Improper Fractions & Mixed Numbers — KS3 Maths

An improper fraction is top-heavy — its top number is bigger than its bottom, like $\tfrac{7}{3}$. A mixed number writes the same value as a whole number plus a fraction, like $2\tfrac{1}{3}$. They're two ways of saying exactly the same amount, and one short method converts each way.

What is an improper fraction?

A fraction is improper when its top number is at least as big as its bottom. That sounds like something has gone wrong, but nothing has — $\tfrac{7}{3}$ just means seven thirds, and since three thirds make a whole, seven of them make more than one whole.

Picture three-slice pizzas. Seven slices fills two whole pizzas with one slice left over — which is exactly $2\tfrac{1}{3}$. Both notations describe the same seven slices.

Improper fractions are easier to calculate with; mixed numbers are easier to picture. That's why you need both.

How do you turn an improper fraction into a mixed number?

Divide the top by the bottom. The answer is the whole number, the remainder is the new top, and the bottom never changes.

Worked example

Convert 7/3 to a mixed number

Divide the top by the bottom:

$7 \div 3 = 2 \text{ remainder } 1$

The 2 is the whole number, the remainder 1 goes on top, and the bottom stays 3:

$\tfrac{7}{3} = 2\tfrac{1}{3}$

Worked example

Convert 17/5 to a mixed number

$17 \div 5 = 3 \text{ remainder } 2$

$\tfrac{17}{5} = 3\tfrac{2}{5}$

Worked example

Convert 8/4 to a mixed number (no remainder)

$8 \div 4 = 2 \text{ remainder } 0$

There's nothing left over, so there's no fraction part at all:

$\tfrac{8}{4} = 2$

A remainder of zero means the answer is a whole number — write $2$, not $2\tfrac{0}{4}$.

How do you turn a mixed number back into an improper fraction?

Times the whole number by the bottom, then add the top. The bottom stays the same.

Worked example

Convert 2 3/5 to an improper fraction

The whole number is 2 and the bottom is 5, so the two wholes are worth $2 \times 5 = 10$ fifths. Add the 3 fifths already there:

$2 \times 5 + 3 = 13$

$2\tfrac{3}{5} = \tfrac{13}{5}$

Check it by going back the other way: $13 \div 5 = 2$ remainder $3$, giving $2\tfrac{3}{5}$. Every conversion can be checked by reversing it.

Where you'll use this next

• Adding and subtracting fractions often produces a top-heavy answer that should be written as a mixed number.
• Multiplying and dividing fractions need mixed numbers converted to improper first — the methods don't work on mixed numbers.
• Measurements and recipes are usually written as mixed numbers, so you convert in both directions constantly.

Practice questions

Divide the top by the bottom one way; times and add the other. Then click to check.