Equivalent Fractions Calculator

What this calculator does

So, you have a fraction like 2/3 and you want others that are worth exactly the same. This tool hands you a long list of them: 4/6, 6/9, 8/12, and so on, a hundred fractions that all sit on the same point of the number line as the one you typed.

There is a single text box. You enter the fraction in the usual slash form, like 2/3, and it does the rest.

How to use it

1. Type your fraction as numerator/denominator, for example 2/3.
2. Press Calculate.

It expects positive whole numbers with a slash between them. A whole number on its own can be entered by giving it a denominator of 1, such as 5/1.

What makes two fractions equivalent

Equivalent fractions look different but describe the same amount. 1/2, 2/4, and 50/100 are all one half. They differ only in how finely the whole has been sliced: halves, quarters, hundredths. Cut a pizza into two and take one piece, or cut it into four and take two, and you are holding the same amount of pizza either way.

That is the whole idea the list is showing you. Every fraction in it points at the same place; only the numbers used to name it change.

How it builds the list

The tool takes your numerator and denominator and multiplies both by the same number, first by 1, then by 2, then by 3, all the way up to 100. Multiplying the top and the bottom by the same number never changes a fraction's value, so each result is another fraction equal to the one you started with. For 2/3 that gives 2/3, then 4/6, then 6/9, then 8/12, and it keeps going until it has a hundred of them.

Notice the direction: this always makes the numbers bigger, never smaller. It is climbing up the family of equal fractions, not down toward the simplest one.

A worked example, and the pattern to spot

Enter 3/4 and the list opens 3/4, 6/8, 9/12, 12/16, 15/20, and on it goes.

The pattern is easy to read once you see it. The tops step through the multiples of 3, so 3, 6, 9, 12, 15. The bottoms step through the multiples of 4, so 4, 8, 12, 16, 20. Each fraction is the previous one with both numbers nudged up by one more copy of 3 over 4.

Why they are all the same value

Here is the reason it works, in one line. Multiplying the top and the bottom by the same number, say 5, is the same as multiplying the whole fraction by 5/5. And 5/5 is just 1. Multiplying anything by 1 leaves it unchanged, so the value cannot move, no matter how large the numbers get. That is why 3/4 and 300/400 are the same fraction wearing different clothes.

The opposite trip: reducing to the simplest one

This list runs upward, toward bigger fractions. Every family of equal fractions also has a smallest member, the one in lowest terms, and getting there means dividing rather than multiplying. 6/9 reduces to 2/3, which is the simplest fraction in that whole family. If that is the direction you want, the simplify fractions calculator does it. Scaling up here and reducing there are the same relationship seen from the two ends.

Questions people ask

What makes two fractions equivalent?

They represent the same value. One can be turned into the other by multiplying or dividing the top and bottom by the same number, which is why they land on the same point of the number line.

How many does it give me?

One hundred, from multiplying your fraction by 1 through 100. The first one in the list is the fraction exactly as you entered it.

Does it simplify the fraction for me?

No, it goes the other way, scaling the fraction up into larger equal fractions. To reduce a fraction to its simplest form, use the simplify fractions calculator.

Can I enter a whole number?

Yes, by writing it over 1, such as 5/1. The tool needs a slash, so a bare number without a denominator will not be accepted.

Why multiply the top and bottom by the same number?

Because that is the same as multiplying the fraction by 1, which never changes its value. Multiply only the top or only the bottom and you would change the amount, which is a different thing entirely.

References

On equal fractions. Two fractions are equivalent when they stand for the same rational number, and building one from another by multiplying top and bottom alike is the everyday face of that. The idea sits inside the classical theory of ratios in Euclid's Elements, Book VII, which treats equal ratios and their reduction to lowest terms.

1. Eric W. Weisstein, "Fraction," from MathWorld, a Wolfram resource, on the numerator, denominator, and value of a fraction.
2. Euclid, Elements, Book VII (c. 300 BC), the theory of ratios of numbers and their lowest terms, in David E. Joyce's online edition: Euclid's Elements, Book VII.