SUBTRACTION OF FRACTIONS. 205. Ex. 1. A man bought to of an acre of land, and afterwards sold Zo of it: how much land had he left ? Solution.—7 tenths from 9 tenths leave 2 tenths. Ans. Po of an acre. 2. A laborer having received of a dollar for a day's work, spent of a dollar for liquor: how much money had he left ? Note.— The learner meets with the same difficulty here as in the second example of adding fractions; that is, he can no more subtract fifths from eighths, than he can add fifths to eighths; for, of a dollar taken from of a dollar will neither leave 4 fifths, nor 4 eighths. The fractions must therefore be reduced to a common denominator before the subtraction can be performed. Operation. 7x5=35 the numerators. (Art. 200.) 3X8=24 8X5=40, the common denominator. The fractions become and 24. Now 16444=4 Ans. } 206. From these illustrations we deduce the following 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 addition of fractions. (Art. 198.) EXAMPLES. 3. From š take . Ans. Zo 9. From take $3. 10. From jy of take of . 11. From of take ţ of 5. 12. From of 40 take of 20. 13. From of of 4 takes of . Quest.-206. How is one fraction subtracted from another? with componind fractions ? Obs. What is to be done 207. Mixed numbers may be reduced to improper fractions, then to a common denominator, and be 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. 14. From 97 take 74. First Operation. Second Operation. 9=27 97 7=34 74 Ans. =14, or 14. Ans. 14, or 11. Note.-Since we cannot take 3 fourths from 1 fourth, we borrow a unit in the second operation and reduce it to fourths, which added to the 1 fourth, make 5 fourths. Now 3 fourths from 5 fourths leave 2 fourths: 1 to carry to 1 makes 8, and 8 from 9 leaves 1. 15. From 25 take 133. 17. From 1787; take 56%. 16. From 230775 take 1607. 18. From 76145 take 482-846. 19. From 5 take 3. Suggestion.—Since 3 thirds make a whole one, in 5 whole ones there are 15 thirds; now 2 thirds from 15 thirds leave 13 thirds, Ans. 12, or 43. Hence, 208. 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. 197. Obs. 2.) 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. 69. Obs. 1.) 20. From 20 take Ans. 19%. 21. From 135 take 97. 26. From 729 take 12543. 22. From 263 take 2412. 27. From 1000 take 251. 23. From 168 take 305. 28. From 563 take 56245. 24. From 567 take 100%4. 29. From 9263 take 9994.. 25. From 634 take 342;. 30. From 857 take 78574. QUEST.-207. How are mixed numbers subtracted ? 208. How is a fraction subtracted from a whole number? MULTIPLICATION OF FRACTIONS. 209. We have seen that multiplying by a whole number, is taking the multiplicand as many times as there are units in the multiplier. (Art. 82.) 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, Multiplying by }, is taking 1 half of the multiplicand once. Thus, 12x2=6. Multiplying by }, is taking 1 third of the multiplicand once. Thus, 12x}=4. Multiplying by , is taking 1 third of the multiplicand twice. Thus, 12 X 3=8. Hence, 210. Multiplying by a fraction is taking a certain PORTION of the multiplicand as many times, as there are like portions of 6 unit in the multiplier. Obs. If the multiplier is a unit or 1, the product is equal to the multiplicand; if the multiplier is greater than a unit, the product is greater than the multiplicand; (Art. 82 ;) and if the multiplier is less than a unit, the product is less than the multiplicand. CASE I. 211. To multiply a fraction and a whole number together. Ex. 1. If 1 man drinks šof á barrel of cider in a month, how much will 5 men drink in the same time? Analysis.-Since 1 man drinks š of a barrel, 5 men will drink 5 times as much; and 5 times 2 thirds are 10 thirds; that is, $X5=1, or 33. (Art. 196.) Ans. 3} barrels. Ex. 2. If a pound of tea costs of a dollar, how much will 4 pounds cost ? Solution.-*X4=20; and 20=24, or 21 dolls. Ans. Or, since dividing the denominator of a fraction by any number, multiplies the value of the fraction by that number, (Art. 189,) Quest.–209. What is meant by multiplying by a whole number? 210. What is meant by multiplying by a fraction ? 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? if we divide the denominator 8 by 4, the fraction will become 1, which is equal to 27, the same as before. Hence, 212. To multiply a fraction by a whole number, 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. 189.) Obs. 1. A fraction is multiplied into a number equal to its denominator by canceling the denominator. (Ax. 2.) Thus 4x7=4. 2. On the same principle, a fraction is multiplied into any factor in its denominator, by canceling that factor. (Art. 189.) Thus, 15 X3= 3. Since multiplication is the repeated addilion of a number or quantity to itself, (Art. 80,) the student sometimes finds it difficult to account for the fact that the product of a number or quantity by a proper fraction, is always less than the number multiplied. This difficulty will at once be removed by reflecting that multiplying by a fraction is taking or repeating a certain portion of the multiplicand as many times, as there are like portions of a unit in the multiplier. (Art. 210.) EXAMPLES. 3. Multiply it by 15. Ans. 315, or 107. 4. Multiply 17 by 8. 9. Multiply by 165. 5. Multiply by 30. 10. Multiply 45 by 100. 6. Multiply s by 27. 11. Multiply 735 by 530. 7. Multiply 15t by 45. 12. Multiply it by 1000. 8. Multiply } by 100. 13. Multiply 761 by 831. 14. Multiply 127 by 8. Operation. 123 8 times į are , which are equal to 5 and 3. 8 Set down the }. 8 times 12 are 96, and 5 (which Ans. 1013. arose from the fraction) make 101. Hence, 213. To multiply a mixed number by a whole one. Multiply the fractional part and the whole number separately, and unite the products. Quest.-212. How multiply a fraction by a whole number? Obs. How is a fraction multiplied by a number equal to its denominator? How by any factor in its denominator ? 113. How is a mixed number multiplied by a whole one ? 15. Multiply 455 by 10. Ans. 451%. 214. Multiplying by a fraction, we have seen, is taking a certain portion of the multiplicand as many times, as there are like portions of a unit in the multiplier. Hence, To multiply by : Divide the multiplicand by 2. 215. Hence, to multiply a whole number by a fraction. Divide the multiplicand by the denominator, and multiply the quotient by the numerator. Or, multiply the given number by the numerator, and divide the product by the denominator. Obs. 1. When the given number cannot be divided by the denominator without a remainder, the latter method is generally preferred. 2. Since the product of any two numbers is the same, whichever is taken for the multiplier, (Art 83,) the fraction may be taken for the multiplicand, and the whole number for the multiplier, when it is more convenient. 22. If 1 ton of hay costs 21 dollars, how much will of a ton cost? Operation. Analysis.-Since 1 ton costs 21 dollars, of 4)21 a ton will cost as much. Now, I fourth of 21 57 is *; and of 21 is 3 times as much; but 3 Ans. 154 dolls 4 23. Multiply 136 by . Ans. 45 . 24. Multiply 432 by 1. 26. Multiply 360 by j. 25. Multiply 635 by š. 27. Multiply 580 by . 21 QUEST.--215. How is a whole number multiplied by a fraction ? 216. How find a fruc tional part of a number ? |