Equivalent fractions: examples and practice

What equivalent fractions are

Equivalent fractions name the same amount. 1/2, 2/4, 3/6 and 4/8 look different, but each one is exactly half of the whole. The numbers change because the whole has been cut into a different number of pieces, not because the amount has changed.

Picture a bar cut into 2 equal parts with 1 part shaded. Now cut each part in half again. The bar has 4 parts and 2 are shaded, but the shaded length is the same. The number of pieces doubled and each piece became half as big, so 1/2 = 2/4. That is why the rule works: multiplying the numerator and denominator by the same number cuts every piece into smaller equal pieces without changing the shaded amount.

Another way to see it: multiplying by 3/3 is multiplying by 1, and multiplying by 1 never changes a number. So 2/5 × 3/3 = 6/15 is the same amount as 2/5.

Equivalent fractions on a number line

Draw a line from 0 to 1 and mark halves on it. Underneath, draw a second line the same length marked in quarters, and a third marked in eighths. 1/2, 2/4 and 4/8 all line up at exactly the same point. That is how the grade 3 standards define it: two fractions are equivalent if they are the same point on a number line.

The number line also shows wholes in disguise. 2/2, 4/4 and 8/8 all sit at 1, and 6/4 sits at the same point as 3/2. When you count along a number line, count the jumps between the marks, not the marks themselves.

Equivalent fractions examples

• 1/3 = 2/6 = 3/9 = 4/12, multiplying the top and bottom by 2, then 3, then 4.
• 3/4 = 6/8 = 9/12 = 15/20.
• 2/5 = 4/10 = 40/100, a useful link to decimals, because 40/100 is 0.40.
• 10/15 = 2/3, found by dividing the top and bottom by 5. Going this way is called simplifying.
• 3/5 = __/20: the denominator was multiplied by 4, so the numerator is 3 × 4 = 12, giving 12/20.
• Which is equivalent to 3/5: 6/8, 6/10 or 3/10? The answer is 6/10, because both parts were multiplied by 2. 6/8 came from adding 3 to both parts, and adding changes the amount.

Equivalent fractions explained for kids

Imagine two pizzas of the same size. One is cut into 4 slices, the other into 8. Eating 1 slice of the first is the same amount of pizza as eating 2 slices of the second. The second person ate more slices, but each slice was half the size. So 1/4 = 2/8.

Paper folding works too. Fold a strip in half and shade one half. Fold it in half again and open it out: there are now 4 parts and 2 are shaded. Fold once more and there are 8 parts with 4 shaded. The shading never moved, so 1/2 = 2/4 = 4/8.

How it is taught in school

In the US, Common Core starts equivalent fractions in grade 3, where children recognize and make simple equivalents like 1/2 = 2/4 and explain them with pictures (3.NF.A.3), using denominators of 2, 3, 4, 6 and 8. In grade 4 they explain why a/b is equivalent to (n × a)/(n × b) and use that to generate equivalent fractions (4.NF.A.1).

In England, Year 3 pupils show equivalent fractions with small denominators using diagrams. Year 4 covers families of common equivalent fractions, and Year 5 asks pupils to identify, name and write equivalent fractions, including tenths and hundredths. Everything that follows, from comparing fractions to adding fractions with unlike denominators, depends on this skill.

Equivalent fractions practice at home

• Start with pictures, not the rule. A child who has seen two quarters cover one half will remember why the rule works.
• Play same or different: write 2/3 and 4/6, or 2/3 and 3/4, and ask your child to prove the answer with a drawing.
• Build a family: start with 1/3 and see how many equivalent fractions your child can write in a minute.
• Try the practice set below. It asks for a missing numerator or denominator, or for the equivalent fraction from a list. Then print a fresh worksheet with an answer key for another day.