Math Core

In third grade you saw that $\dfrac{1}{2}$ and $\dfrac{2}{4}$ name the same amount. Now you will learn why that works, and a rule that lets you build as many equivalent fractions as you want, with any denominator you need.

Cutting the pieces smaller

Start with a rectangle split into 3 equal columns. Shade 2 of them. That is $\dfrac{2}{3}$ of the rectangle.

Now draw 3 lines across the rectangle, so each column is cut into 4 equal parts. Nothing new is shaded. The shaded amount did not change, but look at the pieces:

Left: 2 of 3 columns are shaded. Right: the same rectangle cut into 12 parts, with 8 of them shaded.

The whole now has $3 \times 4 = 12$ pieces.
The shaded part now has $2 \times 4 = 8$ pieces.

So $\dfrac{2}{3} = \dfrac{8}{12}$ . Cutting every piece into 4 made 4 times as many pieces in the whole and 4 times as many pieces in the shaded part. Each piece is smaller, but you have more of them, so the amount stays the same.

Multiply the top and bottom by the same number

To make an equivalent fraction, multiply the numerator and the denominator by the same whole number (not 0):

\frac{a}{b} = \frac{n \times a}{n \times b}

For example, $\dfrac{2}{3} = \dfrac{4 \times 2}{4 \times 3} = \dfrac{8}{12}$ .

Grouping pieces together

The rule works backward too. In $\dfrac{6}{8}$ , you can group the eighths in pairs. Each pair of eighths makes one fourth. There are $6 \div 2 = 3$ pairs shaded out of $8 \div 2 = 4$ pairs in all, so

\frac{6}{8} = \frac{6 \div 2}{8 \div 2} = \frac{3}{4}

You can divide the top and bottom by the same number, as long as it divides both evenly.

Equivalent fractions on a number line

Equivalent fractions land on the same point of a number line. Here, fifths and tenths:

Each fifth splits into 2 tenths, so $\dfrac{3}{5} = \dfrac{3 \times 2}{5 \times 2} = \dfrac{6}{10}$ . The two dots are at the same spot.

Worked example: Find the missing numerator

Fill in the blank: $\dfrac{3}{4} = \dfrac{?}{12}$ .

Ask: what times 4 gives 12? $4 \times 3 = 12$ , so the denominator was multiplied by 3. Do the same to the numerator:

\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}

The missing number is 9.

Worked example: Find the missing denominator

Fill in the blank: $\dfrac{5}{6} = \dfrac{10}{?}$ .

The numerator went from 5 to 10, so it was multiplied by 2. Multiply the denominator by 2 as well: $6 \times 2 = 12$ .

\frac{5}{6} = \frac{10}{12}

Worked example: Tenths and hundredths

Fill in the blank: $\dfrac{40}{100} = \dfrac{?}{10}$ .

The denominator went from 100 down to 10, so it was divided by 10. Divide the numerator by 10 too: $40 \div 10 = 4$ .

\frac{40}{100} = \frac{4}{10}

Worked example: Make a list

Write three fractions equivalent to $\dfrac{2}{5}$ .

Multiply the top and bottom by 2, then 3, then 4:

\frac{2}{5} = \frac{4}{10} = \frac{6}{15} = \frac{8}{20}

Common mistake

Adding the same number to the top and bottom does not make an equivalent fraction. $\dfrac{1}{2}$ plus 1 on top and bottom gives $\dfrac{2}{3}$ , and $\dfrac{2}{3}$ is more than half. Always multiply (or divide) both numbers by the same amount.

Tip

To check two fractions, ask: "Did the top and bottom both get multiplied by the same number?" For $\dfrac{3}{8}$ and $\dfrac{6}{16}$ : $3 \times 2 = 6$ and $8 \times 2 = 16$ . Yes, they are equivalent.

Practice

Practice 1

Fill in the missing numerator: $\dfrac{1}{2} = \dfrac{?}{12}$ .

Type a number.

Practice 2

Fill in the missing numerator: $\dfrac{4}{5} = \dfrac{?}{10}$ .

Type a number.

Practice 3

Fill in the missing denominator: $\dfrac{2}{3} = \dfrac{6}{?}$ .

Type a number.

Practice 4

Which fraction is not equivalent to $\dfrac{3}{4}$ ?

Practice 5

Fill in the missing numerator: $\dfrac{30}{100} = \dfrac{?}{10}$ .

Type a number.

Practice 6

Fill in the missing numerator: $\dfrac{8}{12} = \dfrac{?}{3}$ .

Type a number.

Practice 7

A rectangle is split into 6 equal columns, and 5 columns are shaded. Then every column is cut into 2 equal parts. Which fraction names the shaded part now?

Practice 8

Ben's recipe uses $\dfrac{3}{5}$ of a liter of juice. His measuring cup is marked in hundredths of a liter. How many hundredths should he pour? (Fill in: $\dfrac{3}{5} = \dfrac{?}{100}$ .)

Type a number.