uFigure 4-4

Figure 4-4. -Equivalent fractionsThe markings on a ruler show equivalentfractions, The major division of an inch dividesit into two equal parts. One of these partsrepresentsThe next smaller markings dividethe inch into four equal parts. It will be noted thattwo of these parts represent the same distance asthat iBequalsAlso. the next smallermarkings break the inch into B equal parts, Howmany of these parts are equivalent to g inchThe answer is found by noting that s equalsPractice problems. Using the divisions on aruler for reference, complete the followingexercise:

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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-handfraction by the same number. In each case thenumber may be determined by dividing the denominator of the right-hand fraction by the denominator of the left-hand fraction. Thus inproblem 1, both terms of were multiplied by 2.3, both terms were multiplied by 4.that multiplying both terms of a fracsame number does not change thhe fractionSince equals the reverse must also betrue; that is must be equal to s. This canlikewise be verified on a ruler. We have alreet: thesameas2是uand a equals=. We see that dividing both termsof a fraction by the same number does notchange the value of the fractionFUNDAMENTAL RULE OF FRACTIONSThe foregoing results are combined to forme fundamental rule of fractions, which isstated as follows: Multiplyiding bothterms of a fraction by the gtot change the value of theing e duveThis igone of the most important rules used in dealingwith fractionsThe following examples show how the fundanental rule ig used1. Change 1/4 to twelfths. This problem is setup as follow目:

The first step is to determine how many 4'sre contained in 12. The answer is 3, so weknow that the multiplier for both terms of thefraction is 3, as follows:

2. What fraction with a numerator of 6 is equalto3/4?

We note that 6 contains 3 twice: therefore weneed to double the numerator of the right-handfraction to make it equivalent to the numeratorof the fraction we seek. We multiply both termsof 3/4 by 2, obtaining 8 as the denominator ofthe new fraction, as follows:

3. Change 6/16 to eighths.

We note that the denominator of the fractionwhich we seek is 1/2 as large as the denominator of the original fraction. Therefore the newfraction may be formed by dividing both termsof the original fraction by 2, as follows:

Practice problems, Supply the missing num-ber in each of the following

REDUCTION TO LOWEST TERMSIt 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 processis called REDUCTION, ThuB, a reduced tolowest termB I8Reduction can be accom.plished by finding the largest factor that iscommon to both the numerator and denominatorand dividing both of these terms by it, Dividingboth terms of the preceding example by 6 reduces the fraction to lowest terms, In computa-tion, fractions should usually be reduced tolowest terms where possibleIf the greatest common factor cannot readilyDe found, any common factor may be removedand the process repeated until the fraction is inlowest terms: Thus, r could first be dividedby 2 and then by 3

Practice problems, Reduce the followingfractiona to lowest terms

IMPROPER FRACTIONSAlthough the "improper" fraction is reallyquite "proper"mathematically, it is usuallycustomary to change it to a mixed number. Arecipe may call for 1a cups of milk, but wouldnot call for s cups of milkSince a fraction is an indicated division, amethod is already known for reduction of im-proper fractions to mixed numbers. The im-proper fraction T may be considered as the division of 8. by 3. This division is carried outas follows:

The truth of this can be verified another way:m1gul,tem2mul号,rmu,

These examples lead to the following con-clusion, which is stated as a rule: To changean improper fraction to a mixed number, dividethe numerator by the denominator and write thefractional part of the quotient in lowest termsPractice problems, Change the followingfractions to mixed numbers:

OPERATING WITH MDXED NUMBERSIn computation, mixed numbers are often un-leldy. As it is possible to change any im-proper fraction to a mixed number, it is like-wise possible to change any mixed number to animproper fracton. The problem can be reducedto the fnding of an equivalent fraction and agimple addition.EXAMPLE: Change 22 to an Improper fraction.SOLUTION:Step 1: Write 2- as a whole number plus afraction, 2+Step 2: Change 2 to an equivalent fractionwith a denominator of 5, as follows:

In each of these examples, notice that themultiplier used in step 2 is the same number asthe denominator of the fractional part of theoriginal mixed number. This leads to the following conclusion, which is stated ag a rule:To change a mixed number to an improper fraction, multlply the whole-number part by thedenominator of the fractional part and add thenumerator to thisproduct. The result is thenumerator of the improper fraction; its denomInator is the same as the denominator of thefractional part of the original mixed number.Practice problems, Change the followingmlxed numbers to improper fractions: