Finding Equivalent Fractions
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Example 1

Let us look at the fraction strips for a half.

A half is worth half of a whole. If we split each of the halves into two equal pieces, we end up with fourths or a whole split up into 4 equal parts.

Each half has been split into two-fourths (as there are two parts out of 4 parts shaded), so we have the equivalent fraction:

One half is equivalent to (or equal to) two-fourths.

In other words, \[ {1 \over 2} = {2 \over 4} \]

If we split each half into three equal pieces instead of 2, we get 3 strips shaded (the numerator) out of a total of 6 strips (the denominator), giving the fraction three-sixths.

This gives us: \[ {1 \over 2} = {2 \over 4} = {3 \over 6} \]

We can repeat this by splitting the same half into 4, 5, 6, etc. pieces, ending up with:

\[ {1 \over 2} = {2 \over 4} = {3 \over 6} = {4 \over 8} = {5 \over 10} \]

This series of fractions equivalent to a half could be continued for ever.

It also makes sense in context, for example, if you got 5 out of 10, or five-tenths, in a spelling test correct, then you would have got half the answers correct and half wrong.

This process can be repeated for any fraction.

Example 2

Let us look at the fraction two-thirds.

If we split each of the thirds into two equal parts, we get a total of 6 pieces with 4 pieces shaded, so we now have 4 out of 6 shaded instead of 2 out of 3 shaded.

In other words, we have doubled the number of shaded parts (numerator) and also doubled the number of total parts (denominator).

So now we have: \[ {2 \over 3} = {4 \over 6} \]

Again, this can be repeated and the fraction two-thirds could be split into any number of equal parts.

This gives us the equivalent fraction sequence: \[ {2 \over 3} = {4 \over 6} = {6 \over 9} = {8 \over 12} = {10 \over 15} \]

Each time the fraction is split, both the denominator and the numerator are multiplied by the split. This gives us an equivalent fraction.

In other words, if you multiply the numerator and denominator by the same number, you get the same (or equivalent) fraction.

And by using the same logic, if we divide the numerator and denominator by the same number, you get an equivalent fraction.

Example 3) Take the fraction \[ {2 \over 3} \]

If we multiply the numerator and denominator both by 2, we get

\[ {2 \times 2 \over 3 \times 2} = {4 \over 6} \]

These two fractions are equivalent.

Example 4) Take the fraction \[ {4 \over 8} \]

If we divide the numerator and denominator both by 2, we get

\[ {4 \div 2 \over 8 \div 2} = { 2 \over 4 } \]

These two fractions are equivalent.