149. To subtract a proper fraction or a mixed number from a whole number. Ex. 1. From 16 take 24. OPERATION. From 16 £ Rem. 13 simple numbers. Ans. 132. Since we have no fraction from which to subtract the, we add 1, equal to , to the minuend, and say from leaves. We write the below the line, and carry 1 to the 2 in the subtrahend, and subtract as in subtraction of 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. 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 94 take 3g. FIRST OPERATION. = From 94939 Take 33 Rem. 83 53 2 X 5=10 7x5=35 ; Ans. 5. We first reduce the fractional parts to a common denominator by multiplying the terms of the fraction by 5, the denominator of the other, thus: and then the terms of the fraction by 7, the denominator of the first, thus: Now, since we cannot take from 18, we add 1, equal to, to the 39 in the minuend, and obtain. We next subtract from 4, 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. 3 x7=21 5x7=35° SECOND OPERATION. = 18 From 9455 = 325 126 199 In this operation, we reduce the mixed numbers to improper fractions, and these fractions to a common denominator. We then subtract the less fraction from the greater, and, reducing the remainder to a mixed number obtain 53, 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 denominator, and subtract the less fraction from the greater. 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 common denominator? How do you then proceed? What other method of subtracting mixed numbers? How may all like examples be performed? 17. From a hogshead of wine there leaked out 12 gallons; how much remained? Ans. 50 gallons. 18. From $10, $24 were given to Benjamin, $3 to Lydia, $1 to Emily, and the remainder to Betsey; what did she receive? Ans. $3. 151. To subtract one fraction from another, when both have 1 for their numerator. Ex. 1. What is the difference between and ? Ans. 1. 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. 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. 152. Multiplication of Fractions is the process of taking one number 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. FIRST OPERATION. 7 × 4 = 28 = 3 Ans. 34. In the first operation we multiply the numerator of the fraction by the whole number, and obtain 3 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. SECOND OPERATION. 7×4=7=31 as before. In the second operation we divide the denominator of the fraction by the whole number, and obtain 3 for the answer, 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 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. Multiply the numerator of the fraction by the whole number. Divide the denominator of the fraction by the whole number, when it can be done without a remainder. 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? 10. If a man receive of a dollar for one day's labor, what will he receive for 21 days' labor? 11. What cost 56lb. of chalk at of a cent Ans. $77. per lb.? Ans. $ 0.42. 12. What cost 3961b. of copperas at of a cent per lb.? Ans. $.3.24. 13. What cost 79 bushels of salt at 7 of a dollar per bushel? Ans. $691. 154. To multiply a whole number by a fraction. Ex. 1. Multiply 15 by 2. FIRST OPERATION. 5) 15 3X3 9 SECOND OPERATION. 15 3 4559 Ans. 9. In the first operation we divide the whole number by the denominator of the fraction, and obtain of it. We then multiply this quotient by 3, the numerator of the fraction, and thus obtain g of it, which is 9. In the second operation we multiply the whole number by the numerator of the fraction, and divide the product by the denominator, 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. 154. How do you multiply a whole number by a fraction, according to the first operation? How by the second? What inference is drawn from the operation? The rule for multiplying a whole number by a fraction? |