It will be recalled thatí any number dividedby itself is 1. For example, l/l, 2/2, 3/3, 4/4, and all other numbers formed in this way, have the value 1. Furthermore, any number multi- plied by 1 is equivalent to the number itself. For example, 1 times 2 is 2, 1 times 3 is 3, 1 times l/2 is l/2, etc.
These facts are used in changing the formof a fraction to an equivalent form which is more convenient for use in a particular problem.
For example, if 1 in the form 2/2 is multiplied by3/5, the product will still have a value of 3/5 but will be in a different form, as follows:
Figure 4-4 shows that 3/5 of line a is equal to 6/10 of line b where line a equals line b. Line ais marked off in fifths and line b is marked off in tenths. It can readily be seen that 6/10 and 3/5 measure distances of equal length.
Figure 4-4.-Equivalent fractions.
The markings on a ruler show equivalentfractions. The major division of an inch divides it into two equal parts. One of these parts represents 1/2 The next smaller markings divide the inch into four equal parts. It will be noted that two of these parts represent the same distance as 1/2; that is, 2/4 equals 1/2.. Also, the next smaller markings break the inch into 8 equal parts. How many of these parts are equivalent to 1/2 inch? The answer is found by noting that 4/8 equals 1/2. Practice problems. Using the divisions on a ruler for reference, complete the following exercise:
A review of the foregoing exercise will reveal that in each case the right-hand fractioncould be formed by multiplying both the numerator and the denominator of the left-hand fraction by the same number. In each case the number may be determined by dividing the denominator of the right-hand fraction by the denominator of the left-hand fraction. Thus in problem 1, both terms of 1/4 were multiplied by 2. In problem 3, both terms were multiplied by 4. It is seen that multiplying both terms of a fraction by the same number does not change the value of the fraction.
Since 1/2 equals 2/4, the reverse must also betrue; that is 2/4 must be equal to 1/2. This can likewise be verified on a ruler. We have already seen that 4/8 is the same as 1/2 12/16 equals 3/4 and 2/8 equals 1/4. We see that dividing both terms of a fraction by the same number does not change the value of the fraction.
FUNDAMENTAL RULE OF FRACTIONS
The foregoing results are combined to formthe fundamental rule of fractions, which is stated as follows: Multiplying or dividing both terms of a fraction by the same number does not change the value of the fraction. This is one of the most important rules used in dealing with fractions.
The following examples show how the fundamental rule is used:
1. Change 1/4 to twelfths. This problem is setup as follows:
The first step is to determine how many 4ísare contained in 12. The answer is 3, so we know that the multiplier for both terms of the fraction is 3, as follows:
2. What fraction with a numerator of 6 is equalto 3/4?
We note that 6 contains 3 twice; therefore weneed to double the numerator of the right-hand fraction to make it equivalent to the numerator of the fraction we seek. We multiply both terms of 3/4 by 2, obtaining 8 as the denominator of the new fraction, as follows:
3. Change 6/16 to eighths.
We note that the denominator of the fractionwhich we seek is l/2 as large as the denominator of the original fraction. Therefore the new fraction may be formed by dividing both terms of the original fraction by 2, as follows:
Practice problems. Supplythe missing number in each of the following:
REDUCTION TO LOWEST TERMS
It is frequently desirable to change a fraction to an equivalent fraction with the smallestpossible terms; that is, with the smallest possible numerator and denominator. This process is called REDUCTION. Thus, 6/30 reduced to lowest terms is 1/5 Reduction can be accomplished by finding the largest factor that is common to both the numerator and denominator and dividing both of these terms by it. Dividing both terms of the preceding example by 6 reduces the fraction to lowest terms. In computation, fractions should usually be reduced to lowest terms where possible.
If the greatest common factor cannot readilybe found, any common factor may be removed and the process repeated until the fraction is in lowest terms: Thus 18/48 could first be divided by 2 then by 3
Practice problems. Reduce the following fractions to lowest terms: