43 OPERATION. EXAMPLES FOR PRACTICE. 2. From Ty take an Ans. 726 3. From {: take 15. Ans.. 4. From 11 take 270 Ans. 5. From 1 take to Ans. 6. From t take . Ans. Mas 7. From $ take it: Ans. 287 8. From itt take it Ans. 3:83. 9. From to take totoo Ans. 18 10. From of it take of 7. Ans. 124 11. From of take 1 of 14. Ans. I jo 12. From f of 12 take of 915. Ans. 4. 149. To subtract a proper fraction or a mixed number from a whole number. Ex. 1. From 16 take 21. Ans. 13. Since we have no fraction from which to subFrom 16 en tract the d, we add 1, equal to $, to the minuTake 21 end, and say from & leaves i. We write the Rem. 13 below the line, and carry 1 to the 2 in the subtrahend, and subtract as in subtraction of simple numbers. The same result will be obtained, if we Subtract the number denoting the numerator from that denoting the denominator, and under the remainder write the denominator, and carry 1 to the integral part of the subtrahend before subtracting it from the minuend. Note. – When the subtrahend is a mixed number, we may reduce it to an improper fraction, and change the whole number in the minuend to a fraction having the same denominator, and then proceed as in Art. 148. EXAMPLES FOR PRACTICE. 6. From 1 2 19 17 Take 43 33 613 Ans. 71 154 311 54 1011 7. From 23 take 133. Ans. 9. 8. From 47 take js. Ans. 460. 9. From 139 take 7511. Ans. 6314 149. How do you subtract a proper fraction or mixed number from a whole number? The reason for this rule ? 150. To subtract one mixed number from another. Ex. 1. From 9 take 3. Ans. 534. FIRST OPERATION. common We first reduce the fractional parts to a From 94 = 93% Take 3 = 335 denominator by multiplying the terms of the fraction by 5, the denominator 2X5=10 Rem. 5 5 of the other, thus : and then the 7x5= 35 ; terms of the fraction by 7, the denominator of the first, thus : 3x7=21 5x7= 35 Now, since we cannot take f} from }}, we add 1, equal to us, to the }; in the minuend, and obtain. We next subtract from *, and write the remainder, , below, and carry 1 to the 3 in the subtrahend, and subtract from the 9 above, as in simple whole numbers. SECOND OPERATION. 325 126 From 94 In this operation, we reduce Take 33 the mixed numbers to imRem. :53 proper fractions, and these frac tions to a common denominator. We then subtract the less fraction from the greater, and, reducing the remainder to a mixed number obtain 5%, as before. Hence, in performing like examples, Reduce the fractional parts, if necessary, to a common denominator, and subtract the fractional part of the subtrahend from that of the minuend, as in Art. 147; remembering to increase the fractional part of the minuend, when otherwise it would be less than that of the subtrahend, before subtracting, by as many fractional units as it takes to make a unit of the fraction (Art. 131), and carry 1 to the whole number of the subtrahend before subtracting it. Or, Reduce the mixed numbers to improper fractions, then to a common dem nominator, and subtract the less fraction from the greater. EXAMPLES FOR PRACTICE. 6. 103 Take 511 37 1075 Ans. 333 317 393 53 From 97 150. How do you reduce the fractions of the mixed numbers to a common denominator? How does it appear that this process reduces them to a com. mon denominator ? How do you then proceed? What other method of subtracting mixed numbers? How may all like examples be performed ? 193 15% 7. 11. From 1 23 1611 971 8713 197 Take 94 181 5$ Ans. 213 1034 345 6 792 Ans. 1138 12. From 19 take 71 Ans. 636 13. From 157 take 87. Ans. 53 14. From 91 take 315. Ans. 576 15. From 711 take 1379. Ans. 27435. 16. From 6111 take 33 11. 17. From a hogshead of wine there leaked out 12% gallons ; Ans. 50% gallons. how much remained ? 18. From $ 10, $ 21 were given to Benjamin, $34 to Lydia, $ 11 to Emily, and the remainder to Betsey ; what did she re Ans. $35. ceive? 151. To subtract one fraction from another, when both have 1 for their numerator. and 7? Ans. . • Ex. 1. What is the difference between OPERATION. 4 Difference of the denominators, 7 — 3 Product of the denominators, 7 X 3 =21 We first find the product of the denominators, which is 21, and then their difference, which is 4, and write the former for the denominator of the required fraction, and the latter for the numerator. By this process the fractions are reduced to a common denominator, and their difference found. Hence, to find the difference of two fractions of this kind, Write the difference of the denominators over their product. EXAMPLES FOR PRACTICE. 2. Take ţ from $, 4 from 1, it from , + from f. 3. Take from , g from +, from }, from 4. Take from from }, I from, it from . 5. Take 4 from }, { from t, it from $, 7 from . 6. Take } from , j from $, Ił from , from f. 7. Take } from }, from , f from 5 il from t. 151. How do you subtract one fraction from another when both fractions have a unit for a numerator? What is the reason for this process ? MULTIPLICATION. FIRST OPERATION. 152. Multiplication of Fractions is the process of taking one gumber as many times as there are units in another, when one or both of the numbers are fractions. 153. To multiply a fraction by a whole number. Ex. 1. Multiply } by 4. Ans. 34. In the first operation we multiply the * X 4 = = 3} numerator of the fraction by the whole number, and obtain 31 for the answer. It is evident that the fraction is multiplied by multiplying its numerator by 4, since the parts taken are 4 times as many as before, while the parts into which the number or thing is divided remain the same. Therefore, Multiplying the numerator of a fraction by any number multiplies the fraction by that number. In the second operation we divide the X 4 =1=31 denominator of the fraction by the whole number, and obtain 31 for the answer, as before. It is evident, also, that the fraction is multiplied by dividing its denominator by 4, since the parts into which the number or thing is divided are only t as many, and consequently 4 times as large, as before, while the parts taken remain the same. Therefore, Dividing the denominator of a fraction by any number multiplies the fraction by that number. RULE. Multiply the numerator of the fraction by the whole number. Or, Divide the denominator of the fraction by the whole number, when it can be done without a remainder. SECOND OPERATION. EXAMPLES FOR PRACTICE. 2. Multiply by 9. 3. Multiply 5 by 5. 4. Multiply 24 by 3. 5. Multiply they by 85. Ans. 64. Ans. 2. Ans. 1}. Ans. 49. 152. What is multiplication of fractions ? 153. How is a fraction multiplied, by the first operation? The reason of the operation? What inference is drawn from it?' How is a fraction multiplied, by the second operation ? The reason of the operation? What inference is drawn from it? The rule for multiplying a fraction by a whole number? FIRST OPERATION. 6. Multiply ti by 83. Ans. 7612 7. Multiply by 189. Ans. 16614 8. Multiply 11by 365. Ans, 352614 9. Multiply s7 by 48. Ans. 431 10. If a man receive of a dollar for one day's labor, what will he receive for 21 days' labor ? Ans. $71. 11. What cost 56lb. of chalk at of a cent per lb. ? Ans. $ 0.42. 12. What cost 396lb. of copperas at ii of a cent per lb. ? Ans. $ 3.24. 13. What cost 79 bushels of salt at } of a dollar per bushel ? Ans. $ 69 154. To multiply a whole number by a fraction. Ex. 1. Multiply 15 by . Ans. 9. In the first operation we divide the whole 5) 15 number by the denominator of the fraction, and obtain of it. We then multiply this 3 X 3 = 9 quotient by 3, the numerator of the fraction, and thus obtain of it, which is 9. In the second operation we multiply the 15 whole number by the numerator of the frac3 tion, and divide the product by the denomi5 9 nator, and obtain 9 for the answer, as before. Therefore, Multiplying by a fraction is taking the part of the multiplicand denoted by the multiplier. RULE. Divide the whole number by the denominator of the fraction, when it can be done without a remainder, and multiply the quotient by the numerator. Or, Multiply the whole number by the numerator of the fraction, and divide the product by the denominator. EXAMPLES FOR PRACTICE. 2. Multiply 36 by 7. Ans. 28. 3. Multiply 144 by 15. Ans. 88. 4. Multiply 375 by 18. Ans. 325. 5. Multiply 2277 by yg. Ans. 1610. 6. Multiply 376 by 17. Ans. 24317 SECOND OPERATION. 45 154. How do you multiply a whole number by a fraction, according to tho first operation ? How by the second ? What inference is drawn from the operation? The rule for multiplying a whole number by a fraction? |