Ans. 32 2. Reduce of a penny to the fraction of a pound. { x 12 x zo = , the Answer. 3. Reduce il to the fraction of a penny. Ans. d. 4. Reduce şq to the fraction of a pound. Ans. Tuo 5. Reduce cwt to the fraction of a lb. 6. Reduce dwt to the fraction of a lb troy. 7. Reduce crown to the fraction of a guinea. 8. Reduce half-crown to the fract. of a shilling. Ans. *1. 9. Reduce 2s 6d to the fraction of a £e. Ans. 1. 10. Reduce 175 7d 339 to the fraction of a te. Ans. co. ADDITION OF VULGAR FRACTIONS. Ir the fractions have a common denominator; add all the numerators together, then place the sum over the common denominator, and that will be the sum of the fractions required. * If the proposed fractions have not a common denominator, they must be reduced to.one. Also compound fractions must be reduced to simple ones, and fractions of different denominations to those of the same denomination. Then add the numerators as before. As to mixed numbers, they may either be reduced to improper fractions, and so added with the others; or else the fractional parts only added, and the integers united afterwards. can. * Before fractions are reduced to a cominon denominator, they are quite dissimilar, as much as shillings and pence are, and there. fore cannot be incorporated with one another, any more than these But when they are reduced to a common denominator, and made parts of the same thing, their sum, or difference, may then be as properly expressed by the sum or difference of the numerators, as the sum or difference of any two quantities whatever, by the sum or difference of their individuals. Whence the reason of the Rule is manifest, both for Addition and Subtraction. When several fractions are to be collected, it is commonly best, first to add two of them together that most easily reduce to a common denominator; then add their sum anıl a third, and so on. EXAMPLES EXAMPLES. 1. To add and together. Here + 4 = = 1}, the Answer. • 2. To add } and together. t + f = 8 + = f = 1, the Answer. 3. To add and 7; and of together, +7+ofj={+'*+={+ + ==81. 4. To add and together. Ans. 13. 5. To add and together. Ans. 17. 6. Add and together. Ans. 7. What is the sum of į and and ? Ans. 11os 8. What is the sum of į and } and 27? Ans. 30 9. What is the sum of } and of }, and 936? Ans. 107. 10. What is the sum of į of a pound and į of a shilling? Ans. Poss or 13s 10d 29. 11. What is the sum of } of a shilling and its of a penny? Ans. Td or 7d 119. 12. What is the sum of 4 of a pound, and of a shilling, and of a penny? Ans. tags or 3s ld 1119. SUBTRACTION OF VULGAR FRACTIONS. PREPARE the fractions the same as for Addition, when necessary; then subtract the one nunierator from the other, and set the remainder over the common denominator, for the difference of the fractions sought. EXAMPLES, 1. To find the difference between and %. ś == , the Answer. *-= - 38 = , the Answer. 3. What Ans. Ans. s 3. What is the difference between 1 and 1? 4. What is the difference between 1 and ? 5. What is the difference between fand ? Ans. It 6. What is the diff. between 51 and 4 of 4+? Ans. 475 7. What is the difference between of a pound, and of of a shilling? Ans. Poss or 10s 7d 19. 8. What is the difference between 5 of 53 of a pound, and of a shilling? Ans. 10371 or 1/8s 11 d. MULTIPLICATION OF VULGAR FRACTIONS. * Reduce mixed numbers, if there be any, to equivalent fractions; then multiply all the numerators together for a numerator, and all the denominators together for a denominator, which will give the product required. EXAMPLES 1. Required the product of and ;. Here *} = %= }, the Answer. . 2. Required the continued product of 1, 3, 5, and of Ź 13 3 13 x 3 39 Here = 4, Ans. 3 4 4 5 4 x 2 3. Required the product of į and k. 1. Required the product of 4 and Ans. 1. 5. Required the product of }, $, and in Ans. 25 Х Х 1 Ans. as * Multiplication of any thing by a fraction, implies the taking some part or parts of the thing; it may therefore be truly expressed by a compound fraction; which is resolved by multiplying together the numerators and the denominators. Note, A Fraction is best multiplied by an integer, by dividing the denominator by it; but if it will not exactly divide, then multiply the numerator by it. 6. Required Ans. 276. 6. Required the product of }, }, and 3. Ans. 1: 7. Required the product of , }, and 4tr 8. Required the product of , and of . Ans .. 9. Required the product of 6, and of 5. Ans. 20. 10. Required the product of of }, and of 33. Ans .. 11. Required the product of 3 and 4**. Ans. 1413 12. Required the product of 5, į, of , and 45. Ans. 275. DIVISION OF VULGAR FRACTIONS. * PREPARE the fractions as before in Multiplication; then divide the numerator by the numerator, and the denominator by the denominator, if they will exactly divide: but if not, then invert the terms of the divisor, and multiply the dividend by it, as in Multiplication. LXAMPLES. Ans. 4. 1. Divide by s. Here ”f=í = lý, by the first method. 2. Divide by **s. Heref=ts = { x = {x{== 45. 3. It is required to divide us by t. Ans. 7. 4. It is required to divide to by . 5. It is required to divide in by Ans. 1 6. It is required to divide { by '. 7. It is required to divide }} by. 8. It is required to divide , by Ans. 13. Ans. 4. Ans. IO * Divison being the reverse of Multiplication, the reason of the Rule is evident. Note, A fraction is best divided by an integer, by dividing the numerator by it; but if it will not exactly divide, then multiply the denominator by it. • 9. It is required to divide is by 3. Ans. Po Ans. o Ans .. Ans. 171 RULE OF THREE IN VULGAR FRACTIONS. MAKE the necessary preparations as before directed; then multiply continually together, the second and third terms; and the first with its parts inverted as in Division, for the answer * EXAMPLES. : Х Х 5 3 1. If y of a yard of velvet cost į of a pound sterling; what will is of a yard cost ? 3 2 5 8 2 E B =6s 8d, Answer. 8 16 Ans. 1/ 1s. 4 d. 3. If is of a ship be worth 273/2s 6d; what are of her worth? Ans. 2277 12s Id. 4. What is the purchase of 12301 bank-stock, at 108 per cent. ? Ans. 13361 1s 9d. 5. What is the interest of 2731 15s for a year, at 3 per cent. ? Ans. 81175 114d. 6. If of a ship be worth 73/ 1s 3:1; what part of her is worth 2501 10s? 7. What length must be cut off a board that is 7 inches broad, to contain a square foot, or as much as another piece of 12 inches long and 12 broad ? Ans, 1844 inches. 8. What quantity of shalloon that is of a yard wide, will line 95 yards of cloth, that is 2 yards wide? Ans. 31; yds. Ans.. * This is only multiplying the 2d and 3d terms together, and dividing the product by the first, as in the Rule of Three in whole numbers. VOL. I. 9. If |