CASE III. To subtract a proper fraction or a mixed one from a whole number. 1. From $7 take $35. OPERATION. 7 3 To subtract in this example we must borrow 1 from the 7 in the minuend, and reduce it to eighths (g) and the must be taken from them; § from $33 Ans. leaves g. To pay for the 1 which was borrowed, 1 must be added to the 3 in the subtrahend, 1 + 3 4, and 4 taken from 7 leaves 3, and the placed at the right hand of it gives the answer $33. By adopting the following rule, the same result will be ob tained. RULE. Subtract the numerator from the denominator of the fraction, and under the remainder write the denominator, and carry one to the whole number of the subtrahend to be subtracted from the minuend. If it be required to subtract one mixed number from another mixed number, the following method may be adopted. 14. From 8 take 4. OPERATION. 83/ 44 Ans. 3. In this question, we multiply the 3 and the 7, the 815 numerator and the denominator of the fraction in the minuend, by 5, the denominator of the fraction 33 in the subtrahend, and we have a new fraction 1, which we write at the right hand of the 8, thus, 815. We then multiply the numerator and denominator of the subtrahend by 7, the denominator of the minuend; and we have another new fraction, g, which we place at the right hand of the 4, thus, 48. It will now be perceived, that we have changed the fractions 83 and 44 to other fractions of the same value, having a common denominator. We now subtract as in question 1, by adding 1 (=) to, which makes, and ; thus, 58-38-38. We then from this we subtract carry the 1 we borrowed to the 4, 1+ 4-5, which we take from 8, and find 3 remaining. The answer, then, is 333. If the fraction in the subtrahend be less than the fraction in the minuend, we proceed as in the following problem. 15. From 94 take 3. OPERATION. 94 31 91 Ans. 61. Having reduced the fractions to a common denominator, as in the last problem, we subtract 35, the numerator of the subtrahend, from 48, the numerator of the minuend, and the remainder 13 we write over the common denominator, which we annex to the difference between 9 and 3,6; thus, 612. 3 612 18. 21. Miles. 84 16. cwt. 17. 19. 20. 28. From a hhd. of wine there leaked out 7 quantity remained ? gallons; what Ans. 55 gal. was absent 512 Ans. 24 da. 29. A man engaged to labor 30 days, but days; how many days did he work? 30. From 144 pounds of sugar there were taken at one time 171⁄2 pounds, and at another 28 mains? pounds; what quantity reAns. 971lb. 31. A man sells 97 yards from a piece of cloth containing 34 yards; how many yards remain ? Ans. 244yd. 32. The distance from Boston to Providence is 40 miles. having set out from Boston, has travelled of the distance; and B. having set out at the same time from Providence, has gone of the distance; how far is A. from B. ? CASE IV. To subtract one fraction from another, when both fractions have a unit for a numerator. The student will perceive, that this operation reduces the fractions to a common denominator. RULE. Write the difference of the denominators over their product. 2. Take from 1, 1, 1, †, t, †, §; 2b from To, TT, 12, 15. 3. Take from †, 1, 1, 1, t, t, 4; 1 from 1, 16, 15. from 1,,,; from 1, 4, 1, 4, §. from 1, 1, 1 ; † from 1, 1, 1, 1. 4. Take 5. Take 6. Take from 1, 1, 1, 1, 1, 8, 7, §, §. 7. Take from ; from, ; from . 8. Take 9. Take 10. Take NOTE. -- from †, 1, 4, t, t, t, d, d, to, tr If the numerators of the given fractions be alike, and more than a unit, multiply the difference of the denominators by one of the numerators for a new numerator, then multiply the denominators together for a new denominator. 19. Take 18 from 42; 18 from 19; 19 from 19; 19 from 19. 20. Take 2 from 12; 12 from 12; 12 from 12; 13 from 12. NOTE.-The above questions, and those of a similar kind, may readily be performed mentally. CASE V. To subtract compound numbers. 1. From of a £. take of a £. OPERATION. £. = 12s. 8d. (33 common denominator Ans. 8s. 313d. 11 13 To perform this question we find by Case XI., Sect. XVI., the value of 7£.—12s. 133 88d.; and also of £. 4s. 54d.; we then find a com = mon denominator of the fractional part, by multiplying together their denominators, 11 × 3 = 33. We then proceed as in Case II., Sect XVIII. This question can be performed by first subtracting the fraction of a £. from of a £., and then reducing the remainder by Case XI., Sect. XVI.; thus: cwt. to the fraction of a ton by Case IX., Sect. XVI., and subtracting it from of a ton, and then reducing the remainder to its proper terms by Case XI., Sect. XVI. Thus: 2% of a ton = 1qr. 26lb. 14oz. 10,82 dr. Ans. = RULE. — Find the value of the fractions in integers; then subtract as in the foregoing rules. 3. From of an ell English take 4. From of a mile take 5. From of a degree take of a yard. Ans. 3qr. Ona. 2in. of a furlong. Ans. 1fur. 5rd. 10ft. 10in. of a mile. Ans. 49m. Ofur. 13rd. 11ft. 9in. 14bar. 6. From of an acre take of a rod. Ans. 1R. 17p. 22yd. 2ft. 108in. 7. From of a cord take of a cord. Ans. 91ft. 16021gin. 8. From of a hhd. of wine there leaked out of it; what remained? Ans. 6gal. 3qt. Opt. 17gi. 9. From Boston to Concord, N. H., the distance is 72 miles; of this distance, how much remains ? Ans. 30m. 6fur. 34rd. 4ft. 8in. 11⁄2bar. of a week. having travelled 10. From of a year take 11. From of an acre take Ans. 101da. 5h. 54m. 174sec. of a foot. Ans. 1R. 18p. 5yd. 4ft. Oin. SECTION XIX. MULTIPLICATION OF VULGAR FRACTIONS. CASE I. To multiply a simple fraction by a simple fraction. 1. Multiply by %. OPERATION. This process may be understood by sup× Ans. posing a man to have found of a dollar, and that he gave of it to his wife, and that he wished to ascertain what part of a dollar his wife received. If of a dollar be divided into 5 equal parts, one of these parts will be of a dollar. And, if of of a dollar be of a dollar, ofwill be 7 times as much, and 7 times are If then, of be, of will be 3 times as much, and 3 times are. The wife will therefore receive of a dollar. RULE. - Multiply the numerators together for a new numerator, and the denominators for a new denominator. The fraction should then be reduced to its lowest terms. |