PRINCIPLE. To subtract one number from another, whether whole or fractional, they must be reduced, if not already so, to like units. 189. CASE I.-When the denominators or fractional units are alike. 1. Subtract from f ANALYSIS. Since the numbers to be subtracted, viz: 3 tenths and 9 tenths, have the same unit, 1 tenth, we subtract as in whole numbers, and obtain 6 tenths, or f=§, Ans. Hence, the RULE.-Take the less numerator from the greater, and place the difference over the common denominator. 2. What is the difference between 3. What is the difference between How much more is: and ? and ? 190. CASE II.-When the denominators or fractional units are unlike. 1. From take EXPLANATION.-Since 3 fifths and 1 half are unlike, they can not be subtracted in their present form. Reducing them to a common denominator, by Art. 183, we obtain 6 tenths and 5 tenths, the difference between which is 1 tenth, or . Hence, the RULE.-Reduce the fractions to a common denominator, and proceed as in Case I. EXPLANATION.-We first reduce the fractions to a common denominator, then take their difference, and unite it to the difference of the whole numbers. Thus, from leaves; 5 from 8 leaves 3; now uniting the 3 and 12, we get 312. From: 17. 12 take 7. Ans. 18. 13 take 81. Ans. 19. 35 take 223. Ans. 13. 23. 1953 take 126. 20. 27 take 23. Ans. 45. 24. 2005 take 85,7. Ans. 3811. Ans.? Ans.? 57. 21. 45 take 71. Ans.? 25. A rope was 483 feet long, but 17% feet were cut off; how long was the rope then? 26. From 8 take §. EXPLANATION.-8 is equal to 7 and one, or, reducing one to ninths, 7 and 3. Hence, we write 8 under the form of 7%, and subtract § as in the preceding examples. How much more is: 27. 12 than ? Ans. 114. 28. 13 than? Ans. 12161. 29. 14 than ? Ans. 13g. 30. 43 than? Ans. 421. OPERATION. 8=73 = 73. 35. From 7 take 54. EXPLANATION.-We reduce the fractions to a common denominator, and as we can not subtract from, we take one, or 12, from the 7, and add it to, making 19. Then we say, from 11⁄2 leaves, and 5 from 6 (7 less one) leaves 1. What is the value of: 36. 81-37? Ans. 4§. 37. 163 — 123? Ans. 311. OPERATION. 7}=712 5=512 17. 39. 688-497? Ans. 1843. 40. 100-6511? Ans. ? 38. 431-18? Ans. 241.41. 1463-861? Ans. ? 42. A man had $5 and paid for a knife $; how much did he have left? Ans. $4. 43. One melon weighs 25 pounds, and another pounds; how much heavier is one than Ans. 7 pounds. weighs 17 the other? 44. Frank and John went fishing; Frank walked 18 miles and John 2 miles; how much further did Frank walk than John? Ans. 15148 miles. 45. A farmer sold 15 acres from a field containing 37% acres; how many acres had he left in the field? Ans. 351 acres. 46. One bale of cotton weighs 463 pounds, and another bale weighs 17 pounds less; what does the lighter bale weigh? Ans. ? 191. In the following examples the whole and one or more parts are given, and the c part required. See Art. 89. 47. The whole is 3854 days, and one part 671 days; what is the other or c part? Ans. 317 days. 48. A man had $16734 and spent $3564; how many dollars had he left? Ans. $131627. 49. A man had $540 and spent $271 for sheep, and $180 for hogs; how much money did he have left? Ans. $88. 50. A flag-pole, standing in the water, is 100 feet long; 72 feet of its length are above the water, and 12 feet are in the mud below the water; how deep is the water? Ans. 1433 ft. 51. Two boys, Charles and Henry, are 400 feet apart; if Charles goes towards Henry 1675 feet, and Henry goes towards Charles 2074 feet, how far apart will they then be? Ans. 245 ft. PRINCIPLE. To multiply two abstract numbers, whether whole or fractional, we may multiply the numbers regarded as units together for a new unit, and the numerators, or numeral factors, together for a new numerator. 193. CASE I. To multiply a fraction by a whole number. EXPLANATION. - The multiplicand is 5 sixths and the multiplier 3. 3 times 5 sixths are 15 sixths, or 15 = 21. Instead of multiplying the factor 5 by 3, we may multiply the unit factor 1 sixth by 3. By Art. 173, 3 times 1 sixth is 1 half. Hence, 5 sixths X3=5 halves, or {2}. Hence, the = RULE.-Multiply the numerator by the whole number, or divide the denominator by the whole number when it can be done without a remainder. 15. If 1 bushel of potatoes cost of a dollar, what will 8 bushels cost? 16. If 1 basket holds of a bushel of bushels will 8 baskets hold? Ans. ? corn, how many Ans. 6 bushels. |