Equivalent Fractions Calculator

The calculator finds equivalent fractions of given fractions, integers, and mixed numbers. Input values can be positive or negative. To find equivalent fractions of integers and mixed numbers, the calculator will first convert them into fractions. If the input value is already a fraction, this calculator can be used as a fraction-to-fraction converter.

Directions for use

To use the calculator, enter the given value and press “Calculate.”

Input value limitations

The calculator accepts the following numbers as inputs:

1. Proper fractions. For example, \$\frac{1}{3}\$ or \$-\frac{16}{32}\$. Note, that the fractions do not have to be simplified.
2. Improper fractions. For example, \$-\frac{5}{2}\$ or \$\frac{16}{8}\$.
3. Mixed numbers. When entering a mixed number, separate the whole number part from the fractional part with a space. For example, \$2\frac{2}{3}\$ or \$5\frac{9}{2}\$. Note that the fractional part of a mixed number can be proper or improper.
4. Integers, with the exception of zero. For example, 92 or -1.

Definitions

Equivalent fractions – are fractions describing the same value, but consisting of different numbers. For example, \$\frac{1}{2}\$ is equivalent to \$\frac{4}{8}\$, even though they consist of different numbers.

How to find equivalent fractions

To find equivalent fractions, multiply or divide the numerator and the denominator of the given fraction by the same number. The process should be performed only when both resulting numbers (numerator and denominator) are whole (not decimals and not fractions).

For example, to find equivalent fractions of \$\frac{1}{2}\$, you can continuously multiply the numerator and the denominator by ANY number, as long as both resulting numbers (numerator and denominator) are whole.

Let’s write equivalent fractions of \$\frac{1}{2}\$ by multiplying by 4:

\$\frac{1}{2}\$ = \$\frac{1 × 4}{2 × 4}\$ = \$\frac{4}{8}\$ = \$\frac{16}{32}\$ = \$\frac{64}{128}\$ …

Since the process of multiplication can continue infinitely, each fraction has an infinite number of equivalent fractions.

It is important to note, that since equivalent fractions are calculated by multiplying or dividing the numerator and the denominator of the given fraction with the same number, the simplest form of all equivalent fractions is the same.

It is also obvious that two different fractions in their simplest form can never be equivalent.

Checking if two fractions are equivalent

To check if two fractions are equivalent, calculate their cross products. The fractions are equivalent, if their cross products are equal.

Example 1

Let’s check if \$\frac{1}{3}\$ and \$\frac{4}{11}\$ are equivalent. To find cross products of two fractions, multiply the numerator of the first fraction with the denominator of the second fraction, and the denominator of the first fraction with the numerator of the second fraction:

$$\frac{1}{3}\ and\ \frac{4}{11}$$

The cross products of these two fractions are (1 × 11) = 11 and (3 × 4) = 12. 11 ≠ 12, therefore, \$\frac{1}{3}\$ ≠ \$\frac{4}{11}\$, and the given fractions are not equivalent.

Example 2

Which fraction is equivalent to \$\frac{2}{3}\$: \$\frac{12}{18}\$ or \$\frac{12}{19}\$?

To answer this question, we need to check the cross products of two pairs of fractions:

$$\frac{2}{3}\ and\ \frac{12}{18}$$

$$\frac{2}{3}\ and\ \frac{12}{19}$$

Cross products of \$\frac{2}{3}\$ and \$\frac{12}{18}\$ are (2 × 18) = 36, and (3 × 12) = 36. The cross products are equal, therefore, \$\frac{2}{3}\$ and \$\frac{12}{18}\$ are equivalent fractions.

Cross products of \$\frac{2}{3}\$ and \$\frac{12}{19}\$ are (2 × 19) = 38 and (3 × 12) = 36. 38 ≠ 36, therefore, \$\frac{2}{3}\$ and \$\frac{12}{19}\$ are not equivalent.

Calculation example

In real life finding equivalent fractions is very useful, when we have to add, subtract or compare fractions with different denominators, or fractions and mixed numbers or integers.

Cutting the pizza

Let’s demonstrate an easy example of pizza cutting. Imagine you and your friend ordered a pizza, but it was delivered uncut. You want to share the pizza equally between the two of you, but of course, cutting it in two pieces and eating half of the pizza is not very convenient. In how many pieces can you cut the pizza, and how many pieces should each of you eat?

Solution 1

It is obvious that each of you should eventually eat half of the pizza, so \$\frac{1}{2}\$. To answer the given questions, we should find some fractions, equivalent to \$\frac{1}{2}\$. Let’s first do that by repeatedly multiplying the numerator and the denominator of \$\frac{1}{2}\$ by 2. We will get:

\$\frac{1}{2}\$ = \$\frac{1 × 2}{2 × 2}\$ = \$\frac{2}{4}\$ = \$\frac{4}{8}\$ = \$\frac{8}{16}\$ …

That means you can cut the pizza into 4 slices, in which case each of you can eat 2 slices. Or you can cut the pizza smaller, into 8 slices, in which case each of you can eat 4 slices. Or you can cut it into 16 slices, in which case each of you can eat 8 slices. Cutting the pizza into more than 16 pieces would be inconvenient, so we'll stop there.

Solution 2

Note that you can solve the given problem by multiplying the original fraction with a different number each time:

\$\frac{1}{2}\$ = \$\frac{1 × 2}{(2 × 2}\$ = \$\frac{2}{4}\$ = \$\frac{1 × 2}{2 × 2}\$ = \$\frac{3}{6}\$ = \$\frac{1 × 3}{2 × 3}\$ = \$\frac{4}{8}\$ = \$\frac{1 × 4}{2 × 4}\$ = \$\frac{5}{10}\$ = \$\frac{1 × 5}{2 × 5}\$ = \$\frac{6}{12}\$ = \$\frac{1 × 6}{2 × 6}\$ = \$\frac{7}{14}\$ = \$\frac{1 × 7}{2 × 7}\$ = \$\frac{8}{16}\$ = \$\frac{1 × 8}{2 × 8}\$ …

In this case, some of the obtained fractions will be the same as fractions from Solution 1, but some will be different. Here, we get the same options of \$\frac{2}{4}\$, \$\frac{4}{8}\$, and \$\frac{8}{16}\$, but we also get additional options of \$\frac{3}{6}\$, \$\frac{5}{10}\$, \$\frac{6}{12}\$, and \$\frac{7}{14}\$.

This means, that you can also cut the pizza in 6 pieces, while each of you can have 3; or cut it in 10 pieces, while each of you can have 5; or cut it in 12 pieces, while each of you can have 6, etc. Again, this process can continue infinitely, but we only list options that seem reasonable for cutting a pizza.

Answer

\$\frac{1}{2}\$ = \$\frac{2}{4}\$ = \$\frac{3}{6}\$ = \$\frac{4}{8}\$ = \$\frac{5}{10}\$ = \$\frac{6}{12}\$ = \$\frac{7}{14}\$ = \$\frac{8}{16}\$ …

In these equivalent fractions the denominators represent the total number of pieces, while the corresponding numerators represent the number of pieces each of you can eat.