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. 5. 35. From g of 24 take § of 27. MULTIPLICATION OF FRACTIONS. MENTAL EXERCISES. 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 spend 7 times; and × 7 is. Ans. of a dollar. 2. If a man spends of a dollar for rum in 1 week, how much will he spend in 4 weeks. Ans. 28 or 34 dolls. 3. If 1 man drinks of a barrel of beer in a month, how much will ten men drink in the same time? 4. What will 4 yards of cloth cost, at 2 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 barrels of peanuts, at 3 dollars a barrel? 6. What cost 10 pounds of tea, at 4 shillings a pound? 7. If 1 drum of figs cost 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 travels 50 miles in a day, how far will he travel in 2 fifths of a day? 3 fifths? 4 fifths? 10. Henry's kite line was 90 feet long, but getting entangled 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.) If, therefore, the multiplier is only a part of a unit, it is plain we must take only a part of the multiplicand. For example, to multiply by, we must take 1 half of the multiplicand once; to multiply by, we must take 1 third of the multiplicand once; to multiply by 3, we must take 1 third of the multiplicand twice, &c. Thus 6× 6-2, or 3; 6×6÷3, or 2; 6×32 times 1 third of 6, or 4, &c. (Art. 104. Obs.) Hence, 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. 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 cost of a dollar, how much will 5 bushels cost? 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? Solution. 5 bushels will cost 5 times as much as 1 bushel. Now×5=5, or 21; that is, 5 times halves, equal to 2 and 1 half are 5 Ans. 2 dollars. 17. If a pound of tea cost 6 shillings, how much will of a pound cost? Solution. Multiplying by, is taking 1 third of the multiplicand twice. (Art. 132.) Now 1 third of 6 is the same as 6 thirds of 1, or §; and 2 thirds of 6 must be 2 times as much; that is, §×2, or 12; and 12=4. Ans. 3 Note. Since the product of any two numbers will be the same, whichever is taken for the multiplier, (Art. 47,) the fraction may be taken for the multiplicand, and the whole number for the multiplier, when it is more convenient. Thus,×6=12, or 4, which is the same result as be fore. 18. Multiply 12 by . Ans. 3. 19. Multiply 10 by . 20. Multiply 15 by 7. 21. Multiply by 2. Ans. §×2=10, or 14. Suggestion. Dividing the denominator of a fraction by any number, multiplies the value of the fraction by that number. (Art. 114.) Now, if we divide the denominator 8 by 2, the fraction will become, which is equal to 14, the same as before. Hence, 133. To multiply a fraction and a whole number together. Multiply the numerator of the fraction by the whole number, and write the product over the denominator. Or, divide the denominator by the whole number, when this can be done without a remainder. (Art. 114.) QUEST.-133. How multiply a fraction and a whole number together? Obs. 1. A fraction is multiplied into a number equal to its denominator by canceling the denominator. (Art. 89.) Thus, 4×7=4. 2. On the same principle, a fraction is multiplied into any factor in its denominator, by canceling that factor. (Arts. 91, 114.) Thus, 135x3=3. 23. Multiply by 9. 25. Multiply 36 by 12. 27. Multiply 359 by 25. 29. Multiply 9 by 5. Operation. 9 5 Ans. 47 5 times are, which are equal to 2 and 1. Set down the . 5 times 9 are 45, and 2 (which arose from the fraction) make 47. Hence, 134. To multiply a mixed number by a whole one. Multiply the fractional part and the whole number separately, and unite the products. 30. Multiply 15 by 7. Ans. 1101. 31. Multiply 25 by 10. 32. Multiply 4810 by 8. 33. Multiply 24 by 31. Operation. 2)24 31 72 12 Ans. 84 We first multiply 24 by 3, then by, and the sum of the products is 84. Multiplying by is taking one half of the multiplicand once. (Art. 131.) But to find a half of any number, we divide the number by 2. (Art. 104. Obs.) Hence, 134. a. To multiply a whole by a mixed number. Multiply first by the integer, then by the fraction, and add the products together. QUEST. Obs. How is a fraction multiplied by any factor in its denominator? How by a number equal to its denominator? 134. How is a mixed number multiplied by a whole one? 134. a. How is a whole number multiplied by a mixed number? 34. Multiply 27 by 31. 35. Multiply 63 by 10%. Ans. 90. 36. Multiply 75 by 123. CASE II. 37. A man owning of a ship, sold of what he owned. What part of the ship did he sell? Analysis of is; for, multiplying the denominator by any number, divides the value of the fraction. (Art. 113.) Now 2 thirds of is twice as much; that is, 12=1, which, reduced to its lowest terms, is . Ans. 3 6 Again, the example may be performed thus: Since he owned 3, and sold of what he owned, it is evident he sold of of the ship. Now of is a compound fraction, whose value is found by multiplying the numerators together for a new numerator, and the denominators for a new denominator. (Art. 123.) Operation. =15, or 3. Ans. Hence, 135. To multiply a fraction by a fraction. Multiply the numerators together for a new numerator, and the denominators together for a new denominator. OBS. It will be seen that the process of multiplying one fraction by another, is precisely the same as that of reducing compound fractions to simple ones. 38. Multiply by 3. 39. Multiply by 3. 41. Multiply by §. Ans.. 40. Multiply by §. 136. In multiplication of fractions, the operation may often be shortened by canceling the factors common both to the numerators and denominators, as in reducing compound fractions to simple ones. (Art. 123.) QUEST.-135. How is a fraction multiplied by a fraction? Obs. To what is the process of multiplying one fraction by another similar? 136. How may the operation in multiplication of fractions be shortened? |