Simplifying fractions

24/36 and 2/3 are the same number. Dividing the top and bottom by 12 turns one into the other.

What simplest form means

A fraction is in simplest form, also called lowest terms, when its numerator and denominator have no common factor other than 1. 2/3 is in simplest form, because no whole number bigger than 1 divides both 2 and 3. 24/36 is not, because 2, 3, 4, 6 and 12 all divide both numbers. Simplifying a fraction means rewriting it as the equivalent fraction in simplest form.

Simplifying changes how the fraction looks, never how big it is. Each division by a common factor merges groups of small pieces into larger pieces, exactly the reverse of the folding in the last unit. A simplified answer is easier to read, easier to compare with other fractions and easier to check, which is why many questions ask for it.

Method 1: divide by small factors, step by step

Look for a small number that divides both parts, divide, and repeat until nothing more works. For 24/36, both numbers are even, so divide by 2: 12/18. Still both even: 6/9. Now both are divisible by 3: 2/3. Neither 2 nor 3 has a common factor left, so 2/3 is the simplest form. This method needs no planning, only patience.

Quick tests help you spot factors. A number is divisible by 2 if its last digit is even, by 5 if it ends in 0 or 5, and by 10 if it ends in 0. It is divisible by 3 if the sum of its digits is divisible by 3, and by 9 if the digit sum is divisible by 9. For 51/87: 5 + 1 = 6 and 8 + 7 = 15, both divisible by 3, so 51/87 = 17/29.

Method 2: divide once by the greatest common factor

The greatest common factor (GCF) of two numbers is the largest number that divides both. Dividing by it simplifies the fraction in one step. To find it, list the factors of each number. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36. The largest one in both lists is 12, and 24 ÷ 12 = 2, 36 ÷ 12 = 3.

For larger numbers, prime factors are faster. Write each number as a product of primes: 84 = 2 × 2 × 3 × 7 and 126 = 2 × 3 × 3 × 7. The common primes are one 2, one 3 and one 7, so the GCF is 2 × 3 × 7 = 42. Then 84/126 = 2/3. You can also cancel the shared primes directly from the top and bottom, which gives the same result.

A third way to find the GCF uses remainders and needs no factor lists. Divide the larger number by the smaller and keep the remainder: 126 ÷ 84 leaves 42. Then divide 84 by 42, which leaves 0. When the remainder reaches 0, the last divisor, 42, is the GCF. This method is the one computers use, because it stays quick even for very large numbers.

Special cases

Improper fractions simplify the same way: 22/8 = 11/4, dividing both parts by 2. A fraction with numerator 0, such as 0/7, equals 0 whatever the denominator. When the numerator divides the denominator exactly, the result is a unit fraction: 7/35 = 1/5. When the denominator divides the numerator exactly, the result is a whole number: 35/7 = 5.

A common slip is cancelling digits instead of factors. In 16/64 it is tempting to strike out the two sixes and write 1/4. The answer happens to be right, but the method is wrong: try it on 13/39 and you get 1/9, while the true value is 1/3. Only whole factors that divide both numbers can be cancelled.

Example. Simplify 45/75. Both end in 5, so divide by 5: 9/15. Both are divisible by 3: 3/5. Check with the GCF: the factors of 45 and 75 in common are 1, 3, 5 and 15, so the GCF is 15, and 45 ÷ 15 = 3, 75 ÷ 15 = 5.