Mixed Numbers and Improper Fractions

Change improper fractions to mixed numbers and back, and show that a subtraction, multiplication or division with mixed numbers is correct.

In the exam for Edexcel IGCSE Maths 4MA1, Cambridge IGCSE Maths 0580, Cambridge O Level Maths 4024 · 29 real past-paper questions below

اقرأ هذا الدرس مشروحاً بالعربي ←

What Mixed Numbers Are

A mixed number has a whole number and a fraction, like $2 \frac{3}{4}$ . An improper fraction has a top as big as, or bigger than, its bottom, like $\frac{11}{4}$ .

They can be the same amount. Two wholes are 8 quarters, and 3 more quarters make 11 quarters.

Mixed number	Improper fraction
$2 \frac{3}{4}$	$\frac{11}{4}$
$1 \frac{1}{2}$	$\frac{3}{2}$

Key idea

Mixed numbers are easy to read. Improper fractions are easy to calculate with. You need to switch both ways.

Key idea

An improper fraction is not badly behaved. It is just top-heavy! Help $\frac{9}{2}$ stand up straight and it becomes $4 \frac{1}{2}$ .

Improper to Mixed

The 1 mark question: "Write $\frac{29}{6}$ as a mixed number." Divide the top by the bottom.

Worked example

1
$29 \div 6 = 4 rem 5$ — Divide the top by the bottom to see how many whole sixes fit in 29. Four 6s make 24, so 4 wholes with 5 left over.
2
$\frac{29}{6} = 4 \frac{5}{6}$ — The 4 is the wholes. The 5 left over are still sixths, so they go on top of the 6.

The 2 mark question adds "in its simplest form".

Worked example

1
$46 \div 10 = 4 rem 6$ — Four 10s make 40, so there are 4 wholes and 6 tenths left over.
2
$\frac{46}{10} = 4 \frac{6}{10}$ — The 6 leftover pieces are tenths, so the bottom stays 10.
3
$= 4 \frac{3}{5}$ — 6 and 10 are both even, so divide both by 2. 3 and 5 share no factor, so this is the simplest form.

Common mistake

Putting the remainder over the old top, like $4 \frac{5}{29}$ . The pieces left over are sixths, so the bottom stays 6.

Mixed to Improper

Before you calculate with a mixed number, make it improper. Turn the wholes into parts, then add the parts you already have.

Worked example

1
$3 \frac{2}{5} : 3 \times 5 = 15$ — Each whole is 5 fifths, so 3 wholes are 15 fifths. Multiply the whole number by the bottom.
2
$15 + 2 = 17$ — Add the 2 fifths you already had. That makes 17 fifths in all.
3
$3 \frac{2}{5} = \frac{17}{5}$ — The pieces are still fifths, so the bottom stays 5. Only the top changes.
4
$17 \div 5 = 3 rem 2 ✓$ — Check by going back: 3 wholes and 2 left over is $3 \frac{2}{5}$ again.

Common mistake

Writing $\frac{3 \times 2 + 5}{5}$ . Multiply the whole number by the bottom, not by the top.

Show That: Subtracting Mixed Numbers

The 3 mark question: "Show that $6 \frac{1}{4} - 2 \frac{2}{3} = 3 \frac{7}{12}$ ." The answer is given, so the marks are for the steps.

Worked example

1
$6 \frac{1}{4} = \frac{25}{4}$ — Improper first: 6 × 4 = 24 quarters, plus 1 more is 25 quarters.
2
$2 \frac{2}{3} = \frac{8}{3}$ — Same for the second: 2 × 3 = 6 thirds, plus 2 more is 8 thirds.
3
$4, 8, 12 and 3, 6, 9, 12$ — You cannot take thirds from quarters, so find the same bottom. 12 is the first number in both lists.
4
$\frac{25}{4} = \frac{75}{12}$ — Multiply top and bottom by 3 so the bottom is 12. The size does not change.
5
$\frac{8}{3} = \frac{32}{12}$ — Multiply top and bottom by 4 so this bottom is 12 too.
6
$\frac{75}{12} - \frac{32}{12} = \frac{43}{12}$ — The bottoms match, so take the tops away: 75 − 32 = 43. The bottom stays 12.
7
$\frac{43}{12} = 3 \frac{7}{12}$ — 43 ÷ 12 = 3 remainder 7. That is the given answer, so you have shown it.

Common mistake

Doing 6 − 2 = 4 and $\frac{1}{4} - \frac{2}{3}$ apart. The fraction part is $\frac{3}{12} - \frac{8}{12} = - \frac{5}{12}$ , a minus. It is easy to lose that minus and write $4 \frac{5}{12}$ , which is wrong. Make them improper first.

Key idea

Show every line. The answer is given, so all the marks are for the working: the improper fractions, the same bottom, and the take-away. Writing only the answer scores nothing.

A Mixed Number and a Plain Fraction

Some questions mix a mixed number with a plain fraction, and the answer is less than 1. The method is the same: improper first.

Worked example

1
$1 \frac{3}{10} = \frac{13}{10}$ — Make the mixed number improper: 1 × 10 = 10 tenths, plus 3 is 13 tenths.
2
$\frac{13}{10} - \frac{5}{6}$ — Write the question again. $\frac{5}{6}$ has no wholes, so it stays as it is.
3
$10, 20, 30 and 6, 12, 18, 24, 30$ — Find the same bottom. 30 is the first number in both lists.
4
$\frac{13}{10} = \frac{39}{30}$ — Multiply top and bottom by 3 so the bottom is 30.
5
$\frac{5}{6} = \frac{25}{30}$ — Multiply top and bottom by 5 so this bottom is 30 too.
6
$\frac{39}{30} - \frac{25}{30} = \frac{14}{30}$ — Take the tops away: 39 − 25 = 14. The bottom stays 30.
7
$\frac{14}{30} = \frac{7}{15}$ — Simplest form: 14 and 30 are both even, so divide both by 2. 7 and 15 share no factor, so stop.

Key idea

The top, 7, is smaller than the bottom, 15, so the answer is less than 1. It stays a fraction. There is no mixed number to write.

Dividing works the same way, and the answer can also be less than 1:

Worked example

1
$1 \frac{1}{5} = \frac{6}{5}$ — Improper first: 1 × 5 = 5 fifths, plus 1 is 6 fifths.
2
$4 \frac{1}{2} = \frac{9}{2}$ — 4 × 2 = 8 halves, plus 1 is 9 halves.
3
$\frac{6}{5} \div \frac{9}{2} = \frac{6}{5} \times \frac{2}{9}$ — Keep the first, flip the second, and change ÷ to ×.
4
$= \frac{12}{45}$ — Multiply the tops (6 × 2 = 12) and the bottoms (5 × 9 = 45).
5
$= \frac{4}{15}$ — Simplest form: 12 and 45 are both in the 3 times table, so divide both by 3.

Common mistake

Stopping at $\frac{14}{30}$ . The question says simplest form, so check whether the top and bottom still share a factor.

Show That: Multiply or Divide

The 3 mark question: "Show that $3 \frac{1}{3} \times 1 \frac{4}{5} = 6$ ." Again, improper first.

Worked example

1
$3 \frac{1}{3} = \frac{10}{3}$ — 3 × 3 = 9 thirds, plus 1 more is 10 thirds.
2
$1 \frac{4}{5} = \frac{9}{5}$ — 1 × 5 = 5 fifths, plus 4 more is 9 fifths.
3
$\frac{10}{3} \times \frac{9}{5} = \frac{2}{3} \times \frac{9}{1}$ — Cancel before you multiply: 10 and 5 are both in the 5 times table, so divide both by 5.
4
$\frac{2}{3} \times \frac{9}{1} = \frac{2}{1} \times \frac{3}{1}$ — Now 3 and 9: divide both by 3. Small numbers are easier to multiply.
5
$= 6$ — 2 × 3 = 6 on top and 1 × 1 = 1 on the bottom. That is 6, as required.

To divide, keep the first, flip the second, multiply:

Worked example

1
$2 \frac{1}{2} = \frac{5}{2}$ — Improper first: 2 × 2 = 4 halves, plus 1 is 5 halves.
2
$1 \frac{7}{8} = \frac{15}{8}$ — 1 × 8 = 8 eighths, plus 7 is 15 eighths.
3
$\frac{5}{2} \div \frac{15}{8} = \frac{5}{2} \times \frac{8}{15}$ — Keep the first, flip the second, and change ÷ to ×. Dividing by a fraction is the same as multiplying by it upside down.
4
$= \frac{40}{30}$ — Multiply the tops (5 × 8 = 40) and the bottoms (2 × 15 = 30).
5
$= \frac{4}{3}$ — Simplify: 40 and 30 can both be divided by 10.
6
$= 1 \frac{1}{3}$ — 4 ÷ 3 = 1 remainder 1, so 1 whole and 1 third.

Common mistake

Multiplying the wholes and the fractions apart: $3 \times 1 = 3$ and $\frac{1}{3} \times \frac{4}{5} = \frac{4}{15}$ gives $3 \frac{4}{15}$ , not 6. Make them improper first.

Which Method? Read the Question

The words in the question tell you what to do.

You see	You do
Write as a mixed number	divide: wholes, then remainder on top
...in its simplest form	then cancel the fraction part
A mixed number in a sum	improper first
Show that ... $-$ mixed numbers	improper, same bottom, take away
A mixed number and a plain fraction	make only the mixed number improper; if the top ends smaller than the bottom, leave it as a fraction
Show that ... $\times$ or $\div$	improper, cancel, multiply (for ÷, flip the second first)
Adding mixed numbers, or fractions with different bottoms	Addition and Subtraction of Fractions

Key idea

Check your answer size. $6 \frac{1}{4} - 2 \frac{2}{3}$ must be a bit less than 4.

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