Mixed Numbers and Improper Fractions

What you will learn

• Tell proper fractions, improper fractions and mixed numbers apart.
• Change an improper fraction into a mixed number.
• Change a mixed number into an improper fraction.

Before you start: you should know what a fraction is and be able to divide with a remainder. See Fractions Explained and Short and Long Division.

The idea

A fraction can be more than one whole. Seven quarters is more than four quarters, and four quarters is one whole.

• A proper fraction is less than one whole: the numerator is smaller than the denominator, like $\frac{3}{4}$.
• An improper fraction is one whole or more: the numerator is the same as, or bigger than, the denominator, like $\frac{7}{4}$.
• A mixed number shows whole numbers and a fraction together, like $1\frac{3}{4}$.

Improper to mixed

Seven quarters fill one whole bar of four quarters, with three quarters over.

Divide the numerator by the denominator. The answer is the whole number and the remainder is the new numerator:

$\frac{7}{4} = 7 \div 4 = 1 \text{ remainder } 3 = 1\tfrac{3}{4}$

Mixed to improper

Count every whole as a full set of parts, then add the extra parts. For $2\frac{1}{3}$, two wholes are $2 \times 3 = 6$ thirds, and one more third makes $7$ thirds.

Worked examples

Example 1: Improper to mixed

Write $\frac{17}{5}$ as a mixed number.

1. Divide: $17 \div 5 = 3$ remainder $2$.
2. Three wholes, and $\frac{2}{5}$ left over.
3. Check: $3 \times 5 + 2 = 17$.

Answer: $3\frac{2}{5}$

Example 2: Mixed to improper

Write $4\frac{2}{3}$ as an improper fraction.

1. Multiply the whole number by the denominator: $4 \times 3 = 12$.
2. Add the numerator: $12 + 2 = 14$.
3. Keep the denominator: $\frac{14}{3}$.

Answer: $\frac{14}{3}$

Example 3: Place it on a line

Between which two whole numbers is $\frac{11}{4}$?

1. $11 \div 4 = 2$ remainder $3$, so $\frac{11}{4} = 2\frac{3}{4}$.
2. That is between $2$ and $3$, closer to $3$.

Answer: between $2$ and $3$.

Tip: a fraction like $\frac{8}{4}$ with no remainder is a whole number: $8 \div 4 = 2$.

Why it matters

We say “two and a half hours” rather than “five halves of an hour”. Mixed numbers suit everyday talk, while improper fractions are easier to calculate with. Being able to switch is a skill you need when adding, subtracting, multiplying and dividing fractions.