ART. 147. To subtract fractions that have a common denominator. Ex. 1. From 3 take . 7-2 Ans. §. OPERATION. In this operation we take the less numerator from the greater, and write the difference over the common denominator. Hence the following 9 9 p RULE. Subtract the less numerator from the greater, and write the difference over the common denominator, and reduce the fraction if necessary. ART. 148. To subtract fractions that have not a common denominator. Having found the common denominator and new numerators as in Art. 141, we subtract the less numerator from the greater, and place the difference over the common denominator. RULE. Reduce the fractions to a common denominator, then write the difference of the numerators over the common denominator. NOTE. If the minuend or subtrahend, or both, are compound fractions, they must be reduced to simple ones. QUESTION. Art. 147. What is the rule for subtracting fractions having a common denominator? - Art. 148. What is the rule for subtracting fractions not having a common denominator? If the minuend or subtrahend is a compound fraction, what must be done? ART. 149. To subtract a proper fraction or a mixed num ber from a whole number. Ex. 1. From 16 take 24. OPERATION. From 16 Take 21 Ans. 131. Since we have no fraction from which to subtract the 4, we must 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 simple numbers. The same result will be obtained, if we adopt the following RULE.Subtract the numerator from the denominator of the fraction, and under the remainder write the denominator, and carry one to the subtrahend to be subtracted from the minuend. NOTE. If the subtrahend is a mixed number, we may, if we choose, 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. QUESTIONS. Art. 149. What is the rule for subtracting a proper fraction or mixed number from a whole number? Give the reason for this rule. ART. 150. To subtract a mixed number from a mixed number. Ex. 1. From 94 take 33. FIRST OPERATION. From 94 Take 33 Rem. 2 X 5=10 thus, 7 X 5=35 Ans. 53. In this example, we first reduce the fractions to a common denominator by multiplying the terms of the upper fraction, by 5, the denominator of the lower, and then the terms of the lower fraction, 3 X 7 = 21 by 7, the denominator of the upper, thus, 5x7=35 since we cannot take from 18, we add 1, equal to, to the in the minuend, and obtain . We next subtract from 4, and write the remainder, , below the line, and carry 1 to the 3 in the subtrahend, and subtract as in simple numbers. SECOND OPERATION. = 325 From 94 6,5 126 35 Rem. = Now, In this operation, we first reduce the mixed numbers to improper fractions, and these 1995 fractions to a common denominator, as in the first opera tion. We then subtract the less fraction from the greater, and, reducing the remainder to a mixed number, obtain the same answer as before. RULE I. Reduce the fractions, if necessary, to a common denominator, and if the lower fraction is greater than the upper, subtract the numerator of the lower fraction from the common denominator, and to the remainder add the numerator of the upper fraction. Write this sum over the common denominator, and carry 1 to the subtrahend, and subtract as in simple numbers. But if the upper fraction is greater than the lower, subtract the less from the greater, and the whole numbers as before. Or, RULE II. - Reduce the mixed numbers to improper fractions, then to a common denominator, and subtract the less fraction from the greater. Write the remainder over the common denominator, and if the fraction is improper, reduce it to a whole or mixed number. QUESTIONS..-Art. 150. How do you reduce the fractions of 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 ? What are the two rules? 16. From 6114 take 331. 17. From a hogshead of wine there leaked out 123 gallons; how much remained? Ans. 50 gallons. 18. From 10, $21 were given to Benjamin, $3 to Lydia, $1 to Emily, and the remainder to Betsey; what did she receive? Ans. $34. ART. 151. To subtract one fraction from another, when both fractions have a unit for a numerator. Ex. 1. What is the difference between and ✈ ? Difference of the denominators 7 OPERATION. Ans.. 3 = 4 Product of the denominators 7 × 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. QUESTIONS. Art. 151. What is the rule for subtracting one fraction from another when both fractions have a unit for a numerator? What is the reason for the rule ? . MULTIPLICATION OF VULGAR FRACTIONS. ART. 152. MULTIPLICATION of Fractions is the process of multiplying fractions together, or whole numbers and fractions into each other. و ART. 153. To multiply a fraction by a whole number. Ex. 1. Multiply 7 by 4. FIRST OPERATION. = Ans. 31. In the first operation, we multiply the 3umerator 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 8÷4 7 = 31 In the second operation, we divide the denominator of the fraction by the whole number, and obtain 3 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 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 I.— Multiply the numerator of the fraction by the whole number, and under the product write the denominator. Or, RULE II. — Divide the denominator of the fraction by the whole number, when it can be done without a remainder, and write the quotient under the numerator. QUESTIONS. Art. 152. What is multiplication of fractions?-Art. 153. How is a fraction multiplied by the first operation? Give the reason of the operation. What inference is drawn from it? How is a fraction multiplied by the second operation? What is the reason of the operation? What inference is drawn from it? What is the first rule for multiplying a fraction by a whole number? The second? |