continued product is 3456; one of the numbers is 12, another is 18; what is the other number? 21. The sum of two numbers is 1296, their difference is 144; what are the numbers-? 22. The sum of three numbers is 640, the difference between the least and the greatest is 220, and the difference between the middle number and the slm is 460; what are the numbers? 23. A grocer bought a hogshead of molasses, containing 133 gallons, at 27 cents a gallon; but 19 gallons having leaked out, he sold the remainder at 39 cents a gallon; did he gain or lose, and how much? 24. What is the difference between 1:9 times 144 and 6732 divided by 17? EXAtMPLES IN THE PRECEDING RULES, WITH SIGNS. The signs of addition, multiplication, subtraction, and division have already been explained, Art. 27. 43. A vinculum, - or parenthesis, is used to collect several quantities into one. Thus 5+ 4 X 6, or (5 + 4) X, 6, signifies that the sum of 5 and 4 is to be multiplied by 6, which is 54. Without the vinculum What is a vinculum? 410 DIVISIBILITY OF NUMBERS. 5+-4X6, only 4 is multiplied by 6, and the 5 is added to the product, making 29. Also, 12 —3X3, or (12-3)X3, signifies that the diffirence between 12 and 3 is to be multiplied by 3, which is 27. Without the vinculum, the product of 3 by 3 is subtracted from 12, which leaves 3. 25. 6+5X4_? 4 32. 84X6+8X9. 12=? 26. 6+5x4 —? 33. 64-14X17-12 -5-? 27. 7+6-3 X 5-? 34, 96. 16 X 18 —12-? 28. 7+6-3X5-? 35. 144X4- - 4X6-? 29. 24x6X9 - 3-? 36. 16-9+24 -2x3 — 30. 36 —-12X4 — 8 —? 37. 4X8-12 X 4-2 — 31. 96 16 X55+4=? 38. 64X6-7+8+9=? 39. Write the sum of the products of 8 into 9, and 7 into 4. 40. Write the difference of the products of 8 into 9, and 7 into 4. 41. Write the product of the sum of 8 plus 9, and 7 plus 4. 42. Write the product of the difference of 9 minus 8, and 7 minus 4. 43. Write 24 divided by the product of 4 into 2. 44. Write the quotient of 24 for a dividend, and 3 for a divisor. 45. Write the quotient of 24 divided by the product of 3 and 2. THE DIVISIBILITY OF NUMBERS. 44. The divisibility of a number is its property of being exactly divided by another. Thus 36 is divisible by 2, 3, 4, 6, 9, 12, and 18, because each of these numbers will divide it without a remainder. What is the divisibility of a number? DIVISION. 41 45. Every number divisible by another is also divisible by each one of its factors. Thus 48 is divisible by 12, also by 4 and 3, and 2 and 6, factors of 12. 46. Every number divisible by another is called a multiple of that number. If it be divisible by two or more numbers, it is a common multiple of those numbers. Thus 24 is the common multiple of 3, 4, 6, 8, and 12. 47. The least number divisible by two or more numbers is the least common multiple of those numbers. Thus 12 is the least common multiple of 3, 4, and 6. 48. Every number that will divide exactly another number is called the measure of that number. Thlus 4 is the measure of S. 49. Every number that will divide exactly two or snore numbers is their common measure. Thus 6 is the common measure of 12 and 18. 50. The greatest number that will divide exactly two or more numbers is their greatest common measure. Thus 8 is the greatest common measure of 24 and 56. 51. Every number that will divide exactly another number will also divide any multiple of that. numnber. Thus 6 will divide 12; it will also divide 24, 36, 48, &c. 52. Every number that will divide exactly two other numbers will also divide their sum, their difference, and their product. Thus 3 will divide 6 and 15; it will also divide 6+-15=21, 6X15-90, and i 5-6=9. 53. Every number divisible by 2 is called an even. number; when not divisible by 2, an odd number. 54. Every number is divisible by 3 when the sum What is a multiple? What is a common multiple? What is the least common multiple? What is the measure of a number? What is the common measure? What is the greatest common measure? What is an even number? What is an odd number? What numbers are divisible by 3? 42 DIVISION. of its figures, considered as units, is divisible by 3. Thus 321 is divisible by 3, since 3+2+1 —6 is divisible by 3. 55. Every number is divisible by 4 when its two right hand figures are divisible by 4. Thus 112, 1,16, 128, are divisible by 4, since 12, 16, 28, are divisible by 4. 56. Every number is divisible by 5 when its right haltd figure is 0 or 5. Thus 10, 15, 20, 25, are all divisible by 5. 57. Every even nnumber is divisible by 6 which is divisible by 3. Thus 12, 18, 24, 30, are all divisible by 6. 58. Every nunmber is divisible by 9 when the swum of' its fiAucres, considered as units, is divisible by 9. Thus 8172 is divisible by 9, since 8+1+7+2=18 is divisible by 9. PRIME AND COMPOSITE NUMBERS. 59. Whole numbers are either prime or composite. 6O. A prime number is one which cannot be formed bly multiplying together any two or more whole numbers greater than a unit, as 1, 2, 3, 5, 7, 11, 13. 61. A composite number is one. which may be formed by multiplying together two or more whole numbers greater than a unit, as 4, 6, 9, 8, 10, 12. OBS. A prime number cannot be divided by another numbei, except itself and a unit. 62. Every composite number may be resolved into prime numbers, which are called prime factors. 63, To find the prime factors of any composite number, --- RULE. Divide the given numbers in succession by What numbers are divisible by 4? What by 5? What by 6? WVhat by 9? What are prime numbers? What are composite numbetrs? What are prime factors? DIVISION. 43 the least prime number greater than a unit that will divide themn without a remainder. The last quotient and the several divisors will be all its prime.factors. 64. The principle of this rule is evident from the fact that the divisors are all prime numbers, and the last quotient also must be a prime number, since it cannot be divided by any number but itself and a unit. 1. What are the prime factors of 96? 96' 2-48. 48 2 —24. 24 2 = 12. 12 - 2 = 6. 6 2 -- 3. 2 X 2 X 2 X 2 X 2 X 3 -- 96. The given number, 96, is divided in succession by 2, the least prime number. The several prime factors are 2, 2, 2, 2, 2, 3. EXAbMPLES. What are the prime factors of the following numbers? 2. 12 and 14 9. 18 and 20 16. 24 and 27 3. 15 " 21 10. 28 " 30 17. 32 " 34 4..33 " 35 11. 36 " 40 18. 38 "c 42 Z. 44 " 48 12. 45 " 50 19. 49 51. 6. 54 " 56 13. 58 " 63 20. 64 " 72 7. 76'S80 14. 84" 88 21. 90" 96 8. 99" 102 15. 104'. 108 22. 112 " 115 23. What are the prime factors of 128 and 132? 24. What are the prime factors of 136 and 144? 25. What are the prime factors of 156 and 160? 26. What are the prime factors of 164 and 168? 27. What are the prime factors of 176 and 192? 28. What are the prime factors of 216 and 224? 29. What are the prime factors of 234 and 336? 30. What are the prime factors of 375 and 450? 31. What are the prime factors of 470 and 540? 32. What are the prime factors of 560 and 680? 44 DIVISION. A Table of Prine and Comnposite Numbers. No. F'actors. No. Factors. No. Factors. 1 prime. 34 2.17 67 prime. 2 prime. 35 5.7 68 2.2.17 3 prime. 36 2.2.3.3 69 3.23 4 2.2 37 prime. 70 2.5.7 5 prime. 38 2.19 71 prime. 6 2.3 39 3.13 72 2.2.2.3.3 7 prime. 40 2.2.2.5 73 prime. 8 2.2.2 41 prime. 74 2.37 9 3.3 42 2.3.7 75 3.5.5 10 2.5 43' prime. 76 2.2.19 11 prime. 44 2.2.11 77 7.11 12 2.2.3 45 3.3.5 78 2.3.13 13 prime. 46 2.23 79 prime. 14 2.7 47 prime. 80 2.2.2.2.5 15 3.5 48 2.2.2.2.3 81 3.3.3.3 16 2.2.2.2 49 7.7 82 2.41 17 prime. 50 2.5.5 83 prime. 18 2.3.3 51 3.17 84' 2.2.3.7 19 prime. 52 2.2.13 S5 5.17 20 2.2.5 53 prime. 86 2.43 21 3.7 54 2.3.3.3 87 3.29 22 2.11 55 5.11 88 2.2.2.11 23 prime. 56 2.2.2.7 89 prime. 24 2.2.2.3 57 3.19 90 2.3.3.5 25 5.5 58 2.29 91 7.13 26 2.13 59 prime. 92 2.2.23 27 3.3.3 60 2.2.3.5 93 3.31 28 2.2.7 61 prime. 94 2.47 29 prime. 62 2.31 95 5.19 30 2.3.5 63 3.3.7 96 2.2.2.2.2.3 31 prime. 64 2.2.2.2.2.2 97 prime. 32 2.2.2.2.2 65 5.13 9S 2.7.7 33 3.11 66 2.3.11 99 3.3.11 DIVtISION. 45 No. Factors. No. Factors. No. Factors. 100 2.2.5.5 135 3.3.3.5 170 2.5.17 101 prime. 136 2.2.2.17 171 3.3.19 102 2.3.17 137 prime. 172 2.2.43 103 prime. 138 2.3.23 173 prime. 104 2.2.2.13 139 prime. 174 2.3.29 105 3.5.7 140 2.2.5.7 175 5.5.7 106 2.53 141 3.47 176 2.2.2.2.11 107 prime. 142 2.71 177 3.59 108 2.2.3.3.3 143 11.13 178 2.89 109 prime. 144 2.2.2.2.3.3 179 prime. 110 2.5.11 145- 5.29 180 2.2.3.3.5 111 3.37 146 2.73 181 prime. 112 2.2.2.2.7 147 3.7.7 182 2.7.13 113 prime. 148 2.2.37 183 3.61 114 2.3.19 149 prime. 184 2.2.2.23 115 5.23 150 2.3.5.5 185 5.37 116 2.2.29 151 prime. 186 2.3.31 117 3.3.13 152 2.2.2.19 187 11.17 118 2.59:153 3.3.17 188 2.2.47 119 7.17 154 2.7.11 189 3.3.3.7 120 2.2.2.3.5 155 5.31 190 2.5.19 121 11. 11 156 2.2.3.13 191 prime. 122 2.61 157 prime. 192 2.2.2.2.2.2.3 123 3.41 158 2.79 193 prime. 124 2.2.31 159 3.53 194 2.97 125 5.5.5 160 2.2.2.2.2.5 195 3.5.13 126 2.3.3.7 161 7.23 196 2.2.7.7 127 prime. 162 2.3.3.3.3 197 prime. 128 2.2.2.2.2.2.2 163 prime. 198 2.3.3.11 129 3.43 164 2.2.41 199 prime. 130 2.5.13 165 3.5.11 200 2.2.2.5.5 131 prime. 166 2.83 201 3.67 132 2.2.3.11 167 prime. 202 2.101 133 7.19 168 2.2.2.3.7 203 7.29 134 2.67 169 13.13 204 2.2.3.17 46 DIVISION. CANCELLATION. 65. Cancelling is the process of abridging arithmetical operations by striking out factors common to the divisor and dividend..RULE. Write the numbers forming the dividend above a horizontal line, and the numbers forming the divisor below.it'; cancel all the factors common to the dividend and divisor. Then multiply and divide with the remaining figures. Ons. When any entire number is cancelled, either in the dividend or divisor, 1. must be written in its place. 66. This rule depends upon the principle illustrated in (ART. 41,) viz., that by multiplying or dividing the dividend and divisor by the same number, the quotient will not be changed. 1. Divide 24 X 6 by 12 X 3. 2X2 x x 4 4 4x 1 1 24 multiplied by 6 is divided by 12 multiplied by 3. Since 12 and 2 are factors of 24, cancel 12 in the divisor and in the dividend, and write the other factor, 2, over the 24. Since 3 is a factor of the divisor, and also of 6 in the dividend, cancel 3 in the divisor and dividend, and write 2, the other factor, over the 6. The product of the remaining figures, 2 into 2, will be 4, which being divided by 1, will remain unchanged. EXAMPLES. 2. Divide 48 X 15 by 16 X 3. 3. Divide 72 X 8 by 12 X 9. What is cancelling? Recite the rule. Upon what does the rule depend? GREATEST COMMON MEASURE. 47 4. Divide 144 X 9 by 24 X 18. 5. Divide 342X8 by 42X6X4. 6. Divide 396 X 12 by 18 X 6 X 4. 7. Divide 484 X 16 by 8 X 6 X 12. 8. Divide 724 X 4 by 8 X 4x 6. 9. Divide 365X5by 55X X5. 10. Divide 426 X 6 by 4 X 3 X 8. 11. Divide 828X9 by 18X9X4. 12. Divide 936 X 24 by 36 X 9. 13. Divide 96 X 9 X 8 by 8X 12. THE GREATEST COMMON MEASURE. 67. The greatest common measure of two or more numbers has been shown to be (ART. 50) the greatest number that will divide them without a remainder. 68. To find the greatest common measure of two or more numbers, - When the numbers are small, the greatest commion nmeasure may readily be determined by inspection, or by dividing the numbers by the largest number that will divide them without a remainder. Thus the greatest common measure of 12 and 16 is evidently 4. OBs. When there are more than two numbers, first find the greatest common measure of any two of them, then of this common measure and one of the other numbers. Proceed thus with all the numbers. What is the greatest common measure or factor of the following numbers? 12 and 18. 14 and 35. 15 and 20. 1S and 27. 21 and 28. 24 and 30. 16 and 28. 27 and 81. 39 and 52. 28 and 63. 32 and 48. 35 and 63. 42 and 77. 56 and 84. 28 and 126. 56 and 84. 21 and 126. 26 and 117. 42 and 112. 32 and 144. 69. When the numbers are large,What is the greatest common measure? How may it be found when the numbers are small? 48 GREATEST COMMON MEASURE. RULE. Divide in succession the greater number by the less, and that divisor by the last remainder, till nothing remains. The last divisor will be the greatest common measure. 70. This rule depends upon the principle illustrated in (ART. 52,) viz., that any number that measures any other number will also measure their sum, their difference, and any mulliple of those numbers. 1. What is the greatest common measure of 720 and 612? 612 ) 720 (1 612 108 )612( 5 540 72 )108 (1 72 36 ) 72 ( 2 72 The greatest number being divided by the less, and the last remainder by the last divisor, gives 36 as the greatest common measure. Since 36 measures 72, it will also measure 108, the sum of 36 and 72; it will also measure 612, a multiple of 36. EXAMPLES. 2. What is the greatest common measure of 252 and 348? 3. What is the greatest common measure, of 493 and 899? 4. What is the greatest common measure of 208 and 648? Recite the rule, when the numbers are large. LEAST COMMON MULTIPLE. 49 5. What is the greatest common measure of 825 and 960? 6. What is the greatest common measure of 5184 and 6912? 7. What is the greatest common measure of 3242 and 7564? S. What is the greatest common measure of 972 and 1468? 9. What is the greatest common measure of 656, 864, and 976 10. What is the greatest common measure of 896, 764, and 938? 11. What is the greatest common measure of 372; 964, and 704? THE LEAST COMMON MULTIPLE. 71. The least number divisible by two or more numbers is the least common multiple of those numbers, (ART. 47.) 72. To find the least common multiple of two or more numbers, — RULE. First cancel every number that will measure any other given number,;'Then- cancel the largest factor common to the two largest numbers, and all the factors of the other numbers that are contained in the uncancelled factors. The continued product of the remaininng figures will be the least common multiple. O3s. 1. The numbers should be written in order in ahorizontal line, beginning with the least on the left, and endingo with the largest on the right. OBs. 2. The largest common measure and the least common multiple should not be confounded. The largest common measure is the largest number that will divide two or more numbers. The least common multiple is the least number that is divisible by two or more numbers. 73. It is evident that the least common multiple of two numbers is the product of the numbers after striking What is the least common multiple? What is the rule? E 4 50 LEAST COMMON MULTIPLE. out their greatest common divisor, or the largest factor common to them, as it is necessary that a number be contained in the product as a factor only once. The principle is the same for finding the least common multiple for three or more numbers. 1. What is the least common multiple of 6, 8, 12, 16, 20, 24? ~, Sf, AX, X~, 1, 24. 2 X 5 X 24-240. First cancel 6, 8, 12, since they measure 24. Then cancel 4 in 20, since it is the largest factor common to 20 and 24, and write underneath the other factor, 5. Cancel 8 in 16, as it is the largest factor contained in 24. The other factors, 2 X 5 X 24=240, the least common multiple. 2.'What is the least common multiple of 14, 18, 21, 27, 28, 1267? X~, 15, d;, #7, #S, 126. 3 2 X 126 —756. Cancel 14, 18, 21, since they measure 126. Cancel also 14 in 28, the largest factor common to 28 and 126, and also 9 in 27, the largest factor common to the uncancelled factors in 2 and 126. The product of 3 X 2 X 126 756, the least common multiple. EXAMPLES. 3. What is the least common multiple of 4, 6, 8, and 10? 4. What is the least common multiple of 3, 4, 7, and 14? 5. What is the least common multiple of 5, 7, 10, and 21? 6. What is the least common multiple of 6, 9, 18, and 24? Upon what principle does the rule depend? LEAST COMMON MULTIPLE. 51 7. What is the least common multiple of 8, 12, 16, and 20)? S. VWhat is the least common multiple of 10, 12, 16, 18, and 20? 9. What is' the least common multiple of 2, 3, 4, 5, 6, 7, 8, 9? 10. What is the least common multiple of 18, 36, 72, and 108? 11. W7Vhat is the least common multiple of 14, 21, 28, and 35? 12. What is the least common multiple of 15, 20Q 35, 45, and 60? 13. What is the least common multiple of 23, 46, 69, and 92? 14. What is the least common multiple of 34, 51, 68, and 85? 15. What is the least common multiple of 37, 74, 11, and 148? 16. What is the least common multiple of 41, 82, 123, and 164? 17, What is the least common multiple of 25, 60, 75, and 100? 18. What is the least common multiple of 30, 70, 90, and 140? 19. What is the least common multiple of 65, 75, 95, and 125? 20. What is the least common multiple of 136, 144, and 284? 21. What is the least common. multiple of 272, 384, and 756? 22. What is the least common multiple of 320, 450, and 640? 23. What is the least common multiple of 426, 560, and 846? 24. What is the least common multiple of 576, 646, and 736? 52 FRACTIONS. FRACTIONS., SECTION VII. 74. A FRACTION is one or more equal parts of a unit. 75. Fractions are of two kinds, commrnon and decimal. A common fraction is composed of two terms, one written above the other, with a line between them. Thus, 4 3 The term below the line is called the denominator, and shows the number of equal parts into which the unit is divided. The term above the line is called the nugnerator, and shows how many parts are expressed by the fraction. Bss. The term nzumzerator is from a Latin word, signifying to nuamher; the de-torminator from a Latin word, signifying to name. 76. A fraction may also be considered as the quotient resulting from division, the numerator being the dividend, and the denominator the divisor. Thus 4 is the quotient resulting from 4 being divided by 7, and may be read one seventh of four, or four sevenths of one. In the same manner,. is read one third of two, or two thirds of one. A- is read one eighth of five, or five eighths of one. - is read one seventh of nine, or nine sevenths of one. 77. A proper fraction is one whose numerator is less than its denominator, as 2, 3. 78. An ivtproper fraction is one whose numerator is equal to or greater than its denominator, as 9-, -6. OBS. Proper and improper fractions are also called simple firactions. What are fractions? I-low many kinds of fractions? Of what is a fraction composed? How are fractions written? What are the terms of a fraction called? What does the term below the line show-? What does the term above the line show? How may a fraction be considered? How are fractions read? What is a proper fraction? Anl inproper fraction? A simple fraction? FRACTIONS. 53 79. A mixed number consists of a whole number and a fraction, as 51, 30%4 1680. A comgpound fraction is a fraction of a fraction, or any number of fractions connected by the word of: Tpl1us 3 of - of 4. 8, A conmplex fraction is one which has a fraction or mixed number for its numerator or denominator, or both. OBs, Any whole number may be considered as the fraction of another. Thus 4 is four twelfths of 12, 5 is five sevenths of 7. 82. A fraction is multiplied by any?umber by,multiplying its nugmerator or dividing its denominator by that number. 8$3 A fraction is divided by any number by dividinJig its numerator or multiplying its denominator by that number. 84. The value of a fraction is not changed by inultiplying or dividing both of its termns by the same number. Thus 3- is equat in value to 8, 12 20 c.; -? is also equal to ~. 85. To reduce a fraction to its lowest terms,-. RULE. Divide both terms by any numbers in succession that will divide them both without a remainder. Or divide both terms by their greatest common?measure. 1. Reduce -39 to its lowest terms. 3 3 33 -1 9 - Dividing both terms by 3 twice gives -I, whose terms are the lowest. The same result is obtained by dividing both terms by 9, their greatest common measure. W\That is a mixed number? A compound fraction? A omnplex ~raction? 54 FRACTIONS. Oms. 1. A fraction is reduced to its lowest terms awhen no number greater than a unit will divide both of its terms without a rermainder. OBs. 2. The value of a fraction is not changed by reducing it to its lowest terms, (AaRT. 84.) EXAMPLES. 2. Redulce 26 to its lowest terms. 3. Reduce 1 11. Reduce _124 4. Reduce 42 12. Reduce 2.36 - 64 I 4 9 47 5. Reduce A 6 13. Reduce.76 6. Reduce 6 4 14. Reduce 97 4. 7. Reduce 96 15. Reduce 15. Reduce 6 8. Reduce 10lt 16. Reduce 22346 112Red~ce 8648' 9. Reduce 1o10 17. Reduce.366 10. Reduce l. I 18. Reduce 457 55. 86. To reduce an improper fraction to an equivalent whole or mixed number, — RULE. Divide the numerator by the denominator, and write the remainder, if there be any, over the denominator. This rule depends upon the fact stated in ART. 76, that the denominator of a fraction is the divisor, and the numerator- the dividend. 19. Reduce 2-9- to a whole or mixed number. 5 ) 29 5 Dividing the numerator, 29, by 5, its denominator gives 54, a mixed number. What is the rule for reducing an improper fraction to a whole or mixed numnber? FRACTIONS. 55 Reduce the following examples to whole or mixed aumbers: — 20. Reduce 27 26. Reduce 76 21. Reduce37. 27. Reduce 91 22. Reduce — L. 28. Reduce 108 23. Reduce 5-. 29. Reduce 116 24. Reduce 64 30. Reduce -251l-. 25. Reduce 71 31. Reduce 4561 87. To reduce a mixed number to an equivalent improper fraction, - RULE. Multiply the whole number by the denominator of the fraction, and to the product add the numerator, and, write the sum17 over the denominator. 32. Reduce 72 to its equivalent improper fraction. Multiplying the 7 by 3, and adding the 2, gives 23 for a numerator; which written over the denominator, 3, gives -233- for the improper fraction. This rule is the converse of the preceding one. Ons. 1. Converse means performed in an opposite order. OBS. 2. A whole number may be changed to an improper fraction by writing 1 under it for a denominator. OBs. 3. A whole number may be changed to an improper fraction of a given denominator by multiplying the whole number by the proposed denominator for a numerator. Reduce the following examples to their equivalent improper fractions. 33. Reduce 84. 37. Reduce 16-. 34. Reduce 9a. 38. Reduce 301. 35. Reduce 10~. 39. Reduce 272k. 36. Reduce 125. 40. Reduce 69. What is the rule for reducing a mixed number to an improper fraction? 5G FRACTIONS. 41. Reduce 73492O. 43. Reduce 90101~9. 42. Reduce 6730 4. 44. Reduce 701 145. Reduce 5, 6, 7, 8, 9, 11, to improper fractions. 46. Reduce 36 to thirds, 42 to fourths, 54' to sixths. 47. Reduce 64 to sevenths, 95 to sixteenths, 112 to twelfths. 88. To reduce compound fractions to simple frac tions, - RULE. Multiply the numerators together for a numerator, and the denominators for a denominator. OBs. 1. All factors common to the numerator and denominator should be cancelled before multiplying. OBs. 2. If a part of a compound fraction be a mixed or whole number, it must first be reduced to an improper fraction. 48. Reduce A of 5 to a simple fraction. ~ of - is evidently 53; is 3 times 5-, which is 45 the simple fraction. 49. Reduce.- of 6 of 4 to a simple fraction. 5 5 4 12 3 Cancelling the factors 2 and 3 in the numerators, and also 2 and 3 in the denominators, gives 5 for a numerator, and the product of 3 and 4 for a denomijiator, which is -5, the simple fraction. EXAMPLES. 50. Reduce 4 of 6 of 7 to a simple fraction. 51. Reduce 3 of A of 14 to a simple fraction. 52. Reduce 2 of 3 of 3 of of 7 to a simple fraction. 53. Reduce - of 215 of -L of 22 to a simple fraction. 9 25 1 23 What is the rule for reducing compound fractions to simple fractious? FRACTIONS. 57 54. Reduce - of 26 of 9 of 39 to a simple fraction. 55. Reduce 4 of 27 of I of 39 to a simple fraction. 56. Reduce 4- of 3 of 7 of 8 to a simple fraction. 57. What is a of 7 of 10k? 58. What is 3 of 1-1 of 91~? 59. What is of f of 16? 60. What is -9 of 39 of 19-? 61. What is 7of of 112? 62. What is 6 of 21 of 30? 63. What is -90 of 20 of 100? 64. What is - of 12 of 260? 89. To change fractions which have different denominators to other equivalent fractions which shall have a common denominator,RULE. Multiply both terms of each fraction by the product of the denominators of the other fractions. Or find the least common multiple' of the denominators for a common denominator, (ART. 71.) Then vmTultiply each numerator by that number which denotes the number of times its denominator is contained in the common multiple. 65a. Reduce 2 3, and 45 to a common denominator. X 5 3X 5X 2 4 X 4 X 2 2 X 5 4 X 5 X 2 5 X 4 X 2 4 It is obvious that the value of the fractions has not been changed, since the numerators and denominators of each have been multiplied by the same number) (ART. 84. ) OBS. It is generally best to find the least common multiple of the denominators, as this reduces the fractions to their lowest terms. Recite the rule for changing fractions to a common denominator. 58 FRACTIONS. EXAMPLES. 66. Reduce, 3 4 and 6 to a common denominator. 67. Reduce 6, 8, 8, and -9- to a common denominator. 68. Reduce 1, 12, l and 14 to a common denominator. 69. Reduce 2 5 17 and 37 to a common denominator. 70. Reduce 4, 3, and 7 to a common denom2) 42 19i 24 inator. 71. Reduce I of 4 6 of 8, and I of 161 to a colnmon denominator. 72. Reduce of,9 o f 4 and 9 to a common denominator. 73. Reduce 2 3 9 and 4 6. to a common denominator. 74. Reduce of 2, 294, and -v to a common denominator. 75. R'educe, 1, and 191 to a common denominator. 76. Reduce 4, 8 7 and -9 to a common denominator. 77. Reduce 6 of 9 and 35 of 24 to a common denominator. 78. Reduce 4 -7 8 anld 9 to a common denominator. 79. Reduce 11, 6, and I to a common deuominator. 80. Reduce -4 8 17 and 19 to a common denonminator. 81. Reduce 6 of 71 and 3 of 91 to a common denominator. FRACTIONS. 5 9 90. To reduce one fraction to another of the same value, having a given numerator,RULE. Multiply both terms of the fraction by the proposed numerator, and divide both terms by the numerator of the given fraction. 82. Reduce - to a fraction with 11 for its numerator. 7 X 11 77 77 7 11 9 X 11 99 99 7 14l Since both terms are multiplied and divided by the same numbers, the value of the fraction is not changed. EXAMPLES. 83. Reduce 1 to a fraction with 15 for a numerator. 84. Reduce l4l to a fraction with 21 for a numerator. 85. Reduce r to a fraction with 31 for a numerator. 86. Reduce 19 to a fraction with 21 for a numerator. 87. Reduce 2 9 to a fraction with 41 for a numerator. 91. To reduce one fraction to another of the same value, having a given denominator,RULE. Multiply both terms of the fraction -by the proposed denominator, and divide both terms by the denominator of the given fraction. 88. Reduce -jr to a fraction with 13 for a denominator. 7 X 13 91 91 1 11 S8 11 X 13 143 143 + 11 13 Since both terms are multiplied and divided by the same numbers, the value of the fraction is not changed. What is the rule for reducing one fraction to another of a given numerator? What for reducing one to a given denominator? 60 ADDITION OF FRACTIONS. EXAMPLES. 89. Reduce 9- to a fraction with 19 for a denominator. 90. Reduce 1-T to a fraction with 21 for a denominator. 91. Reduce 23 to a fraction with 19 for a denominator. 92. Reduce ~ to a fraction with 45 for a denominator. 93. Reduce 6- to a fraction with 17 for a denominator. 94. Reduce. to a fraction with 23 for a denominator. 95. Reduce Ad to a fraction with 96 for a denominator. ADDITION OF FRACTIONS. 92. RULE. Reduce the fractions, when necessary, to their least common denominators; add their numerators, and write their suvm over the common denominator. Ons. 1. All whole and mixed numbers must first be changed to improper fractions, and compound fractions to simple fractions. Ohs. 2. In mixed and whole numbers, the whole numbers and fractions may be added separately, and their sums united. 96. WThat is the sum of I, [, and? X4X5 5 1 X3 X 5 2X4X 4 X 3 Ug X 4 X 35U 3 X4 XX 5 4 X 3 X 5 X 4X 3 20 1- 24:559 2 H OF + B - WEu The fractions, reduced to a common denominator, are -,, r2, 24; which being added together are na. 93. It is evident that parts of a number of the same kind can be added in the same manner as whole numbers of the same kind. Thus, one sixtieth added to one sixtieth is two sixtieths, and so of any number of sixtieths. Recite the rule for the addition of fractions. MULTIPLICATION OF FRACTIONS. 61 EXAMPLES. 97. What is the sum of:, I, and R? 98. What is the sum of 2, 4, and 4? 99. What is the sum of 2, 4, and I? 100. What is the sum of I; -, and i? 101. What is the sum of 4, 4, and -9T? 102. What is the sum of I, 14, and 2? 103. What is the sum of Ad, 4 3 and 4? 104. What is the sum of of 4 and of of and of? 105. What is the sum of: of Ad and 17 of? 106. What is the sum of 4 of 4 and - of 9? 107. What is the sum of 3~, 9Y,' 74, and 8? 108. What is the sumn of I of 5, and 6 of 111? 109. WThat is the sumof of of nd 4- of 25? 110. What is the sum of 44, 4, _, ~ and d? 111. What is the sum of 4, 4, 4,, and 2-? 112. What is the sum of 214, 144, and 4 of 40? 113. What is the sum of, -1, to, and I 1 Gs T'T T a) ~U 114. What is the sum of 1, 14, 4d, and 5? 115. What is the sum of of 10~ and X of 30?1 116. What is the sum of 3 of 9} and 5 of 64-? MULTIPLICATION OF FRACTIONS. 94. RULE. Multiply the numerators together for a numnerator, and the denominators for a denominator~ OBS. 1. Whole and mixed nnmbers must first be changed to imn proper fractions. OBS. 2. Cancel all factors common to the numerators and denominators. The principle of this rule is the same as that for the reduction of cornpound fractions. What is the rule for the multiplication of fractions? SUBTRACTION OF FRACTIONS. 117. Multiply 4 by -. First multiply 4 by 4, which is iX 4; but since the multiplier is not 4, but - of 4, the product is 9 times too large. This product must therefore be divided by 9. 7 26 118. Multiply - by 2. 2 X Ax $ 7 7 Cancel the two factors common to the numerator and divisor; the product of the remaining figures is 2. EXAMPLES. 19,. Multiply 4 by I. 131. Multiply 7 by -A. 120. Multiply - by I-. 132. Multiply 19r by T a. 121. Multiply 1, by 4+. 133. Multiply 19 by +4. 122. Multiply 5- by 5~. 134. Multiply 3S by 20. M23. Multiply 16~ by 16}. 135. Multiply 36 by 71. 124. Multiply 301 by 301. 136. Multiply 42 by 94. 125. Multiply 25 by 2~. 137. Multiply 75 by 5. 126. Multiply 24 by 62. 138. Multiply 64 by 8+. 127. Multiply 7{ by 49. 139. Multiply A by 40. 128. Multiply 62~ by 75. 140. Multiply R by 675. 129. Multiply 764 by T1. 141. Multiply 975 by A.!30. -Multiply 849 by ~. 142. Multiply 964 by 11. SUBTRACTION OF FRACTIONS. 95. RULE. Reduce the fractions as in addition, subtract the less num, erator from thIe greater, and twrite the difference over the common denomindtor. OBs. 1. All whole and mixed numbers must first be reduced to improper fractions, and compound fractions to simple ones. What is the rule for the subtraction of fractions? DIVISION OF FRACTIONS. 63 OBS. 2. In whole and mixed numbers, the whole numbers may be subtracted separately, and then their difference united to the fraction. 143. From 4 take i. 3X5 2X4 15 -- 5 4X5 5 X 4 Change ~ and - to a common denominator, which is SE and 2%. 8 from 15 leaves 7, which, written over the common denominator, 20, gives -72 for the answer. It is obvious that parts of numbers of the same kind can be subtracted in the same manner as whole nIumbers. EXAMPLES. g44. From ~ take 2. 152. From 271 take 92. 145. From 4 talke.. 153. 19 take 7r.8 1.46. From take 154. From 33~ take 29td 147. From - take 4. 155. From 9 take 31. 148. From jeT take -1 156. From 163 take 11. 149. From 74 take R. 157. From 226 take 37-9, 150. From 2' take 1. 158. From 297 take 9.Tr 151. From 164~ take 5E. 159. From 201 take 944. 160. From -a of 96 take X of 36. 161. From 9 of 874 take i of 32. 162. From the sum of i, 4, and I take - of 34. 163. From the sum of 371 and 654 take:~ of 832. 164. From the sum of 3364 and 21l9 take 1i of 671. 165. From the sum of 9737 and 264 take 2759 and 26.7 DIVISION OF FRACTIONS. 96. RULE. Invert the divisor, and proceed as -in multiplication. OBs. 1. The divisor is inverted by writing the denominator for a numerator and the numerator for a denominator. What is the rule for the division of fractions? 64 FRACTIONS. OBs. 2. WVhole and mixed numbers must first be changed to im. proper fractions, and compound fractions to simple fractions. Cancel when possible. 166. Divide " by y. Dividing 4- by 4 gives 4 XE; but since the divisor, 4, is divided by 9, and dividing the divisor is the same as multiplying the quotient, (ART. 40,) the quotient X T must be multiplied by 9, which gives 5 X 9 -~. The principle of this rule has already been explained in ART. 83. EXAMPLES. 167. Divide 7 by 3. 178. Divide 304 by 5~. 168. Divide $ by 4. 179. Divide 2724 by 161. 169. Divide -T by {4. 180. Divide 242 by 691. 170. Divide IT by 9-. 181. Divide 345 by 5~y. 171. Divide 4- by T. 182. Divide 176 by 272+. 172. Divide 24 by T. 183. Divide 464 by 30X. 173. Divide 1 by 36. 184. Divide 45 by 2-. 174. Divide I by 780. 185. Divide 764 by 312. 175. Divide 640 by gT,. 186. Divide 841 by 1. 176. Divide 780 by T7. 187. Divide 765 by 3~.. 177. Divide 91 by 846~. 188. Divide TO-T by 849. OBS. 1. The reciprocal of any number is 1 divided by that number. Thus the reciprocal of 4 is -. Ons. 2. Dividing by any number is the same as multiplying by its rceciprocal. Thus dividing 9 by 4 is 9. Multiplying 9 by R is the saine, 4. 97. A complex fraction is only one form of the division of one fraction by another fraction, (ART. 81.) 98. To reduce complex fractions to simple ones,RULE. Prepare the fractions, and proceed as in the division of fractions. What is the rule for reducing complex fractions? FRACTIONS. 65 This rule depends upon the fact that the numerator of every fraction is a dividend and the denominator a divisor, (ART. 76.) 18S9. Reduce i-to a simple fraction. XA — a Dividing - by ~ gives a for the simple fraction. Reduce the following complex fractions to simple ones. EXAM3PLESe 190. Reduce - 197. Reduce T4' 7 13 191. Reduce 2. 198. Reduce -. 9 192. Reduce 199. Reduce e Y9 7 4' 193. Reduce:6 200. Reduce 12' 27 2 -7;r 194. Reduce 2 201. Reduce {5'3 9 8 195. Reduce WT 202. Reduce -;. 74* 11 11 196. Reduce 203. Reduce 1s 240' 8R 99. Complex fractions, after being reduced to simple ones, may be added, mrultiplied, subtracted, and tditide in the same manner as simple fractions. Ohs. Whenever in the reduction of fractions there are factors cormon to the dividend and divisor, they should always be cancelled. How may complex fractions be added, &c.? r ~ 5 66 PRACTICAL QUESTIONS. PRACTICAL QUESTIONS. _12 19 204. Add - j +- 208. Subtract - from - 19 i - 71 9 d4 9_ 14_ 205. Add --. 209. Divide - by 6k 11. 71 k206. 5 30 4 19 206. Add -+ - 210. Multiply - by 5S 30 4?3+ 9 27-1 21 207. Add -+-. 211. Multiply -by - 97/i9 9 7T k' 212. Divide - of by of 10. - 214 213. Multiply ~ of by- of -- 64 341 4t- 93 7. 4214. Multiply - of - by -of - 71 1 1- 21 94 74 94 6 215. Divide 97 of -- by --- of. 831 7T 171 13k 8k 216. Divide -- of 7 by - of 13k. 16i 121 217. What is the sum of 5-, 4, 4, and ~7-? 218. What is the sum of. of 71 and 5 of 24? 219. What is the product of 7 of'1 multiplied. by {q of 29? 220. What is the product of' of 9 multiplied by jj of 633? 221. What is the difference between ~ of 8 and S of 5? PRACTICAL QUESTIONS. 67 222. If a bushel of oats cost T of a dollar, what is of a bushel worth? 223. If 4T of a yard of cloth is worth 9-19 of a dollar, what is one yard worth? 224. A owns ~ of a ship, B 5, C', and D the remainder. What part of the ship.belongs to D? 225. What will 7- yards of cloth cost at 5-3 dollars per yard? 226. What will 137 barrels of flour cost at 6U dollars per barrel? 227. What will 654~ feet of land cost at 97- cents. per foot? 228. What will A9 of an acre of land cost at 176 dollars per acre? 229. What will 137r9 tons of hay cost at 132 dollars per ton? 230. How many yards of cloth are there in three pieces, the first piece containing 19-, the second 27A and the third 413 yards. 231. A boy gave ~ of an orange to one companion,. and - of the remainder to another. How much did he keep for himself? 232. Add together the sum, difference, and product of l and A9. 233. If 25 dollars be paid for 2- tons of hay, what is the price of a ton? 234.' A merchant sold i of - of a ship to one person, and 2 of j- to another. What part of the whole ship had he left? 235. How much greater is the sum of 9~4x and 6ythan their difference? 68 PtRACTICAL QUESTIONS. 236. A gentleman bequeathed - of his estate, which was worth 36000 dollars, to his son, 2 of what reImained to his daughter, and the rest to his widow. What sum of money did each receive? 227. If the ciicumference of a wheel of a locomotive be 7? feet, how many revolutions would it make between Boston and Providence, a distance of 41~ miles, there being 5280 feet in a mile? 238. A lady's fortune was 8000 dollars, which was c of 3 of her brother's. How much was her brother's portion? 239. What number is that to which if you add ~ of ~ of 4 the sum will be g? 240. A grocer sold 23W pounds of tea for 1754%-9 cents. How much was that a pound? 241. If one acre acre of land will produce 27 bushels of corn, how much will 192 acres produce? 242. A merchant sold 372 yards of cloth for 2968 dollars. How much was that a yard? 243. If a barrel contain- 24 bushels of apples, how many barrels would contain 962 bushels?' 244. How many pounds of sugar, at 64 cents per pound, can be bought for 960~ cents? 245. How many coats, containing 14 yards, can be made from 164 yards? How much cloth will be left? 246. A man paid 3643 dollars for Mocha coffee, at -,of a dollar a pound. How much coffee did he-buy? 247. A gentleman who owns 4 of a copper. imine sells - of his share for 1640 dollars. What was the mine worth? 248. If the greater of two fractions be l2, and their difference r~, what is the smaller fraction? PRACTICAL QUESTIONS. 69 249. If the smaller of two fractions be _-, and their difference 9r, what is the larger fraction? 250. If the sum of two fractions be,64- and their difference 3, what are the two fractions? (See ART. 42. ) 251. What number is that, from which if you take f of itself, the remainder will be 15? 252. What number is that, to which if you add, of 2 of itself, the sum will be 21? 253. What number is that which, being divided by a, and multiplied by w, will give 20 for a product? 254. A merchant bought 7 of a ship, and afterwards sold ~ of his share. What part did he keep for himself? 255. Howa many barrels of flour can be purchased for 7355 dollars, at 4] dollars per barrel? 256. If there are 5~ yards in a rod, how many rods are there in 17659- yards? 257. If there are 16~ feet in a rod, how many rods are there in 18364 feet? 258. In an academy there are 240 pupils; ] of them are studying arithmetic; t- of the remainder are studying grammar, and the rest are studying geography. What is the number in each study? 259. A gentleman left to his elder son -X of his property, to the younger 2f of the remainder, and the rest to his wife. The elder son had 2400 dollars more than the younger. How much did the wife receive, and how much was left to the family? 260. Three persons enter into trade A, B, and C. A advances T as much as B, C advances 4 as much as B, which is 1200 dollars. The loss sustained by them was j~ of the sum advanced. How much will each have left'? 70) r>DECIMAL FRACTIONS. DECIMAL FRACTIONS. 100. A decimal fraction is one or more parts of a unit, divided into tenths, hundredths, thousandths, &c. OBS. Decimnal is from the Latin word. decem, signifying ten. 101. Instead of writing the denominator, a point is placed at the left of the numerator, called the deciveal point, which separates the decimal from the whole number. 102. The denominator of a decimal fraction, when written, is 1, with as many ciphers annexed as there are figures in the numerator. 1103. The first place at the right of the decimal point is the place of tenths, the second the place of hundredths, the third the place of thousandths, the fourth the place of ten thousandths, &c. 104. The same figure, in decimals, represents a value ten times greater in the first place than in the second, and ten times greater in the second place than in the third, &c., and by each removal of any figure to the next place on the right, its representative value is diminished ten times. 105. Ciphers are used to keep significant figures in their proper places, and are written where no value is to be expressed. 106. As the law of the increase in the representative value of figures from right to left is the same in decimals as in whole numbers, they may be written together by placing the decimal point between them. WVhat are decimal fractions? How are decimals written? What is the denominator of a decimal? What is the first place of decimals called? the second? the third? the fourth? &c. For what are ciphers used, and where are they written? What is the law of increase in decimals? DECIMAL FRACTIONS. 71 107. Decimals may be added, multiplied, subtracted, and divided in the same manner as whole numbers. OBs. A whole number and decimal written together is called a nzmied number. 108. Annexing ciphers to decimals does not change their value; as the significant figures retain the samne places as before Thus,.5,.50,.500, are decimals of the same value, for -- o =-50-0. 109. Prefixing ciphers to decimals diminishes their value ten times for every cipher prefixed, for each cipher removes the significant figures one place to the right. Thus,.4,.04,.004, are respectively equal to 4 4 js TS TGOOOBS. Ciphers are annexed when they are placed on the right of the decimal; they are prefixed when placed on the left of the decimal, and on the right of the decimal point. 110. A decimal may be enmultiplied by 10, 100, 1000, &c., by removing the decimal point as many places towards the right as there are ciphers in the multiplier. If there be not as many places, make the number equal by-annexing ciphers. Thus.244 multiplied by 10, 100, 1000, &c., is 2.44, 24.4, 244, 2440, &c. 1 1. A decimal may be divided by 10, 100, 1000, &c., by removing the decimal point as many places towards the left as there are ciphers in the divisor. If there be not as many, make the number equal by prefixing ciphers. Thus.244 divided by 10, 100, 1000, &c., is.0244,.00244,.000244. What is annexing ciphers? What is prefixing ciphers? What is their effect? How may decimals be multipled by 10, &c.? IHow may they be divided by 1.0, &c.? 72 DECIMAL FRACTIONS. The names of the different orders of decimals, and their relation to whole nllmbers, inay be learned froin the following table: — o fi'*0 -e. P uC~.Z. ~~, Q, o -" W IIOLE NUMBERS. D ECIMAL$. The whole nrumbers and decimnals in the above table are expressed in words thus: Eight million, seven hundred and sixty-five thousand, four hundred and thirty-two, and ninety-one million, two hundred and thirty-four thousand, five hundred and sixty-seven hundred millionths. 11 2. To read decimals, begin at the first figure on the left of the decimal point, and read the figures as it they were whole numbers, and to the last add the name of its order. Thus:-.5 is read 5 tenths..05 is read 5-hundredths..236 is read 236 thousandths..45678 is read 45678 hundred thousandths..365366 is read 365366 millionths. Repeat the table, I1-lw r' e decbials read&. REDUCTION OF DECIMALS. a, 1. Read the following numbers:-.4.60 4.0101.04.050 2.0129.004.0600 16.2121.0004.6060 8.3004.00004.0606 42.0009.000004.00606 36.10009.0000004.000606 45. 100009 Write the following numbers in figures:2. Seven hundred, and five tenths. 3. Seven, and one ten thousandth. 4. Seven hundred and five, and five millionths. 5. Two hundred and two, and two hundred thousandths. 6. Twenty-five million, and twenty-five ten millionths. 7. Three hundred and forty thousand, and three tenths. 8. Sixty-five thousand, and sixty-five ten thousandths. 9. Forty-four million, and forty-four ten millionths. 10. Five hundred thousand, and five hundred thousandths. 11. Sixty-four million, and sixty-four millionths. REDUCTION OF DECIMALS. 113. To reduce a common fraction to a decimal, - RULE. Annex ciphers to the numerator, and dividei by the denominator. OBS. The number of decimal places in the quotient will be equal to the number of ciphers annexed. When there are not as many, make the number equal by prefixing ciphers. Recite the rule for reducing a common fraction to a decimal. G 7 1 REDUCTION OF DECIMALS. 12. Reduce ~ to a decimal. 8 ) 3.000.375 As three ciphers were annexed to the numerator, three figures must be pointed off for decimals ill the quotient. This rule depends upon the principle stated in ART. 39, that if a number be multiplied and divided by the same number, its value remains unchanged. 13. Reduce the following fractions to decimals:i i 3'7~ z1,fi 1s 1 4 i T7 5 114. Any common fraction can be expressed exacty in decimals whose denominator has no other factor but 2 and 5. If the denominator contains other factors, it cannot be expressed exactly in decimals. 1 15. To reduce a decimal fraction to a common fraction, - RULE. WTrite the denominator of the decimal under the numerator, and then reduce it to its lowest terms. 14. Reduce.625 to a common fraction. 652 5 The denominator of.625 is 1000. (ART. 102.) -6s What is the rule for reducing a decimal to a common fraction? ADDITION OF DECIMALS. to reduced to its lowest terms, (ART. 85,) gives -E for a common fraction. EXAMPLES. 15. Reduce.125 to a common fraction. 16. Reduce.725 to a common fraction. 17. Reduce.925 to a common fraction. 18. Reduce.1125 to a common fraction. 19. Reduce.1265 to a common fraction. 20. Reduce.1475 to a common fraction. 21. Reduce.0045 to a common fraction. 22. Reduce.0208 to a common fiaction. 23. Reduce.0046 to a common fraction. ADDITION OF DECIMALS. 1 16. RULE. Write the nuzrmbers of the same order under each other, so that the decimal points shall be in the same vertical column. Add as in whole numbers, and point of in the sum from the right as many places for decimals as are equal to the greatest number of decimal places in any of the given numbers. PROOF. Addition of decimals may be proved in the same manner as addition of whole numbers. The principle of this rule is the same as that in the addition of whole numbers, (ARTS. 9, 1.7.) 24. Add together 3.014, 3014, 30.14,.003014, and.04045. 3.014 3014. 30.14.003014.04045 3047.197464 Wrrite the numbers of the same order under each What is the rule for the addition of decimals? 76 ADDITION OF DECIMALS. other, units under units, tens under tens, and tenths under tenths, hundredths under hundredths, &c., and add as in whole numbers, and point off from the right for decimals as many figures as are equal to the greatest number of decimal places in any of the given numbers. The greatest number of decimals in the above example is six; six figures must therefore be pointed off for decimals. OBS. 1. The decimal point will always be exactly under the decimal points in the given numbers. OBs. 2. The learner must be very careful in placing the decimal points according to the rule. EXAMPLES. 25. Add 4.032, 64.5010, 96.081, 310.018, 1.0012. 26. Add 63.03036, 73.46030,.090345, 41.23101, 1.0109. 27. Add 340.14561, 84.960,.759112,.000012, 2.0345. 28. Add 9.03456, 1.23456, 12.34567, 123.4567, 1234.567. 29. Add 85.05376, 5.45405, 54.04345, 540.4345, 7.00034. 30. 46.13455+9.73456+.0009345+875. +34.5 —? 31. 9671.03+5.05674+8.7561+750.12+87.34=? 32. 1.45610+67563.1+2.31267+91.234+.679-? 33. Add together 1 tenth, I hundredth, 1 thousandth, and 1 tell thousandth. 34. What is the sum of 5 hundredths, 5 ten thousandths, and 5 millionths? 35. What is the sum of 95 millionths, 1 hundred thousandth, and 1 tenth? 36. What is the sum of 784 thousandths, 347 millionths, 75 ten thousandths, and 99 hundredths? 37. What is the sum of 465. 78 hundred thousandths, 9 millionths, 99 ten thousandths, and 9 tenths? 38. What is the sum of 475, 9 ten thousandths, 83 hundred thousandths, and 9 tenths? MULTIPLICATION OF DECIMALS., 7 MULTIPLICATION OF DECIMALS. 117. RULE. Multiply as in whole numbers, and point off from the right qof the product as many figures as there are decimal places in both factors. PRoOF. Mu'ltiplication of decimals may be proved in the same manner as- multiplication of whole numbers, or by changing them to the form of common fractions. OBs. If there be not as many decimal places in the product as in both factors, make the number equal by prefixing ciphers. Multiply.0075 by-.0025..0075.0025 375 T w X TfWTWU W'''n' 150.00001875 As there are eight decimal places in both factors, and only four in the product, four ciphers must therefore be prefixed to the product.'The reason of this rule will be evident by changing, the decimal multiplicand and multiplier to the form of a common fraction, and then multiplying. 1 18. When the multiplier is a whole number, the multiplicand is taken as many times as there are units in the multiplier; but when the multiplier is a fraction, only such parts of the multiplicand' are taken as are indicated by the multiplier. When the multiplier is less than a unit, the product will be less than the multiplicand. What is the rule for the multiplication of decimals? When the multiplier is less than a unit, what is the product? GE ,83 SUBTRACTION OF DECIMALS. EXAMPLES. Multiply the following decimals:39. 45.601 x 7.456. 46. 67.456 X 1.245. 40. 31.735X9.735. 47. 96.314X2.103. 41. 41.244 x 1.642. 48. 814.21X 36.142. 42. 784.67 X 9.641. 49. 204.101 X 8.9614. 43. 46.043 X.0009. 50. 56.421 X 96.463. 44. 966.43 X.0061. 51.,42.001 X.00234. 45. 56.300 X.0312. 52. 84.241. X.00085. 53. Multiply 2 hundredths by 9 millionths. 54. Multiply 44 thousandths by 4 ten thousandths. 55. Multiply I ten millionth by 9 hundred thousandths. 56. Multiply 9999 by 9 hundred millionths. 57. Multiply 6 hundred millionths by 9 millionths. SUBTRACTION OF DECIMALS. 1 19. RULE. Write the less number under the greater, units under units, tenths under tenths, ~'c., so that the decimal points shall be exactly under each other. Subtract as in whole numbers, and point o/f from the right of the remainder as mnany places for decimals as are equal to the greatest number of places in either of the given numbers. PRoor. Subtraction of decimals may be proved itn the same manner as subtraction of whole numbers. The principle of this rule is the same as that in subtraction of whole numbers. 58. From 961.345 take 2.456789. 961.345000 2. 456789 958.888211 In this example three ciphers are annexed to the What is the rule for the subtraction of decimals? DIVISION OF DECIMALS. 79 minuend, to make the number of decimals equal to the number in the subtrahend, which does not change'the value of the minuend, (ART. 108.) EXAMPLES, 59. From 103.013 take 95.0134. 60. From 96.401 take 65.12034. 61. From 965.14 take 9.45614. 62. 641.34- 56.345. 65. 86.67- 9.8675. 63. 7561.2 9.6456. 66. 101.1- 90.9014. 64.:961.62-54.645. 67. 970.2 -84.3456. 68. From I take 1 millionth. 69. From 9999 take 9 hundred thousandths. 70. From 5 take 555 ten millionths. 71. From 222 take 22 hundred thousandths. 72. From 99 and 9 hundredths take 9 ten thousandths. 73. From 1 million take I hundred thousandth. DIVISION OF DECIMALS. 120. RULE. Divide as in whole numbers, acnd point off from the right of the quotient as many fig.ures for decimals as thle decimal places in the dividendc exceed those in the divisor; and if there be not as many, make the number equal by prefixing ciphers to the quotient. PROOF. Division of decimals may be proved in the same manner as division of whole numbers, or by commoQn fractions. OBS. 1. When the decimal places in the divisor and dividend are equal, the quotient will be a whole number. OBs. 2. When there are not as many decimal places in the dividend as in the divisor, ciphers may be annexed, and the division continned indefinitely. The ciphers thus annexed must be considered as decimal places. What is the rule- for the division of decimals? 430 DIVISION OF DECIMALS. Os. 2. i Unless great accuracy is required it is not necessary to have more than five places of decimals in the quotient. 74. Divide:000288072 by 3.6. 3.6 ).000288072 (.00008002 288 072 72 Since the dividend has nine decimal places, and the divisor but one, the quotient must have eight decimal places; four ciphers must therefore be prefixed to the quotient. Oss. When there is a remainder after division, the sign + should be annexed to the quotient, to denote that the division may be continued farther. EXAMPLES. Divide the following decimals - 75. 16.440 — 85. 83. 845.6701 3.02. 76. 184;20 9.6. 84. 9.,781234,.246. 77. 2345.6 —.54. 85. 45.67801 ~.008. 78. 3.6751+.04. 86. 56789.1.4.1 01 79..0456-.09. 87. 4.56.407.845. 80. 7645 90. 88. 4702.41~ 24.1. 81. 47.588.4. 89. 57.3745 —3.61. 82. 674.5 —.62. 90. 8.74562 —,096. 91. Divide one hundred and twenty-five, and nine ten thousandths, by six, and fifty-four ten thousandths. 92. Divide four hundred and eleven, and seven hundred and six millionths, by fifty-five, and ninety-three ten thousandths. 93. Divide seven million and one hundred and ten thousand, and ninety-four millionths, by eight hundred and forty-five ten thousandths. 94. Divide two hundred and twenty-four, and nine ten millionths, by three hundred and twenty hundred anilliolnths. PRACTICAL QUESTIONS. 8 PRACTICAL QUESTIONS. 95. What is the sum of 25.104, 25104,.6456, 45.52, 67.84, 96045.1,.8456, 75.104? 96. What is the sum of 4.6789, 567801.1, 5.0014, 375.27, 640.36, 8457.2? 97. What is the suni of 1 tenth, 1 hundredth, 1 thousandth, 1 ten thousandth, 1 hundred thousandth and 1 ten millionth? 98. Multiply six tenths by seven hundred ten thousandths. 99. Multiply four hundred, and four ten thousandths by ninety-seven millionths. 100. What cost nine tenths of a gallon of molasses, at three tenths of a dollar a gallon? 101. What cost 75.4 pounds of sugar, at.08 of a dollar a pound? 102. What is the difference between nine ten thousandths and nine hundred thousandths? 103. What is the difference between 444 thousandths and 4 one hu. ndred millionths? 104. What is the difference between 9 and.8, and 6 and.08? 105. In one rod there are 16.5 feet. How many rods are there in 4:78964.54 feet-? 106. Divide two hundred and fifty-six ten thousandths by four millionths. 107. If a pound of flour cost.05 of a dollar, how many pounds can be bought for 9650 dollars? 108. What would 1 bushel of oats cost, if 69.5 bushels cost 30.45 dollars:? 109. How much hay can be purchased for 960.6 dollars, if 1 ton cost 12.8 dollars? 110. How many barrels of flour can be bought for 750.5 dollars, at 4.25 dollars a barrel? 111. How mlany gallons -of molasses, at 33.5 cents per gallon, can be bought for 75.5 dollars? 6 82 CONTRACTIONS IN CONTRACTIONS IN MULTIPLICATION AND DIVISION. 121. When the multiplier or divisor is an exact measure of 10, 100, 1000, &c., it may be changed to the form of a common fraction before multiplying or dividing, as in-the following table - 3 —-:1 61 —=-Q 3163 1060 5= — 3 132 25 — 4 ~10,252100 3313 —-~-~iJ -a --- 1220 To multiply by any of the numbers in the preceding table,RULE. Annex as many ciphers to the multiplicand as there are ciphers in,the numerator of the fraction, and divide by the, denominator. EXAMPLES. l. Multiply 4563 by 3t. 6. Multiply 7348 by 61. 2. Multiply 8844 by 81-. 7. Multiply 8416 by 121. 3. Multiply 9366 by 162. 8. Multiply 9264 by 5. 4. Multiply 6372 by 33k4. 9. Multiply 6428 by 25. 5. Multiply 8193 by 333I. 10. Multiply 9416 by 125. 123. When one part of the multiplier is a multiple of the other, the process of multiplication may be shortened by the following rule:RULE. Multiply by the unit figure, and thent multiply this product by that number which denotes the number of times that this figure is contained in the two left hand figures of the mnultiplier. What is the rule for multiplying when the multiplier measures 10, 100, 1000, &c.? -low may the process be shortened, when one part of the multiplier is a multiple of the other? MULTIPLICATION AND DIVISION. 83 11. Multiply 46564 by 243. 46564 243 1.39692 3 times the multiplicand. 1117536 240 times the multiplicand. 11315052 Multiply first by the unit figure, 3. Then, as 24, the remaining figures in the multiplier, is a multiple of 3, multiply the first product, 139692, by 8, which is 11-17536. The unit figure of this product being written in the place of tens, nakes the whole of this last product the same as 240 times the multiplicand. The sum of the partial products is 243 times the multiplicand. EXAMPLES. 12. Multiply 428677 by 328. 13. Multiply 567894 by 427. 14. Multiply 678969 by 369. 15. Multiply 456784 by 728. 16. Multiply 734567 by 639. 124. To divide by any of the numbers in the preceding table - RULE. Multiply by the denominator of the fraction, and point off from, the right of the quotient for a remainder as many figures as there are ciphers in the numerator. EXAMPLES. 17. Divide 4675 by 3-. 22. Divide 8796 by 61. 18. Divide 5789 by 8. 23. Divide 9234 by 12~. 19. Divide 6542 by 162. 24. Divide 9541 by 5. 20. Divide 7345 by 33-. 25. Divide 9645 by 25. 21. Divide 8456 by 333~. 26. Divide 9846 by 125. WVhat is the rule for dividing when the divisor m easures 10, &c.? 84 FEDERAL MONEY. FEDERAL MONEY. SECTION VIII. 125.- THE currency of the United States is styled federal money. 126. The denominations of federal money are eagles, dollars, dimes, cents, and mills. 127. The coins are made, of gold, silver, and copper. 12 S. The gold coins are the double eagle, the eagle, the half-eagle, the quarter-eagle, and the dollar. 129. The silver coins are the dollar, the half-dollar, the quarter-dollar, the dime, and the half-dime. 130. The copper coins are the cent and the halfcent. OBs. Coin is a piece of metal stamped with certain figures or characters by the authoriry of government. TABLE OF FEDERAL MONEY. 10 mills make 1 cent. 10 cents' 1 dime. 10 dimes " 1 dollar, (f.) 10 dollars " 1 eagle. It is now almost the universal practice in the United States to keep accounts in dollars, cents, and mills. OBS. The mill is seldom regarded, except when great accuracy is required. 131. As the dollar is considered the unit or whole number, and cents and mills decimals, federal money may be added, multiplied, subtracted, and divided, in the same manner as decimals. What is the currency of the United States styled? What are its denominations? What are the coins? Of what is each made? REDUCTION OF FEDERAL MONEY. 85 REDUCTION OF FEDERAL MONEY. 132. RULE. Dollars may be reduced to cents by annexing TWO ciphers. Dollars may be reduced to mills by annexing THREE ciphers. Cents may be reduced to mills by annexing ONE cipher. Dollars and cents may be reduced to cents by removing the decimal point to the left of the dollars. Cents may be changed to dollars by pointing off the TWO right hand figures. Mills may be changed to dollars by pointing off the THREE right hand figures. Mills may be changed to cents by pointing off ONE figure on the right. 133. As pointing off two figures is the same as dividing by 100, and pointing off one figure the same as dividing by 10, the figures pointed off will be the same as the dividend, (ART. 31.) EXAMiPLES. 1. Reduce $75 to cents. 1 3. Reduce $110 to mills. 2. Reduce $96 to cents. 4. Reduce $871 to mills. 5. Change 144 cents to dollars. 6. Change 1605 cents to dollars. 7. Change 6456 mills to dollars. 8. Change 1790 mills to dollars. ADDITION OF FEDERAL MONEY. 134. RULE. Write the numbers under each other, dollars under dollars, cents under cents, Arc., and add as in whole numnbers. PROOF. The same as in addition of whole numbers. How are dollars reduced to cents? How to mills? How are cents reduced to mills? How are cents changed to dollars? How are mills changed to dollars? What is the rule for addition of federal money? 86 MULTIPLICATION OF FEDERAL MONEY. Oss. As cents occupy the places of tenths and hundredths, when there is but one figure expressing cents, a cipher must be prefixed to it. EXAMPLES. 9. What is the sum of $84 and 15 cts., $97 and 9 cts., $137, 12 cts., and 9 mills? 10. What is the sum of $19; 6 cts., and 4 mills; $76, 7 cts., and 8 mills; $275, 10 cts., and 1 mill? 11. What is the sum of $1756, 4 cts., and 9 mills; $196, 7 cts., and 8 mills; $550, 5 cts., and 5 mills; $730, 17 cts., and 7 mills? 12. What is the sum of $1375, 10 cts., and 1 mill; $13, 75 cts., and 3 mills; $7, 3 cts., and 6 mills;. $14, 8 cts., and 2 mills? MULTIPLICATION OF FEDERAL MONEY. 135. RULE. Multiply as in whole numbers, and point off from the right of the product as in mnltiplication of decimal fractions. PRooF. The same as in multiplication of whole numbers. EXAMPLES. 13. What will 75 yards of cloth cost at $5.50 per yard? 14. What will 156 barrels of apples cost at $2.375 per barrel? 15. What will 344 pounds of cheese amount to at 61 cts. per pound? 16. What cost 436 bushels of wheat at $11 per bushel? 17. What cost 5750 feet of land at 73 cents per foot? 18. What cost 76 bushels of potatoes at 33~1 cents per bushel? 19. What cost 86 acres of land at $45.375 per acre? What is the rule for the multiplication of federal money? SUBTRACTION OF FEDERAL MONEY. 87 SUBTRACTION OF FEDERAL MIONEY. 136. RULE. Write the numbers under each other, dollars under dollars, cents under cents, Arc., and subtract as in whole numbers. PROOF. The same as in subtraction of whole numbers. EXAMPLES. 20. FroIn $96.125 take $75.374. 21. From $116.013 take 99.101. 22. From $137, 6 cts., take $39, 7 cts., and 4 mills. 23. From $561, 9 cts., take $396, 14 cts., and 9 mills. 24. From $856, 10 cts., take $703, 9 cts., and 7 m-ills. 25. From $937, 5 cts., take $364, 15 cts., and 5 mills. DIVISION OF FEDERAL MONEY. 1370 RULE. Divide as in whole numbers, and point off from the. right of the'quotient, as in division of decimal fractions. PRooF. The same as in division of whole numbers. EXAMPLES. 26. If 27 pounds of sugar cost $1.635, what will 1 pound cost? 27. If 56 barrels of flour cost $275.75, what will 1 barrel cost? 28. If 3242 feet of land cost $2825.75, what xvill 1 foot cost? 29. At $0.40 per yard, how many yards of calico can be bought for $15.56? What is the rule for the subtraction of federal money? What is the rule for division? 88 PRACTICAL QUESTIONS. 30. At $0.37~ per yard, how many yards of gingham can be bought for $25.75? 31. How many times is $0.06 contained in $240.36? 32. How many times is $0.05 contained in, $425.75? 33. How many times is $0.0875 contained in $58.645 PRACTICAL QUESTIONS. 34. What is the sum of twenty-five dollars and nine cents; thirty-seven dollars, seven cents, and five mills.; fifty-eight dollars, ten cents, and three mills? 35. What is the sum of one hundred and tell dollars, one cent, and nine mills; four hundred and forty dollars, six cents, and eight mills? 36. What cost 25.5 pounds of sugar at 121 cts. per pound? 37. What cost 441 pounds of tea at 56~ cts. per pound? 38. What cost 85~ pounds of beef at 7- cts. per pound? 39. What cost 4567 feet of land at 8A cts. per foot? 40. What is the difference between seven hundred and forty dollars, sevuen cents, and nine mills, and three hundred and thirty dollars, nine cents, and seven mills? 41. From one hundred dollars take nine cents and nine mills. 42. From one thousand dollars and ten cents take ninety dollars, nine cents, and nine mills. 43. How much must be added to four dollars, four cents, and four mills, to make five hundred dollars? 44. A man bought a farm for $7560.87, and sold it for $8050.50. How much did he gain? 45. If a man pay $375.50 for 85 barrelh of flour, how much does he give a barrel? 46. If a man pay $42.50 for 271 pounds of tea, how much does he give a pound? 47. If $425.50 were paid for 4651 bushels of corn, what would one bushel cost? BILLS OF PARCELS. S9 BILLS OF PARCELS. 138. A bill of parcels is an account given by the seller to the buyer of the articles purchased, with the price of each. 48. ~ Boston, August 10, 1850. Mr. James Blackington Bought of William Jones 15 yards of broadcloth, at $3.75 per yard, 12~ yards of cassimere, at $2.371 per yard, 7- yards of satin, at $4.12~ per yard, 5a yards of linen cambric, at $0.371 per yard. 49. Boston, Sept. 1, 1850. Mr. George Bright Bought of J. D. Williams & Co. 4 boxes of sugar, of 240~ pounds each, at 6cts. per pound, 61 bags of coffee, of 110k pounds each, at 101cents per pound, 15 casks of rice, of 2971 pounds each, at 31 cts. per pound, 24 chests of tea, of 75~ pounds each, at 671 cts. per pound, 19 barrels of Genesee flour, at $57 per barrel, 50. Boston, August 20, 1850. Mr. Benjamin Southey Bought of Wm. J. Reynolds & Co.. 40 Webster's Dictionary, at $4.75, 75 Worcester's History, at $0.62~, 450 Colburn's Arithmetic, at $0.12., 84 Smellie's Philosophy, at $0.67., 230 Bibles, at $0.87~, Hi @' 90) DENOMINATE NUMBERS. DENOlIMINATE NUMBERS. SECTION IX. 139. DENOMINATE NUMBERS express things of different kinds, or denominations, as 5 pounds, 6 shillitlgs, 7 pence, &c..TABLES OF MONEY, WEIGHTS, AND MEASURES. English Money. 4 farthings (qr.) 1 penny,.. d. t2 pence - 1 shilling,. s. 20 shillings _ pound,.. ~. 21. shillings 1 guinea. qr. d. 4- - =! s. 48 - 12 1 960 -240 -- 20 I 1 Ors. A farthing is often expressed by { d., 2 farthings by. d., 3 farthings by.a d. Troy Weig ht. 24 grains (gr.) I pennyweight, dwt. 20 pennyweights -- 1 ounce,.. oz. 12 ounces I- pound,.. lb. gr. dwt. 24 --- 1 OZ. 480 20 = 1 lb. 5760 240 - 12 I- 1 OBs. t. This weight is used in weighing gold, silver, and precious stones. It is also used in philosophical experiments. OnS. 2. Troy weight may easily be changed into Avoirdupois, and Avoirdupois into Troy. 1 lb. Troy - 14 lb. Avoirdupois. 1 oz. Troy 192 oz; Avoirdupois. DENOMINATE NUMBERS. 91 Avoirdupois Weight. 16 drams (dr.) 1 ounce,... oz. 16 ounces - 1 pound,.. lb. 28 pounds 1 quarter,.. qr. 4 quarters - 1 hundred weight, cwt. 20 hundred weight 1 ton,.... T. dr. oz. 16 -- 1 - lb. 256 - 16 -- 1 qr. 7168 - 448 - 28 - 1 cwt. 28672 - 1792 - 112 - 4 - I T. 573440 - 35840 -2240 - 80 -20 = 1 OBS. By this weight all commodities are weighed except gold, silver, and precious stonies. Apothecariies' Weight. 20 grains (gr.) 1 scruple, sc., or D. 3 scruples - 1 dram, dr., or 7. 8 drams- - 1 ounce, oz., or 3. 12 ounces - 1 pound, m. 20 -. 60 3 3 1. 80 - 24 8 - 1.. 5760 - 288 - 96 - 12 - 1 OBs. By this weight apothecaries mix their meclicines. Cloth M.Zeasure. 2- inches (in.) - I nail,.. na. 4 nails - I quarter of a yard, qr. 4 quarters 1 yard,. yd. in. na. 2j, - qr. 9 - 4 -- 1 ydl. 36 - 16 - 4 - 1 Oas. The Flemish ell is 3 quarters, the English ell is 5 quarters, and the French ell is 6 quarters. 92 DENOMINATE NUMBERS. Lonzg Measure. 12 inches (in.) - 1 foot,... ft. 3 feet -- 1 yard,.. yd. 6 feet = 1 fathom,. fath. 51 yards, or 161 feet, 1 rod,... r. 40 rods = I furlong,.. fur. 8 furlongs 1 mile,... m. 3 miles = I league,.. lea. in. ft. 12 - 1 yd. 36 - 3 1 198 = 161z 5 — 1 fur. 7920 - 660 - 220 40 1 m. 63360 - 5280 - 1760 =- 320 - 8 - lea. 190080 15840 - 5280 - 960 24 - 3 - 1 OBs. 1. Long measure is used in measuring distances, &c. The rod is sometimes called perch, or pole. OBs. 2. The circumference of the earth is measured by degrees of latitude. 60 geographical miles, or 69- English miles, form 1 degree, and 360 degrees form the circumference. Square ]Measure. 144 square inches (sq. in.)) 1 square foot, sq. ft. 9 square feet - 1 square yard, sq. yd. 30} sq. yds., 272k sq. ft., 1 square rod, sq. r. 40 square rods - 1 rood,.. R. 4 roods 1 acre,.. ac. 640 acres 1 square mile, sq. nm. sq. in. sq. ft. 144 - 1 sq. yd. 1296 9 1 sq. r. 39204 - 272 - 30 - 1 R. 1568160 - 10890 - 1210 - 40 = - a1. 6272640 - 43560 - 4840 = 160 - 4 - 1 OBs. Square measure is used in measuring surfaces. DENOMINATE NUMBERS. 93 Cubic Measure. 1728 cubic inches (cub. in.) 1 cubic foot, cub. ft. 27 cubic feet - 1 cubic yard, cub. yd. 16 cubic feet make one foot of wood, and 8 feet of wood one cord; or 128 cubic feet make one cord. 40 feet of round and 50 feet of hewn timber were formerly considered a ton, but timber is now chiefly sold by board measure. cub. in. cub. ft. 1728 -- 1 cub. yd. 46656 - 27 1 OBs. Cubic measure is used in measuring solid bodies. Liquid Measure. 4 gills (gi.) 1= pint,.. pt. 2 pints - 1 quart,... qt. 4 quarts - 1 gallonl,.. gal. 31. gallons = 1 barrel,... bar. 63 gallons = 1 hogshead,. hhd. gi. pt. 4 - 1 qt. 8 - 2 - 1 gal. 32 -- 8 - 4 - 1 bar. 1008 - 252 - 126 - 311 - 1 laid. 2016 - 504 - 252 63 =2 - OMs. 1. The English wine gallon contains 231 cubic inches. In some places, milk and malt liquors are sold by a measure which contains 282 cubic inches in a gallon. OBs. 2. Only a few articles are bought and sold by the barrel. Hogsheads are used only in estimating the contents of cisterns, wells, and other large bodies of water. The terms pipe and butt are never used as exact measures of quantity, but simply designate casks of a certain shape or form. OBs. 3. A gallon is divided by apothecaries into pints, fluid ounces, fluid drams, and minims. A pint of water is estimated at a pound. 9 DENOMINATE NUMBERS. Dry Measure. 2 pints (pt.,) = 1 quart,.. qt. 8 quarts - 1 peck,...pk. 4 pecks = 1 bushel,.-. bu. 36 bushels - 1 chaldron,... ch. pt. qt. 2 - 1 pk. 16 8 - bu. 64 32 -- 4 - 1 ch 0304 1152 144 36 1 Tiime. 60 seconds (sec.) 1 minute,. min. 60 minutes = I hour,.. hr. 24 hours - 1 day,.. day. 7 days = 1 week,.. wk. sec. min. 60 - 1 hr. 3600 - 60 - I day. 86400 - 1440 - 24 - 1 wk. 604800 - 10080 - 168 - 7 - 1 OlBS. Leap years are those which are exactly divisible by 4. Thus 1840, 1844, &c., are leap years. A common month consists of 4 weeks. Circular Measure. 60 seconds (//) -= minute,.. 60 minutes - 1 degree,. 0 30 degrees - 1 sign,.. s. 12 signs, or 360 degrees,- 1 circle,.. cir. 60- 1. 3600 - 60 - 1 108000 -- 1800 30 - 1 cr. 1296000 - 21600 - 360 - 12 REDUCTION OF DENOMINATE NUMBERS. 95 REDUCTION OF DENOMINATE NUMBERS. 140. Reduction is the process of changing the denomination of any quantity without changing its value. Thus 6 pounds being changed into shillings, becomes 120 shillings. This change is produced by multiplying 6 by 20, the number of shillings equal to a pound. 60 inches being changed to feet, becomes 5 feet. This change is produced- by dividing 60 inches by 12, the'number of inches equal to a foot. 141.O To reduce denominate numbers from a higher to a lower denomination, - RULE. Multiply the highest denomination by that 1nunmber which denotes how many units of the next lower denomination make ONE unit of the higher, and to the product add the next lower denomination. Proceed in this manner with each denomaination. 1. Reduce ~14 12 s. 9 d. 2 qr. to farthings. ~. s. d. qr. 14 12 9 2 20 292 12 593 292 3513 4 14054 14 pounds are reduced to shillings by multiplying by 20, because 20 shillings make one pound, and adding the 12 shillings makes 292 shillings. This being What is reduction:? What is the rule for reducing denominate numbers from a higher to a lower denomination? 96 REDUCTION OF DENOMINATE NUMBERS. multiplied by 12, because 12 pence make one shilling, and adding the 9 pence, makes 351.3 pence. This being multiplied by 4, because 4 farthings make one penny, and adding the 2 farthings, makes 14054 farthings. 142. To reduce denominate numbers from a lower to a higher denomination,RULE. Divide the given denomination by that number which denotes how many units of this denomination make one of the next higher. Proceed in this manner with each denomination. 2. Reduce 14054 farthings to pounds. 4 ) 14054 12 ) 3513, 2 retm. 20 ) 292, 9 rem. 14, 12 rem. ~14, 12s. 9d. 2qr. Divide the farthings by 4, because 4 farthings make one penny. The quotient will be 3513 pence and 2 farthings over. Divide the pence by 12, because 12 pence make one shilling. The quotient will be 292 shillings and 9 pence over. Divide the shillings by 20, because 20 shillings make one pound. The quotient will be 14 pounds and 12 shillings over. OBs. 1. Denominate numbers are changed from a higher to a lower by multiplication, from a lower to a higher by division. The forrier has been called reduction descending, the latter, reduction ascending. OBs. 2. The addition and subtraction of denominate numbers is the same as the addition and subtraction of simple numbers, except in the subdivisions of the unit. English Money. 3. In ~20, 15 s. 9 d. 3 qr. how many farthings? 4. In ~40, 10 s. 8 d. 2 qr. how many farthings? What is the rule for reducing denominate numbers from a lower to a higher denomination? REDUCTION OF DENOMINATE NUMBERS. 97 5. In ~240, 19s. how many pence? 6. In 144560 farthings how many pounds? 7. In 176842 farthings how many pounds? 8. In 6845 pence how many pounds? Troy Weight. 9. In 10 lbs. 8 oz. how many grains? 10. In 9 lbs. 6 oz. 18 dwt. how many grains? 11. In 42645 grains how many pounds? 12. In 6456 pennyweights how many pounds? Avoirdupois Weight. 13. In 4 qrs. 26 lbs. how many ounces? 14, In 2 cwt. 3 qrs. 20 lbs. how many drams? 15. In 5 tons, 2 cwt. 3 qrs. 6 lbs. how many drams? 16. In 7500 ounces how many hundred weight? 17. In 6456780 drams how many tons? Cloth Measure. 18. In 45 yards how many nails? 19. In 65 yards how many inches? 20. In 45 yards how imany inches? 21. In 244 inches how many yards? 22. In 614 inches how many yards? Long 11 easure. 23. In 7 furlongs, 30 rods, 5 yards, how many feet? 24. In 2 miles, 6 fiurlongs, 14 rods, 12 feet, how many inches? 25. In 4 miles, 7 furlongs, 20 rods, 16 feet, how many inches? 26. In 324560 inches how many miles? 27. In 456740 inches how many miles? 28. In 675670 inches how many miles? 29. In 360 degrees how many inches? I 7 98 REDUCTION OF DENOMINATE NUMBERS. Square Mreasure. 30. In 64 square rods and 20 square yards how many feet? 31. In.72 roods,. 30 square rods, 30 square yards, how many inches? 32. In 2- acres, 3 roods, 25 square rods, 5 square yards, how many inches? 33. In S8756704 square inches how many acies? 34. In 9567805 square inches how many acres? Cubic Measure. 35. In 56 cubic feet how many inches? 36. Ill 75 cubic feet how many inches? 37. In 14560 inches how many cubic feet? 38. In 234 cords of wood how many cubic feet? 39. In 27560 cubic feet of wood how many cords? Dry Measure. 40. In 24 bushels, 3 pecks, how many quarts? 41. In 29 bushels, 2 pecks, how many pints? 42. In 3694 pints how many bushels? 43. In 4675 pints how many pecks? Liquid Measure. 44. In 13 hhds. of wine how many pints? 45. In 42 hogsheads how many gills? 46. In 1345 pints how many hogsheads? 47. In 1464 gills how many hogsheads? Time. 48. In 24 days, 10 hours, and 25 minutes, how many seconds? 49. In 365 days, 6 hours, how many seconds? 50. In 40345600 seconds how many months? 51. In 784560 minutes how many weeks? ADDITION OF DENOMINATE NUMBERS. 99 Circular Measure. 52. In 50 degrees how many seconds? 53. In 6 signs, 10 degrees, how many seconds? 54. Ill 756700 seconds how many degrees? 55. In 967500 minutes how many signs? ADDITION OF DENOMINATE NUMBERS. 143. RULE. Write the numbers of the same denomination directly under each other. Add the ntmbers in the lowest denomination, and finzd, by division, how many units of the next higher denomination are contained in their sum. Write the remainder under the column or colunzns of the lowest denomination, and add the quotient to the column of the next higher denomination. Proceed thus to the end. PROOF. The same as in addition of simple numbers. 56. ~. s. d. qr. 20 14 10 2 16 12 8 3 14 16 7 3 19 18 11 2 72 3 2 2 The sum of the column of farthings is 10 farthings, which is 2 pence and 2 farthings. Write the 2 farthings underneath, and add the 2 pence to the next column, whose sum is 38 pence, which is 3 shillings and 2 pence. Write the 2 pence underneath, and add the 3 shillings to the next column, whose sum is 63 shillings, which is 3 pounds and 3 shillings. Write the 3'shillings underneath, and add the 3 pounds to the next column, whose sum is 72, which write underneath. What is the rule for the addition of denominate nu1nh1er.: 100 ADDITION OF DENOMINATE NUMBERS. EXAMPLES. English MIoney. 57. 58. 59. ~. s. d. qr. ~. s. d. qr. ~. s. d. qr. 16 15 10 3 36 19 11 3 34 13 10 2 21 12 9 2 75 15 10 2 59 14 11 1 -41 17 11 3 64 13 9 1 67 19 9 3 37 18 8 1 55 12 8 3 34 16 8 2 Troy Weight. 60. 61. 62. lb. oz. dwt. gr. lb. oz. dwt. gr. lb. oz. dwt. gr. 12 10 16 23 16 1015 18 25 10 16 22 13 9 18 20 15 9 19 17 27 11 13 20 14 11 17 19 17 8.18 12 30 9 14 16 20 10 15 17 19 11 16 21 42 8 12 15 Avoirdupois Weight. 63. 64. ton. cwt. qr. lb. oz. dr. ton. cwt. qr. lb. oz. dr. 8 15 3 24 14 12 4 12 2 16 13 14 7 16 2 21 15 13 5 17 1 14 12 15 9 18 1 20 12 10 9 16 3 10 11 12 6 13 3 19 11 14 8 13 I 12 15 10 Cloth Measure. 65. 66. 67. yd. qr. na. yd. qr. na. yd. qr. na. 17 3 2 24 2 3 35 2 3 18 2 3 27 1 2 34 3 1 19 1 2 25 3 1 39 2 2 20 3. 2 -34 2 3 40 1 2 ADDITION OF DENOMINATE NUMBERS. 101 Long ]Measure. 68. 69. m. fur. rd. yd. ft. in. m. fur. rd. yd. ft. in. 4 7 30 4 2 8 11 6 24 3 1 9 5 6 29 3 1 9 13 7 26 4 2 8 6 3 25 2 2 7 16 5 33 2 1 7 9 4 31 1 1 6 18 4 37 3 2 11 Square 3Measure. 70. 71. ac. R. sq. r. sq. yd. sq. ft. ac. R. sq. r. sq. yd. sq. ft. 40 3 27 25 8 56 3 20 8 8 37 2 25 - 22 7 64 2 16 7 7 45 1 29 19 6 75 1 15 6 5 54 2 30 18 5 84 2 14 5 6 Cubic Measure. 72. 73. ft. in. ft. in. 139 1425 116 1235 134' 1634 108 1475 135 1560 116 1573 238 1140 127. 1680 Liquid Mfleasure. 74. 75. hhd. gal. qt. pt. hhd. gal. qt. pt. 43 50 3 1 42 60 3 1 22 45 1 0 23 54 2 0 33 46 3 1 13 40 1 1 22 34 2 0 12 36 3 1 102 MULTIPLICATION OF DENOMINATE NUMBERS. Dry Measure. 76. 77. ch. bu. pck. qt. pt. ch. bt. pk. qt. pt 6 28 3 7 1 5 30 2 6 1 7 25 2 6 0 4 18 3 7 0 9 30 2 5 1 3 15 2 5 1 8 31 3 4 1 6 14 1 4 0 Time. 78. 79. y. d. h. m. s. y. d. h. m. a. 46 260 20 54 40 75 164 14 56 34 36 160 S1 44 36 80 175 16 42 37 74 214 17 38 27 77 164 17 37 48 65 196 16 24 17 64 178 15 53 27 Circular Measure. 80. 81. S. 0 i t! 0 1,, 24 26 54 48 46 24 55 49 16 25 46 37 35 27 57 44 14 24 42 59 37 22 35 42 34 27 16 35 32 20 54 26 MULTIPLICATION OF DENOMINATE NUMBERS. 1L44. RULE. Write the multiplier under the lowest denomination in the multiplicand. Multiply each denomination of the multiplicand, beginning with the lowest, and find how many units of the next higher denomination are contained in the product. Write underneath the remainder, and add the quotient to the product of the next higher denomination. Proceed thus to the end. What is the rule for the multiplication of denominate numbers? MULTIPLICATION OF DENOMINATE NUMBERS. 103 PRooF. The same as in multiplication of simple numbers. The principle of this rule is the same as that of multiplication of simple numbers. OBs. When the multiplier is a composite number, multiply by each factor of the multiplier in succession. 82. Multiply ~9 10 s. 8 d. 3 qr. by 8. ~. s. d. qr. 9 10 8 3 8 76 5 10 0 8 times 3 farthings are 24 farthings, which are 6 pence, and no remainder. 8 times 8 pence are 64, and 6 added to the product make 70 pence, which are 5 shillings and 10 pence. Write the 10 underneath, and add the 5 to the next product, which is 80, making 85 shillings, which are 4 pounds and 5 shillings. Write the 5 underneath, and add the 4 to the next product which is 72, making 76, which write underneath. EXAMPLES, 83, 84. 85. ~. s. d. qr. ~. s. d. qr. ~. s. d. qr. 4 5 4 2 8 7 11 3 9 16 11 1 2 3 4 86. 87. 88. a. s d. ~. S. d. ~. s. d. 12 17- 9 - 21 17 103 19 12 8t 6 8 9' 89. 90. 91. ~. S. d. ~. s. d. ~. s. d 987 19 11 9[5 17 81 966 13 9j 12- 83 42 104 SUBTRACTION OF DENOMINATE NUMBERS. 92. What is the price of 27 lbs. of tea, at 5 s. 61 d. per lb.? 93. What is the price of 47 lbs. of sugar, at 81d. per lb.? 94. What is the price of 63 tons of coal, at 13 s. 6 d. per ton? 95. What is the weight of 47 pieces of lead, each piece weighing 25 lbs. 6 oz. 12 dr.? 96. A farm consists of 9 fields, each 12 ac. 1 R. 32 rd. What is the extent of the farm? 97. What is the value of 576 lbs. of iron, at 3] d. per lb.? 98. What is *the value of 15 pairs of shoes, at 7 s. 6d. per pair? 99. How much sugar in 15 boxes, each box containing 5 cwt. 3 qr. 15 lb.? 100. How much wood in 16 piles, each pile containing 10 C. 7 cub. ft. 15 cub. in.? 101. How many yards in 16 pieces of cloth, each piece containing 10 yds. 2 qr. 3 na.? 102. How many gallons of molasses in 75 pipes., each pipe containing 120 gal. 3 qt. 2 pt.? 103. How many bushels in 120 bbls. of potatoes, each barrel containing 2 bu. 1 pl.? 104. If a man travel 30 rn. 7. fur. 35 rd. in 1 day, how far would he travel in 12 days? 105. If 1 acre produce 275 bu. 3 pk. of potatoes, how many bushels will 9 acres produce? 106. How many bushels of wheat in 84 sacks, each sack containing 3 bu. 2 pk. 1 qt.? SUBTRACTION OF DENOMINATE NUMBERS. 145. RULE. Write the less nutmber under the greater, so that numbers of the same denomination shall be directly iunder' each other. Begin to subtract with the What is the rule for the subtraction of denominate numbers? SUBTRACTION OF DENOMINATE NUMBERS. 105 lowest denomination, and take each number in the lower line fromn the numnber above it, and write the remainder underneath. If the numvber in the lower line be greater than the number above it, add to the qpper number as many units as mnake one of the next higher denomination, and then szubtract, and add one to the next lower line before subtracting. Proceed thus to the end. PROOF. The samaJe as in subtraction of simyle numbers. 107. From ~20, 14s. 9d. 2qr. take ~14, 15s. 10d. 3 qr. S. s. qr. 20 14 9 2 14 15' 10 3 5 18 10 3 Beginning with the lowest denomination, subtract. As 3 farthings cannot be taken froin 2, add to it 4 farthings, making 6; 3 from 6 leaves 3. Since 4 fatrthings Were added to the farthings in the upper line, 1 penny, its equal, must be added to the 10 pence in the lower line, making 11 pence. As 11 cannot be taken from the 9, 12 pence must be added to the 9, making 21 pence; II from 21 leaves 10. As 12 pence were added to the upper line, 1 shilling, its equal, must be added to'15 in the lower line, making 16 shillings. As 16 cannot be taken from 14, 20 shillings must be added, makling 34 shillings; 16 from 34 leaves 18. As 20 shlillings were added to the upper line, 1 pound, its equal,niust be added to the 14 in the lower line, ma'kiig 15; 15 friom 20 leaves 5. EXAMPLES. 108. What is the difference between ~3, 5s. 6d. and ~2, 3 s. 2d.? 109. What is the difference between ~25, 12 s. 9 d. and ~22, 13 s. 9 d.? 106 SUBTRACTION OF DENOMINATE NUMBERS. 110. A merchant cuts 5 yds. 3 qr. 2 na. of cloth from a piece containing 21 yds. 2 qr. 2 na. How many yards remained? 111. The latitude of the Cape of Good Hope is 330 56' 13" S., and that of Cape Horn 55~ 58' 40" S. What is the difference of latitude between the two places? 112. The latitude of Boston is 420 21' 23/" N.; that of New Orleans 29~ 8' 32" N. How much farther south is New Orleans than Boston? 113. The latitude of Paris is 4S~ 50' 13" N.; that of New York 40~ 42' 35," N. How many degrees farther north is Paris than New York? How many degrees farther north is Paris than Boston? 114. The latitude of Quebec is 46~ 49' 12". How many degrees farther north is Paris than Quebec? 115. The latitude of Charleston is 320 46' 33" N.; that of Cincinnati 390~ 5 54" N. How many degrees farther south is Charleston than Cincinnati? 116. The longitude of Boston is 71~ 4' 20/; that of the city of Mexico 990 5'. How many degrees farther west than Boston? 117. The longitude of Chicago is 870 30' 30". How many degrees farther west is Chicago than Boston? 118. The longitude of St. Louis is 90~ 15' 10". How many degrees farther west is St. Louis than Boston? 119. The latitude of Constantinople is 410 0' 16" N. How many degrees farther south is Constantinople than Boston? 120. The latitude of St. Augustine is, 29~ 48' 30" N. How many.degrees farther south is St. Augustine than Boston? 121. The latitude of Burlington, Vt., is 44~ 27'; that of Washington, D. C., 38~ 53' 33". How many degrees farther south is Washington than Burlington? DIVISION OF DENOMINATE NUMBERS. 107 DIVISION OF DENOMINATE NUMBERS. 146. RULE. Divide each denomination of the dividend, beginning at the left, by the divisor, and write the result in the quotient. Reduce the remainder, if there be any, to the next lower denorinInation, and add it to the number of the same denomination in the dividend. Continue the division, and write the result i n the quotient, as before. PROOF. The same as in division of simple numbers. Os.. 1.. When the divisor exceeds 12, and is a composite number, divide by each factor in succession, as in simple division. OBS. 2. Division of denominate numbers is precisely the same in principle as simple division. Reducing the remainder to the next lower denomination, and adding it to the next number, is the same as prefixing the remainder to the next figure. 122. Divide ~266, 12 s. 93 d. into 6 equal parts. ~. s. d. 6 ) 266 12 93 44 8 9, 3 qr. rem. 6 in 26 four times and 2 over. Write the 4 in the quotient, and prefix the 2 to the next figure, making 26. 6 in 26 four times and 2 over. Write the 4 in the quotient, and reduce the 2 pounds to shillings, which are equal to 40 shillings, and add them to the next number, 12 shillings, making 52 shillings. 6 in 52 eight times and 4 remainder, which reduced to pence, and added to the 9, make 57 pence. 6 in 57 nine times and 3 over, which reduced to farthitngs, and added to the 3, make 15 farthings. 6 in 15 two times and 3 over. What is the rule for the division of denominate numbers? 108 DIVISION OF DENOMINATE NUMBERS. EXAMPLES. 123. Divide ~329, 17 s. 6 d. 2 qr. by 14. 124.'Divide ~735, 13 s. 7 d. 3 qr. by 25. 125. Divide ~964, 2s. 1 d. 1 qr. by 27. 126. Divide 324 cwt. 2 qr. 3 lb. by 93. 127. Divide 312 cwt. 1 qr. 21 lb. by 63. 128. Divide 67 tons, 13 cwt. 2 qr. by 97. 129. Divide 365 days,5 h. 48 m. 51 s. by 12. 130. If 36 cwt. of cheese cost ~80, 2 s., how much will 1 cwt. cost? 131. If a gentleman's income be ~548 a year, how much is his income for a month? 132. If 19 parcels of tea contain 332 lb. 8 oz., what is the weight of one parcel? 133. If 9 cows eat 21 tons; 7 cwt. 2 qr. 12 lb. of hay in a year, bho much will 1 cow eat in the same time? 134. If 42 acres produce 567 bu. 3 pk. of oats, how much will 1 acre produce? 135. If i lb. of silver be worth $15.50, what would be the weight of $500,000 in silver? 136. If 1 ounce of pure gold be worth $18.60, what would be the weight of 1 million dollars in gold? 137. A farmer puts 1350 bushels of apples into 600 barrels. How many does he put in each barrel? 138. How many spoons, weighing 18 dwt. each, can be made of 5 lb. 9 oz. 12 dwt. of silver? 139. How many bottles, containing 1 pt. 2 gi. each, wvill hold 32 gal. 2 qt. of cider? 140. At 1 s. 6 d. per gal., how many gallons of io-,asses can be bought for ~35, 18s.? 141. If 39 hboxes of oranges cost ~202, 12s. 6d,, what was the price per box? 142. If a man travel 21 m. 5 fur. in a day, how many days will it take him to travel 207 m. 3 fur.; PRACTICAL QUESTIONS. 109 PRACTICAL QUESTIONS. 143. How much would 16 boxes of sugar cost, at 6- cents per lb., if each box contain 4 cwt. 3 qr. 18 lb.? 144. If a pile of wood be 140 ft. long, 3 ft. 6 in. wide, how high must it be to contain 18 cords? 145. If a gentleman's annual income be ~1(000, and his daily expenses ~1, 17s. 31 d., how much does he save in 9 years?. 146. If a gentleman receive ~1, 9 s. 9 d. per week, what is his annual income? 147. If the distance between London and Edinburgh be 389 m. 6 fur. 20 rods, how long would a person be in walking from one place to the other, at the rate of 27 m. 6 fur. 30 rods per day? 148. How many yards of cloth are there in 21 pieces, each piece containing 15 yd. 3 qr. 2 ia.? 149. What will 25 doz. of knives and forks cost, at Ils. 6d. per doz.? 150. If 36 doz. of knives and forks cost ~9, 10s. 9 d., what costs 1 doz.? 151. If 45 yds. of superfine broadcloth cost ~43, 0 s., how m1uch is that a yard? 152. If 27 pairs of silk hose cost ~8, Os. 10~ d.7 how rmuch is that per pair? 153. If 11 cwt, of sugar cost ~22, 14s. Sd., what will I cwt. cost? 154. How long will a person be in saving ~150, if he save 2 s. 6 d. per week? 155. If 16 acres produce 1246 bu. 3 p1c. 6 qt. of cornl, how munch will 1 acre produce? 156. If the circumference of a wheel be 8 ft. 3 in., how many times will it turn in a distance of 184 miles? 157. How many dozen of tea-spoons, each spoon weighing 1 oz; 3 dwt., can be made out of 25 lb. 10 oz. 10 dwt. of' silver? J 110 DENOMINATE NUMBERS. 147. To find the difference of time between two dates, - RULE. Write the first date under the last, the years on the left, and the number of the month in order next, and the day of the month on the right and then subtract. OBs. January is reckoned the first month, February the second, March the third, &c. The difference of time between February 3, 1845, and September 19, 1846, is i year, 7 months, 16 days. y. mo. d 1846 9 19 1845 2 3 1 7 16 Ons. In finding the difference between two dates, each month is usually reckoned 30 days. 158. What is the difference of time between September 4, 1845, and July 6, 1848? 159. What is the difference of time between October 9, 1844, and August 1, 1812? 160. What is the difference of time between-November 29, 1842, and April 4, 1846? 161. What is the difference of time between May 18, 1845, and December 2, 1847? 162. What is the difference between June 6. 1845, and February 4, 1846? 163. What is the difference between January 1, 1840, and December 31, 1844? 164. What is the difference between August 1, 1842, and November 16, 1846? 148. To find the difference in time and longitude between different places,What is the rule for finding the time between two dates? DENOMINATE NUMBERS. 111 As the earth passes through 360~. in 24 hours, in 1 hour it passes through 150, and in 1 minute -it passes through 161 of 150, or T of a degree, or 15 geographical miles; and in one second it will pass through t of a geographical mile, Therefore, by,multiplyingl the difference of longitude between two places, expressed in degrees, and minutes, and seconds, by 4, will give the difference of time in minutes, and seconds, and parts of a second. Thus, if the difference of longitude between two places be 77~, the difference of time will be 77X 4-308 minutes, equal to 5 hours, S minutes. If the difference of longitude be 2,40 12' 20/, the difference of time will be 240 12' 20"X 4=96 minutes, 49 seconds, and An, or a, of a second, equal to 1 hour, 36 minutes, 49~ seconds. OBs. It is obvious that when it is noon at any particular place, at any point east of that place it is after noon, and at any point west of that place it is before noon. 165. If the difference of longitude between Boston and London be 700 58' 35"//, what time is it in London when it is noon in Boston, and what time is it in Boston when it is noon in London? 166. If the longitude of Boston be 710 4' 20", and that of Chicago 780 30' 30", what is the difference in the time? 167. What time is it iln Boston when it is noon at Chicago? 168. What time is it in Chicago when it is noon in Boston? 169. If a gentleman travel from Boston to Louisville, and his watch keeps accurately the Boston time, will his watch be too fast or too slow, on arriving at Louisville, and how much, allowing the difference of longitude to be 140 25'? What is the rule for finding the difference of time between wo places? 112 DENOMINATE NUMBERS. 170. When it is noon at London, what time will it be at the mouth of Columbia River, which is 1200 west of London? 149. When the difference of time between two places is known, the difference of longitude may be found by dividing the minutes and seconds by 4; the quotient will be the difference of longitude in degrees and Ininutes. 171. What is the difference of longitude between two places, if the difference of time be 5 h. 20 nm. 16 sec.? 172. What is the difference of longitude between two places, if the difference of time be 2 h. 24 m. i5 sec.? 173. What is the difference of longitude between two places, if the difference of time Ibe 7 h. 40 m. 20 sec.? 174. If a vessel sail from Boston for Europe, and, after a number of' days, the captain finds, by taking an observation of the sun, that the difference of time, compared with his chronometer, which gives Boston time, is 2h. 40 m., how many degrees is he east of Boston? 150. To reduce denominate numbers to equivalent decimals of a higher denomination, - RULE. Divide the lowest denomnination by that?nugnber which nzakes one of3 the nex.t hiEgher denomination, and annex the quotient to the next hi g-her denomination, and divide as before. Proceed thus throurgh all the denominatiozns to the last. OBs. 1. The denominate numbers may first be reduced to a common fraction, (ARnT. 158,)anld then reduced to a decimal. 1What is the rule for finding the difference of longitude? What is the rule for reducing denominate numbers to equivalent decimals of a higher denomination? DENOMINATE NUMBERS 113 OBs. 2. The numbers should be written one above the other, the lowest denomination at the top and the highest at the bottom, and ciphers annexed when necessary. 175. Reduce ~15 10 s. 9d. to the decimal of a ~. 12 ) 9.00 20 ) 10.7500 15. 5375 The 9 pence is divided by 12, because 12 pence make 1 shilling. The quotient, 75, is annexed to the 10 shillings. This is divided -by 20, because 20 shilhings make ~1. EXAMPLES. 176. Reduce 16 s. 9T d. to the decimal of a pound. 177. Reduce 17 s. 5 d. to the decimal of a pound. 178. Reduce 18 s. 7~ d. to to the decimal of a pound. 179. Reduce ~58 12s. 62 d. to the decimal of a pound. 180. Reduce 2 oz. 14 dwt. 12 gr. to the decimal of a pound. 181. Reduce 2qr. 17 lb. to the decimal of a hundred weight. 1S2. Reduce 1 R. lords. to the decimal of an acre. 183. Reduce 365 d. 5 h. 48 m. 51 sec. to the decimal of a day. 184. Reduce 3 qr. 3 n. to the decimal of a yard. 185. Reduce 20 fur. 4 yds. to the decimal of a mile. 186. Reduce 26 sq. rd. to the decimal of an acre. 187. Reduce 3 plis. 7 qts. to the decimal of a bushel. 188. Reduce 20 min. 35 sec. to the decimal of a degree. 189. Reduce 4 h. 25 min. 34 sec. to the decimal of a day. 151. Shillings, pence, and farthings may also be reduced to a decimal of a pound in the following manner: Write half the number of shillings in the place Ad* 8 114 DENOMINATE NUMBERS. of te:ths, and reduce the given pence and farthiings to farthings, and if they amount to 24 or mnore, increase them by adding 1, and if there is an odd shilling, this sum is to be increased by 50, which must be written in the place of hundredths and thousandths. Thus 12 s. 9 d., reduced to the decimal of a pound, is.637. The half of 12 is 6, and 9d., changed to farthings, gives 36, which being increased by 1, because it exceeds 24, is made 37. Thus written, the places of hundredths and thousandths give.637. The reason of the above rule is obvious from the fact, that as there are 20 shillings in a pound, half of the number of shillings will express the tenths; and as 1 farthing is wow of a pound, and 960, increased by Ad of itself, is 1000, any number of farthings, increased by 9v part of itself, will express so many thousandth parts of a pound. When the. number of shillings is odd, 50 must also be added to the farthings, as I shilling is'-~" of a pound. ORs. When great accuracy is required, as many twenty-fourths,of the farthings should be added tothe farthings as there are farthings. 7 s. 6 d. = —.375 18 s. 4d. —.916 14s. 3 d. =.712 3 s. 4 d.-.166 17 s. 9 d.-.887 12 s. 7d..629 4 s. 6 d. -.225 15 s. 6 d..775 19s. 1d. =.954 i52, To change decimals to denominate numbers, - RULE. Aultiply the decimal by that number which is required of the next lower denomination to make a unit of the higher, and point off as in decimal fracti;ons. Proceed thus with the decimal in each product. The figures oia the left of the decimal point in the.several products will be the denonminate numbers. OBs. If there be a decimal in the last product, it should be changed to a common fraction, which will denote a fractional part of the lowest denonrinate number. WVhat is the rule for changing decimals to denominate numbers? DENOMINATE NUMBERS. 15 190. Reduce ~.840625 to denominate numbers..840625 20 16.812500 ~. 12 9.750000 s. 4 3.000000 d. Multiply the decimal by 20, because 20 shillings make 1 pound. Point off fiom the right of the product six figures, as in the multiplication of decimals. Miultiply the next decimal product by 12, because 12 pence make 1 shilling, and point off as before. Multiply the last decimal product by 4, because 4 farthings make I penny. The figures in the several products, at the left of the decimal point, are the denominate numbers, viz.: ~~ 16 9 s. 3 d. EXABIPLES. 191. Reduce ~.78125 to denominate numbers. 192. Reduce ~.61.925 to denominate numbers. 193. Reduce ~. 15625 to denominate numbers. 194. R6duce.728125 of a pound Troy to denominate numbers. 195. Reduce.9642857 of a month to denominate numbers. 196. What is the value of:45 of a bushel? 197. What is the value of.4 of a yard? 198. What is the value of.96 of a cord? 199. What is the value of.864 of a rod? 200. What is the value of.765 of a mlile? 201. What is the value of.965 of a ton? 202, What is the value of.875 of an hour? 115 DENOMINATE FRACTIONS. DENOMBINATE FRACTIONS. SECTION X. 153. DENOMINATE FRACTIONS may be changed from a higher denomination to a lower, and from a lower to a higher, in the same manner as whole numbers. 1 54. To reduce denominate fractions from a higher denomination to a lower, - RULE. iMutltiply the fraction by the same numbers that are required to reduce a whole number of the same denomination as the fraction, to the lower denomination required, (ART. 141.) OBS. The numbers used as multipliers may first be changed to the form of an improper fraction, and factors common to the numerators and denominators may be cancelled. 1. Reduce ~~1, to the fraction of a penny. 3 xI 0 x _ 3 0s 1' 1 4 4 The fraction is multiplied by 20 and 12, the same that are required in whole numbers, and the common factors are cancelled. EXAMPLES. 2. Reduce ~-T51 to the fraction of a farthing.d 3. Reduce ~Z to the fraction of a shilling. 4. Reduce +TX~U of a lb. Troy to the fraction of a grain. WVhat is the rule for reducing denominate fractions from a higher denonmination to a lower? DENOMINATE FRACTIONS. 117 5. Reduce TW of a lb. Avoirdupois to the fraction of a dram. 6. Reduce Ta4 of a cwt. to the fraction of a lb. 7. Reduce 5d~ of a ton to the fraction of a lb. 8. Reduce,-, of an acre to the fraction of a rod. 15'5. To change denominate fractions from a lower denomination to a higher, - RULE. Divide the fraction by the same numbers that are required to change a whole number of the same denomination as the fraction, to the higher denomination required. OBs. The numbers used as divisors may be changed to improper fiactions, and factors common to the numerators and denominators may be cancelled, 9. Reduce 5 of a farthing to the fraction of a L. $ 1 1 1 X — X. X -- 8 4 12 M0 1536 4 Dividing j of a farthing by 4, 12, and'20, the same numbers that are required to change farthings to pounds, 5 is cancelled in the numerator, and as a factor of 20 in the denominators. EXAMPLES. 10. Reduce I of a penny to the fraction of a ~. 11. Reduce ] of a farthing to the fraction of a ~. 12. Reduce 2 of a grain to the fraction of a lb. Troy. 13. Reduce 3 of a dram to the fraction of a cwt. 14. Reduce of a nail to to the fraction of a yard. 15. Reduce ] of a minute to the fraction of a day. 16. Reduce 4 of a foot to the fraction of a mile. What is the rule for changing denominate fractions from a lower denomination to a higher? 118 fDENOMINATE FRACTIONS. 156. To reduce a fraction of any denomination to denominate numbers, - RULE. Multiply the numerator by the number required of t.he next lower denomnination to make a unit of the deornmination of the fraction, and divide the product by the denominator. If there be a remainder, Multiply and divide in the same manner to the lowest denomination. The several quotients will be the dle nominate numbers required. 17. What is the value of ~7? 3 2() 7 ) 60 ( 8s 56 4 12 7 )48 ( 6d. 42 6 ) 24 ( 3 r. 21 3 Multiplying the numerator, 3, by 20, and dividing by the denominator, 7, gives 8 s. and 4 remainder. Multiplying the remainder by 12, and dividing agaia by 7, gives 6 d. and 6 remainder. Multiplying the remainder, 6, by 4, and dividing again by 7, gives 3 qr. and 3 remainder, which is a. AiWhat is the rule for reducing a fraction of any denominnatiouc to dnorllinate inumbers DENOMINATE FRACTIONS, 119 EXAMPLES. 18. What is the value of ~-8? 19. What is the value of j~ of a cwt.? 20. What is the value of - of a lb. Troy? 21. What is the value of -7L of a yard? 22. What is the value of -9 of a mile? 23. What is the value of 2 of a ton? 24. What is the value of -LT of a year? 25. What is the value of 3 of a cord? 26. What is the value of ~ of X of a bushel? 27. What is the value of - of 5 of a mile? 28. What is the value of.825 of a cwt.? 29. What is the value of.475 of a ton? 30. What is the value of.625 of a lb. Troy? 157. To reduce denominate numbers to equivalent common fractions,RULE. Reduce the denominate number to the lowest denomination it contains, for a numrerator, and a unit of the denomination of the required fraction to the same denomination as the numerator, for a denominator. 31. Reduce 6 s. 9 d. 3 qr. to the fraction of a ~. 6. 9. 3. ~1=20 12 12 81 240 4 4 327 960 960 6 s. 9 d. 3 qr. reduced to farthings are 327 farthings. ~1 reduced to the same denomination is 960. 3 is the fraction required. What is the rule for reducing denominate numbers to equivalent common fractions? 120 DENOMINATE FRACTIONS. EXA1MPLES. 32. Reduce 9 s. 11 d. 2 qr. to the fraction of a ~. 33. Reduce 6 oz. 10 dwt. to the fraction of a lb. 34. Reduce 14 dwt. 12 gr. to the fraction of an oz. 35. Reduce 14 lb. 6 oz. 10 dr. to the fraction of an cwt. 36. What part of a ton. is 4 qr. 16 lb. 8 oz.? 3Z, What part of a mile is 5~ yd. and 9 ft? 38. What part of a mile is 4 fur. 30 rd. 16 ft.? 39. What part of a day is 4 h. 26 min.? 40. What'part of a gallon is 2 qt._ 3 gi.? 41. What part of a day is 5 of an hour? 42. What part of a mile is i of a rod? 43. What part of a yard is a of a nail? OBs. When the lowest denomination is expressed in the form of a fraction, the unit of the higher denomination must be reduced to esuch parts of the lowest denomination as are expressed by the denominator of the fraction. 1P58. To change one denominate number to the fractional part of any other denominate numberRULE. Reduce the given numbers to the same denomination, and write the numiber which is to be- the fractional part for the numerator, and the other number for the denomninator of the fraction, and reduce the fractionz to its lowest terms. 44. What part of a ~ is 16 s. 8 d.? 16s. 8d. ~1 x 20 X 12-240 12 200 5 240 6 Reduce 16 s. 8 d. and E~ to pence. The fraction What is the rule for reducing one denominate number to the fractional'part of any other denominate number? ADDITION OF DENOMINATE FRACTIONS. 121 will be 2o$, which reduced to its lowest terms will give x. EXAMPLES. 45. What part of a ~ is 9 s. 6 d.? 46. What part of a lb. Troy is 4 oz. 16 dwt.? 47. What part of a cwt. is 12 lb. 12 oz.-? 48. What part of ~4 is 10s. 9d.? 49. What part of 2 ton is 4 cwt. 3 qr. 0O lb.? 50. What part of 4 bushels is 4 pt.? 51. What part of a mile is W of a foot? 52. What part of 7 hours is 4 of a minute? 53. What part of 3 gallons is ~ of a gill? 54. What part of 2 cord is 104 cubic feet? 55. What part of 2 furlongs is 154 feet? 56. What part of 3 miles is 44 inches? ADDITION AND SUBTRACTION OF DENOMINATE FRACTIONS. 159. RULE. Reduce each fraction to equivalent denominate numbers, and add or subtract, as in ARTS. 143, 145. Or reduce each fraction to the same denomination, and add or subtract, as in common fractions. 57. Add ~ ]- and - s. ~4 -= 13s. 4d. 2s. = 4d. 3-qr. 13s. 8d. 3 qr. X W ~or-I +,z -1 zo -13s. Sd. 3- qr. EXAMPLES. 58. Add I4, s., and 4 d. 59. Add ~E, s., and 4d. 60. Add 5 lb. Troy and 4 dwt. What is the rule for adding and subtracting denominate fractions? K 122 PRACTICAL QUESTIONS. 61. Add - ton and ] lb. 62. Add 7 gal. and 2 pt. 63. Add mrn. and 16~ ft. 64. Add y ac. and - rd. 65. From A~ take 4- s. 66. From 4 lb. Troy take 4 oz. 67. From I% ton take i lb. 68. From 4 ac. take 4 rd. 69. Fronl 4 m. take 9 rd. 70. From 7 yd. take 4 na. Oas. Denominate fractions may also first be reduced to decimals of the' same denomination, and then added and subtracted as decimals.PRACTICAL QUESTIONS. 71. What is the sum of ~ 5 and 1 4s. 9 d. 3qr. 72. What is the sum of 4 lb. Troy and 4 oz. 12 dwt.? 73. What is the sum of 4 tons, 2 qr. 12 lb. and 3.0245 tons? 74. What is the difference between 25 gal. 3 qt. and 14.0246 gal.? 75. If 4 cwt. 14 lb. of sugar cost $30.50, what cost i lb.? 76. If.375 of a yard of cloth, cost ~4, what will 225 yards cost? 77. If 24 yd. 2 qr. cost $125.50, what cost I yard? 78. If 674 gallons of molasses cost $22.50, what cost 1 qt.? 79. If 1 gallon of molasses can be bought for ~4, how many gallons can be bought for ~20? 80. If 1 bushel of wheat cost 6s. 9d., how many bushels can be bought for 15, 16 s.? 81. How many square feet in a board that is 9 ft. 4 in. long and 1 ft. 6 in. wide? 82. Bought 161 bushels of salt at 75 cents per bushel, and sold it at 23 cents per peck. What was the gain? PRACTICAL QUESTIONS. 123 83. A grocer bought a pipe of molasses, containing 124 gallons, at 33:;cents per gallon; 19 gallons having leaked out, he sold the remainder at 40 cents per gallon. Did he gain or lose, and how much? 84. A farmer bought 28~ cords of wood, at $3.50 per cord; he sold 13 cords at $4.30 per cord, and the remainder at $3.20 per cord.. Did he gain or lose, and how much? 85. Bought - of an acre of land for $325.50, and sold it at 2 cents per foot? What was the gain or loss? 86. How many yards of carpeting, ~ of a yard wide, will be required to carpet a floor that is 23 ft. 9 in. long, 16 ft. 8 in. wide? 87. How many square feet of paper will be required to cover the walls of a room that is 16 ft. 4 in. long, 15 ft. 6 in. wide, and 10 ft. 3 in. high, after deducting 116 ft. for windows and doors? 88. If the cargo of a ship be worth ~8000, and if of - of _ of the ship be worth of R of g of the cargo, what is the value of both ship and cargo? 89. A grocer bought 15 cwt. 3 qr. 21 lb. of coffee at $9.50 per cwt., and sold it at 121 cents per lb. WThat did he gain on the whole? 90. A farmer purchased 4 acres of woodland, at $150 per acre; he paid for cutting 300 cords of wood $} per cord; for carting the same, $1- per cord; he sold the 300 cards of wood at $4~ per cord. Did he gain or lose by the bargain, and how much? 91. A farmer paid $160.50 for the lease of a farin for one year; he sold 16 tons of hay, at $13 per ton; 150 bushels of potatoes, at 33- cents per bushel; 155 bushels of corn, at 75 cents per bushel; 62 bushels of oats, at 42 cents per bushel; 202- barrels of apples, at $1.75 per barrel; and 1505 lb. of cheese, at 9 cents per pound. He paid for expenses of family and for labor $345. Did he gain or lose, and how much? 124 PERCENTAGE. PERCENTAGE. SECTION XI. 160. PERCENTAGE, or PER CENT., denotes any num-, ber of hundredths of a given sum. Ohs. These terms are from two Latin words, per and centtum, which signify by the hLzcndred. 161. The per cent. is sometimes called the rate and is expressed in decimals, thus: — 1 per cent. is written.01 2 per cent. "C ".02 r per cent. "c ".03 4 per cent. o ".04 5 per cent. "'".05 6 per cent. "'.06 7 per cent. " I.07 8 per cent. " ".08, &c. ~ of I per cent. " ".005 4 of 1 per cent." ".0025 ~ of I per cent.' ".00125 Oas. WVhen the parts of 1 per cent. cannot be expressed exactly in decimals, they should be written in the form of a common fraction, thus: 41 per cent. =.04~. 162. To find the per cent. of any number, - RULE. lMgultiply the given number by the per cent., and point off the product, as in multiplication of decimals. The principle of this rule is the same as that in the multiplication of decimals. What does percentage, or per cent., denote? What are the terms derived from? What is the per cent. sometimes called, and how is it expressed? How are the parts of 1 per cent. written? PERCENTAGE. 125 1. What is 4A per cent. of 144? 144.041 576 48 6.24 Multiplying 144 by 4~, and- pointing off according to the rule in the multiplication of decimals, gives 6.24. EXAMPLES. 2. What is 6 per cent. of $34.556? 3. What is 5 per cent. of.$90.755? 4. What is 4 per cent. of $875.25? 5. What is 3 per cent. of $456.675? 6. What is 2 per cent. of $640.205? 7. What is i per cent. of $735.75? 8. What is 10 per cent. of $905.60? 9. What is 64 per cent. of $45.305? 10. What is 4- per cent. of $734.02? 11. What is 1~ per cent. of $560.24? 12. What is R per cent. of $84.50? 13. What is ~ per cent. of $93.60? 14. What is -per cent. of $56.49? 15. What is ~. per cent. of $81.90? 16. A man paid 6~ per cent. for the use of $175.50 for 1 year. How much did he pay? 17. A man bought 15 shares of railroad stock, at $100 a share, and sold them at 9 per cent. advance? What did he gain? 18. A merchant bought 450 barrels of flour, at $4.37~ per barrel, and gained 12 per cent. in the sale of it. What did he receive? 19. A man's income is $1500 a year; he saves 12} per cent. of it. How much does he spend? K* 1 26 PERCENTAGE. 163. To find what per cent. any number is of another given number; — RULE. Annex two ciphers to the number whose per cent. is sought, and divide by the number of which it as,sought. ART. 21. OBS. OBs. The numbers must have the same number of decimal places, or be of the same denomination. 20. What per cent. of $50 is $3? 50 ) 300 ( 6 300 Annexing two ciphers to the $3, and dividing by $50, gives 6 as the per cent. OBS. This rule is evidently the converse of the preceding. Thus 6 per cent. of $50 is $3. 50 X.06 = 3.00, 21. What per cent. of $150 is $4? 22. What per cent. of $240 is $6? 23. What per cent. of $48 is $30? 24. What per cent. of $6250 is $.3125? 25. What per cent. of $840 is $648? 26. What per cent. of $56 is $.08? 27. What per cent. of $96 is $84? 28. What per cent. of $85 is $.065? 29. What per cent. of $506 is $.084? 30. What per cent. of $364.40 is $9? 31. What'per cent. of $896.50 is $24? 32. What per cent. of $560 is $76.20? 33. What percent. of $240 is $46.1.0? 34. What per cent. of $360 is $32.40? 35. What per cent. of $60.50 is $18? 36. What per cent. of ~9 10s. is ~2 4 s.'9 d.? 37. What per cent. of ~40 6 s. 9 d. is ~5 10 s. 6 d.? What is the rule for finding what per cent. any number is of another given number? SIMPLE INTEREST. 127' SIMPLE INTEREST. SECTION XII. 164. INTEREST is a certain per cent. of a sum of motey, paid for its use. OBs. The per cent. is called the rate, and is always reckoned per annum, which signifies by the year. 165. The sum on which the interest is computed is called the principal. The principal and interest, added together, is called the amount. 166. The rate per cent. is established by law, and varies in different states and countries. 167. The legal rate'is 6 per cent. in the New Eng-,land States, also in New Jersey, Pennsylvania, Delaware, Maryland, Virginia, North Carolina, Tennessee, Kentucky, Ohio, Indiana, Illinois, Missouri, Arkansas,District of Columbia, and on debts and judgments in favor of the United States; also in Canada, Nova Scotia, and Ireland. 168. The legal rate is 7 per cent. in New York, Michigan, Wisconsin, Iowa, and South Carolina. 1 69. It is 8 per cent. in Georgia, Alabama, Mississippi, and Florida. 170. It is 5 per cent. in Louisiana, also in England and France. 171. RULE. Find the interest of one dollar for the time, and by this multiply the principal; Or remove the decimal point in the principal two places to the left, and multiply by one half of the X umber of months, and one sixtieth of the number of' days. What is interest? What is the per cent. called? What is the meaning of per annum? What is the principal? What is the amnount? How is the rate determined:? What is the rule for casting interest? 128 SIMPLE INTEREST. Ons. If the principal contain only dollars, point off two figures on the right for cents. It is evident that the interest of one dollar for twelve months, at 6 per cent., is six cents.; for two months, or sixty days, one cent; and the interest of one dollar, for any number of months, is one cent. for every two months, or one half as many cents as there are months; and the interest of one dollar for one day is one sixtieth of a cent, or one sixth of a mill, and the. ilnterest of one dollar for any number of days is one sixth as many mills as there are days. 1. What is the interest of $1 for 1 year, 9 months, and 5 days? Interest for 1 year is =.06 cc ~" 9 months is =-.045 " " 5 days is -.0005.105-% 2. What is the interest of $1 for 3 years, 5 months, and 3 days? Interest for 3 years is -.18 " " 5 months is -.025 " 3 days is=.000,.205% OBs. By the preceding rule, the interest for days is found for as many 360ths of one year's interest as there are days, which is evidently-, or too much. When great accuracy is required, as many 365ths of one year's interest must be taken as there are days, or As of the interest for days, found by the preceding rule, must be de. ducted. Some states require this deduction to be made. EXAMPLES. 3. What is the interest of $1 for 7 months and 9 days? 4. What is the interest of $1 for 10 months and 4 days? SIMPLE INTEREST. 129 5. What is the interest of $1 for 13 months and 1 day r 6. What is the interest of $1 for 17 months and 29 days? 7. What is the interest of $1 for 19 months and 2 days? 8. What is the interest of $645.60 for 2 years, 4 months, and 5 days? 645.60 2 years - 24 months..1405 4 2582400 2 )28 6456 53800 90.92200. 4o In 2 years and 4 months there are 28 months. The interest of i dollar for 28 months is 14 cents; the interest for 5 days, - of a mill, which added to 14 cents, gives.140k, the principal being multiplied by which gives $90.922. Ons. When there are even months, and the days are less than 6, a cipher must be written in the place of mills. 9. What is the interest of $1557.56 for 3 years, 7 months, and 5 days? 10. What is the interest of $763.54 for 4 years, 9 months, and 11 days? 11. What is the interest of $351.67 for 3 years, 4 months, and 2 days? 12. What is the interest of $454.75 for 5 years, 5 months, and 11 days? 13. What is the interest of $570.60 for 6 years, 7 months, and 3 days? 14. What is the interest of $674.375 for 7 years, 3 months, and 27 days? 9 130 SIMPLE INTEREST. 15. What is the interest of $735.75 for 8 years, 2 mnolths, and 4 days? 16. What is the interest of $84.901 for 3 years, 1 nonth, and 6 days? 17. What is the interest of $936.40 for 4 years, 2 monlths, and 2 days? 18. What is the interest of $124.50 for 2 years, 4 aonths, and 5 days? 17n2. To find the amount of any sum of money for a given time,RULE. Find the interest for the rate per cent. and time, and add it to the principal. Or multiply the principal by the amount of $1 for the tinte. 19. What is the amount of $160 for 4 months and 12 days? 160.022 160 320 1.022 320 3.520 320 160 160 $163.520 $160.520 Interest of $1 for 4 months and 12 days is 2 cents and 2 mills; the amount of $1 for the time is $1.022. EXAMPLES. 20. What is the amount of $75.60 for 2 years, 11 months, and 1 day? 21. What is the amount of $324.60 for 3 years, 9 nmonths, and 18 days? 22. What is the amount of $450.30 for 19 months and 29 days? 23. What is the amount of $675.80 for 1 year, S months, and 24 days? What is the rule for finding the amount? SIMPLE INTEREST. 131 24. What is the amount of $735.75 for 2 years, I I months, and 5 days? 25. What is the amount of $936.60 for 3 years, 4 months, and 2 days? 26. What is the interest of $84.54 for 7 years, 6 months, and 6 days? 27. What is the interest of $96.40 for 1 year, 9 months, and 5 days? 28. What is the amount of $244 for 4 years, 8 months, and 11 days? 29. What is the amount of $735.36 for 5 years, 2 months, and 17 days? 30. What is the interest of $560.25 f)r 2 years, 11 months, and 16 days? 31. What is the amount of $435.70 for 6 years, 10 months, and 14 days? 173. To find the interest for any sum of money, wvhen the rate is any other than 6 per cent, — RULE. Find the interest at 6 per cent. for the given time, and divide the interest thus found by 6, vwhich will give the interest for 1 per cent. Thenz multiply this quotient by the number denoting the per cent. sought. 32. What is the interest of $344.40 for 3 years, 4 months, and 5 days, at 7 per cent.? $344.40.200o 6888000 28700 6) 6916700 1152783 7 $80.69481 What is the rule for finding the interest when the rate is any other than 6 per cent.? 132 SIMPLE INTEREST. 3 years and 4 months are 40 months; interest for 40 months is 20 cents; interest for 5 days is ~ of a mill; 20 cents plus 6 of a mill is.200k, interest of $1 for the given time. EXAMPLES, 33. What is the interest of $256.10 for 1 year, 9 months, and 3 days, at 7 per cent.? 34. What is the interest of $295.80 for 3 years, 7 months, and 5 days, at 8 per cent.? 35. What is the interest of $376.94 for 2 years, 3 months, and 2 days, at 5 per cent.? 36. What is the interest of $565.30 for 4 years, 5 months, and 4 days, at 4 per cent.? 37. What is the interest of $756.45 for 9 months and six days, at 3 per cent.? 38. What is the interest of $96.75 for 11 months and 29 days, at 2L per cent.? 39. What is the amount of $739.40 for 2 years, 1 month, and 11 days, at 7 per cent.? 40. What is the interest of $84.20 for 3 years, 5 months, and 12 days, at 5 per cent.? 41. What is the amount of $96.30 for 5 years, 7 months, and 15 days, at 4~ per cent.? 42. What is the amount of $2452.06 for 7 months and 9 days, at 5~ per cent.? 43. What is the amount of $3764.08 for 11 months and 29 days, at 61 per cent.? 44. What is the amount of $2460.90 for 93 days, at 7~ per cent.? 45. What is the amount of $643.73 for 63 days, at 3k per cent.? 46. What is the amount of $960.40 for 25 days, at 7T per cent.? 47. What is the amount of $65735 for 6 months and 3 days, at 8 per cent.? 48. What is the interest of $245.60 from July 2, 1848, to June 20, 1849? SIMPLE INT'iREST. 133 49. What is the interest of $36.40 from August 10, 1,847, to September 8, 1849? 50. What is the interest of $760.30 from October 8, 1846, to November 5, 1848? 51. What is the amount of $470.90 from May 6, 1844, to April 9, 1846? 52. What is the amount of $48.64 from March 1, 1846, to February 28, 1848? 53. What is the amount of $276 from January 1, 1848, to December 30, 1849? 54. What is the amount of $8650 from January 4, 1844, to September 6, 1847, at 51 per cent.? 55. What is the interest of $96.50 firom February 7, 1843, to July 21, 1850, at 6~ per cent.? 56. What is the interest of $84.60 from June 1, 1847, to May 29, 1849-? 174. Interest may also be computed by the follow, ing rule - RULE. Multiply the principal by the rate per cent., an d the product will be the interest for 1. year; and nultiply this product by the number of years for which the interest is required. "For qnonths, take such a fractional part of the interest for one year as is denoted by the number of 1months. For days, take such a fractional part of the interest for one month as is denoted by the number of days. OBS. Some accountants find the greatest number of whole months between the two dates, and the remaining time is reckoned in days, allowing as many days for each month'as there are in each calendar month. Thus from January 15 to June 6 there are 4 months and 22 days. From January 15 to May 15 there are 4 months; allowing 31 days in May, there are 16 days remaining, which, added to the 6 in June, make 29 days. What is the second rule for casting interest? L 134 SIMPLE INTEREST. 57. What is the amount of $764.20 for 4 years, 5" months, and 8 days? $764.20.06 Interest for 1 year, 458520.4 Interest for 4 years, 183.4080 Interest for 4 mos., (~ of 1 year,) 15.2840 Interest for 1 mo., (o of 4 mos.,) 3. 210 Interest for 6 days,(1- of 1 mo.,) 7642 Interest for 2 days, (1 of 6 days,) 2547 203.5319 764.20 Amount, $967.7319 OBs. 1. Great care must be used in pointing off the decimals, and in writing dollars under dollars, cents under cents, &c. OBs. 2. When the per cent. is not specified, it is always considered to be 6 per cent. EXAMPLES. 58. What is the interest of $456.84 for 7 years, 7 months, and 5 days? 59. What is the interest of $78.50 for 6 years, 9 months, and 12 days?; 60. What is the interest of $96.60 for 5 years, 3 months, and 25 days? 61. What is the interest of $8735.69 for 1i years, 7 months, and 27 days? 175. RULE. To find the interest on pounds, shillings, pence, and farthings, first reduce the shillings to the decimal of a pound, and proceed as in federal noney. What is the rule for finding the interest on pounds, shillings, pence, and farthings? SIMPLE INTEREST. 135 EXAMPLES. 62. What is the interest of ~25, 6s. 11 d. for 2 years, 7 months, and 10 days? 63. What is the interest of ~33, 11s. 4-d. for 3 years, 7 months, and 18 days? 64. What is the amount of ~121, 15s. 9.d. for 4 years, 9 months, and 13 days? 65. What is the amount of ~130, 19s. 6{-d. for 1 year, 11 months, and 25 days? 66. What is the interest on a note of six hundred and seventy-five dollars and fifty cents from May 4, 1844, to February 14, 1846? 67. What is the interest on a note of three hundred and forty-eight dollars and thirty cents from November 17, 1843, to September 12, 1846? 68. What is the interest on a note of fifty-seven dollars and sixty-four cents from November 16, 1843, to April 6, 1846, at 7 per cent.? 69. What is the amount due on a note of two hundred and forty-three dollars and twenty-nine cents from October 4, 1842, to March 18, 1845, at 5 per cent.? 70. What is the amount due on a note of four hundred and fifteen dollars and twenty cents from August 5, 1846, to January 24, 1849, at 7 per cent.? 71. What is the interest on a note of one hundred and eighteen dollars and twenty cents from February 28, 1844, to July 17, 1847, at 7 per cent.? 72. What is the interest on a note of seventy-six dollars and forty-four cents from June 12, 1845, to October 16, 1847? 73. What is the amount of a note of thirty-four dollars and ninety cents from October 4, 1843, to December 15, 1847? 74. What is the amount of a note of ninety-six dollars and forty cents from August 10, 1845, to July 15, 1848? 136 PARTIAL PAYMENTS. PARTIAL PAYMENTS. 176. The following rule has been adopted by the Supreme Court of the United States, and by many of the states, for computing interest on bonds and notes when partial payments have been endorsed on them'RULE. " The rule for casting interest, when partial payments have been made, is to apply the payment, in the first place, to the discharge of the interest then due. If the payment exceeds the interest, the surplus goes towards discharging the principal, and the subsequent interest is to be computed on the balance of principal remaining due. " If the payment be less than the interest, the surplus must not be taken to augment the principal; but interest continues on the former principal until the period when the payments, taken together, exceed the interest due, and then the surplus is to be applied towards discharging the principal, and interest is to be computed on the balance, as aforesaid." $400. PROVIDENCE, April 12, 1840. 75. For value received, I promise to pay to James Fenner, or order, four hundred dollars, on demand, with interest. ALFRED BLODGET. On this note were received the following endorsements: - March 4, 1842, received forty-five dollars. May 10, 1843, received sixty dollars. June 6, 1845, received ninety dollars. What was due January 12, 1847? What is the rule for casting interest when partial payments are made? PARTIAL PAYMENTS. 137 Principal, $400 Interest to 1st payment, (22 m. 22 d.) 45.466 Payment being less than the interest, interest to the 2d payment, on same principal, (14 m. 6 d.) 28.400 Amount to 2d payment, 473.866 Sum of both payments, 45+60- 105.000 368.866 Interest to the next payment, (24 m. 26 d.) 45.862 414.728 Last payment, 90.000 324. 28 Interest to January 12, 1847, (18 m. 6 d.) 31.173 Balance remaining due January 12, 1847, $355.901 $600.75. NEW YORK, M7ay 4, 1S40. 76. For value received, I promise to pay to James Cunningham, or order, six hundred dollars and seventyfive cents, on demand, with interest. BENJAMIN TUCKER. On this note were endorsed the following paymnents - Oct. 6, 1841, received sixty-four dollars. July 8, 1842, received forty-eight dollars and fifty cents. Nov. 20, 1844, received two hundred dollars and sixty cents What was due May 10, 1846? $565.90. BUFFALO, June 12, 1840. 77. For value received, I promise to pay William Bliss, or order, five hundred and sixty-five dollars and ninety cents, on demand, with interest. GEORGE PACKARD. L * 138 PARTIAL PAYMENTS. On this note were endorsed the following payments: - Jan. 6, 1841, received forty-five dollars. March 6, 1843, received sixty-eight dollars. Sept. 9, 1844, received thirty-seven dollars. What was due December 30, 1846? $340.00. PROVIDENCE, July 9, 1841. 78. For value received, I promise to pay to Henry Blackstone, or order, three hundred and forty dollars, on demand, with interest. NOAH CURTIS. On this note were endorsed the following payments: -- May 4, 1842, received sixty-six dollars and ninety cents. August 10, 1843, received twenty-nine dollars and four cents. Oct. 12, 1844, received forty-two dollars and six cents. What was due November 20, 1845? $609.65. PHILADELPHIA,, June 8, 1845. 79. For value received, I promise to pay to Benjamin Tucker, or order, six hundred and nine dollars and sixty-five cents, in six months, with interest afterwards. ARTEMAS SMITH. On this note were endorsed the following payments: - Oct. 4, 1846, received twenty-five dollars. March 15, 1847, received sixteen dollars and twenty-five cents. August 24, 1848, received thirty-six dollars and fifty-six cents. What was due December 19, 1849? PARTIAL PAYMENTS. 139 $874.95. NEW YORK, Mray. 9, 1843. 80. For value received, I promise to pay to Horatio Tremnlet, or order, eight hundred and seventy-four dollars and ninety-five cents, in three months, with interest afterwards. THOMAS CHEEVER. On this note were received the following endorsements: - April 12, 1844, received fifty-six dollars and thirty cents. July 14, 1845, received twenty-fouir dollars and eighty cents. Sept. 18, 1846, received two hundred and forty dollars and sixty cents. What was due February 10, 1848? 177. The following, though not a legal rule, is adopted by many for computing interest when the note on which partial payments have been made is settled within a year from the time the interest commenced: - RULE. Find the amount of the nzote for the whole time. Then find the amount of each paymnent, from the time it was endorsed, to the time of settlement. Subtract the amount of the several payments frjom the amount of the note. Ois. This rule is adopted in Vermont for any period of time. $240. PROVIDENCE, May 4, 1845. 81. For value received, I promise to pay to Joshua Bent, or order, on demand, two hundred and forty dollars, with interest. H IRAM BRADLEY. What is the rule for finding the interest on notes settled within a year? 140 PARTIAL PAYMENTS. On this note were received the following endorsemients - Sept. 10, 1845, received sixty dollars. Jan. 16, 1846, received ninety dollars. What was due May 4, 1846? 1st payment, $60 2d payment, $90 Principal, $240 Int. 7m. 24d., 2.34 Int. 3m. 18d., 1.62 lit. 12m., 14.40 Amount, 62.34 91.'62 254.40 62.34 153.96 Amount of payment, 153.96 Balance, 100.44 $460. CINCINNATI, Septemnber 1.0, 1846. 82. For value received, I promise to pay to Joseph Hovey, or order, on demand, four hundred and sixty dollars, with interest. PHrNEAs BRowN. On this note were received the following endorsements - Jan. 4, 1847, received two hundred dollars. May 15, 1847, received sixty-five dollars. What is due September 10, 1847? $340. PORTLAND, June 16, 1848. 83. Three months after date, I promise to pay to Jacob Appleton, or order, three hundred and forty dollars, with interest. WILLIAM MORSE. On this note were received the following endorsements: - Oct. 14, 1848, received eighty-six dollars. Feb. 12, 1849, received forty dollars. What was due August 10, 1849? PARTIAL PAYMENTS. 141 $400. HARTFORD, September 12, 1848. 84. Four months after date, I promise to pay to Charles Newcomb, or order, four hundred dollars, with interest. HENRY GAY. On this note were received the following endorsements: Dec. 12, 1848, received one hundred and ten dollars. March 16, 1849, received eighty-six dollars. What is due October 9. 1849? 178. The following is the Connecticut rule: — " Compute the interest on the principal to the time of the fJirst payment; if that be one year or more fromn the time the interest commenced, add it to the principal, and deduct the payment from the sum total. If there be after payments made, compute the interest on the balance due to the next payment, and then deduct the payment as above, and in like manner from one payment to another, till all the payments are absorbed, provided the time between one payment and another be one year or more. "If any payments be made before one year's interest has accrued, then compute the interest on the principal sum due on the obligation for one year, add it to the principal, and compute the interest on the sumn paid, from the time it was paid, up to the end of the year; add it to the sum paid, and deduct that sum from the principal and interest, added as above.' If a year extends beyond the time of payment, then find the amount of the principal remaining unpaid up to the time of settlement, likewise the amount of the endorsements, from the time they were paid to What is the Connecticut rule? 142 PROBLEMS IN INTEREST. the time of settlement, and deduct the sunm of these several amounts front the amount of the principal. If any payments be made of less sumr than the interest arisen at the time of such payment, no interest is to be computed, but only on the principal su~ m for any period."' OBs. 1. Cast the interest on all of the notes by each of the preceding rules. OBs. 2. In some states when a note is written with " interest annually," interest is computed on the principal to the time of settlement, and on each year's interest after it is due, and the sum of the interests is added to the amount of the principal. PROBLEMS IN INTEREST. 179. The interest, time, and rate per cent. being given, to find the principal, - RULE. Divide the interest by the product of the rate per cent. and time, and the quotient will be the principal. OBs. 1. It must be remembered that the rate per cent is a decimal, and expresses a certain number of hundredths. The quotient must therefore be pointed off according to the rule of division of decimal fractions. OBs. 2. Months and days must be reduced to a decimal of a year, or to a fractional part of a year. 85. What principal will gain $78.40 in 2 years and 4 months, at 6 per cent.? 78.40 - 560. 2A X.06 Divide the interest, $78.40, by the product of the rate and time. The time, 2 years and 4 months, is 2 years, which, multiplied by.06, is.14. As 6 is a decirnal denoting hundredths, the product of 21 by.06 will be 14 hundredths. 78.40 divided by.14 will be What is the rule when the interest, time, and rate are given? PROBLEMS IN INTEREST. 1]43 $560, since there are as many decimal places in the dividend as divisor. EXAMPLES. 86. What sum must be invested to gail $450 at 6 per cent. in 9 months? 87. What principal will gain $750 at 6 per cent. in 1 year and 3 months? 88. What sum must be invested at 7 per cent. to gain $760 in 8 months? 89. What sum will pay a semi-annual dividend of $640 at 8 per cent.? 90. What sum invested at 7 per cent. will produce $900 a year? 180. The principal, interest, and rate per cent. being given, to find the time; RULE. Divide the interest by the product of principal and rate, and the quotient will be the time. 91. In what time will $640 gain $102.40 at 6 per cent.? 102.40 -__- _ 2~- years - 2 years and 8 months. 640 X.06 Dividing the interest, $102.40, by the product of the principal and rate, gives 2. years for the time. 640X.06=-38.40. *Two figures for decimals, as 6 per cent. is 6 hundredths. As there are the same number of decimal places in the dividend and divisor, the quotient will be a whole number. EXAMPLES. 92. In what time will $500 gain $500 at 6 per cent.? 93. In what time will $460 gain $230 at 6 per cent.? WVhat is the rule when the principal, interest, and rate are given? 14 1 PROBLEMS IN INTEREST. 94. In what time will $330 gain $110 at 6 per cent.? 95. In what time will $68.50 gain $34.25 at 6 per cent.? 96. How long will it take $600 to gain $600 at 5 per cent.? 97. How long will it take $800 to gain $800 at 7 per cent.? 98. How long will it take $400 to gain $400 at 10 per cent.? 181. The principal, interest, and time being given, to find the rate per cent., RULE. Divide the interest by the product of the principal and time, and the quotient will be the rate per cent. 99. The interest of $400 for 3 years and 6 months is $84. What is the rate per cent.? 84.00.06, or 6 per cent. 400 x 3Divide the interest, $84, by the product of the time and principal. The product of 400 and 31 is 1400. Since 1400 is not contained in 84, two ciphers must be annexed for decimals. As the dividend contains two more decimal places than the divisor, there must be the same number pointed off in the quotient. The quotient is therefore 6 one hundredths, or 6 per cent. EXAMPLES. 100. At what rate per cent. will $840 gain $49 in 2 years- and 2 months? 101. A man loaned $750 for 3 years and 4 months, and received $160.50 for the use of it. What did he receive per cent.? Wilat is the rule when the principal, interest, and time are given? COMPOUND INTEREST. 145 102. A man deposited in a savings bank $84, for which he received $2.10 for 6 months. What per cent. did he receive? 103. If $500 be received as a semiannual dividend on an investment of $100,000, what per cent. is the dividend? OBs. These rules may be concisely represented by the following formulas: Let p represent the principal, t the time, r the rate per cent., andi the interest, as follows:p = principal. t = time. - rate per cent. i interest. The first rule may be represented by the formula - p. The second rule by the formula. pxr The third rule by the formula r. pXt COMPOUND INTEREST. SECTION XIII. 1882. COMPOUND INTEREST is the interest on the principal, and on the interest added to the principal after it beconies dule. RULE. Find the interest on the principal to the time the interest becolmes due, and add it to the principal. Then find the interest on this sum for the next period, and add the interest as before. Proceed in this manner wvith each successive period at which the interest becomes dale. Subtract the first principal fron, the last suin, and the remainder twill be the compound interest. What is the rule for compouncd interest M 10 1 46 COMPOUND INTEREST. OBs. WMhen there are months and days, find the interest for them on the principal for the last period. 1. What: is the compound interest of $300 for 3 years, at 6 per cent.? $300 Interest for one year, 18 Principal for the second year, 318 Interest for the second year, 19.08 Principal for the third year, 337.08 Interest for the third year, 20.2248 357.3048 The first principal subtracted, 300 Compound interest for 3 years, $57.3048 EXAMPLES. 2. What is the compound interest of $600 for 2 years, to be paid semiannually? 3. What is the compound interest of $320 for 2~ years, at 7 per cent.? 4. What is the compound interest of $540 for 3 years, at 5 per cent.? 5. What is the compound interest of $840 for 2 years, at 7 per cent., to be paid quarterly? 6. What is the compound interest of $460 for 3 years, 4 months, and 10 days? 183. The process of computing compound interest may be much shortened by the following table, in which the amount of $1 is computed for 30 years, at 4, 5i, 6, and 7 per cent. 184. To find the amount of any sum by the table, - RULE. Miultiply the given sumr by the amount of $1 for the time, as found in the table. COMPOUND INTEhREST. 147 TABLE, Shlowing the amoznt of I dollar, or 1 2pound, for any number of years under 30, at 4, 5, 6, and? 7 per cent., compound interest. Years. 4 pzr cent. 5 per cent. 6 per cent. 7 per cent. 1 1.040000 1.050000 1.060000 1.070000 2 1. 181600 1.102500 1.123600 1.144900 3 1.124864 1. 157625 1.191016 1.225043 4 1.169858 1.215506 1.262476 1.310795 5 1.216652 1.276281 1.338225 1.402552 6 1.265319 1.340095 1.418519 1.500730I 7 1.315931 1.407100 1.503630 1.605781 8 1.368569 1.477455 1.593848 1.7181$; 9 1.423311 1.551328 1.689478 1.8384859 10 1.480284 1.628894 1.790847 1.96TI5,1 11 1.539454 1.71.0339 1.898298.2. 1 04852 12 1.601032 1.795856 2.01.2196.252'191 13 1.665073 1.885649 2.132928' 2.4.00845 14 1.731676 1.979931 2.260903 2.5785-4 15 1.800943 2.078928 2.396558 2.759032 16 1.872981 2.182874 2.540351 2.952164 17 1.947900 2.292018 2.692772 3.158815 18 2.025816 2.406619 2.854339 3.379932 19 2.106849 2.526950 3.025599 3.616528 20 2.191123 2.653297 3.207135 3.869685 21 2.278768 2.785962 3.399563 4.140563 22 2.369918 2.925260 3.603537 4.430403 23 2.464715 3.071523 3.819749 4.740530 24 2.563304 3.225099.4.048934 5. 072367 25 2.665836 3.386354 4.291870 5.427434 26 2.772469 3.555672 4.549382 5.807352 27 2.883368 3.733456 4.822345 6.213868 28 2.998703 3.920129 5.111686 6.648838 29 3.118651 4.116135 5.418387 7.114257 30 3.243397 4.321942 5.743491'7.612255 1 liS DISCOUNT. EXAMPLES. 7. What is the amount of $900 for 5 years, at 7 per cent.? 8. What is the amount of $640 for 3 years and 6 months, at 5 per cent.? 9. What is the amount of $10,000 for 30 years, at 6 per cent.? 10. What is the amount of 1$5000 for 20 years, at 5 per cent.? DISCOUNT. SECTION XIV. 185. DISCOUNT is a deduction of a certain pet cent. from a sum of money, when paid before it is due. 186. The sum paid is called the present worth. 187. To find the present worth of any sum of money, and its discount for a given time, - RULE. Divide the sum7 by the amount of $1 for the rate and time, and the quotient will be the present worth. Subtract the present worth from the sum, andc the remainder will be the discount. OBS. This rule may be expressed by the following formula:The present worth kp) is equal to the given sum or amount (a) dividecl by 1 plus the product of the time (t) and the rate (r). Thus p a a - p - the discount.!+tXr The reason of this rule is obvious. Since $1 is the present worth of its amount, the present worth of any sum will be as many dollars as the amount of 1 dollar is contained in it. What is, the rule to find the present worth of any sum of money,,ad its discount for ally given time? DISCOUNT. 149 1. What is the present worth of $336, payable in I0 months, when money is worth 6 per cent.? The amount of $1 for 10 months is $;1.05. $336' 1.05 = $320, present worth. EXAMPLES. 2. What is the present worth of $464, payable in 16 months, when money is worth 6 per cent.? 3. What is the present worth of $936, payable in 1 year and 10 months, when money is worth 7 per cent.? 4. WIVhat is the discount on $3200, payable ill 9 months, at 6 per cent.? 5. \What is the discount on $840, payable in I year and 3 months, at 6 per cent.? 6. A merchant bought a quantity of goods for $560.40 cash, and sold them the same day for $640.80 on 9 months' credit, when money was worth 6 per cent. How much did he gain on the goods? 7. What is the difference between the interest and the discount of $5000 for 1 year and 6 months? 188. It is now almost the universal practice of merchants and others to find the discount by the following rule -- RULE. Deduct the simple interest of the given sum for the rate and tihne, the difference will be the present worth. EXAMIPLES. 8. What is the present Wtorth of $90 for 2 years and 4 months? 9. What is the present worth of $175.60 for 3 years and 9 months, at 7 per cent.? m *-~ 150. BANK DISCOUNT. 10. What is the present worth of $840 for 6 months and 10 days? 11. What is the present worth of $56.75 for 2 years, at I per cent. a month? 12. A merchant sold $1500 worth of goods, one half to be paid in 6 months, the other half in 9 months. What sumn must he receive for themn iln cash after deducting 1~ per cent. a month? BANK DISCOUNT. 189. Bank discount of a note, &c., is the interest of the sumn specified in the note from the time the note is discounted to the time it becomes due, with three days additional, called days of grace. 190. To find the bank discount on a note, draft, &c., -- RULE. Find the interest on the sum specified ila the note for the ti me, including the three days of grace. Subtract the discount from the sum contained in the note, and the remainder -will be the present worth. 13. What is the present worth of $400 for 90 days, at 6 per cent.?.0155 400.00 400 6.20 $6.2000, bank discount. $393.80, present worth. The interest of $1 for the 93 days is.0155. $400 multiplied by this interest gives the bank discount, $6.20. This, subtracted from $400, gives the present worth. What is the rule to find the bank discaullt on a not, dcraft, &c. BANK. DISCOUNT. 151 EXAMPLES. 14.. What is the bank discount onl a note for $350 Df 60 days? 15.'What is the bank discount on a note for $560 of 90 days? 16. What sum will be received from a bank for a note of $640 for 4 months? 17. What sum will be received from a bank for a note of $2500, payable in 6 months? 18. What is the bank discount on a note for $5000, payable in 9 months? 191. To find the sum for which a note must be given at a bank, in order to obtain a certain sumn for a given time, - RULE. Divide the sum to be obtained by the present worth of $1 for the given rate-and time, at bank discount, alnd the quotient will be the sum required. It is evident, since the present worth of $1 requires $1 to be discounted, as many dollars will be required to be discounted for any sum as the present worth of $1 is contained in this sum. 19. For what sum must a note be given, to be paid in 60 days, in order to obtain $460 from a bank, at 6 per cent.? $1.0(000 Interest of $1 for 63 days,.0105 Present worth of $1,.9895 $460 -.9895=$464.881. The interest of $1 for 63 days, subtracted from $1, gives the present worth of $1. $460, the sum required, divided by this, gives the sum for which the note must be given. What is the rlule to find the sum for which a note must be given at a bank to obtain a certain sum for a given time? 1 52' COMMISSION. EXAMPLES. 20. For what sum must a note be written, in order to receive from a bank $500 for 6 months? 21. For what sum must a note be written, in order to receive from a bank $1000 for 9 months? 22. For what sum must a note be written, in order to receive from a bank $400 for 90 days? 23. A merchant bought a quantity of goods for $600. For what sum must he write his note, to be discounted at the bank for 6 months? 24. A farmer bought a farm for $5000 cash, anld having only half of the sum on hand, he wishes to obtain the balance from the bank. For what sum must a note be written, to be discounted for 9 months? CO M[ IS SI ON. SECTION XIV. 1 92. COMMISSION is the compensation paid to agents for their services in transacting business for others. It is usually estimated at a certainm per cent. of the money employed in the transaction. OBs. The agents are styled factors, brokers, commission nmerchants, correspondents, &c. 193. To find the commission on any sum of money,RULE. Ilmultiply the given sumn by the per cent., and point off as in decimals. EXAMPLES. 1. What is the commission on $450.60, at 2~ per cent.? 2. What is the commission on $886.50, at 3~ per cent.? WVhat is commission? Recite the rule. COMMISSION. 153 3. What is the commission on $630.40, at 4~ per cent.? 4. A commission merchant sold goods amounting to $6560.75, at 11 per cent. What was his commission? 5. An auctioneer sold goods at auction, amounting to $376.40, at 2~ per cent. What was his commission? 6. An auctioneer sold goods at auction, amounting to $640.50, at 23 per cent. What was his commission? 7. A collector was entitled to 4k per cent. onithe money collected. How much commission did he receive on $2060? 8. A commission merchant sold goods amounting to $75,600, at 3k per cent. What was his commission? 194. To find the commission when it is to be deducted from the given sum, and the remainder to be invested,RULE. Divide the given sum by $1 plus the given rate of commission, and the quotient will be the sum to be invested. Subtract the sum to be invested from the given sum, and the remainder will be the discount. O3s. This rule is the same as the rule for discount, with the exception of the time. It is evident that the given sum must contain $1 plus the commission as many times as there are dollars to be invested. EXAMPLES. 9. A gentleman sent to a broker $1218, to be invested in railroad stock, after deducting his commission of 1 - per cent. How much did the broker receive for commission, and how much did he invest? What is the rule to find the commission on a given sum, to be invested? .154 STOCKS. 10. A merchant sent to his agent $2440, to purchase goods, after deducting his commission of 21 per cent. How much was his commission, and what sum did he spend for goods? 11. A merchant sent to his agent in Buffalo $5049, to purchase flour. How many barrels of flour did he purchase, at $41'per barrel, after deducting his commission of 2 per cent.? 12. A broker negotiated a bill of exchange of $6015 for X per cent. How much commission did he receive,? STOCKS. SECTION XV. 195. THE money employed by companies in trade is styled stock, as bank stock, railroad stock, manufacturing stock, &c.; also government bonds and funds are styled stocks. 196. The whole amount invested by any company or corporation is called the capital stock, which is usually divided into shares. 197. The first cost of a share is called its par value. OBS. Par is from a Latin word, signifying equal. 198. When stocks will sell for their first cost, they are said to be at par. When they will sell for more than their first cost,'they are said to be above par, or at a premium. When they will not sell for their first cost, they are said to be below par, or at a discoungt. 199. The profits of stock are called dividends, which are paid at regular periods to the owners or stockholders. What is stock? What is capital stock? What is par va'ue? When are stocks said to be at par? When above par?'When below par? What are the profits of stock called? INSURANCE. 155 200. To find the value of stocks,RULE. WIhen the stocks are above par, rulliply the sum invested by $1 plus the per cent., and the product will be their value. When the slocks are below par, multiply the stlnl invested by $1 minnus the per cent., and the prodZuct twill be their value. EXAMPLES. 1. What is the value of $900 in stock, at 6 per cent. above par? 2. What is the value of $1000 in stock, at 3~- per cent. below par? 3. What is the value of $500 in stock, at 10~ per cent. below par? 4. What sum must be paid for 20 shares in the Western Railroad stock, at 5 per cent. above par, the par value of each share being $100? 5. What surn must be paid for 50 shares in the Worcester Railroad, at 33 per cent. below par, the par value of each share being $100? INSURANCE. SECTION XVI. 201. INSURANCE is a security given to pay a certain sum on ships, houses, or property of any kind that may be destroyed by fire, accidents, or at sea. 202. The sum paid for the insurance is called the premium, and is estimated at a certain per cent. of the value of the property insured. 203. The written agreement or certificate is called the policy. What is the rule to find the value of stocks? What is insurance? What is the premium? What is the policy? 1.O56 INSURANCE. 204. The company or persons insuring are called underwriters. Ons. Property is seldom insured for its whole value. 205. To find the premium on any amount of property insured, - RULE. M1iultiply the suin to be insured by the pet cent., and the product vwill be the premitnum. This rule is the same in principle as percentage. EXAMPLES. 1. What is the premium for insuring $4000 on a house, at 1-4 per cent.? 2. What preumium must be paid for insuring $600 on a barn, at 2j per cent.? 3. What premium must be paid for insuring $9000 on a store, at 1I per cent.? 4. What premium must be paid for insuring a cargo of cotton from New Orleans to Liverpool, valued at $9600, at 1 per cent.?,5. What premium must be paid for insuring a cargo of sugar from Havana to St. Petersburg, valued at.25,000, at 2{ per cent.? 6. If a vessel and cargo, valhed at $65,000, and insured at 41 per cent., were lost, what would be the loss to the underwriters? 7. What sum must be insured on a vessel and cargo, valued at $40,000, at 51 per cent, in order to include the premium and the property insured? S. What sum must be insured on $70,000, to lnelude the premium of 4~ per cent. and a commission on' the property insured of i per cent.? What are underwriters? Recite the rule to find the premium on property insured. PROFIT AND LOSS. 157 PROFIT AND LOSS. SECTION XVII. 206. PROFIT AND Loss treats of the gain and loss in trade, and shows how the price of goods must be adjusted, to gain or lose a certain per cent. 207. The price at which goods are bought is called the first or prime cost; that at which they are sold, the -selling price. When the selling price is greater than the prime cost, there is a gain; when the selling price is less than thle prime cost, there is a loss. 208. The gain or loss is always reckoned at a certain per cent. on the prime cost. 209. To find the gain or loss per cent., when the selling price and prime cost are known, — RULE. Divide the gain or loss by the prime cost, and the quotient,?multiplied by 100, will be the gain or loss per cent. OBS. The gain or loss is the difference between the prime cost and the selling price. The reason of this rule is obvious. The gain or loss, divided by the sum on which it was gained or lost, will give the gain or loss on 1 dollar, &c.. This, multiplied by 100, will evidently be the gain or loss on 100, which will be the per cent. 1. What is the gain per cent. on cloth bought at $5 per yard and sold at $6 per yard? 6 —5 =1. _I x L0 = 20 per cent. The difference between the prime cost and selling What is profit and loss?, What is the prime cost? What is the selling price. On what is the gain or loss per cent. always reckoned? What is the rule for finding the gain or loss per cent., when the sexling price and prime cost are known. N 158 IPROFIT AND LOSS. price is $1. This, divided by 5, and multiplied by 100, gives 20 per cent. EXAMPLES. 2. Bought sugar at 6~- cents per pound, and sold it at 8 cents. What was the gain or loss per cent.? 3. Bought cloth at $4.75 per yard, and sold it at $5 per yard. What was the gain or loss per cent.? 4. Bought cloth at $5 per yard, and sold it at $4.75 per yard. What was the gain or loss per cent.? 5. Bought molasses at 28 cents per gallon, and sold it at 31 cents per gallon. What was the gain or loss per cent.? 6. Bought corn at 65 cents a bushel, and sold it at 62 cents per bushel. What was the gain or loss per cent.? 7. Bought 7~ cords of wood at $3.50 per cord, and sold it at $3.75 per cord. What was the gain or loss per cent.? S. Bought butter at 17 cents per pound, and sold it at 20 cents per pound. What was the gain or loss per cent.? 9. A broker bought stocks at $96' per share, and sold them at $102 per share. What did he gain per cent.? 210. To find the prime cost, when the gain or loss per cent. and the selling price are known, - RULE. Divide the selling price by 1 plus the gain r' minus the loss per cent.,i and the quotient will be the prine cost 10. A merchant sold cloth at $6 per yard, and gained 20 per cent. What was the prime cost? 1+.2.201.20 ) 6.00 ( $5 6.00 What is the rule for finding the prime cost, when the selling price and the gain per cent. are known? PROFIT AND LOSS. 159 The selling price, $6, divided by 1+.20, which is 1.20, gives $5 as the prime cost. EXAMPLES. 11. A farmer sold -wood at $5.40 per cord, and gained 8 per cent. What was the prime cost? 12. A farmer sold hay at $14 per ton, and gained 25 per cent. What was the prime cost? 13. A merchant sold nails at 5. cents per pound, and lost 10 per cent. What was the prime cost? 14. If 20 per cent. be gained on raisins, at $2.75 per box, what is the prime cost? 15. If 15 per cent. be'gained on rice, at 4~ cents per pound, what is the prime cost? 16. If 12y per cent. be gained on potatoes, at 48 cents per bushel, what is the prime cost? 17. If 9 per cent. be gained on cheese, at 10 cents per pound, what is the prime cost? 18. A merchant sold sugar at 6~ cents per pound, which was 10 per cent. less than it cost him. What was the prime cost? 19. A gentleman sold land at $175 per acre, which was 25 per cent. less than it cost him. What was the prime cost? 20. A merchant sold coal at $5~ per ton, which was 8 per cent. less than it cost him. What was the prime cost? 2 11. To find at what price goods must be sold, in order to gain or lose a certain, per cent., - RULE. Find the required per cent. of the prime cost, and add this per cent. to it, if there is to be a gain; but subtract this per cent. from the prime cost7 if there is to be a loss. What is the rule for finding the selling price to gain a certain per cent.? 160 PROFIT AND LOSS. 21. Bought flour at $4 per barrel. At what price must it be sold to gain 20 per cent.? 4 X.20 =.80..80 +4 z 4.80. 20 per cent. of $4 is 80 cents, which, added to $4, gives $4.80, the selling price. EXAMPLES. 22. A merchant bought cloth at $5.00 per yard. At what price must it be sold, to gain 25 per cent.? 23. A merchant bought sugar at 7} cents per potund. At what price must it be sold, to gain 15 per cent.? 24. A merchant bought molasses at 28 cents per gallon. At what price must it be sold, to gain 12% per cent.? 25. A merchant bought cotton at 9 cents per pound. At what price must it be sold, to lose 20 per cent.? 26. A farmer bought apples at 42 cents per bushel. At what price must they be sold, to lose 25 per cent.? 27.- Bought cotton at $275,per bale. For how much must it be sold per bale, to gain 30 per cent.? 28. A merchant bought flour at $4.50 per barrel. At what price must it be sold per barrel, to lose 15 per cent.? 29. Bought molasses at 28 cents per gallon. At what price must it be sold per gallon, to lose 121 per cent.? 212. To find the gain or loss per cent. at any proposed price, when the selling price and the gain or loss per cent. is known,RULE. First find the prime cost, and then the gain or loss per cent. at the proposed price. Vhat is the rule for finding the gain or loss per cent. at any proposed price, when the selling price and the- gain and loss are known? PROFIT AND LOSS. 161 EXAMPLES. 30. A merchant sold sugar at 8 cents per pound, and gained 10 per cent. What per cent. would he have gained if he had sold it at 9 cents per pound? 31. A farmer sold corn at 65 cents per bushel, and gained 5 per cent. What per cent. would he have gained if he had sold the corn at 70 cents per bushel? 32. A farmer sold rye at 95 cents per bushel, and gained 8 per cent. What would he have gained or lost per cent. if he had sold the rye at 80 cents per bushel? 33. A man sold his farm for $4560, which was 10 per cent. more than it cost him. What would he have gained or lost per cent. if he had sold it for $400o? 34. A farmer sold land at 5 cents per foot, andgained 25 per cent. more than it cost him. What would he have gained or lost per cent. if he had sold it at 3} cents per foot? 35. A grocer sold tea at 45 cents per pound, and gained 10 per cent. What would he have gained per cent. if he haid sold it at 50 cents per pound? 36. A merchant sold broadcloth at $4.75 per yard, and gained 12~ per cent. What would he have gained per cent. if he had sold it at $5.25 per yard? 37. A farmer sold oats at 371 cents per bushel, and lost 14 per cent. What would he have gained or lost per cent. if he had sold them at 48 cents per bushel? 38. A merchant sold coffee at 11 cents per pound, and gained 10 per cent. What would he have gained per cent. if,he had sold it at 12~ cents per pound? 39. A farmer sold potatoes at 35 cents per bushel, and lost 12% per cent. What would he have gained or lost per cent. if he had sold them at 40 cents per bushel? N~ ~.i.t 162 PRACTICAL QUESTIONS. PRACTICAL QUESTIONS. 40. A man bought 12 acres of land at 3 cents per foot, and after keeping it 10 years, sells it at 20 per cent. advance. Allowing money to be worth 6 per cent., does he gain or lose, and how much? 41. A merchant bought 500 barrels of flour in Chicago, at $4 per barrel; he paid for freight to Boston 65 cents, and for truckage, 7 cents per barrel; he sold it in Boston at $5I per barrel. How much did he gain per cent.? 42. A merchant bought at New Orleans 500 bales of'cotton, of 300 pounds each, at 61 cents per pound; he paid for freight to Liverpool 1~ cents per pound; for wharfage and truckage, $50; he sold the cotton for 9 cents:per pound. Did he gain or lose, and how much per cent.? 43. A merchant bought in Maine 1.50,000 feet of lumber, at $9 per 1000 feet; he paid for freight, truckage, and wharfage, $4860. At what price per foot must lie sell the lumber, to gain 20 per cent.? 44. A merchant bought in Vermont 6000 bales of wool, of 100 pounds each, at 40 cents per pound; he paid for freight and truckage to Boston 65 cents per bale. At what price per pound must he sell the wool, to gain 25 per cent.? 45. A merchant sold flour at $5~ per barrel, and thereby gained 121- per cent. What would he have gained per cent. had he sold the flour at $6 per barrel? 46. A grocer bought 10 boxes of Havana sugar, of 400 pounds each, at 61 cents per pound; he paid for freight, truckage, &c., $1.75 per box; he gained 2 per cent. on the weight of the sugar; he sells it at 71 cents per pound. How much does he gain per cent.? 47. A merchant sold tea at 45 cents per pound, and gained 12. per cent. What would he have gained per cent. if he had sold the tea at 54 cents per pound? RATIO. 163 RATIO. SECTION XVIII. 213. RATIO is the relation which one quantity bears to another of the same kind with respect to magnitude. 214. The ratio of two numbers is the quotient resulting from dividing the first by the second. Thus the ratio of 12 to 4 is 3; the ratio of 30 to 5 is 6; since 12 divided by 4 is j:1 -3, and 30 divided by 5 is %-Q —6. 215. The two numbers are called the terms of the ratio. The first is called the antecedent, the second, the consequent, and may either be expressed in the form of a fraction, -the antecedent for the numerator, and the consequent for the denominator, -as -a-, or by placing two points between them, as 12: 4. OBs. Both of the numbers must either be abstract numbers or of the same kind. 216. If both terms of the ratio be multiplied or divided by the same number, the ratio will not be changed. Thubs 12: 4 is the same ratio as 6: 2, 24: 8, and 36: 12. This is evident fiom the fact that the terms of a ratio are the terms of a fraction, wvhich may be multiplied or divided without changing its value, (ART. 84.) 217. An inverse or reciprocal ratio is the ratio of the consequent to the antecedent, and is expressed by changing the order of the terms, or by inverting the fraction. Thus the ratio of 3 to 6, or -, is a direct ratio; the ratio of 6: 3, or a, is an inverse or reciproWhat is ratio? What is the ratio of two nitubers? ~What are the numbers called? HIow may a ratio be expressed? WVhat is an inverse ratio? 161 RATIO cal ratio, and is always the same as the ratio of the reciprocals of those numbers. Thus -: ~. OBS. For definition of reciprocal, see Obs. 1, Art. 166. 218. A compound ratio is composed of two simple ratios. Thus, The ratio of 5: 20 is 4. " " 6: 36 is 6. The ratio of 5 X 6: 20 X 36 is 24. As the terms of a ratio are the same as the terms of a fraction, they may be treated as such in every respect. Ons. The question, What is the ratio of one number to another is the same as, What part of one number is another EXAMPLES. 1. What is the ratio of 8: 4? 2. What is the ratio of 9: 6? 3. What is the ratio of 12: 16? 4. What is the ratio of 27: 30? 5. What is the ratio of 39: 48? 6. What is the ratio of 64: 72? 7. What is the ratio of 84: 104? 8. What is the ratio of 96: 112? 9. What is the ratio of 102: 28? 10.'What is the ratio of 148: 24? 11. What is the ratio of a: 2? 12. What is the ratio of 5:'? 13. What is the ratio of':? 14. What is the ratio of 7: -~?9 15. What is the ratio of 5~: 1-6? 16. What is the ratio of 71: 15k? 17. What is the ratio of 3: 191? 18. What is the ratio of,5y: 233? 19. What is the ratio of 7: 11? PROPORTION. 165 PROPORTION. SECTION XIX. 219. PROPORTION is the union of two equal ratios. Thus, 6:12: 4: 8, or -a-:. OBs. Proportion is expressed by four dots between the ratios. Thus, 2: 4:: 3: 6. 220. The first and fourth terms are called the extremes;' the second and third, the means. Ons. The first and third terms are sometimes called the first and second antecedent; and the second and fourth, the first and second consequent. 221. In any proportion, if the first termn be greater or less than the second, or equal to it, the third term will also be greater or less than the fourth, or equal to it. 222. In every proportion the product of the extremnes is equal to the product of the means. Thus in the proportion, 20: 5:: 36: 9, the product of 20 X 9180, 5X36=180. 223. Four nunmbers are in proportion when the product of the extremes is equal to the product of the meaLns. 224. If any three terms of a proportion are known, the other may easily be found. 225. The product of the second and third terms, divided by the first, will give the fourth. OBs. Let the unknown term be represented by u. 3 3 8:6:: 12: u. 6X( =9 -zu. What is proportion? What are the first and fourth terms called What are the second and third called? 166 PROPORTION. 4: 9: 7:u. 9 15 -= u. 4 226. The product of the second and third terms, divided by the fourth, will give the first. 2X4 u:4::2 6. 2 u. 6 12X8': 8:: 12: 7. - - 13 = u. 7 227. The product of the first and fourth termns, divided by the third, will give the second. 3XIS 3'U' 12 18. - - =-u. 12 6:'':: 5: 14. _ 16 1 --. 5 228. The product of the first and fourth terms, divided by the second, will give the third. 5:3::': 15. 15X5 _-25_u. 3 7X4 7: 10:: u:4. 2- U 10 229. If four quantities are in proportion, they will remain so if the extremes are put in the place of the means, and the means in the, place of the extremes. The proportion will also remain the same if the meahs and the extremes are interchanged, as in the following examples:12: 4::15: 5 4:12:: 5: 15 12: 15:: 4: 5 4: 5:: 12: 15 5: 4:: 15 12 15: 12:: 5: 4 5 15: 4 12 15: 5: 12 4 PROPORTION. 167 In each of these proportions the product of the extremes is equal to the product of the means. As a proportion is composed of two simple ratios, it may always,be changed to the form of fractions, and treated the same as simple ratios. 230. When three terms are given or known, they may be arranged in a proportion by the following rule - RULE. Write that number for the third termn which is of the same kind as the fourth term. If the fourth term must be greater than the third term, write the greater of the other two numbers for the second term, and the less for the first. If the fourth termn,must be less than the third, write the greater of the other two numbers for the first term, and the less for the second. The product of the second and third termns, divided by the first, will give the fourth term. Ous. 1. Factors which are common to the dividend and divisor should be cancelled. Ons. 2. If either of the terms be a denominate number, they must be reduced to the lowest denomination mentioned in either. OBS. 3. It may readily be perceived, from the conditions of the question, whether the fourth term be greater or less than the third. 1. If 12 yards of cloth cost $42, what will 16 yards cost? 4 14 12: 16:: 42: u. 56. Make $42, the same kind as the required or fourth term, the third term. It is evident that 16 yards must cost more than 12 yards; 16 yards must there' fore be the second term, and 12 yards the first. The product of the second and third, divided by the first, gives $56. The 12 is cancelled,'because its factors, 3 and 4, are also factors of 16 and 42. What is the rule for finding the fourth term when the first three terms axe known? 168 PROPORTION. EXAMPLES. 2. If 6 yards of cloth cost $30, what will 9 yards cost? 3, If 12 bushels of wheat cost $15, how many bushels can be bought for $75? 4. If 14 pounds of flour cost 68 cents, what will 196 pounds cost? 5. If 9 cords of wood cost $30, how many cords can be bought for $156? 6. If 12 men can cut 49 cords of wood in a day, how many cords can 20 men cut in the same time? 7. If 12 pounds of rice cost $25, how many pounds can be bought for $56~? 8. If 24 acres cost $160, what will 164 acres cost? 9. If 12 men can perform a piece of work in 60 days, how many men would perform the same work in one third of the time? 10. If 48 mnen can build a wall in 36 days, how many men wilI be required to do the same in 72 days? 11. If 160 barrels of flour can be bought for $640, how many barrels can be purchased for $2240? 12. If 10 tons of hay can be purchased for $96; how many tons can be purchased for $240? 13. How far can a man travel in 16 days, if he travel 960 miles in 12 days? 14. If 9 men can reap 12 acres of rye in 15 days, how much would 15 men reap in the same time? 15. If A can do a piece of work in 9 days, and B can do the same in 12 days, what part of it can both do in 3 days? 16. If a person walk 396 miles in 14 days, of 1 hours each, in how many days, of 9 hours each, can he walk the same distance? 17. A hare, pursued by a dog, was 96 yards before him at starting. The dog ran 7 yards while the hare PROPORTION. 169 ran 5. How far did the dog run before overtaking the hare? 18. $540 were divided between three persons, A, B, and C, in the following proportion: A received $5 as often as B received 6, and C 7. How mlch did each receive? 19. There are two numbers in proportion as 4 to 9, the larger of which is 117. What is the smaller? 20. There are two numbers in proportion as 9 to 7, the smaller of which is 126. What is the larger? 21. If a staff 3 feet, long cast a shadow 5 feet in length, what is the height of a tower whose shadow at the same time is 175 feet? 22. If a reservoir containing 5690 gallons have two pipes, one of which discharges 50 gallons a minute, and the other admits 45 gallons a miniute, in how long time, when it is full, will it be emptied? 23. If 36 yards of carpeting, I of a yard wide, will cover a floor, how many yards, 1- wide, will be reqnired to cover the same floor,? 24. If v of a ship be worth $12,000, how much will 8 of the.same be worth? 25. If 9. yards of cloth are worth $57, what are 12}- yards worth? 26. If 16 men can perform a piece of work in 10 days, how many men can perform a piece of work 6 times as large in ~ of the time? 27. If 9 men can mow 2t acres in 5 days, how many men would be required to mow 40 acres in the same time? 28. If A can cut 2 cords of wood in 12~ hours, and. B dan cut 3 cords in 17~ hours, how Inany cords can they both cut in 24i hours? 29. A, B, and C start at the same time and the same place to travel round an island 75 miles in circumference. A goes 5 miles a day, B 8, and C 10. In what time will they all be together? 0 170 ANALYSIS. 30. If 69 yards of carpet, v of a yard wide, will cover a floor 18 feet wide, what is the length of the roorn? 31. At what time, between 3 and 4 o'clock, are the hands of a watch together? 32. If a board be 9 inches wide, what must its length be to contain 6 square feet? 33. Two ships sailed together fromn Boston to San Francisco. One sailed at the rate of 6 miles an hour, and the other 5~, on an average, the whole distance. The first arrived in 165 days. In how many days did the other arrive? 34. If the circumference of the wheel of a railroad car be 7 feet, and it make 5 revolutions in a second, in how long time will the car run from Boston to Providence, a distance of 42 rniles-? 35. If the diameter of a wheel be 4 feet, and it make 445 revolutions in a mile, what would be the diameter of a wheel which makes 593~ revolutions in the same distance? ANALYSIS. 231. The preceding examples may also be performed by analysis. 232. Analysis is a process of fidlding the value of a unit, or any part of a unit, of a given number, and from this, finding the value of any proposed number of, units, or parts of units. Thus, if 3 yards of cloth cost $12, what will 5 yards cost? If 3 yards cost $12, 1 yard will cost ~ of $12, or $4. If 1 yard cost $4, 5 yards will cost 5 times $4, which is $20. 36. If 25 bushels of corn cost $1.50, what will 7~ bushels cost? In 2~ bushels there are 5 halves'of a bushel. If 5 halves of a bushel cost $1.50, 1 What is analysis > COMPOUND PROPORTION. 171 half will cost 30 cents. If 1 half cost 30 cents, 71, which is 15 halves, will cost 15 times 30 cents, which is $4.50. 233. When the numbers are large, cancellation may be used with advantage in the analytical method. 37. If 15 cords of wood cost $52.50, how much will 20 cords cost? 17.50 4 X - 70.00. 1 If 15 cords of wood cost $52.50, 1 cord will cost 9; of $52.50, which may be expressed by writing $52.50 for a numerator, and 15 for a denominator. 20 cords will cost 20 times as much, which may be expressed by a fraction and by the sign of multiplication. The factors of 15 are 3 and 5, which are commoll to the numerator, 52.50, and to 20. COMPOUND PROPORTION. SECTION XX. 234. COMPOUND PROPORTION is composed of a com2pound and a simple ratio. 235. A compound proportion may be resolved into as many simple proportions as there are pairs of terms, or may be combined into one. 236. In every compound proportion, one of the given numbers is always of the same kind as the required number. 237. RULE. Write that numberfor the third term What is compound proportion? Into what may a compound proportion be resolved? What-is of the same kind as the required term? What is the rule for compound proportion 172 COMPOUND PROPORTION. which is of the same kind as the required terme. Then take two numbers of the same kind for the first and second terms, and arrange them as in simple proportion. - Proceed in this manner with each pair of simnilar terms, and write them under the former. The continued product of the third and all the second terms, divided by the continued product of all the first terms, will give the required term. 1. If 16 horses eat 9 bushels of oats in 6 days, how many horses will eat 24 bushels in 8 days? 8 2 2 9:24:: 16 x x X2 8: 6 - 32. Make 16 the third term, as it is of the same kind as the required term. As more horses are required to eat 24 bushels than 9, in the same time, make 24 the second term, and 9 the first, of the two remaining terms. Make 8 days the first, and 6 days the second term, as a less number of horses is required to eat the same quantity in 8 days than in 6 days. The continued product of all the second and the third terms, divided by the continued product of the first terms, gives 32 horses as the required term.'BS. 1. The numbers to be multiplied and divided should be written as directed in the rule for cancellation, (AnT. 65.) OBs. 2. The first and second terms must be of the same denomination, and if the third term be a denominate number, it must be reduced to the lowest denomination mentioned in it. OBs. 3. The conditions of every question should be thoroughly understood before attempting to arrange the numbers in proportion. EXAMPLES. 2. If 6 cows produce 56 gallons of milk in 4 days, how many gallons will 26 cows produce in 7 days? 3. If 5 furnaces consume 30 tons of coal in 6 days, how many tons, at the same rate of consumption, will 6 furnaces consume in 40 days? COMPOUND PROPORTION. 173 4. If a person travel 120 miles in 4 days, by walking 9 hours a day, what time will be required to travel 386 miles, by walking 7 hours a day? 5. If the freight of 2 tons, 12 cwt. 20 lb. for 42 miles be $8.50, what would be the freight of 16 tons, 10 cwt. 16 lb. for 140 miles? Questions in compound proportion may be most intelligibly performed by analysis and cancellation. 6. If 32 horses eat 24 bushels of oats in 8 days, how many bushels will 16 horses eat in 6 days? 3 4x xt6 X 18 2- ~ 9 bushels. If 32 horses eat 24 bushels in 8 days, 1 horse will eat I of 24 in 8 days; and in 1 day l horse will eat S as much as in 8 days. 16 horses will eat 16 times as much as 1. horse in 1 day, and 6 times as much more in 6 days. 24 is therefore divided by 32 and 8, and multiplied by 16 and 6. The common factors being cancelled, the required term is 9 bushels. 7. If 6 men earn $150 in 4 weeks, working 6 days a week, how much would 10 men earn in 7 weeks, working 5 days a week? 8. If 84 men mow 72 acres of grass in 15 days, how many acres will 96 mlen mow in 12 days? 9. If 27 men build 54 rods of wall in 26 days, how many rods will 32 men build in 39 days? 10. If $300 gain $16 in 12 months, what principal will gain $10 in 9 months? 11. If 18 compositors can set up 24 sheets in 8 days, how many sheets would 45 compositors set up in 14 days? 12. If 5 persons can be maintained 30 days for $60, o0X 174 PARTNERSHIP. how much money would be required to support 24 persons 365 days? 13. If 12 men can build 18 rods of wall in 30 days, working 9 hours a day, how many days would be required for 18S men, working 8 hours a day, to build a similar wall, 52 rods long? 14. If 16 men can build a wall 40 rods long, 4 feet high, and 3 feet thick, in 16 days, working 8 hours a day, in how many days will 20 men, working 9 hours a day, build a similar wall, 160 rods long, 6 feet high, and 5 feet thick? 15. If 1080 bricks, 8 inches long, and 2 inches wide, are required for a walk 20 feet long, and 6 feet wide, how many bricks will be required for a walk 100 feet long, and 4 feet wide? PART NERSH IP SECTION XXI. 238. PARTNERSHIP is the process of ascertaining the gain or loss of partners in trade. 239. The money invested is called the capital or stock. 240. The profit to be divided is called the dividend. 241. To find each partner's share of the profit or loss, when there is no reference to time, — RULE. Divide the whole gain or loss by the amount of the whole stock, or the number of shares in trade, and multiply the quotient by each man's stock, or share of the stock. What is partnership? What is the money invested called? What is called' the dividend? TPARTNERSHIP. 1 7Z It is evident that the whole gain or loss, divided by the sum on which it wvas gained or lost, will give the gain or loss on $1. This, multiplied by each partner's stock or interest, will give the gain or loss of each partner. J1. Two persons, A and B, trade together. A puts in $600, B $800. They gain in one year $280. What is each man's share of the profit? 600 +- 800 - 1400. 20 6. x $120, A's share. X400 1 20) 8 0 X -00 $160, B's share. it00 1 Thire whole gain, $280, divided by the amount cf the whole stock, $1400, and multiplied by each partners'share, gives $120 for A's share, and $160 foi B's share. EXAMPLES. 2. Three persons, A, B, and C, enter into partnership. A advances $400, B $500, and C $600. They gain by trade $640. What is each person's share of the profit? 3. A, B, C, and D, purchase a ship. A pays for 6 shares, B for 5, C for 4, and D for 3. They receive net freight $4560. What sum ought each to receive? 4. An insolvent debtor owes to one of his creditors $450, to another $560, to a third $840. His property amounts to only $1250. How much will each of his creditors receive? 5. A gentleman left his estate in his will to his four sons, A, B, C, and D, as follows: To A $1600, to B $1500, to C 1860, and to DI 2000. But his whole 176 PARTNERSHIP. property, after his debts were paid, amounted only to $4840. How much will each son receive? 6. A, B, and C enter into partnership. A's stock was $800, B's $900, and C's $1000. They lose 10 per cent. of the whole stock. What was each man's share of the loss 7. Two persons engage in trade. The whole sum invested is $5000. They gain $600. A put's in.5 of -5 of { of the whole, and B puts in the remainder. What was each man's share of the gain, and what did each put in? 8. A, B, and C owned a ship and cargo, valued at $75,000, and insured for $60,000, which w.as a total loss. A owned ~, B -, and C the remainder. How much of the insurance ought each to receive, and what was each man's share of the loss? 9. A and B purchase a lot of land for $4500. A pays ~ of the price, B the remainder. They gain by the sale of it 20 per cent. of the first cost.' What is each man's share of the gain? 10. A, B, and C purchase a lot of woodland. A pays $200, B $300, and C $400. A works 40 days in cutting the wood, B 30 days, and C 2.0 days. A pays for cutting $75, B $50, and C $20. They sell 400 cords of wood, at $4 a cord. Allowing each man's labor to be worth $ t a day, what ought each man to receive? 11. The suni of $5000 is to be divided among 4 persons as follows: A is to receive -, B a, and C is to receive $5 as often as D receives $4. How much ought each man to receive? 242. To find each partner's share of the gain or loss, when the capital is invested for different periods, RULE. Multiply each manl's stock by the time of What is the rule for finding each partner's share of the gain or loss, when the capital is invested for different periods? PARTNERSHIP. 177 its continuance in trade, and divide the whole gain or loss by the sum of the several products, and multiply the quotient by the product of each man's stock and tinme, which will be each partner's gain or loss. Oias. The principle of this rule may be applied to the solution of a great variety of questions, whose conditions are similar in tLeir nature. 12. Three merchants, A, B, and C, entered into partnership. A put in $600 for 4 months, B $750 for 3 months, and C $900 for 5 months. They gained $488. What is each man's share of the gain? 600 X 4=- 2400 750 X 3 - 2250 900 X 5 - 4500 9150 32 488 4 0400 488.. -- X — 00 $128, A's share. 9150 10 1 30 4- X $120, B's share. 60 4- X 000 $240, C's share. Dividing the whole gain, 488, by the stun of the products, gives the fraction 1-%,s which, reduced toits lowest terms, is -. This, multiplied by the product of each man's stock and time, gives $128 for A's share, $120 for B's share, and $240 for C's share. 75 being a factor of 2400, 2250, and 4500, the numerators of the several fractions, and also in the denominator, it can be cancelled in each. 12 178 PARTNERSHIP. EXAMPLES. 13. A, B, and C entered into partnership. A put in $1000 for 4 months, B $900 for 5 months, C $1200 for 3 months. They lost $600. What was each partner's share of the loss? 14. A, B, and C contract to perform a certain piece of work. A employs 40 men for 4~ months, B 45 men for 3A months, C 50 men for 2& months. They gain, after paying all expenses, $850. What part of the gain belongs to each? 15. A commenced business January 1, with a capital of $10,000. April I he admits B as a partner, with a capital of $5000. September 1 they receive C as a partner, with a capital of $3000. At the end of the year they had gained $2600.'What was each man's share of the gain? 16. Four men hired a pasture for $175. A puts in 16 cows for 8 months, B puts in 12 cows for 9 months, C puts in 10 cows for 10 months, D puts in 8 cows for 12 months. 2How much ought each to pay? 17. Three persons, A, B, and C, form a partnership for 1 year, commencing January 1. A puts in $4000, B $3000, and C $2500. April 1, A withdraws $500 and B withdraws $600. June 1, C puts in $S00 more. September 1, A fuirnishes $700 more, and B $400 more. At the end of the year they find they have gained $1500. What is each person's share of the gain? 18. A commenced business on the first day of January with a capital of $25,000. On the first of April he admits B as a partner, who furnishes $5000 capital. On the first of June they admit C as a partner, with a capital of $9000. At the end of two years they dissolve partnership, and find they have gained $8000. Each partner received his share of the stock and prof its. What did each receive? EQUATION OF PAYMENTS. 179 EQUATION OF PAYMENTS. SECTION XXII. 243. EQUATION OF PAYMENTS is the process of determining the average time for the payment of several suims, due at different periods. 244. RULE. Multiply each suvm by the time before it becomes due, and divide the suimn of the products by the sumn of the payments. OBS. 1. This rule has been considered by many as incorrect; but it is as correct as any rule founded upon simple interest can be. If A owes B $200, $100 of which are to be paid in cash, and the other $100 to be paid in two years, the average imne for the payment of the rwhole, according to the rule, would be one year) which is correct; for it is evident that if the money be paid according to the agreement, B would have at the end of two years $200, and the interest of $100 {for two years. If the whole be paid in 1 year, B would have at the end of two years $200, and the interest of $200 for one year, which is the same as the interest of $100 for two years. OBS. 2. If any sum be paid on the day from which the time is reckoned, it will have no product, but it must be added with the others in finding the sum of the payments. OBS. 3. It is customary with merchants to add three days grace. This rule is founded upon the principle that the use o)f $1 for any number of days is equivalent to the use of as many dollars as there are days for 1 day. Thus, the use of $1 for 1.50 days is equivalent to the use of $150 for 1 day. 1. A gentleman owes $150, payable in 60 days, 8200, payable in 90 days, $400 in 120 days. At what time ought the whole to be paid at once? 150X 60_ 9000 200X 90-18000 400 X 120_-48000 75000 =100 days. 750 75000 750 What is equation of payments? What is the rule? ISO EQUATION OFi PAYMENTS. EXAMIPLES 2. A owes B $1500, of which $100 are to be paid in 4 months, $200 in 6 months, and the remainder in 9 months. What is the average time of payment? 3. A merchant sold the following bills of goods, on a credit of 6 months: May 10, a bill of $600; June 12, a bill of $450; September 20, a bill of $900.- At what time will the whole become due? 4. A gentleman has to pay $5000 as follows: $540 in 4 months;, 2400 in ~8 months, $600 in 10 months, the remainder in 6 months. What is the average time for the payment of the whole sum? 5. A owes B $1200. $400 are due at the present time, $300 are due in 8 months, and $500 are due in 12 months. At what time ought the whole to be paid? 6. A gentleman left his son $1500, to be paid as follows: - in 3 months, - in 4 months, {- in 6 months, and the remainder in 8 rmonths. At what time ought the whole to be paid at once? 7. A merchant bought goods amounting to $6000. He agrees to pay $500 in cash, $600 in 6 months, $1500 in 9 months, and the remainder in 10 months. At what time ought he to pay the whole in one payment? 8. A owes B $1600, to be paid in 6 months. A agrees to pay $600 in cash. At what time ought the remainder to be paid? 9. A gentleman purchased a farm for $3600, and agrees to pay $600 down, and the remainder in 5 equal semiannual instalments. At what time may the whole be paid at once? 10. A owes B $5000, ~ to be paid in 30 days, i in 3 months, ~ in 70 days, and the remainder in 9 months. At what tinle ought the whole to be paid at once? 11. A bought goods of B, amounting to $1200. W of the bill was to be paid in cash,. to be paid in 2 months, the remainder in 9 months. At what time ought the whole to be paid? EQUATION OF PAYMENTS. St 245. To find the equated time of payments, when the sumrs due have different dates, - RULE. Find the ti me when each sum becomes due, and multiply each sumt by the timne between it and the first suvm that, is due, and divide the amount of the several products by the amount of the sums due. OBs. As the time at which the first sum is due is the period from Nxhich the average time is computed, the first sum will have no product. The principle of this rule is the saime as that of the preceding, (ART. 244.) 12. What is the equated time for the payment of the following sums — January 10, due $1000 a'' 15,' 2000X 5- 10000 25, " 2500 X 15= 37500 February 26, " 1000X47= 47000 March 27, " 1500 X 76 114000 April 6, - 2000 X 86-172000 10000 ) 380500 ( 38: 30000 80500 80000 38 days from January 10, is February 17, the equated time. EXASMPLES. 13. A merchant sold 484 barrels of rosin, as follos: - February 6, 4 months' cr., 35 barrels, i $3.12~. March 12, " 38 " d $3.00. " v; (" 411 "v $2.621. What is the equated time for the payment of the whole? What is the rule to find the equated time of payments, when the sums due have different dates? Fp 1S2 TAXES. 14. A merchant sold 1650 barrels of flour, as follows - May 6, 3 months' cr., 150 barrels, ~ $4.50. 20, 4 4 " "6 400 " ) 4.75. July 10, 5 4 00 " O 5.00. August 4, 4 4' 9" 600 " ( 4.25. What is the equated tinme for the payment of tlhe whole? 15. A merchant sold 576 barrels of rosin, as follows: May 3, 6 months' cr., 62 barrels, i $2.050. " 10, " " "' 100 " ~ 2.50. " 18, cash, 10 " ~ 2.50. 4 26, 30 days' or., 50 4' f 2.7.5. 4 " 6 months' " 34 5'44 2.50. " 9 G D / 2.00. What is the equated time for the payment of the whole? TAXES. SECTION XXIII. 246. A TAX is a certain sum of money assessed on individuals to pay the expenses of government, or of a corporation, society, district, &c. 247. Taxes for paying the expenses of govermenet are usually assessed on the property of individuals, and on male persons without property, -when of a certain age. The individual tax is called the poll tax. OBss. 1. Poll is from a Dutch word, signifying the head. OBs. 2. Every male individual over 20 years of age, is required by the laws of Massachusetts to pay a poll tax, Oas. 3. Taxable property is of two kinds, viz., real estate and personel property. All property that is unchanging, such as houses, lands, &c., is call.ed re:d estate. All other property, such as monevy What is a talt.? -ow Care taxes nuaually assesed? TAXES. 183 aotes, stocks, mortgages, furniture, cattle, &c., is called personal property. 24.8. To assess a state or other tax, - RULE. Find the amount of all the taxable property, real and personal, and also the number of individuals liable to be taxed. If there be a poll tax, multiply the number of individuals or polls by the amount assessed on each poll, and subtract the p2roduct from the sum to be raised. Divide the remainder by the whole amtount qf taxable property, which will be the tax on one dollar. fMultiply the amount of eachT mtan's property by the tax on one dollar, and the product will be the tax orn his property, which, added to his poll tax, will be his whole tax. 1. A tax of $12,500 is to be assessed upon the inhabitants of a town in which there are 840 ratable polls. The tax on each poll is $1.50. The real and personal property in the town is valued at $5,364,560. What will a man's tax be, whose real estate is valued at $8500, and personal property at $15,000? $12,500-.$1260 for poll taxes-$11,240..53~-4-4=.002095 on a dollar. This sum, multiplie'd by the amount of his property, produces $49.2325-+$ 1.50, the poll tax,-$50.7325, the required sum. EXAMPLES. 2. What is a man's tax whose real estate is valued at $25,000, personal $12,500, and 3 polls, in a town where the property is valued at.$4,278,560, and the amount assessed is $10,000, there being 750 ratable polls? 3. What will be the amount of a man's tax in the same town, whose property is valued at $75,000? 4. What will be the amount of a man's tax in the same town, whose property is valued at $7320.47? What is the rule to assess a state or other tax? 184 DUTIES. DUTIES. SECTION XXIV. 249. THE sum of money imposed by government on most goods imported from foreign countries is called duties. 250. Duties are either ad valorem or specific. 251. Ad valorem duties are a certain per cent. of the value or first cost of the goods. OBS. Ad valorem is from the Latin, signifying according to the vahute. 252. Specific duties are a certain sum imposed on the gallon, yard, hundred weight, ton, &c. OBS. All duties are regulated by government, and are often changed. By the tariff of 1846, ad valorem duties are required to be paid. 25,3. It is usual to make certain deductions, called tare, draft, &c.. before specific duties are imposed. 2B4. Tare is a deduction of a certain per cent. of the weight of the goods, made after the draft, for the box, cask, &c. 255o Draft is a deduction of a certain number of pounds made for Wraste. 2.56. An allowance of 2 per cent. is made for leakage on any liquor in casks, subject to duty by the gallon, and 10 per cent. on all ale, beer, and porter, in bottles, and 5 per cent. on other liquor in bottles; or the duty is computed on the actual quantity, at the option of the importer, to be ascertained by actual measurement. 257. All goods entered at the custom-house are, when bought and sold, subject to the same deductions as are made when they are entered. What are duties? What are ad valorem duties? What are specific? What is tare? What is draft? What allowances are usually made? DUTIES. 185 258. On all goods not entered at the custom-house there is a special agreement respecting tare, draft, &c. 259. To find the ad valorem duty on any rnerchandise, -- RUtE. Multiply the amount of the first cost of the merchandise by the rate -per cent. OBs. In.estimating the first cost of goods, all expenses in the foreign port are included. EXAMPLES. 1. What is the ad valorem duty, at 30 per cent., on glass, china, and stone ware, the invoice amounting to $1260? 2. What is the ad valorem duty, at 30 per cent., on an invoice of cutlery, amounting to $760? 3. What is the ad valorem duty, at 100 per cent., on an invoice of brandy and cordials, amounting to $1560? 4. What is the ad valorem duty, at 30 per cent., on a cargo of Russia iron, amounting to $7560? 5. What is the ad valorem duty, at 40 per cent., on an invoice of raisins, figs, dates, and currants, amounting to $4560? OBs. An invoice is a written account of merchandise, containing the price of each article and the charges of exportation. 260. To find the specific duty on any merchandise, -- RULE. First make all the required deductions for tare, draft, 6'c., and multiply the remainder by the duty on each gallon, yard, pound, a'c. 6. What is the specific duty, at 2 cents per pound,. on 950 bags of coffee, each weighing 200 pounds, tare 2 per cent.? 7. What is the specific duty, at 12 cents per gallon, on 25 pipes of molasses, each pipe containing 120 gallons, allowing 2 per cent. for leakage? What is the-rule for ad valorem duties? for specfic? p qu 186 ALLIGATION MEDIAL. ALLIGATION IIEDIAL. SECTION XXV. 261. ALLIGATION MEDIAL is the process of finditlg the price of a mixture composed of several ingredients of different values. 262. To find the price of a mixture, when the price of each ingredient and the quantity are given,RULE. Ml]ultiply each ingredient by its price, and divide the sum, of the products by the sumt of the iiigredients. The quotient will be the price of the mixture. 1. A grocer mixed 10 pounds of tea worth 60 cents per pound with 15 pounds worth 75 cents per pound, and 25 pounds worth 40 cents per pound. What is the price of one pound of the mixture? 10X60 — 6.00 15 x 75 11.25 25 X 40=10.00 50 27.25 27.25 50 —54.- cents. EXAMPLES. 2. A goldsmith mixes 2 lbs. of gold of 18 carats fine, 3 lbs. of 20 carats fine, and 4 lbs. of 22 carats fine. How many carats fine is the mixture? 3. If 16 bushels of oats at 40 cents per bushel, 10 bushels of corn at 65 cents per bushel, and 12 bushels of barley at 75 cents per bushel, were mixed together, what would be the price of the mixture? 4. A grocer mixes 10 pounds of tea at 40 cents per pound, 20 pounds at 45 cents per pound, 30 pounds at 50 cents per pound. What would a pound of this mixture be worth? What is alligation medial? What is the rule? ALLIGATION ALTERNATE. 187' ALLIGATION ALTERNATE. SECTION XXVI. 263. ALLIGATION ALTERNATE is the process of fitlding what quantity of ingredients, of different prices, must be taken to compose a mixture of a given price. RULE. Find how n?,much is gained or lost by taking, one of each kind of the proposed ingredients. Thetn take one or mnore of the ingredients, or such parts of themn, as will make the gain and loss equal. 1. A trader would mix four sorts of tea, viz., at 4s., 6 s., 9 s., and 12 s. per pound. How many pounds of each sort must be taken, that the mixture may be worth 8 s. per pound? Gain. Loss. Ilb. at 4, 8 - 4 4 1 lb. at 6, 8 - 6 2 l b. at 9, 9 - 8 I 1 llb. at 12, 12 - 8 4 1lb. at 9, 9 -- 8 I 6 6 By taking 1 pound of each ingredient, it is evident there will be a gain of 1 s. By taking 2 pounds at 9 s., the gain and loss will be equal. The same result may be obtained by taking 1-k pounds at 12 s. and 1 pound of each of the other ingredients. EXAMPLES. 2. How much sugar, at 5 cents, at 6 cents, and 9 cents per pound, must be mixed together, that the mixture may be worth 7 cents per pound? 3. In what proportion must sugar, at 4 cents, at 8 What is alligation alternate? What is the rule? ] 88 ALLIGATION ALTEINATE. cents, and 10 cents per pound, be mixed, that the corn pound may be worth 6 cents per pound? 4. In what proportion must gold, valued at 12, 16, 18, and 24 carats fine, be mixed, that the compound may be worth 20 carats fine? 5. In what proportion must grain, valued at 50 cents, 56 cents, 62 cents, and 75 cents per bushel, be mixed together, that the compound may be 62 cents per bushel? 6. In what proportion must corn at 70 cents per bushel, oats at 45 cents per bushel, and rye at 60 cents per bushel, be mixed, that the compound may be worth 55 cents per bushel?'364. When one or more of the ingredients are limited intlulantity, to find the other ingredients, - Rum..'Find how much is gained or lost by taking one of each of the proposed ingredients, in connectionz ~with the ingredient which is limited, and if the gain and loss be not equal, take such of the proposed ingredients, or such parts of them, as will make the. gain and loss equal. 7. How much sugar that is worth 6 cents, 10 cents, and 13 cents per pound, must be mixed with 20 pounds of sugar worth 15 cents per pound, to make a compou:nd worth 11 cents per pound? Gain. Loss. I lb. at 6, 11 - 6 51 lb. at 10, 1 1 -- 10 - 1 1 lb. atl3 13 -11 - 2 20 lb. at 15, 15 11 - 80 6 82 -6 - 76 15 lb. at 6, 11 - 6 75 1 lb. at 10, 11 -- -- 1 82 82 16 lb. at 6 cents, 2 lb. at 10, and 1 lb. at 13 cents. What is the rule when one or more of the ingredients are limited? ALLIGATION ALTERNATE. 189 By taking one of each of the ingredients at their prices, in connection with the limited ingredient, 20 lbs., there is a loss of 76 cents. As there is 5 cents gain on 1 lb. at 6 cents, on 15 lbs. there would be a gain of 75 cents. And as there is 1 cent gain on 1. lb. at 10 cents, by taking 15 lbs. more at 6, and 1 lb. at 10, making 16 lbs. of the one, and 2 lbs. of the other, the gain and loss are equal. EXAMPLES. 8. How much sugar, at 10 cents, at 9 cents, and 8 cents per pound, must be mixed with 30 pounds at 6 cents per pound, that the compound may be worth 8 cents per pound? 9. How much gold of 16 and 18 carats fine must be mixed with 90 ounces of 22 carats fine, that the compound may be worth 20 carats fine? 265. To find the quantity of each ingredient, when the sum of the ingredients and the mean price are given, - RULE. Find the least quantity of each ingredient by ART. 263. Then divide the given amount by this sumn, acid mnutitply the quotient by the quantities found for the least proportional quantities. 10. How much tea, at 25 cents, 35 cents, 50 cents, and 70 cents per pound, must be mixed with 1, pounds, that the mixture may be worth 45 cents' Gain. Loss. lib. at 25, 45 - 25 - 20 1 lb. at 35, 45 - 35 = 10 1 lb. at 50, 50 - 45 =- 5 1 lb. at 70, 70 - 45 - 25 4 30 30 180 4 45 lb. of each. What is the rule when the sum of the ingredients and the mean price are given? 1.90 DUODECIMALS Proof. 45 X 25 = 1125 45 X 35 - 1575 45 X 50 = 2250 45 X 70 - 3150 8100 + 180 - 45 EXAMPLES. 11. How much sugar, at 6 cents, 8 cents, 10 cents, and 12 cents per pound, must be mixed with 200 pounds, that shall be worth 9 cefits per pound? 12. How much gold, of 18, 20, and 22 carats fine, must be mixed with alloy, in order to form a composition of 40 ounces, worth 16 carats fine? DUODECIMALS. SECTION XXVII. g66. DUODECIMALS are a species of denominate numbers. 12 of each lower denomination make one of the next higher. 267. The denominations are feet, inches or primes, seconds, thirds, fourths, &c., which are distinguished from each other by marks called indices. lTable. 12' inches - 1 foot. 12" seconds = 1' inch or prime. 12"' thirds 1" second. 12'"" fourths 1"' third. 1 ft. -— _ 12' inches. 1 ft. - 144" seconds. 1 ft. - 1728"' thirds. 1 ft. - 20736'// fourths. What are duodecimals? What are the denominations, and how are they distinguished? Recite the table. MULTIPLICATION OF DUODECIMALS. 1'91 OBS. Duodecimals is from a Latin word, signifying twelve. 268. Duodecimals may be added or subtracted in the same manner as other denominate numbers. tXAMPLES. 1. What is the sum of 14 ft. 10 in. 8"/ 9/"', 15 ft. 6 in. 7"/ 10'//, 17 ft. 11 in. 4// 6"'//? 2. What is the sum of 20 ft. 10 in. 7// 9"'/, 26 ft. 6 in. 5// 4//', 30 ft. 5 in. 2// 3/', 40 ft. 8 in. 9/ 3"'/? 3. From 24 ft. 7 in. 3// 4/"' take 18 ft. 8 in. 6" 5"'. 4. From 60 ft. 6 in. 8" 9/"' take 56 ft. 7 in. 9// 11/'. MULTIPLICATION OF DUODECIMALS. 269. RULE. Write the corresponding denominalions of the multiplicand and nmultiplier under each other. Multiply each denornination in the qultiplicanzd in succession by each denomination in the multiplier, and for'every twelve in the product add one to the next product, and write the remainder under its corre.sponding denomination. Add together the partial products, and for every twelve add one to the next colu.mn. 270. The foot being considered the unit or whole number, the multiplication of duodecimals is the same ill principle as, the multiplication of common fractions. Thus 6 ft. X 6 ft. - 36 square feet. 6 ft. X 6 in. is the same as 6 ft. X ~Z ft. - - -36 in. 6 ft. X 6/ is the same as 6 ft. X T - T- - 36/. 6 ft. X 6"// is the same as 6 ft. X T-=- - wre ft. = 36"'. And ( in. X 6 in. is the same as -6, ft. X -T6 = - ft. 36/'. 6 in. X 6" is the same as'6f X T -- Tr3 ft. -36"//. 6'" X 6f" is the same as -TA1 ft. X T --,_3 -6.3 ft. -- 36 "", &c. How may duodecimals be added and subtracted? What is thei rule for the multiplication of duodeoimals? 19'2 MULTIPLICATION OF DUODECIMALS. It is evident from the preceding illustration thatFeet X by feet produceesq. ft. Feet X by inches produce inches. Feet X by seconds produce seconds. Feet X by thirds produce thirds, &c. Inches X by inches produce seconds. Inches X by seconds produce thirds. Inches X by thirds produce fourths. Seconds X seconds produce fourths, &c. Ons. 1. The product of any two denominations will have as many indices as there are in both of the denominations. Ons. 2. Many mechanics divide the foot into tenths, hundredths, &c.; and it is to be regretted that this is not the universal.practice, as the calculations would be much more simple and easy. 5. Multiply 10 ft. 6 in. 8" by 8 ft. 4 in. 7". ft. in. "t /ft tlit ft. in. ft rtt lit 10 6 8 10 6 8 8 4 7 8 4 7 84 5 4 6 1 10 8 3 6 2 8 3 6 2 8 6 1 10 8 84 5 4 ft. 88 5' 8" 6"' 8'"" ft. 88 5' 8" 6"' 8"" Ons. It is obvious from the preceding examples that it is not important whether the lowest denomination in the multiplicand be multiplied first by the highest or lowest denominations in the multiplier, provided the remainder be placed directly under its corresponding denomination. E XAMPLESS; 6. Mulltiply 15 ft. 9 in. 8" by 14 ft. 6 in. 9". 7. How many square feet in a board 12 ft. 4 in. long and 10 inches wide? 8. How many square feet in a board 15 ft. 4 in. long and 1 ft. 3 in. wide? 9. How many cubic feet of wood in a load 8 ft. 3 in. long, 3 ft. 9 in. high, and 3 ft. 10 in. wide? 10. How many cubic feet in a pile of wood that is DUODECIMALS. 1193 50 ft. 9 in. long, 4 ft. 3 in. high, and 3 ft. 10 in. wide? 11. How many square feet in a room that is 16 ft. 4 in. long and 18-ft. 8 in. wide? 12. How many cubic feet in a block of granite 15 ft. 6 in. long, 3 ft. 4 in. wide, and 2 ft. 9 in. thick? 13.' How many cubic feet in a load of gravel that is 5 ft. 4 in. long, 3 ft. 9 in. wide, and 11 in. high? 14. How many cubic feet in a cellar that is 40 ft. 9 in. long, 36 ft. 8 in. wide, and 6 ft. 4 in. deep? 15. How many cubic feet in a stone wall that is 3 ft. 4 in. high, 2 ft. 9 in. thick, and 120 ft. long? 16. How many yards of carpeting, a: yard wide, wvill be required to cover a floor that is 24 ft. 6 in. long and 18 ft. 6 in. wide? 17. How many yards of carpeting,: yard wide, will be required to cover a floor that is 15 ft. 4 in. long and 16 ft. 6 in. wide? 18. How many feet of wood in a pile that is 60 ft. 9 in. long, 4 ft. 3 in. high, and 3 ft. 10 in. wide? OBs. Cellars and other excavations are generally estimated by sqluares, 216 cubic feet being considered a square. 19. How many squares in a cellar that is 50 ft. 6 in. long, 36 ft. wide, and 6 ft. 4 in. deep? 20. How many squares in a trench that is 4 ft. deep, 3 ft. 6 in. wide, and 100 ft. long? 21. How many square yards of plastering in a room 16 ft. 8 in. long, 15 ft. 6 in. wide, and 9 ft. 10 in. high? 22. How many bushels of corn will a bin hold that is 20 ft. long, 4 ft. 6 in. wide, and 3 ft. 8 in. high, there being 2150.4 cubic inches in a bushel? 23. How many hogsheads will a cistern contain that is 13 ft. 6 in. long, 9 ft. 8 in. wide, and 8 ft. high, allowing 231 cubic inches to a gallon, and 63 gallons to a hogshead? 0. 13 194 INVOLUTION. 24. In a room 16 ft. 8 in. long, 15 ft. wide, and 10 ft. high, there are' three doors, each of which is 6 ft. 8 in. high and 2 ft. 8 in. wide; two windows 3 ft. 4 in. wide and 5 ft. 6 in. high; and a fireplace that is 4 ft. high and 4 ft. 6 in. wide. How many yards of plastering are there in the room, and what would it cost to plaster it, at 12 1 cents per yard? 25. How many squares of gravel would be required to raise a lot of land 5 ft. 6 in. that is 100 ft. long and 80 ft. wide; and how many horse-loads, supposing each load to be 5 ft. 6 in. long, 4 ft. 3 in. wide, and 10 iii. high? INV O L UT ION. SECTION XXVIII. 271. INVOLUTION is the process of finding the required power of any number, by multiplying it into itself. 272. The number multiplied is called the root. 273. Powers are of different orders, as the second, the third, the fourth, &c. The second power is also called the square, the- third the cube, the fourth the biquadrate. In the s5econd power the root is used twice as a factor, in the third power it is used three times, in the fourth:four times, &c. 274.'The power of a number and the number of times it is taken as a factor is generally indicated by a small figure called an index, or exponent, placed above.the given umrnber, a little to the right. The index of the second power is 2, and of the third power is 3, &c. 52 =- 5 X 5 — 25 53 _ 5.X 5 X 5 -125 54 - 5 X 5 X 5 X 5 - 625 What is involution? What is the root of a number'? INQVOLUTION. 105 275. To find the required power of any number, PRULE. Multiply t7te given number into itself in succession, till it is taken as a factor as many times as there are units in the index of the required power. OBs. A common fraction is raised to any required power by raising each term to the required power. A mixed'number should first be reduced to an improper fraction, or to a decimal~ and compound and complex fractions to simple ones. EXAMPLES. 1. What is the second power of 6? 2. What is the second power of 9'? 3. What is the third power of 4 4. What is the third power of 7? 5. What is the third power of 8? 6. What is the fourth power of 3? 7. What is the fourth power of 6? S. What is the fifth power of 12? 9. What is the sixth power of 9? Table of Roots and Powers. Roots. 1 2 3 4 5 6 7 2d Power. 1 4 9 16 25 36 49 3d Power. 1 8 27 64 125 216 343 4th Power. 1 16 81 256 625 1296 2401' 5th Power. 1 32 243 1024 3125 7776 16807 6th Power. 1 64 729 4096 15625 46656 117649' 17th Power. 1 128 2187 16384 78125 2799361823543i 276. The product of two powers of the samne nuMmber is equal to that number raised to the power denoted by the sumln of the indices. Thus, 6 X 64 -63+- 4-,67. 63 —216; 64=1296.1296X216-279936-67. 277. The quotient of any power of a number, 7 divided by a less pozwer of the same number, is equal to TWhat is the rule for finding the required power of any number? 196 EVOLUTION. the number raised to the power denoted by the difference of their indices. Thus 67-64_67 —-6 63. 67 279936. 64 - 1296. 279936' 1296 - 216 - 63. 278. Any power of a composite number is equal to the product of the same powers of its factors. Thus, 123 -1728; 43 X 33- 1728. E VOL UT IO N. SECTION XXIX. 279. EVOLUTION is the reverse of involution, and is the process of finding the roots of any given numbers. 280. The square root of any number is designated by the following signs: /, ( )-. Thus, 7/49 and (49)i denote that the square root of 49 is to be extracted.