7. Find the sum of of 16, and of of 27. Thus of 16=16, and of of 2781, then 16 and 8 reduced to a common denominator, will become 128 and 243 which added together, make 37-15 Ans. 24 8. Add together of 25, and 7 of 152. 9. Find the sum of 5 and 2 of 5 of 13. 10. Add together 1, 3, and 7 of 3 of 51. Ans. 20 Ans. 3. Ans. 31. Fractions of different integers (as found in money, weight, &c.) must be reduced to those of the same, before adding; or the value of each fraction may be found by Problem VIII. page 101, and then added together. 347 11. Add of a shilling to of a pound. Thus, s. reduced to the fraction of a pound, by Prob. IV. page 171, equal to £; then, £; and £ added together=144 of a pound, whose value by Prob. VIII. page 94,=4s. 9d. 3 qrs. Ans. Or thus, s.Os. 4d. 2qrs. and £ 4 5 1 Ans. as before 49331 12. Add 1⁄2 of a lb. Troy to 3 of a pwt. Ans. 10oz. Opwt. 9grs. 13. Add together of a hogshead, of a gallon, and á quart. 15. Add of a mile, and 1 16. Add of a week, of a minute together. of Ans. 11gal. Oqt. Opt. of a pound: Ans. 1qr. 4lb. 6oz. 2dr. of a yard together. Ans. 4fur. 98yds. 2ft. 1 in. a day, of an hour, and of Ans. 2d. 2h. 30min. 45sec. SUBTRACTION OF VULGAR FRACTIONS. RULE. Prepare the fractions as in addition; then the difference of the numerators written over the common denominator, will be the difference of the fractions required. EXAMPLES. 1. From take of. Thus, of, then and reduced to their least common denominator by Rule 2, page 169, become 21 and 12, and the difference between the numerators, 21 and 12, is 9, which written over the common denominator, becomes} Ans. Note. To subtract a fraction from a whole number, take the numerator from its denominator, and place the remainder over the denominator, then take 1 from the whole number. To subtract mixed numbers without reducing them to im proper fractions. When the lower fraction is greater than the upper one, subtract its numerator from the common denominator, and to the difference add the upper numerator, then carry 1 to the lower whole number. 8. From 16 subtract. Thus, & taken from & leaves, that is, the numerator 5, subtracted from the denominator 9, leaves 4, which placed over the denominator is, then 1 taken from 16, leaves 15. Ans. 15. Ans. 2413. 9. From 25 take 1. 10. From 29 take 15; thus, and reduced to a common denominator. make and 2 Ans. 131 lower fraction (9) from its denominator (12) and to the difference add the upper numerator (8) makes 11=}}) we then carry 1 to the whole number, and subtract, and the rem. is 1311. 11. What is the difference between 145 and 193. 12. From 36 take 9. Ans: 478. Ans. 26 MULTIPLICATION OF VULGAR FRACTIONS. RULE. Reduce Compound fractions to simple ones, and whole and mixed numbers to improper fractions. Then multiply the numerators together for a new numerator, and the denominators together for a new denominator EXAMPLES. 1. What is the product of 5 multiplied by of? thus, 56, and of, then6x=2. Ans. 2. Multiply by. 3. Multiply by 3. 4. Multiply 4 by 1. Ans. 43=1. Ans. 51 103 Ans. 9 16 5. Multiply of 3 by 3 of 4. Ans. 6. Multiply of 5 by of 8. Product 21. 7. Multiply 7 by 91. Product 693. 8. Multiply 123 by of 7. Product 293. DIVISION OF VULGAR FRACTIONS. RULE. Prepare the fractions as before, then invert the divisor and proceed exactly as in multiplication; and the products will be the quotient required. How many times is contained in § ? Thus, by the rule we invert the divisor, which becomes ; then 18, or 218, the answer. 5 Illustration of the Rule.-Had it been required to divide by 2 a whole number, instead of 2 ninths, it is evident it would have given the quotient 23 eighths, or ; but the divisor, being ninths, will be contained in the dividend 9 times were the whole number is contained 1 time; therefore the quotient is 9 times too small, and must be multiplied by 9, the denominator of the dividing fraction, and 9 18, or 213 the answer as before. This process consists only in multiplying the numerator of the divisor into the denominator of the dividend, and the denominator of the divisor into the numerator of the dividend. 3. Divide of by 3 of 3. 24 Thus, x, Ans. 4. What is the quotient of 75 divided by of 7? 15 5. Divide 9 by 18. 6. Divide 4 by of 4. 7. Divide 5 of 4 by 45. Ans. 488=138. 75 20 Quotient 20 8. How many times is 35 contained in of 27? 39 41 Ans. 12195 9. How many times is contained in 48 ? Ans. 317 10. Divide 19 by 37. Quotient 11. At of a dollar per bushel, how many bushels of oats can be bought for $195, or $112? Ans. 4415bu. 12. At 7 of a dollar per bushel, how much corn can be bought for 27 dollars? Ans. 31 bushels. RULE OF THREE DIRECT IN VULGAR FRACTIONS. RULE. Prepare the fractions as in multiplication: then state the question in the same manner as taught in the Rule of Three in whole numbers; then invert* the first term and multiply all three of the terms continually together, and the product will be the answer in the same name of the second or middle term. EXAMPLES. 1. If of a barrel of flour cost of a dollar, what will of a barrel cost? Thus, :::. Then the divisor or first term being inverted will stand thus &××$120, and $120 = 5dollars, the answer. 249 The same, by analysis.-If cost of a dollar 1 barrel, *The reason of inverting the first term in this rule is very evident, since fractions depend on the same principle as whole numbers, and this produces the same effect as that of multiplying the second and third terms together and dividing by the first; therefore, the first term being a divisor, we invert it as taught in Division of Vulgar Fractions RULE OF THREE DIRECT IN VULGAR FRACTIONS. 177 will cost 8 times as much. Thus, 8 times=40; then of $40-$120, or 5 dollars, the answer as before. 2. If of a bushel of wheat cost of a dollar, what will 8 bushels cost? Thus, :: 2, and 3××11861=$1081, Ans. 3. If 10 dollars will buy 8 bushels of wheat, how much wheat will of a dollar buy? 41 $ bu. $ bu. Thus, 1081861, and 8} = 4; then : , Ans. of an gyd. of of =28, or Ell English. Ell. £ Ell. 9 15 £ 72 S. d. qrs. Then, as:::; and 2×××12-13 8 21 5. If of a cwt. cost of a dollar, what will 15 cwt. come to? Ans. $20, 30c. 8m.+ cost $9, what will 301 Ans. $2041,87. 6. If of of an acre of land acres come to? 7. If her worth? of a vessel cost 1236 dollars, what are 32 of Ans. $206. 8. A merchant sold 5 pieces of cloth, each containing 12 yards, at 9s. d. per yard; what did the whole come Ans. £31 9s. 10d. 3 qrs. 9. At $3 per cwt., what will 93lbs. come to? to? 841 2688' Ans. $, or 31cts. 28m.+ 10. A person owning of a vessel sold of his share for 835 dollars; what was the whole vessel worth at that rate ? Ans. 1565 dollars 62 cents. RULE OF THREE INVERSE IN VULGAR FRACTIONS. RULE. Prepare the given fractions, and state the question as in direct proportion; then invert the third term and multiply all three of the terms together, the product will be the answer in the same name of the middle term. EXAMPLES. 1. How much flannel that is of a yard wide, will line 5 yards of cloth which is 14 yards wide? yds. wide. yds. yards. As 13:5; and 3x23x=345=143 Answer. |