13. From taket. Ans. 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 53. Operation. 8= -25 17 thirds from 25 thirds leaves 8 thirds, which are equal to 23. 5}= Ans. ģ=23. Note.-Since we cannot take 2 thirds from 1 Or thus, 8} third, we may borrow a unit, which, reduced to 53 thirds and added to 1 third, makes 4 thirds. Now 2 thirds from 4 thirds leaves 2 thirds : 1 to Ans. 23 carry to 5 makes 6, and 6 from 8 leaves 2. Ans. 21. 25. From 12 take 71. Ans. 53. 26. From 15 take 9. 27. From 25} take 17. 28. From 374 take 19%. 29. From 2 take s 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. }, for 14. Ans. }, for 1. 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 ž. Ans. 5}. 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 f in 1 day, in 7 days he will spend 7 times $; and $ x 7 is 3. 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. 2 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 25 dollars per yard ? Solution. 4 yards will cost 4 times as much as 1 yard; and 4 times f 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 5ļ barrels of peanuts, at 3 dollars a barrel ? 6. What cost 10pounds of tea, at 4 shillings a pound? 7. If i 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 filths 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 uinths 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 hy $, we must take 1 third of the multiplicand once; to multiply by }, we must take 1 third of the multiplicand twice, &c. Thus 6x1= 6--2, or 3; 6*$=6-3, or 2 ; 6x=2 times I 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 ś? By? Byš? 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 $ x5=, or 2); that is, 5 times are 5 halves, equal to 2 and 1 half Ans. 21 dollars. 12. Multiply & by 5. Ans. , or 3%. 17. If a pound of tea cost 6 shillings, how much will f of a pound cost ? Solution. Multiplying by s, is taking 1 third of the multiplicand twice. (Art. 132.) Now I 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, š x2, or }; and =4. Ans. 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 nuinber for the multiplier, when it is more convenient. Thus, š x6=*?, or 4, which is the same result as besore. 18. Multiply 12 by 4. Ans. 3. 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, 4x7=4. 2. On the same principle, a fraction is multiplied into any factor in its denominator, by cunceling that factor. (Arts. 91, 114.) Thus, 3X3=3 22. Multiply by 5. Ans. 1, or 3. 5 times s are, which are equal to 2 and 1. 9 Set down the d. 5 times 9 are 45, and 2 5 (which arose from the fraction) make 47. Ans. 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 154 by 7. Ans. 1101 31. Multiply 25, by 10. 32. Multiply 4818 by 8. 33. Multiply 24 by 31. Operation. We first multiply 24 by 3, then by 5, and 2)24 the sum of the products is 84. Multiplying 3} by is taking one half of the multiplicand 72 once. (Art. 131.) But to find a half of any 12 number, we divide the number by 2. (Art. Ans. 84 104. 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 fact in denominatör? How by a number equal to its denominator? 134. How is a mised number multiplied by a whole one? 134. a. How is a whole num. ber multiplied by a mixed number? |