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 12 × 16 192 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. 12 = 16. To Find the L. C. D.-If the L. C. D. cannot be easily determined 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, as shown below: 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 left in the last line of quotients is the least common denominator. 17. 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,,, and . Let us take,,, and . We first find the L. C. D., which is 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 shown below. 18. 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 happens to be an improper fraction, it should be reduced to a mixed number. Example: 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 pieces of the following lengths (one piece of each length): 7 in., 57 in., 42 in., 6 in.? Explanation: The sum of the whole numbers is 22. The sum of the fractions is , which reduces to 18 Adding this to the sum of the whole numbers (22), gives 241 as the sum of the mixed numbers. Hence, we must have a bar 24 in. long. 19. 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: 3 How much must be cut from a 15-in. bolt to make it 12 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 subtract the other fraction from it. Example: Subtract 9 from 121. 121 = 11/ 23, Answer. 1 3 Explanation: cannot be subtracted from 8' so we borrow 1(or) from 12 and write the minuend 119. 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. 20. Fractional Dimensions on Drawings. Quite frequently the machinist, pattern maker, or tinsmith is called upon to add or subtract fractions in order to obtain necessary dimensions from his drawings or blueprints. Figure 4 shows a side view and an end view of a machine bolt. The side view shows the bolt as it appears when looked at from the side, and the end view shows the bolt as it appears when looked at from the threaded end. If the machinist were given this drawing and asked to make the bolt, he would be at a loss to know what size of bolt was required, how long it should be, what length of thread to cut, etc. Consequently, dimensions are placed on the drawing, as in Fig. 5, to furnish all necessary information. The long lines with arrowheads at the ends are called "dimension lines," and the distance from the point of one arrowhead to the point of another is written midway between the arrowheads. Thus the drawing in Fig. 5 shows that the bolt has a diameter of 3 in., length of 2 U.S. Std. Thd. FINISH ALL OVER FIG. 5.-Machine bolt with dimensions. head of 21 in., length from the edge of the threads to the head of 9 in., and length of threaded portion of 51⁄2 in. However, the draftsman failed to place the over-all length of the bolt on the drawing. Consequently, the machinist must calculate this length in order to determine what length of rough stock is required. Over-all length = 5 in. +9, in. +21 = in. 18 in. (finished) required = 18,3 in., Answer. |