Equivalent Fractions

Converting one fraction into an equivalent fraction is a very important skill. This skill plays a vital role in the problem of adding or subtracting two fractions when the denominators of the two fractions are different.

Adding or subtracting two fractions is easy when the denominators are the same, for example,

you simply add (or subtract) the numerators, as above. When the denominators are different, such as

the problem is not solved quite so easily.

The strategy for adding or subtracting fractions with different denominators is to replace one or both fractions with equivalent fractions that all have the same denominator. In the above example, it is my idea to write the .rst fraction equivalent to a fraction with a denominator of 6, that way, all fractions will have the same denominator. So I ask myself the question

How do I figure out what the numerator should be? I get an answer of

Now, returning to problem of adding the two fractions given in ( 1), we have

where I have replaced the fraction 2 /3 with the equivalent fraction 4 /6. Once this has been accomplished, the addition problem becomes easy.

In this section, we concentrate on the skill of converting one fraction into an equivalent fraction, this was the skill I used in ( 2) to obtain a fraction with a denominator of 6.

Now, let’s discuss the strategy for writing equivalent fractions. There are two basic methods that we use:

1. We can multiply both numerator and denominator by the same number, and we will create a new fraction equivalent to the original one;
2. we can divide both numerator and denominator by the same number, and we will again create a new fraction equivalent to the original one.

In this lesson, we will use method (1) to create equivalent fractions.

Problem: Convert a fraction to an equivalent fraction having a specified denominator, d. That is, write

the problem is to figure out what the numerator is (the ‘?’).

To solve this kind of problem, most likely we multiply numerator and denominator by some cleverly chosen number.

Example 1. Write the fraction with a denominator of 6, that is,

Solution: We ask ourself, 3 (the denominator we want to change) times what number is equal to 6 (the denominator we want to change to). The answer is 2 since 3 · 2 = 6. So. . .

Rather than straining our brain looking for a number which multiplied by 3 gives 6, we can simply divide. How many times does 3 go into 6, that is and 2 is the number we are looking for.

Example 2. Write .

Solution: We ask ourself, 5 (the denominator we want to change) times what number is equal to 20 (the denominator we want to change to). The answer is 4 since 5 · 4 = 20. So. . .

We could have computed to get the 4 we need.

Let’s reduce our method down to some simple steps.

Problem: Write .

1. Divide the denominator 6 into the denominator 24: .
2. Multiply the numerator of the left-hand numerator by the number, 4, just computed in Step (1), to get the correct numerator of the right-hand side: