7. Reduce 8 to a fraction, whose denominator is 7. EXPLANATION. 8x7=56, numerator; hence, 56, Ans. 8. Reduce 9 to a fraction whose denominator is 11. 9. Reduce 7 to an improper fraction. 10. Reduce 17 to an improper fraction. "CASE 111. § 116. TO REDUCE AN IMPROPER FRACTION TO A WHOLE OR MIXED NUMBER. RULE. Divide the numerator by the denominator. The_quo tient will be the whole number; if there be a remainder, write it over the denominator on the right of the whole number. 1. Reduce 156 to a mixed number. 09 EXPLANATION. 9 ninths 12 1 whole one; therefore, 156÷9-173, Ans. 2. Reduce to a mixed number. 3. Reduce 96 to a whole number. 8 Ans. 6. Ans. 12. § 117. TO REDUCE COMPOUND FRACTIONS TO SIMPLE FRACTIONS. RULE. Multiply the numerators of the several fractions to gether for a new numerator, and the denominators together for a new denominator. The fraction thus produced, when reduced to its lowest terms, (§ 113,) will be the one required. 116. How reduce an improper fraction to a whole or mixed number? 117. How reduce compound fractions to simple ones? 6 1. Reduce of to a simple fraction. To obtain of, we divide by 4; but ÷4=2, (§ 110,) then having of , if we multiply this by 3, we shall obtain of; therefore, X3=18, the Ans. (§ 107.) We have here, merely ,multiplied numerators by numerators, and denominators by denominators. 6 28 2. Reduceof to a simple fraction. Ans. 16. § 118. The solutions of sums of this description may be essentially abbreviated by canceling such factors as are found common both to the numerators and denominators of the several simple fractions. 3. Reduce of of to a simple fraction. of 8 7. Reduce of of of to a simple fraction. 8. Reduce of 1 of 1 to a simple fraction. Whenever any term of a compound fraction is number, reduce it to an improper fraction. (§ 113.) 9. Reduce of of 5 to a simple fraction. EXPLANATION. 5-6. (8 114.) Hence, the proper statement, or 4, Ans. Ans. 16 57 Ans. H. a mixed 7. 6. 16 8. 7. 3 10. Reduce of of 8 to a simple fraction. is Ans. 51=3. (§ 116.) 118. How may such operations be abbreviated? What is to be done if a term of a compound fraction be a mixed number ? CASE V. $119. To REDUCE FRACTIONS TO A COMMON DENOMI NATOR. The value of fractions is not altered by what is here done. All fractions are made to represent the unit as divided into the same number of parts, while the number of these parts differs, in each of the fractions. Thus, the fractions ,,and, differ from each other both in specific value, and in the number of parts; but the fractions,,, and 1, expressing the same value, differ in specific value only. These fractions now, therefore, have the common denominator 8. RULE. Multiply all the denominators together for a common denominator, and each numerator into all the denominators except its own, for a numerator to each new fraction. 1. Reduce, , and to a common denominator. OPERATION. 7X5X11=385, the common denominator. 6×5×11=330, numerator for fraction. 4X7X11=308, numerator for fraction. 8X5X 7=280, numerator for fraction 8 1 Here the required fractions are, for 4, 338; for 4, 388; and for 11,388. 280 2. Reduce,,, to a common denominator. 60' 60 60 The value of these several fractions can, however, be exactly expressed in smaller terms, and retain a common denominator, for 30, 20, 15 and 12 are obviously of the same value, their common denominator being reduced from 120 to 60. It is always desirable to preserve the fractions in the smallest possible terms that will express their true value. 119. Is the value of a fraction altered by the operation of this rule? In what does the change made consist? Illustrate? What is the rule? How may the value of the answers to sum 1st be expressed? Illustrate? To do this, first find the least common multiple of all the denominators, and make this the common denominator. Divide this common denominator by the denominator of each of the given fractions, and multiply the quotient by the numerators of the same. OBS. The least common multiple of two or more numbers is the smallest number that can be divided by each of those numbers without remainder, and is found as follows: Write down the given numbers in a line from left to right, divide by the smallest number that will divide any two of them without remainder, and write the quotient and the undivided numbers in a line below the former numbers. Divide these numbers in turn in the same manner, and so on till no two numbers are left that can be divided by any number greater than 1. The continued product of the divisors and of the undivided numbers of the lowest line, will be the multiple required. ILLUSTRATION. Required the least common multiplier of 4, 6, 8, and 12. 2) 4. 6. 8. 12 2) 2. 3. 4. 6 1. 1. 2. 1 Then 2×2×3×2=24 is the least number that can be divided by each of the given numbers without remainder. 3. Reduce the fractions,,,, and, to their least common denominator. 2) 2. 3. 9. 10 3) 1.3.9. 5 1. 1. 3 5; and 2x3x3x5=90, the least common multiple of the denominators, therefore, How may the least common denominator always be found? What is the least common multiple of two or more numbers? How is it found? 90 2X1=45, 1st numerator, Hence, 45 is the 1st fraction. 90÷3×2=60, 2d numerator, and 68, the 2d fraction. 909x8 80, 3d numerator, and 88, the 3d fraction. 90÷10×3=27, 4th numerator, and 37, the 4th fraction. Hence, 5, 60, 80, and 27, are the fractions required. 90' 90' 90' '6' 4. Reduce, 5 , and, to their least common denominator. 30 16 Ans. 12, 38, 18, and 31. 5. Reduce,, 3, and, to their least common denominator. Ans. 297, 165, 385, and 360 6. Reduce, nator. 495 495' 495' 495. 5 , and, to their least common denomiAns. 10, 120, 120, and 128. 96 42 105 7. Reduce 3 , 2, 9, and 11, to their least common denominator. 8. Reduce,, and, to their least common denomina Ans. 1, 1, 18, and 52. 36 Ans. 37, 38, and 28. least common denominaAns. 39, 3, and 4. 10. Reduce, 12, and, to their least common denominator. 84 18 Ans. 15, 14, and 80. 45 144. ADDITION OF FRACTIONS. § 120. Like things only can be added, so as to form one number. (§ 32, 4.) Hence, before adding fractions they must have a common denominator. RULE. If the denominators of the given fractions are unlike, first reduce them to a common denominator, and, adding the numerators, place their sum over the common denomi nator. 5 6 1. Add the fractions, 8, 8, and together. These fractions have the common denominator 8; hence 3+5+6+7 =21, the sum of the numerators, and 21 is the required answer. 120. What things can be added? What must fractions have before they can be added together? What is the rule? |