Simplify Fractions Calculator

What does it mean to simplify a fraction?

A fraction is in lowest terms (also called simplest form) when its numerator and denominator share no common factor other than 1. The fractions 6/8, 9/12 and 3/4 all describe the same amount, but only 3/4 is fully reduced.

Simplifying never changes the value. Dividing the top and bottom by the same non-zero number is the same as multiplying the fraction by 1, written in disguise as n/n. Reduced fractions are easier to read, to compare and to use in further arithmetic, and teachers and exam boards usually expect answers in this form.

How to use the calculator

1. Type the fraction in the box using a slash, for example 48/180. Do not put spaces around the slash.
2. Press “Simplify fraction”.
3. Read the line “Simplified fraction”. It shows the reduced fraction, a whole number if the fraction divides exactly, a mixed number if the fraction is improper, or a message that the fraction is already in lowest terms.

For the most predictable results use positive whole numbers. Zero in the numerator or denominator gives no result (a zero denominator is undefined, and a zero numerator is just 0). Mixed numbers written as “2 3/6” are not understood: convert them to an improper fraction first, here 15/6.

To simplify a negative fraction, reduce its positive form and attach the minus sign afterwards. A fraction typed with a negative numerator is reported as “cannot be simplified further”, and decimals inside a fraction are not reduced, so work those by hand.

How to simplify a fraction by hand

The reliable route is to divide both parts by their greatest common divisor (GCD), the largest whole number that goes into both.

a/b = (a ÷ gcd) / (b ÷ gcd)

There are two ways to find the GCD. Listing prime factors: 48 = 2⁴ × 3 and 180 = 2² × 3² × 5. The shared part is 2² × 3 = 12, so 48/180 = (48 ÷ 12)/(180 ÷ 12) = 4/15.

The Euclidean algorithm avoids factoring. Divide the larger number by the smaller and keep the remainder, then repeat with the previous divisor until the remainder is 0. The last non-zero remainder is the GCD.

• 180 = 3 × 48 + 36
• 48 = 1 × 36 + 12
• 36 = 3 × 12 + 0, so the GCD is 12.

If you do not want to find the GCD at once, divide by any common factor you can see (2, 3, 5…) and repeat until nothing more divides both numbers. The GCD calculator finds the divisor for you, and the fractions calculator adds, subtracts, multiplies and divides fractions with the same reduction built in.

Worked examples

Improper fractions have a numerator that is at least as large as the denominator. After reducing 45/12 to 15/4, divide 15 by 4: the quotient 3 is the whole part and the remainder 3 sits over the denominator, giving 3 3/4. To go back, multiply the whole part by the denominator and add the remainder: 3 × 4 + 3 = 15, so 15/4.

Once the fraction is reduced

A simplified fraction is the best starting point for conversions. To see it as a decimal use the fraction to decimal calculator; to express it as a percentage use the fraction to percent calculator. Going the other way, the decimal to fraction calculator returns a fraction that you can reduce here.

When two fractions have different denominators, reduce each one first, then find a common denominator with the LCM calculator. Smaller numbers keep the common denominator small.

Common mistakes

• Subtracting the same number from top and bottom. Only division (or multiplication) by the same number keeps the value.
• Stopping too early. Dividing 48/180 by 2 gives 24/90, which is still reducible; dividing by the full GCD finishes the job in one step.
• “Cancelling” digits instead of factors, such as crossing the 6 out of 16/64. The result 1/4 is correct by accident; the method is not.
• Cancelling across a sum. In (2 + 6)/(2 + 8) the 2s cannot be removed.