Equivalent fractions

1/2, 2/4, 3/6 and 50/100 are four names for the same point on the number line.

Same amount, different cuts

Take a sheet of paper, fold it in half and shade one half. Now fold it in half again the other way. The shaded part has not changed, but the paper is now divided into 4 equal parts and the shaded region covers 2 of them. The same amount that was 1/2 is now 2/4. Fold once more and it becomes 4/8. Fractions that name the same amount are called equivalent fractions.

Each fold doubled the number of parts and also doubled the number of shaded parts. That is the rule in one sentence: multiply the numerator and the denominator by the same number, and the value of the fraction does not change. 1/2 becomes 3/6 when both numbers are multiplied by 3, and 50/100 when both are multiplied by 50.

Why the rule works

Multiplying the denominator by 3 cuts every piece into 3 smaller pieces. You now have 3 times as many pieces, and each one is a third of the size it was. Multiplying the numerator by 3 counts 3 times as many of those smaller pieces. The two effects cancel exactly, so the total amount stays the same.

Another way to say it uses the number 1. Any fraction with the same numerator and denominator, such as 3/3 or 50/50, equals 1. Multiplying a number by 1 never changes it. So 1/2 × 3/3 = 3/6 is the same as 1/2 × 1, which is still 1/2. Finding an equivalent fraction is multiplying by a well-chosen form of 1.

The rule also runs backwards. Dividing the numerator and the denominator by the same number gives an equivalent fraction too, as long as both divide evenly. 6/9 divided top and bottom by 3 is 2/3. This direction is called simplifying, and it is the subject of the next unit.

What does not work

Adding the same number to the top and bottom does not give an equivalent fraction. Adding 1 to both parts of 1/2 gives 2/3, and 2/3 is bigger than 1/2. Subtracting does not work either: 3/4 minus 1 on both parts is 2/3, which is smaller. Only multiplying and dividing keep the value fixed, because only they change the number of pieces and the size of each piece in balance.

Finding a missing number

Many problems ask you to complete an equivalent fraction, such as 3/5 = ?/20. Ask what the known part was multiplied by. The denominator went from 5 to 20, which is × 4, so the numerator must also be multiplied by 4: 3 × 4 = 12, and 3/5 = 12/20. For 2/3 = 10/?, the numerator went from 2 to 10, which is × 5, so the denominator is 3 × 5 = 15.

Testing two fractions with cross products

To check whether two fractions are equivalent without drawing anything, multiply each numerator by the other fraction’s denominator. For 6/8 and 9/12: 6 × 12 = 72 and 9 × 8 = 72. The two cross products are equal, so the fractions are equivalent. For 3/7 and 9/20: 3 × 20 = 60 and 9 × 7 = 63, which are different, so the fractions are not equal. The test works because the cross products are the numerators you would get after rewriting both fractions with the shared denominator 8 × 12 or 7 × 20.

Where equivalent fractions are used

Equivalent fractions are what make the rest of this course possible. Comparing 2/3 with 3/4, adding 1/2 to 1/3, and turning 3/4 into a percent all start by rewriting fractions so that they share a denominator. 3/4 = 75/100 is the reason three quarters is 75%. Every recipe that converts cups to half cups and every ruler marked in sixteenths of an inch relies on the same rule.

Example. Write three fractions equivalent to 2/5. Multiply both parts by 2, 3 and 10: 4/10, 6/15 and 20/50. Check one with cross products: 2 × 15 = 30 and 6 × 5 = 30.