Comparing Fractions Calculator

What this calculator does

So, you have two fractions and you need to know which one is bigger, or whether they are equal. This tool takes both and tells you plainly: the first is greater, the first is less, or the two are equal.

Each fraction gets three boxes, a whole number, a numerator, and a denominator, so you can compare mixed numbers like 2 1/2 as well as plain fractions.

How to use it

1. Enter Fraction 1: a whole number if you have one, then numerator over denominator.
2. Enter Fraction 2 the same way.
3. Press Calculate.

You get a verdict in words rather than a number, since the question here is not how much bigger, only which one wins.

Why comparing fractions is not as simple as it looks

With whole numbers, bigger digits mean a bigger number. Fractions do not play that way. Take 5/8 and 3/5. You cannot compare the tops, 5 against 3, and call 5/8 the winner on that alone, and you cannot compare the bottoms either. A larger top does not mean a larger fraction: 1/2 is bigger than 3/8, even though 1 is smaller than 3.

The reason is the one that runs through all of fractions: different denominators mean different-sized pieces, and you cannot rank piece-counts until the pieces are the same size. That is what every comparison method is really sorting out.

The three ways to compare, and the one it uses

There are three standard ways to settle it, and they always agree:

• Common denominator. Rewrite both over the same bottom, then the one with the bigger top is bigger.
• Cross-multiply. For a/b against c/d, compare a × d with c × b. Whichever product is larger belongs to the larger fraction.
• Convert to decimals. Turn each fraction into its decimal value and compare those.

This tool takes the decimal route. It builds each fraction, folding any whole number in first, and compares their values with the help of math.js, a JavaScript maths library, then reports which side is larger.

A worked example, three ways to the same answer

Compare 5/8 and 3/5.

• Common denominator of 40: 5/8 is 25/40, and 3/5 is 24/40. Since 25 beats 24, 5/8 is greater.
• Cross-multiply: 5 × 5 = 25 against 3 × 8 = 24. Again 25 wins, so 5/8 is greater.
• Decimals: 5/8 is 0.625 and 3/5 is 0.6. Once more 5/8 comes out on top.

Three routes, one verdict. Numbers this close, 0.625 against 0.6, are exactly the kind you could not rank by eye, which is the point of having the tool do it.

Mixed numbers, and fractions that turn out equal

Because each fraction has a whole-number box, you can compare mixed numbers directly, such as 2 1/2 against 2 3/5, without converting them yourself first. And the tool is honest about ties. Fractions that look different can be the same value, and it will say so: compare 2/4 with 3/6 and the answer is equal, because both are one half. That is often the most useful thing it tells you, since equal fractions in disguise are easy to mistake for different ones.

Questions people ask

Why can't I just compare the top numbers?

Because the denominators may differ, which means the pieces are different sizes. A bigger numerator over a bigger denominator can still be the smaller fraction, so the tops alone do not decide it.

How does the tool decide which is bigger?

It converts each fraction to its value, with any whole number folded in, and compares those values. Then it reports greater, less, or equal.

Can it compare mixed numbers?

Yes. Use the whole-number box on either fraction, and it takes the whole part into account when comparing.

What if the two fractions are equal?

It says so. 2/4 and 3/6 both come back as equal, since they are the same value, one half, written with different numbers.

What is cross-multiplication?

A shortcut for comparing a/b with c/d: compare a × d against c × b. The larger product marks the larger fraction, without having to find a common denominator first.

References

On comparing and on equal fractions. Ranking two fractions comes down to comparing them as values, and recognising when two are equal comes down to their shared lowest terms, since fractions that reduce to the same simplest form are the same number.

1. Eric W. Weisstein, "Fraction," from MathWorld, a Wolfram resource, on the numerator, denominator, and value of a fraction.
2. Eric W. Weisstein, "Reduced Fraction," from MathWorld, a Wolfram resource, on the lowest-terms form that makes two equal fractions recognisable as equal.