12. of 1, 19, 23. Ans. 30 ANSWERS. 15 32 36 = 401 40 40° 281 281 40 Ans. 38, 18, 15, 18, 88 601 Ans. 120, 128, 128, 188. 90 105 100 For additional problems, see Ray's Test Examples. ART. 147. ADDITION OF FRACTIONS Is the process of uniting two or more fractional numbers. 3 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. += { Ans. ៖ ៖ 3 sixths, and 1 Ans. Ans. 1110. 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 2X3X2=12, Least Com. Mult. Ans. 21-23 21. = = SUGGESTION.—In reducing the fractions to their least common denominator, omit writing the denominator beneath, until the sum REVIEW.-148. Why can not 1-half and 1-third be added, without reducing them to the same denominator? What the rule for addition? 8. Add and§. Ans. 1. 11. Add 1, 7, 11. Ans. 21. Ans. 17. 12. Add 1,,. Ans. 1. 9. Add §, 1. ART. 149. SUBTRACTION OF FRACTIONS Is the process of finding the difference between two fractional numbers. 1. What is the difference between 2 and 1 SOLUTION.-Since the denominators are the same, the express parts of the same size: therefore, subtract from 5 sevenths as you would 2 cents from 5 cents. Σ merators sevenths Thus, 5 sevenths, 2 sevenths, 5 cents, 2 cents, Difference 3 sevenths (3) in one case; 3 cents in the Hence, to find the difference between two fract having a common denominator, Find the difference between their numerators, and write the result over the common denominator. QUESTIONS FOR MENTAL SOLUTION. 2. What is the difference between and ? and ? REVIEW.-148. NOTE. If there are Compound Fractions, what is ro quired? What if each fraction is not in its lowest terms? How &N mixed numbers added? ART. 150. 1. Find the difference between and . 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. Thus, the difference between 1 half and 2 thirds is neither 1 half nor 1 third; but, reduce OPERATION. 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 1. Ans. Ans.. 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? 6=2X3 10=2X5 OPERATION. New 3065, and 5 × 5=25 3X2X5=30, Least Com. Mult. = = 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? ART. 151. MULTIPLICATION OF FRACTIONS. CASE I. To multiply a fraction by a whole number. This operation consists in taking the fraction as many times as there are units in the multiplier. 1. If 1 apple cost of a cent, what cost 3 apples? SOLUTION.-Three apples cost 3 times as much as one; that is, taken 3 times: {+}+}=}×3=3. (Art. 131.) 2. If 1 lemon cost of a cent, what cost 4 lemons? Ans. taken 4 times; +3+3+8=3×4=12=23. REVIEW.-150. Why can not one-half be taken from two-thirds without reducing to the same denominator? What the rule for subtraction? 150. NOTE. What is required if there are compound fractions? What if each fraction is not in its lowest terms? REM. How find the difference between two mixed numbers? Between a whole number and a fraction? |