What are equivalent fractions and how do you reduce one?

The same amount, a different shape

Half a cake and two quarters of a cake are the same amount of cake. We cut each quarter in two, so there are twice as many pieces and each is half the size — and the amount has not moved.

That is exactly what happens with the numbers. Multiply both numerator and denominator by the same number and the count of parts grows while each part shrinks by the same factor, so the value is preserved.

The rule only holds when both numbers are multiplied by the same thing. Multiplying the numerator alone makes the amount larger, multiplying the denominator alone makes it smaller; each on its own changes the fraction, and only both together preserve it.

Expanding and reducing

Expanding is the direction that makes the numbers larger: multiply numerator and denominator, usually to reach a particular denominator you need later.

Reducing is the reverse: divide both by the same number. Look for a number that divides both without remainder, and if you can, keep going until no common factor remains.

Knowing when two fractions are equal

The simple way is to reduce both fully and see whether the same fraction comes out. Two equivalent fractions always reduce to exactly the same thing.

There is also a shortcut: multiply the numerator of one by the denominator of the other and vice versa. If the two products match, the fractions are equal. It always works, and it is convenient when the numbers are large.

Worked examples

1. Expand 2 over 3 to a denominator of 12

   1. Ask what 3 must be multiplied by to give 12: by 4
   2. Multiply the numerator by 4 as well: 2 times 4 is 8
   3. This gives 8 over 12

   Answer: 8 over 12

2. Reduce 18 over 24

   1. Look for a number dividing both: 6 divides both
   2. Divide: 18 over 6 is 3, and 24 over 6 is 4
   3. Check whether 3 and 4 still share a factor: they do not

   Answer: Three quarters

3. Are 4 over 6 and 6 over 9 equivalent?

   1. Reduce the first by 2: two thirds
   2. Reduce the second by 3: two thirds
   3. Both reached the same reduced form

   Answer: Yes, both equal two thirds

Common mistakes

Multiplying only the numerator or only the denominator

The value is preserved only when both are multiplied by the same thing. Changing one alone changes the quantity itself, not merely how it is written.

Adding instead of multiplying

Adding the same number to both changes the value. A half with one added to each gives two thirds, which is not a half, and a single numerical check shows it.

Stopping the reduction too early

After each division, ask again whether a common factor remains. A fraction that looks reduced but is not gives a right answer in the wrong form.

More in Fractions

• What do the numerator and the denominator of a fraction mean?
• What is an improper fraction and how do you convert it to a mixed number?
• How do you compare two fractions and put them in order?
• How do you add and subtract fractions with the same denominator?
• How do you add fractions with different denominators?