With regard to this Operation, the pupil will notice, 1st. When a fraction is in its lowest terms, the denominator of any fraction to which it can be reduced must be a multiple of the denominator of the given fraction: hence, Any denominator common to two or more fractions must be a common multiple of their denominators: therefore, The least common multiple of their denominators, is the least com. denominator to which two or more fractions can be reduced. 2d. The values of the fractions are not altered, for both terms are multiplied by the same number (Art. 135.). Rule for Case VI.-Find the least common multiple of the denominators of the given fractions, and multiply both terms of each fraction by the quotient of the least common multiple, divided by the denominator of the fraction. NOTES.-1. Before commencing the operation, each fraction must be in its lowest terms. 2. Compound must be reduced to simple fractions, and whole or mixed numbers to improper fractions. 3. After the pupil is well acquainted with the nature of the operation, the multiplication of the denominators may be omitted, as the new denominators will be equal to the L. C. M. 4. The object of reducing fractions to a common denominator, is to prepare them for addition and subtraction, which can be performed only when the numbers are of the same unit value. REVIEW.-146. What is the least denominator common to the fractions whose denominators are 2, 3, and 4? How reduce one-half to twelfths? Two-thirds? Three-fourths? Why is the L. C. M. of the denominators, the least common denominator? Why are the values of fractions not altered by the operation? What is the Rule for Case VI? Is the process of uniting two or more fractional numbers. 1. What is the sum of and and ? SOLUTION. Since the denominators are the same, the numerators express parts of the same size: therefore, add Hence, to add fractions having a Com. Denominator, find the sum of the numerators; write the result over the denominator. REVIEW.-146. NOTE 1. Before commencing the operation, what is required? 2. If there are compound fractions or mixed numbers? NOTE 3. What may be omitted? Why? 4. What is the object in reducing fractions to a common denominator? 147. What is Addition of Fractions? How add fractions having a common denominator? Why? ART. 148. 1. What is the sum of and ? 14 SOLUTION. Since the denominators are different, the numerators do not express things of the same unit value, and they can not be added together. The sum of 1 half and 1 third, is neither 2 halves nor 2 thirds. But, reduced to a common denominator, (Art. 144), 1 half third = 2 sixths; their sum is 5 sixths (§). OPERATION. +3= Ans. 3 sixths, and 1 Rule for Addition.-Reduce the fractions to a common denominator; add their numerators together, and place the sum over the common denominator. NOTES.-1. Reduce compound to simple fractions, and each fraction to its lowest terms, before commencing the operation. 2. Mixed numbers may be reduced to improper fractions, and then added; or the fractions and whole numbers may be added separately, then united. 3. After adding, reduce the result to its lowest terms. Art. 138. 1 2 1 12÷6=2, and 2×5=10 2×3×2=12, Least Com. Mult. Ans. 21 = 211⁄2 = 24. SUGGESTION. In reducing the fractions to their least common denominator, omit writing the denominator beneath, until the sum of the numerators is obtained. REVIEW. 148. Why can not 1-half and 1-third be added, without reducing them to the same denominator? What the rule for addition? 1 8. Add and. Ans. 1. 11. Add 1, 7, 11. Ans. 24. 3 Ans. 1. of, and of of. 19. Add of 1, 1 of 21, 7 7 20. Add, 1, 50, 10, and 2800. For additional problems, see Ray's Test Examples. ART. 149. SUBTRACTION OF FRACTIONS Is the process of finding the difference between two fractional numbers. 1. What is the difference between and ? SOLUTION. Since the denominators are the same, the numerators express parts of the same size: therefore, subtract 2 sevenths from 5 sevenths as you would 2 cents from 5 cents. Thus, 5 sevenths, 2 sevenths, 5 cents, 2 cents, Difference 3 sevenths (3) in one case; 3 cents in the other. Hence, to find the difference between two fractions having a common denominator, Find the difference between their numerators, and write the result over the common denominator. REVIEW.-148. NOTE. If there are Compound Fractions, what is required? What if each fraction is not in its lowest terms? How are mixed numbers added? OPERATION. ART. 150. 1. Find the difference between and 3. SOLUTION. Since the denominators are different, the numerators do not express things of the same unit value: hence, one can not be subtracted from the other. Art. 25. -- Ans. Thus, the difference between 1 half and 2 thirds is neither 1 half nor 1 third; but, reduce them to a common denominator (Art. 144), and 2 thirds=4 sixths, and 1 half=3 sixths; their difference being 1 sixth (†). *2. The difference between and 4. Ans. 12. Rule for Subtraction.-Reduce the fractions to a common denominator, find the difference of their numerators, and place it over the common denominator. NOTE.-Reduce compound to simple fractions, and each fraction to its lowest terms, before commencing the operation. After subtracting, reduce the result to its lowest terms. 4. What is the difference between and _3? 10 = 6=2×3 2 X 5 OPERATION. 30÷ 65, and 5 × 5=25) New 3×2×5=30, Least Com. Mult. 9) Nume'rs. Ans. 168. Ans. = Ans. = Ans. 7 13. 20 = Ans. 3'6· 6 3 15 Ans. 18' = REM.-In finding the difference between mixed numbers, either reduce them to improper fractions, and to a common denominator, and then make the subtraction; or, find the difference between the whole numbers and the fractions separately. REVIEW.-149. What is subtraction of fractions? How find the difference between two fractions having a com. denominator? Why? |