Equivalent Fractions - GCSE Maths Definition

Definition

Equivalent fractions are fractions that are written with different values in the numerator and denominator, but represent the same value overall.

Explanation

Two (or more) fractions are equivalent if their value is the same, despite appearing different at first glance. One way to consider equivalent fractions is that If one, both, or all, can be simplified such that the fractions become identical, they are equivalent.

Equivalent fractions are generated by either multiplying the numerator and denominator by the same number, or by dividing the numerator and denominator by the same number.

For example, 

Multiplying: The fraction is equivalent to .

Dividing: The fraction is equivalent to .

You may notice that the fraction simplifies to by dividing both the numerator and denominator by 2.

Equivalent fractions are used when we want to compare the values of different fractions. By using equivalent fractions, we can make their denominators equal and then the comparison is really easy - we just need to consider the values of their denominators.

Equivalent fractions are also useful - almost essential - when there is a need to add or subtract fractions. Adding and subtracting fractions is far easier when their denominators have the same value.

Example 1 - Comparing the size/value of fractions

Question:
Determine which of these fractions is the largest, giving a reason for your answer.

Solution:
First write both fractions with the same denominator - both are already in their simplest form, so we will need to find a number both of the denominators divide into. An obvious choice would be 30.

Step 1:
Multiply the numerator and denominator of by 10 to write it as an equivalent fraction with denominator 30.

Step 2:
Multiply the numerator and denominator of by 3 to write it as an equivalent fraction with denominator 30.

Step 3:
Compare the numerators of the equivalent fractions (20 and 21) to determine which fraction is largest. Remember to give a reason with your final answer!

Answer:
is larger than as when written as equivalent fractions with the same denominator of 30, the numerator of (21) is greater than the numerator of (20).

Example 2 - Adding (and subtracting) fractions

Question:
Find the value of

Solution:

Step 1:
To add and subtract fractions, their denominators need to be equal.
We need to deduce a number that both 9 and 12 divide into.
Considering their times tables, 36 (is the smallest number that) appears in both the 9 and 12 times tables.

Step 2:
Use equivalent fractions to change the numerator and denominator (but crucially not the overall value of) each fraction. We need to multiply ‘9’ by ‘4’ to get ‘36’, and ‘12’ by ‘3’ to get ‘36’

and

Step 3:
Now the adding is straightforward.

Answer:

Common mistakes (and how to avoid them)

Mistake 1: Getting the arithmetic wrong (without a calculator) 

A common mistake, especially when trying to work quickly, is getting the multiplying or dividing wrong, Most students understand the concept of equivalent fractions, but when trying to find or generate them, mistakes often happen with multiplying and dividing.

How to avoid this: Ensure you are familiar with your times tables up to 12 × 12. Although any can be worked out if need be, being familiar with them allows you to work more efficiently so you can concentrate on the fractions, rather than getting bogged down with the arithmetic.

Mistake 2: Not realising the need for equivalent fractions

One of the main uses of equivalent fractions is to be able to easily add and subtract them. A question may ask you to add or subtract fractions, but won’t mention that one, or both, may need to be rewritten using equivalent fractions first.

How to avoid this: Remember that fractions can only be added or subtracted (without a calculator) when the denominators are equal. If they are not, one or both of the fractions should be rewritten as an equivalent fraction so that the denominators are equal

For example,

5 and 15 (the denominators) are not equal. But by multiplying the numerator and denominator of by 3, we can make both denominators 15. The adding process is then easy. Also notice how the final answer could be simplified, too, by dividing the numerator and denominator by 5.

Mistake 3: Not fully simplifying a fraction

In the case of using division to find equivalent fractions, we are often trying to find a fraction in its simplest form. In these cases, a common mistake is to not fully simplify a fraction, and assuming the fraction is simplified because one application of the division process has been completed.

For example,

Whilst this fraction has been simplified by dividing the numerator and denominator by 2, it is not fully simplified. There is still a number (including 2 again) that can be divided into both the numerator and denominator, so it cannot yet be in its simplest form

By dividing again, this time by 6, we do get a fully simplified fraction; no number, other than 1, will divide exactly (evenly) into 6 and 13.

How to avoid this: To fully simplify a fraction in one go, the highest common factor of the numerator and denominator should be found. This can be time-consuming; most GCSE questions will involve common multiples that can be determined by being familiar with the times tables up to 12 × 12.

Mistake 4: Struggling to find a common denominator

When using equivalent fractions to write two or more fractions with the same denominator (for comparison or adding/subtracting purposes), students often struggle to identify a number that all the original denominators will divide into.

How to avoid this: To find the (lowest) common denominator of two or more fractions, we would want to find their lowest common multiple (LCM). This can be time-consuming, most GCSE questions will involve common denominators that can be ‘spotted’ - by being familiar with the times tables up to 12 × 12.

Frequently asked questions

Can I not just use my calculator to find equivalent fractions?

In general, yes, but obviously not on the non-calculator exam! Be aware that whilst a calculator will find equivalent fractions in the simplifying sense, it will not do the opposite. This is because there is no end to the number of equivalent fractions - we can just keep multiplying the numerator and denominator by a different number and generate them all day long!

Other than adding and subtracting, where else might I need equivalent fractions?

Probability. Although adding and subtracting fractions will still be involved, one of the key things to ensure accuracy in probability work is to not simplify fractions in the first place. This makes it much easier to compare probabilities, pick out which probabilities should be used in a particular question (especially if written on a diagram such as a Venn or tree diagram).
(Equivalent) fractions are also linked to ratio and proportion.

Related GCSE Mathematics glossaries

• Fraction

• Simplify 

• Equal

• Lowest common multiple

• Highest common factor

Related GCSE Mathematics Revision Notes

• Equivalent & Simplified Fractions

• Adding & Subtracting Fractions

• Ordering Fractions, Decimals & Percentages

• HCF & LCM