What you need to know

We say that two fractions are equivalent if they are equal. We can determine whether or not two fractions are equal by making their denominators equal and seeing if the numerators match.

To understand how to manipulate fractions, you must understand the golden rule: we can multiply/divide top and bottom of a fraction by the same value without changing the value of the fraction. This is extremely important for manipulating fractions in general and is key for being able to compare them.

Note: when we talk about “cancelling down” fractions, this refers to finding a common factor in the numerator and denominator, and then dividing both numbers by that common factor.

Example: Work out whether or not \frac{12}{42} is equivalent to \frac{2}{7}.

We can make these two fractions have the same denominator in two ways: 1) cancel down the first fraction and see if it equals the second fraction, or 2) change the denominator of the second to make it equal to the first and see if the numerators match.

If we multiply top and bottom of the second fraction by 6, it becomes

\dfrac{2}{7}=\dfrac{2\times 6}{7\times 6}=\dfrac{12}{42}

This matches the second fraction, so they must be equivalent.

Example: Show that \frac{15}{40} and \frac{21}{56} are equivalent.

As in the last question, we can either cancel down the fractions or multiply them up to make them have the same denominator. In this case, both denominators are already quite big, so multiplying them further could be tricky. Instead, we’re going to cancel them down.

\dfrac{15}{40}=\dfrac{3\times 5}{8\times 5}=\dfrac{3}{8}

\dfrac{21}{56}=\dfrac{3\times 7}{8\times 7}=\dfrac{3}{8}

Both these fractions cancel down to \frac{3}{8} so they must be equivalent.

Example: 3 of the 4 fractions shown below are equivalent. Determine which one is not.

\dfrac{96}{108},\,\,\,\,\dfrac{24}{27},\,\,\,\,\dfrac{45}{50},\,\,\,\,\dfrac{8}{9}

The last fraction, \frac{8}{9} cannot be cancelled at all, so we’re going to try to cancel the others down and see if they end equal to it.

\begin{aligned}\dfrac{96}{108}&=\dfrac{48\times 2}{54\times 2} \\ &=\dfrac{48}{54} = \dfrac{8\times 6}{9\times 6} \\ &= \dfrac{8}{9}\end{aligned}

So, that means this is equivalent to \frac{8}{9}. One of the remaining two fraction must not be, so cancelling down the next one we get

\dfrac{24}{27}=\dfrac{8\times 3}{9\times 3}=\dfrac{8}{9}

Therefore, the one that is not equivalent is \frac{45}{50}. Indeed, this cancels down to

\dfrac{45}{50}=\dfrac{9\times 5}{10\times 5}=\dfrac{9}{10}

Example: Show that 5\frac{2}{3} is equivalent to \dfrac{34}{6}

To compare a mixed number to a fraction, we should firstly convert it to a fraction.

5\frac{2}{3}=\dfrac{(5\times 3)+2}{3}=\dfrac{17}{3}

Now, we compare it in the usual way. Cancelling down the second fraction, we get

\dfrac{34}{6}=\dfrac{2\times 17}{2\times 3}=\dfrac{17}{3}

This matches up with our mixed number, and so we have shown they are equivalent.

Example Questions

We’re going to cancel down each fraction and see where we get.

 

\dfrac{16}{20}=\dfrac{4\times 4}{4\times 5}=\dfrac{4}{5}

 

\dfrac{36}{45}=\dfrac{4\times 9}{5\times 9}=\dfrac{4}{5}

 

Both fractions cancel down to the same thing, so must be equivalent.

Let’s cancel down the first fraction to see what we get.

 

\dfrac{16}{34}=\dfrac{2\times 8}{2\times 17}=\dfrac{8}{17}

 

Now, clearly \frac{8}{17} is not equivalent to the second fraction, \frac{8}{16}. One of these two must be the odd one out, so we only need to look at one more fraction to see which one of these two it’s equivalent to. Firstly, though, let’s cancel down fraction number 2:

 

\dfrac{8}{16}=\dfrac{8\times 1}{8\times 2}=\dfrac{1}{2}

 

Now, for fraction number 3 we get

 

\dfrac{36}{72}=\dfrac{36\times 1}{36 \times 2}=\dfrac{1}{2}

 

Therefore, fraction 1 is the odd one out, meaning that (without even checking the last one), we know that

 

\dfrac{8}{16},\,\,\,\dfrac{36}{72},\,\,\,\text{ and }\,\,\,\dfrac{55}{110}

 

are equivalent.

Equivalent Fractions Revision and Worksheets

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