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:
- We can multiply both numerator and
denominator by the same number, and we will create a new
fraction equivalent to the original one;
- 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  .
- Divide the denominator 6 into the denominator 24:
 .
- 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:
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