CHAPTER II ADDITION AND SUBTRACTION OF FRACTIONS 14. Common Denominators.-Fractions cannot be added unless they contain the same kind of parts, or, in other words, have the same denominator. When fractions having different denominators are to be added, they must first be reduced to fractions having a common denominator. A number of fractions are said to have a common denominator when they all have the same number for their denominators. and cannot be added as they stand, any more than can 3 bolts and 5 washers. Both the fractions must be of the same kind, that is, must have the same denominator. may be changed to g. By making this change, the fractions are given a common denominator and can now be added. 6 eighths plus 5 eighths equals 11 eighths, in just the same manner as 6 inches plus 5 inches equals 11 inches. The work of this example would be written as follows: 8 is called the Least Common Denominator (L. C. D.) of and, because it is the smallest number that can be used as a common denominator for these two fractions. In this case, the least common denominator is apparent at a glance; in many other cases it is more difficult to find, especially if there are several fractions to be added. In the case just given, the denominator of one fraction can be used for the common denominator. When we have two denominators like 5 and 8, neither of them is an exact multiple of the other number, and so neither can be the common denominator. In such a case, the product of the two numbers can be used as a common denominator. We can always be sure that the product of the denominators will be a common denominator, to which all the fractions can be reduced, but it will not always be the least common denominator. For example, if we wish to add and, we can use 12X16= 192 for the common denominator, but we readily see that 48 will serve just as well and not make the fractions so cumbersome. In this case 48 is the least common denominator. 15. To Find the L. C. D.-If the L. C. D. cannot be easily seen by examining the denominators, it may be found as follows: Suppose we are to find the L. C. D. of 1, 3, 5, and First place the denominators in a row, separating them by commas. Select the smallest number (other than 1) that will exactly divide two or more of the denominators. In this case, 2 will exactly divide 4 and 16. Divide it into all the numbers that are exactly divisible by it, that is, may be divided by it without leaving a remainder. When writing the quotients below, also bring down any numbers which are not divisible by the divisor and write them with the quotients. Now proceed as before, again using the smallest number that will divide two or more of the numbers just obtained. Continue this process until no number (except 1) will exactly divide more than one of the remaining numbers. The product of all the divisors and all the numbers (except 1's) left in the last line of quotients is the Least Common Denominator. 16. To Reduce to the L. C. D.-Having found the least common denominator of two or more fractions, the next step is to reduce the given fractions to fractions having this least common denominator. Let us take, 1, 1, and . We first find the L. C. D., which turns out to be 120. We next proceed to reduce the fractions to fractions having the L. C. D. Divide the common denominator by the denominator of the first fraction. Multiply both numerator and denominator of the fraction by the quotient thus obtained. Do this for each fraction, as illustrated here. 17. Addition of Fractions.-Addition of fractions is very simple after the fractions have been reduced to fractions with a common denominator. Having done this it is only necessary to add the numerators and place this sum over the common denominator. The sum should always be reduced to lowest terms and if it turns out to be an improper fraction it should be reduced to a mixed number. If there are mixed numbers and whole numbers, add the whole numbers and fractions separately. If the sum of the fractions is an improper fraction, reduce it to a mixed number and add this to the sum of the whole numbers. Example: How long a steel bar is needed from which to shear one piece each of the following lengths: Explanation: The sum of the whole numbers is 22. The sum of the fractions is which reduces to 18 8 4 Adding this to the sum of the whole numbers (22), 1 gives 24 as the sum of the mixed numbers. Hence 4 we must have a bar 24 inches long. 18. Subtraction of Fractions. Just as in addition, the fractions must first be reduced to a common denominator. Then we can subtract the numerators and write the result over the common denominator. In subtracting mixed numbers, subtract the fractions first and then the whole numbers. Example: How much must be cut from a 151⁄2 in. bolt to make it 12 3 16 Sometimes, in subtracting mixed numbers, we find that the fraction in the subtrahend (the number to be taken away) is larger than the fraction in the minuend (the number from which the subtrahend is to be taken). In this case, we borrow 1 from the whole number of the minuend and add it to the fraction of the minuend. This makes an improper fraction of the fraction in the minuend and we can now subtract the other fraction from it. If the minuend happens to be a whole number, borrow 1 from it and write it as a fractional part of the minuend. Then subtract as before. |