NOTES.-1. Before multiplying, reduce all fractions to simple fractions. 2. When the numbers are small, the work may be performed mentally: Thus : 20 1, 1, 3, = 28, 18, 18. NOTE. We may often shorten the work by multiplying the numerator and denominator of each fraction by such a number as will make the denominators the same in all. 11. Reduce and, to a common denominator. ANALYSIS.-Multiply both terms of the first by 3, and both terms of the second by 2. 130. To reduce fractions to their least common denominator. The LEAST COMMON DENOMINATOR is the number which contains all the prime factors of the denominators. 130. What is the least common denominator of several fractions? How do you reduce fractions to their least common denominator? 1. Reduce,, and, to their least common denominator. ANALYSIS. The least common multiple of the denominators will be the least common denominator, and in the example, is 12. We then divide 12 by each denominator, to find the factor by which the corresponding numerator must be multipiied, that the value of the fraction be not changed; and finally, we multiply each numerator by its proper factor. Therefore, the fractions I ,, and 2, reduced to their least common denominator, are least com. denominator. Rule. I. Find the least common multiple of the denominators (Art. 94), which will be the least common denominator of the fractions. II. Divide the least common denominator by the denominator of each fraction, separately; multiply the numerator by the corresponding quotient, and place each product over the least common denominator. NOTE.-Before beginning the operation, reduce every fraction to a simple fraction, and to its lowest terms. Examples. Reduce the following fractions to their least common denominator: ADDITION OF FRACTIONS. 131. ADDITION OF FRACTIONS is the operation of finding the sum of two or more fractional numbers. 1. What is the sum of 1, 3, and 1⁄2? ANALYSIS.-The fractional unit is the same in each fraction, viz.: 1; the numerator of each fraction shows how many such units are taken: hence, the sum of the numerators, written over the common denominator, expresses the sum of the fractions. 2. What is the sum of and ? ANALYSIS. In the first, the fractional unit is, in the second it is. These units, not being of the same kind, cannot be expressed in the same collection. But the , and, in each of which the unit is hence, their sum is 7=1. OPERATION. 1+3+59. Ans. = 4. OPERATION. 号= 3 + 1 = 7 = 1}. NOTE.-Only units of the same kind, whether fractional or integral, can be expressed in the same collection. From the above analysis, we have the following Rule. I. When the fractions have the same denominator, add the numerators, and place their sum over the common denominator. II. When they have not the same denominator, reduce them to a common denominator, and then add as before. 131. What is Addition of Fractions? When the fractional unit is the same, what is the sum of the fractions? What units may be expressed in the same collection? What is the rule for the addition of fractions? 5. Add,, and . 6. Add,,, and . 16. 7. Add 3, 3, 8, and 12. 5 7 8. Add 3,,, and . 9. Add 9,,,, and 10. Add,,,, and . and. 11. Add 1,,, and 3. 12. Add 4,, and . 5 6 13. Add,,, and . 14. Add,,, and 3. 15. What is the sum of 194, 63, and 4? Whole numbers. 19 + 6 + 4 = 29; OPERATION. Fractions. 6 sum 29 +164 = 30,645. 132. NOTE.-When there are mixed numbers, add the whole numbers and fractions separately, and then add their sums. 4. What is the sum of and? Of and ? and? Of SUBTRACTION. 134. SUBTRACTION OF FRACTIONS is the operation of finding the difference between two fractions. 1. What is the difference between and ?? : ANALYSIS. In this example, the fractional unit is there are 5 such units in the minuend and 3 in the subtrahend: their difference is 2 eighths; therefore, 2 is written over the common denominator 8. 2. From 15 take 10. 3. From take 3. 4. From 12 5. I. When the fractions have the same denominator, subtract the less numerator from the greater, and place the difference over the common denominator. II. When they have not the same denominator, reduce them to a common denominator, and then subtract as before. 132. When there are mixed numbers, how do you add? 133. When two fractions have 1 for a numerator, what is their sum equal to? 134. What is Subtraction of Fractions? What is the rule? |