How To Simplify A Mixed Number

A mixed number consists of a non-zero integer like 1, 2, 3 or 4 (or any other higher number, ​or​ any negative version of those numbers) followed by a fractional remainder. Often, a mixed number is the simplest form of expressing a number, so if you're being asked to simplify, there are two things that might be going on: You might be simplifying an improper fraction ​into​ a mixed number, or you might be simplifying the fractional remainder that follows the mixed number.

If you've been given an improper fraction and have been asked to simplify it into a mixed number, all you need is basic division. Note: An improper fraction is a fraction where the numerator, or top number, is larger than the denominator or bottom number. If the numerator is smaller than the denominator, it's a proper fraction and will not yield a mixed number.

1. Divide the Numerator by the Denominator

Divide the numerator of the fraction by the denominator. There's no need to work your answer out to the decimals. Instead, stop once you have a non-zero integer and a remainder. So if you were asked to simplify 13/5, you'd have:

\(13 ÷ 5 = 2 \text{ remainder } 3\)

2. Put the Remainder Over the Denominator

Rewrite your fraction with the non-zero integer (in the example just given, 2) followed by a fraction with the same denominator as the fraction you originally started with. The remainder (in the example just given, 3) goes in the numerator of that fraction. So to continue the example, you'd have this mixed number:

\(2 \,\,\frac{3}{5}\)

In this case the fraction following the mixed number is already in lowest terms, so you can't simplify it any more. If you're not sure whether a fraction is in lowest terms, use the steps in the next section to simplify it (or to see that it's already simplified as much as possible).

If you already have a mixed number and are being asked to simplify it, you may be able to simplify the fraction that follows the mixed number. This only works if the numerator and denominator of the fraction share at least one non-zero factor. For example, if both numbers can be divided by 2, 3, 4 – or any whole number – then you can simplify the fraction. If the only factor they have in common is 1, then the fraction is already in lowest terms and can't be simplified any more.

1. Write out Common Factors

Write out the common factors of the fraction's numerator, and then make a separate list for common factors of the denominator. With practice you'll be able to recognize many of these intuitively, but when you first start out, the lists are very helpful. So if you've been asked to simplify the mixed number 4 15/27, you'd make a list of factors for 15:

\(\text{Factors of 15 }= 1, 3, 5, 15\)

...followed by a list of factors for 27:

\(\text{Factors of 27 } = 1, 3, 9, 27\)

2. Factor out the Greatest Common Factor

Read through the lists you just made and identify the largest non-zero factor that both numbers have in common. In this case, it's 3. Now, factor that number out of both the numerator and the denominator of the fraction. This gives you:

\(\frac{3 × 5}{3 × 9}\)

3. Cancel the Shared Factor

Cancel the shared factor you just identified from both the numerator and denominator of the fraction. In effect, you're dividing both numerator and denominator by 3. This gives you:

\(\frac{5}{9}\)

Because you performed the same division operation on both the numerator and the denominator of the fraction, you haven't actually changed the value of the fraction; you have simplified how it's written. Because the new numerator and denominator don't share any non-zero factors, you can't simplify the fraction any more – but you do need to remember to write back in the whole number or integer that's part of your mixed number. So in truth, your answer is not 5/9 – which was just the fraction part of the mixed number – but 4 5/9.