EXAMPLES FOR PRACTICE. 2. Add 1, 1, 71, fi, fr, and 10 together. 3. Add,,,, and together. 4. Add, 2, 18, and 8 19 together. 5. Add 17, 19, 27, and 4 together. 177 ART. 144. To add fractions that have not a common de nominator. 2 1 6 4x5 20 8 3x3 9 12 2x714 new numerators. Sum of numerators 43 2×3×2×2=24. Com. denominator 24 Having found the common denominator and new numerators, as in Art. 141, we add the numerators together and place their sum over the common denominator, and reduce the fraction. RULE. Reduce mixed numbers to improper fractions, and compound fractions to simple fractions; then reduce all the fractions to a common denominator; and the sum of their numerators, written over the common denominator, will be the answer required. NOTE. In adding mixed numbers, it is sometimes more convenient to add the fractional parts separately, and then to add their sum to the amount of the whole numbers. QUESTIONS. Art. 144. What is the rule for adding fractions not having a common denominator? How may mixed numbers be conveniently added? 9. Add,,, t, §, f, and together. 9 10. Add, fo, ff, 12, 13, 12, and together. Sum of the denominators 4+59 Product of the denominators 4 X 5 = 20 Ans. 20 We first find the product of the denominators, which is 20, and then their sum, which is 9, and write the former for the denominator of the required fraction, and the latter for the numerator. The reason of this operation will be seen, when we consider, that the process reduces the fractions to a common denominator, and then adds their numerators. RULE. Add together the given denominators, and place the sum over their product. EXAMPLES FOR PRACTICE. 2. Add to, to, to, to, to . to 3. Add to, to, to, to T, to T, to .. SUBTRACTION OF VULGAR FRACTIONS. ART. 146. SUBTRACTION of Vulgar Fractions is the process of finding the difference between two fractions of unequal values. QUESTIONS. Art. 145. What is the rule for adding two fractions when the numerators are a unit? What is the reason for this rule? -Art. 146. What is subtraction of vulgar fractions? ART. 147. To subtract fractions that have a common denominator. Ex. 1. From take . OPERATION. 7-2 9 9 RULE. Ans. §. In this operation we take the less numerator from the greater, and write the difference over the common denominator. Hence the following Subtract the less numerator from the greater, and write the difference over the common denominator, and reduce the fraction if 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 number from a whole number. Ex. 1. From 16 take 24. OPERATION. From 16 Take 21 Ans. 134. Since we have no fraction from which to subtract the, 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 92 Rem. thus, 2X5=10 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, 7X5=35; and then the terms of the lower fraction ?, by 7, the denominator of the upper, thus, 5x7=35 Now, since we cannot take from, we add 1, equal to §, to the in the minuend, and obtain . We next subtract from , and write the remainder, 34, below the line, and carry 1 to the 3 in the subtrahend, and subtract as in simple numbers. SECOND OPERATION. From 94 = 5 325 Take 33 Rem. 18 35 126 = 35 = 3 X 7=21 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 i 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? |