Mixed numbers and improper fractions

Most fractions you've met so far are smaller than $1$: $\frac{1}{2}$, $\frac{3}{4}$, $\frac{5}{8}$. But not all amounts in life fit between $0$ and $1$. If you ate one and a half chapatis at lunch, that "one and a half" is a fraction bigger than $1$.

Concept

A proper fraction has the numerator smaller than the denominator: like $\frac{1}{2}$, $\frac{3}{4}$, $\frac{99}{100}$. Its value is less than $1$.

An improper fraction has the numerator equal to or larger than the denominator: like $\frac{5}{4}$, $\frac{11}{3}$, $\frac{8}{8}$. Its value is at least $1$.

A mixed number is a whole number plus a proper fraction: like $1\frac{1}{2}$ or $3\frac{2}{5}$. It is read as "one and one half" or "three and two fifths".

Mixed numbers and improper fractions are two ways of writing the same amount.

Convert mixed to improper

$a \frac{b}{c} = \frac{a \times c + b}{c}$

The idea: $a$ wholes is $\frac{a \times c}{c}$ (each whole is $c$ pieces of size $\frac{1}{c}$), plus $\frac{b}{c}$ on top.

Example: $2\frac{3}{5} = \frac{2 \times 5 + 3}{5} = \frac{13}{5}$.

Convert improper to mixed

Divide the numerator by the denominator. The quotient is the whole part; the remainder is the new numerator; the denominator stays.

Example: $\frac{17}{5}$. $17 \div 5 = 3$ remainder $2$. So $\frac{17}{5} = 3\frac{2}{5}$.

Why use mixed numbers?

For everyday talk and measurement: "I drank $1 \frac{1}{2}$ glasses of water" is more natural than "I drank $\frac{3}{2}$ glasses". But for calculation, the improper form is often easier , no separating wholes from parts.

A common practice: convert mixed numbers to improper for the calculation, then convert back at the end if needed.

Addition and subtraction with mixed numbers

To add $1 \frac{1}{2}$ and $2 \frac{1}{3}$:

• Convert to improper: $\frac{3}{2} + \frac{7}{3}$.
• Common denominator $6$: $\frac{9}{6} + \frac{14}{6} = \frac{23}{6}$.
• Convert back: $23 \div 6 = 3$ rem $5$, so $3 \frac{5}{6}$.

Or you can add the whole parts and fraction parts separately: $1 + 2 = 3$, $\frac{1}{2} + \frac{1}{3} = \frac{5}{6}$, total $3 \frac{5}{6}$.

Worked examples

Example 1. Convert $3 \frac{2}{7}$ to an improper fraction.

• $\frac{3 \times 7 + 2}{7} = \frac{23}{7}$.

Example 2. Convert $\frac{29}{4}$ to a mixed number.

• $29 \div 4 = 7$ rem $1$. So $7 \frac{1}{4}$.

Example 3. Add $2 \frac{1}{4}$ and $1 \frac{3}{8}$.

• Add whole parts: $2 + 1 = 3$.
• Add fraction parts: $\frac{1}{4} + \frac{3}{8} = \frac{2}{8} + \frac{3}{8} = \frac{5}{8}$.
• Total: $3 \frac{5}{8}$.

Example 4. Subtract $4 \frac{1}{3}$ from $7 \frac{1}{2}$.

• Convert to improper: $\frac{15}{2} - \frac{13}{3}$.
• Common denominator $6$: $\frac{45}{6} - \frac{26}{6} = \frac{19}{6}$.
• Convert back: $3 \frac{1}{6}$.

Example 5. A jar holds $2 \frac{3}{4}$ litres. After pouring some out, $1 \frac{1}{2}$ litres remain. How much was poured?

• $2 \frac{3}{4} - 1 \frac{1}{2}$. Improper: $\frac{11}{4} - \frac{3}{2} = \frac{11}{4} - \frac{6}{4} = \frac{5}{4}$.
• Mixed: $1 \frac{1}{4}$ litres poured.

Example 6. Sketch $\frac{7}{4}$ on a number line.

• $\frac{7}{4} = 1 \frac{3}{4}$, so it sits between $1$ and $2$, three-quarters of the way to $2$.

Try it yourself

1. Convert $2 \frac{5}{6}$ to an improper fraction.
2. Convert $\frac{19}{3}$ to a mixed number.
3. Convert $5 \frac{1}{4}$ to an improper fraction.
4. Add $1 \frac{2}{3} + 2 \frac{1}{6}$.
5. Subtract $3 \frac{1}{2} - 1 \frac{3}{4}$.
6. A piece of rope is $5 \frac{1}{2}$ m long. Two pieces of $1 \frac{3}{4}$ m are cut off. How much rope is left?
7. Add $1 \frac{1}{2} + 2 \frac{1}{3} + 3 \frac{1}{6}$.
8. Investigate: is $\frac{8}{8}$ proper, improper, or something else? Explain.

Activity

Recipe maths. A recipe for cake uses $1 \frac{1}{2}$ cups of flour, $\frac{3}{4}$ cup of sugar, and $\frac{1}{4}$ cup of oil. If you triple the recipe to feed more people, how much of each ingredient do you need? Express each answer as a mixed number, and round to the nearest practical fraction for measuring.