30. To add together two or more fractions, the fractional units must be of the same size; in other words, they must be reduced to a common denominator before the addition can be accomplished. For example, suppose it is desired to add the fractions and These may for convenience represent 1 4 3. 1 is well known that a foot contains 12 inches, and the 4 The required sum is then 3 + 4 = 7 inches, or This process is graphically illustrated in Fig. 1. Now if an example be taken in which there are more than two 5 7 fractions, it will be noted that the procedure is the same. + + 8 12 11 24+ 3 4 = ? It may be seen by inspection or by the process already given that 24 is the least common denominator of the several fractions. The next step is to change the numerators of the fractions so that they will express the same value with the common denominator 24 as they now express with their respective denominators. 5 15 = 8 24' numerators and placing the sum over the common denominator the 58 required sum is found to be which is an improper fraction. Re ducing the improper fraction to a mixed quantity gives 214, or finally 2p+ 5p 3p 30p 30p 30p 30p = = 30p 3p 30p❜ If the problem involves letters only, the process is essentially the same. a If- and are to be added, they must first be reduced to n b Р their common denominator, np. As the relation between ap and bn is not known, the sum of the numerators can only be represented by the expression To add mixed quantities, add the entire quantities and the fractions separately, and if the fractional result of this addition is an improper fraction, reduce it to a mixed quantity and add the entire part to the sum of the entire quantities already obtained. Examples. 1. 20 + 131 + 7 = ? SOLUTION. Adding the entire quantities together 40. Reduce the fractions to higher terms having a common denominator and add, gives 20+ 13 + 7 = obtaining + + 3 31 = 24 24 24 24 20 2 = 131 7/1 = = = 201 1311 - 40+1 41 = ab + is 6, and the new 2 b forms of the fractions become and 2ab+ = 6 6 3 PR+MN NP 2ab+ 3ab + 3b 6 26 6 56 6 8. A room is 32 feet long and 29 feet wide. What is the distance around the room? 9. Three castings weigh respectively 2258, 232, and 2401 pounds. What is their total weight? 10. A steel rod is to be cut into five pieces; the first to be 4g inches long, the second 3 inches, the third 5 inches, the fourth 4 inches, and the fifth 115 inches. Find the length of the rod required. 11. A casting weighing 183 pounds has had 2 pounds of metal removed by the planer. How much did the original casting weigh? SUBTRACTION OF FRACTIONS 31. Fractions may be subtracted only when they have a common denominator, and express quantities of like units. Hence to subtract proper fractions reduce the given fractions to their equivalents, having a least common denominator, and write the difference of the numerators over the common denominator. Examples. 1. Find the difference between 5 8 and 6 15 2p 2p 2p 2p 2p To subtract mixed quantities, subtract the fractional and entire parts separately, and add the remainders. If the mixed quantities are small, they may be reduced to improper fractions and subtracted. 14 Examples. 1. From 27 subtract 14§. = SOLUTION. Subtracting the entire quantities, 27 13. Reduce the fractions to higher terms having a common denominator and subtract, obtaining hence 211 = 39. 14' 2 and 1 = = 7 18 the lowest terms by dividing both numerator and 18 14 21 To subtract a fraction or mixed quantity from an entire quantity, or from a mixed quantity in which the fraction of the minuend is less than the fraction of the subtrahend, one unit of the integer in the minuend must be written as a fraction. This is shown by the following example. In the case of letters, subtraction may be made after writing the entire quantity, as shown in the margin. |