2. Mixed numbers may be reduced to improper fractions, then added according to the rule; or, we may add the whole numbers and fractional parts separately, and then unite their sums. 8 Ans. 18, or 1. 13. What is the sum of 4, and 1⁄2? ? SUBTRACTION OF FRACTIONS. MENTAL EXERCISES. Ex. 1. Henry had of a watermelon, and gave away of it: how much had he left? Solution.-3 sevenths from 5 sevenths leaves 2 sevenths. Ans. . 2. John had of a bushel of chestnuts, and gave away how many had he left? 3. If I own of an acre of land, and sell of it, how much shall I have left? QUEST.-Obs. How are mixed numbers added? 4. A man owning the ship had he left? 5. William had many tenths had he left ? of a ship, sold: what part of of a dollar, and spent : how 6. What is the difference between † and 13? 7. What is the difference between 12 8. What is the difference between 1 9. What is the difference between 10. What is the difference between EXERCISES FOR THE SLATE. 11. From take Solution.. Ans. 12. From take 2. and 13? and 2 ? and 48? Suggestion. A difficulty here meets the learner, similar to that which occurred in the 12th example of addition of fractions, viz: that of subtracting a fraction of one denominator from a fraction of a different denominator. He must therefore reduce the fractions to a common denominator, before the subtraction can be per formed. the numerators. (Art. 125.) Also 6×4-24, the common denominator. Ans. 128. From these illustrations we deduce the follow ing general RULE FOR SUBTRACTION OF FRACTIONS. Reduce the given fractions to a common denominator; subtract the less numerator from the greater, and place the remainder over the common denominator. OBS. Compound fractions must be reduced to simple ones, as in ad dition of fractions. (Art. 123.) QUEST. 128. How is one fraction subtracted from another? Obs. What is to be done with compound fractions? 129. Mixed numbers may be reduced to improper fractions, then to a common denominator and subtracted; or, the fractional part of the less number may be taken from the fractional part of the greater, and the less whole number from the greater. 24. From 8 take 5. Operation. 17 thirds from 25 thirds leaves 8 thirds, which are equal to 23. Note. Since we cannot take 2 thirds from 1 third, we borrow a unit, which, reduced to thirds and added to 1 third, makes 4 thirds. Now 2 thirds from 4 thirds leaves 2 thirds: 1 to carry to 5 makes 6, and 6 from 8 leaves 2. 25. From 125 take 74. Ans. 53. 26. From 15 take 92. 27. From 25 take 174. 28. From 374 take 193. 29. From 2 take 3. Suggestion. Since 5 fifths make a whole one, in 2 whole ones there are 10 fifths; now 3 fifths from 10 fifths leaves 7 fifths. Ans. 7, or 13. Hence, QUEST.-129. How are mixed numbers subtracted? 130. How is a fraction subtracted from a whole number? 130. To subtract a fraction from a whole number. Change the whole number to a fraction having the same denominator as the fraction to be subtracted, and proceed as before. (Art. 128.) OBS. If the fraction to be subtracted is a proper fraction, we may simply borrow a unit and take the fraction from this, remembering to diminish the whole number by 1. (Art. 36.) 30. From 6 take . Ans. 54. 31. From 65 take 25. 1. If a man spends of a dollar for rum in 1 day, how much will he spend in 7 days? Suggestion. If he spends in 1 day, in 7 days he will 7 is 7. Ans. of a dollar. spend 7 times; and 2. If a man spends of a dollar for rum in 1 week, how much will he spend in 4 weeks. Ans. 2 or 3 dolls. 3. If 1 man drinks of a barrel of beer in a month, how much will 10 men drink in the same time? 4. What cost 4 yards of cloth, at 24 dollars per yard? Solution.-4 yards will cost 4 times as much as 1 yard; and 4 times is 4 halves, equal to two whole ones: 4 times 2 dollars are 8 dollars, and 2 make 10 dollars. Ans. 4 yards will cost 10 dollars. 5. What cost 5 bushels of peanuts, at 3 dolls. a bushel? 6. What cost 104 pounds of tea, at 4 shillings a pound? 7. If 1 drum of figs costs 16 shillings, what will 3 fourths of a drum cost? Suggestion. First find what 1 fourth will cost. Then 3 fourths will cost 3 times as much. 8. If an acre of land produces 40 bushels of corn, how many bushels will 3 eighths of an acre produce? 9. If a man can travel 50 miles in a day, how far can he travel in 2 fifths of a day? 3 fifths? 4 fifths? 10. Henry's kite line was 90 feet long, but getting en tangled in a tree, he lost 3 ninths of it: how many feet did he lose? 131. We have seen that multiplying by a whole number is taking the multiplicand as many times as there are units in the multiplier. (Art. 45.) On the other hand, If the multiplier is only a part of a unit, it is plain we must take only a part of the multiplicand. That is, 132. Multiplying by a fraction is taking a certain PORTION of the multiplicand as many times as there are like portions of a unit in the multiplier. Multiplying by, is taking 1 half of the multiplicand Thus, 6×1=3. (Art. 104. Obs.) once. Multiplying by 3, is taking 1 third of the multiplicand once. Thus, 6x=2. Multiplying by, is taking 1 third of the multiplicand twice. Thus, 6×3=4. OBS. If the multiplier is a unit, the product is equal to the multiplicand; if the multiplier is greater than a unit, the product is greater than the multiplicand; (Art. 45;) and if the multiplier is less than a unit, the product is less than the multiplicand. EXERCISES FOR THE SLATE. CASE I. 11. If a bushel of corn is worth of a dollar, how much Is 5 bushels worth? QUEST.-131. What is meant by multiplying by a whole number? 132. By a fraction? By By ? By ? By? By ? Obs. If the multiplier is a unit or 1, what is the product equal to? When the multiplier is greater than 1, how is the product, compared with the multiplicand? When less, how? |