Addition and Subtraction of Fractions

Add and subtract fractions and mixed numbers by making the bottoms match, show that a sum is correct without a calculator, and find the fraction that is left.

In the exam for Cambridge IGCSE Maths 0580, Cambridge O Level Maths 4024, Edexcel IGCSE Maths 4MA1 · 51 real past-paper questions below

اقرأ هذا الدرس مشروحاً بالعربي ←

Adding Fractions: Same Size Pieces

The bottom number says the size of the pieces. The top says how many pieces you have.

When the bottoms match, add or take away the tops. The bottom stays the same.

Worked example

1
$\frac{2}{7} + \frac{3}{7} = \frac{5}{7}$ — Both are sevenths, so the pieces are the same size. Count them: 2 + 3 = 5 pieces. The bottom stays 7, because the pieces do not change size.

Rule

When the bottoms are different, change both fractions so the bottoms match first. Then add or take away the tops only.

Show That: Adding Fractions

The 2-mark question: "Show that $\frac{2}{3} + \frac{1}{4} = \frac{11}{12}$ ." No calculator. The answer is given, so the marks are for the working.

Worked example

1
$3, 6, 9, 12 and 4, 8, 12$ — Count up in 3s and in 4s. 12 is the first number in both lists, so 12 is the new bottom.
2
$\frac{2 \times 4}{3 \times 4} = \frac{8}{12}$ — 3 × 4 = 12, so times the bottom by 4. Times the top by 4 too, so the fraction keeps its size.
3
$\frac{1 \times 3}{4 \times 3} = \frac{3}{12}$ — 4 × 3 = 12, so times top and bottom by 3.
4
$\frac{8}{12} + \frac{3}{12} = \frac{11}{12}$ — The bottoms match now, so add the tops: 8 + 3 = 11. This is the answer you were given, so you have shown it.

Key idea

Whatever you times the bottom by, times the top by the same number. Then the fraction is the same size, just cut into smaller pieces.

Common mistake

Adding tops and bottoms: $\frac{2}{3} + \frac{1}{4} = \frac{3}{7}$ is wrong. $\frac{3}{7}$ is smaller than $\frac{2}{3}$ alone!

Show That: Taking Away Fractions

The same 2-mark question with a minus sign. Use the smallest bottom that both numbers go into. It keeps the numbers small.

Show that $\frac{7}{10} - \frac{1}{4} = \frac{9}{20}$ .

Worked example

1
$10, 20 and 4, 8, 12, 16, 20$ — Count up in 10s and in 4s. 20 is the first number in both lists, so 20 is the new bottom.
2
$\frac{7 \times 2}{10 \times 2} = \frac{14}{20}$ — 10 × 2 = 20, so times top and bottom by 2. The fraction keeps its size.
3
$\frac{1 \times 5}{4 \times 5} = \frac{5}{20}$ — 4 × 5 = 20, so times top and bottom by 5.
4
$\frac{14}{20} - \frac{5}{20} = \frac{9}{20}$ — Same bottom, so take away the tops: 14 − 5 = 9. That matches the answer given.

Key idea

Using $10 \times 4 = 40$ also works: $\frac{28}{40} - \frac{10}{40} = \frac{18}{40}$ . But then you must simplify to $\frac{9}{20}$ .

Common mistake

Writing only the answer. In a show that question, write both new fractions with the same bottom, or you lose the marks.

Simplest Form at the End

Most questions end with “give your answer in its simplest form”. So always check your answer at the end. If the top and bottom share a factor, divide both by it.

Worked example

1
$6, 12, 18, 24, 30 and 10, 20, 30$ — Work out $\frac{5}{6} + \frac{7}{10}$ . Count up in 6s and in 10s. 30 is the first number in both lists.
2
$\frac{5 \times 5}{6 \times 5} = \frac{25}{30}$ — 6 × 5 = 30, so times top and bottom by 5.
3
$\frac{7 \times 3}{10 \times 3} = \frac{21}{30}$ — 10 × 3 = 30, so times top and bottom by 3.
4
$\frac{25}{30} + \frac{21}{30} = \frac{46}{30}$ — Same bottom, so add the tops: 25 + 21 = 46.
5
$\frac{46 \div 2}{30 \div 2} = \frac{23}{15}$ — 46 and 30 are both even, so divide top and bottom by 2. 23 is prime, so nothing else divides both.
6
$\frac{23}{15} = 1 \frac{8}{15}$ — The top is bigger than the bottom, so write it as a mixed number: 15 goes into 23 once, with 8 left over.

Key idea

If one bottom goes into the other, use the bigger bottom. For $\frac{2}{5} + \frac{3}{10}$ , 5 goes into 10, so change only one fraction: $\frac{4}{10} + \frac{3}{10} = \frac{7}{10}$ .

Common mistake

Stopping at $\frac{46}{30}$ . It is the right size, but it is not in its simplest form, so you lose the last mark.

A Formula for Any Two Fractions

Some textbooks give one formula that works for any two fractions. It is useful when you cannot spot a smaller shared bottom.

Formula

\frac{a}{b} \pm \frac{c}{d} = \frac{a d \pm b c}{b d}

$a$ and $c$ are the tops, $b$ and $d$ are the bottoms. $a d$ means $a \times d$ . Use + when adding and − when taking away.

Worked example

1
$\frac{5}{6} - \frac{3}{10}$ — The question. $a = 5, b = 6, c = 3, d = 10$ . The sign is minus, so use the − in the formula.
2
$\frac{5 \times 10 - 6 \times 3}{6 \times 10}$ — Put the numbers in. Top: $a \times d$ take away $b \times c$ . Bottom: $b \times d$ .
3
$= \frac{50 - 18}{60}$ — Work out each product: 5 × 10 = 50, 6 × 3 = 18 and 6 × 10 = 60.
4
$= \frac{32}{60}$ — 50 − 18 = 32.
5
$= \frac{32 \div 4}{60 \div 4} = \frac{8}{15}$ — 32 and 60 both divide by 4, so divide top and bottom by 4. 8 and 15 share no factor, so stop.

Key idea

The bottom $b \times d$ is often bigger than you need, so you nearly always simplify at the end. With the smallest shared bottom, 30, it is $\frac{25}{30} - \frac{9}{30} = \frac{16}{30} = \frac{8}{15}$ : the same answer.

The Fraction That Is Left

The 1 or 2 mark question: some fractions of a group are given. What fraction is left? The whole group is 1.

$\frac{1}{6}$ of a class walk to school. $\frac{3}{8}$ come by bus. The rest come by car. What fraction come by car?

Worked example

1
$1 = \frac{24}{24}$ — The whole class is 1. Both 6 and 8 go into 24, so cut the whole class into 24 equal pieces.
2
$\frac{1}{6} = \frac{4}{24}$ — 6 × 4 = 24, so times top and bottom by 4. Walk is 4 pieces.
3
$\frac{3}{8} = \frac{9}{24}$ — 8 × 3 = 24, so times top and bottom by 3. Bus is 9 pieces.
4
$\frac{24 - 4 - 9}{24} = \frac{11}{24}$ — Take walk and bus away from the whole: 24 − 4 − 9 = 11. These 11 pieces come by car.

If it only asks for the fraction who do not walk, just take $\frac{1}{6}$ from 1:

Worked example

1
$1 = \frac{6}{6}$ — The question uses sixths, so write the whole class as 6 sixths.
2
$\frac{6}{6} - \frac{1}{6} = \frac{5}{6}$ — Same bottom, so take away the tops: 6 − 1 = 5. So $\frac{5}{6}$ do not walk.

Common mistake

Stopping after adding $\frac{1}{6} + \frac{3}{8}$ . That is walk and bus. The question asks for the rest.

Mixed Numbers

The no calculator question: "Show that $2 \frac{1}{3} + 1 \frac{3}{4} = 4 \frac{1}{12}$ ." Change each mixed number to a top heavy fraction first.

Worked example

1
$2 \frac{1}{3} = \frac{7}{3}$ — Two wholes are 2 × 3 = 6 thirds. Add the 1 third: 6 + 1 = 7 thirds.
2
$1 \frac{3}{4} = \frac{7}{4}$ — One whole is 4 quarters. Add the 3 quarters: 4 + 3 = 7 quarters.
3
$\frac{7 \times 4}{3 \times 4} = \frac{28}{12}$ — Both 3 and 4 go into 12. 3 × 4 = 12, so times top and bottom by 4.
4
$\frac{7 \times 3}{4 \times 3} = \frac{21}{12}$ — 4 × 3 = 12, so times top and bottom by 3.
5
$\frac{28}{12} + \frac{21}{12} = \frac{49}{12}$ — Same bottom, so add the tops: 28 + 21 = 49.
6
$\frac{49}{12} = 4 \frac{1}{12}$ — 12 goes into 49 four times (4 × 12 = 48), with 1 left over. That matches the answer given.

Taking away works the same way:

Worked example

1
$3 \frac{1}{5} = \frac{16}{5}$ — Three wholes are 3 × 5 = 15 fifths. Add the 1 fifth: 16 fifths.
2
$1 \frac{2}{3} = \frac{5}{3}$ — One whole is 3 thirds. Add the 2 thirds: 5 thirds.
3
$\frac{16 \times 3}{5 \times 3} = \frac{48}{15}$ — Both 5 and 3 go into 15. 5 × 3 = 15, so times top and bottom by 3.
4
$\frac{5 \times 5}{3 \times 5} = \frac{25}{15}$ — 3 × 5 = 15, so times top and bottom by 5.
5
$\frac{48}{15} - \frac{25}{15} = \frac{23}{15}$ — Same bottom, so take away the tops: 48 − 25 = 23.
6
$\frac{23}{15} = 1 \frac{8}{15}$ — 15 goes into 23 once, with 8 left over. 8 and 15 share no factor, so this is the simplest form.

Common mistake

Doing $3 - 1 = 2$ and $\frac{1}{5} - \frac{2}{3}$ separately. The fraction part goes below zero. Top heavy fractions avoid this.

Key idea

A 3-mark question with no calculator gives marks for the working: one for the top heavy fractions, one for the same bottom, one for the answer. Write every line, even when you can do it in your head.

More Than One Operation

Some questions mix × or ÷ with + or −. Use the usual order: brackets first, then × and ÷, then + and −.

Worked example

1
$\frac{7}{9} - \frac{2}{3} \times \frac{1}{2}$ — The question. Times comes before take away, so work out the times part first.
2
$\frac{2}{3} \times \frac{1}{2} = \frac{2}{6}$ — To times fractions, times the tops (2 × 1 = 2) and times the bottoms (3 × 2 = 6).
3
$\frac{2}{6} = \frac{1}{3}$ — Divide top and bottom by 2. Smaller numbers make the next step easier.
4
$\frac{7}{9} - \frac{1}{3} = \frac{7}{9} - \frac{3}{9}$ — Now take away. 3 goes into 9, so times top and bottom of $\frac{1}{3}$ by 3.
5
$= \frac{4}{9}$ — Same bottom, so take away the tops: 7 − 3 = 4. 4 and 9 share no factor, so this is the simplest form.

Halfway between two fractions. Add them, then divide by 2. The bracket shows the adding comes first.

Worked example

1
$(\frac{1}{3} + \frac{1}{2}) \div 2$ — Halfway between $\frac{1}{3}$ and $\frac{1}{2}$ is their mean: add them, then halve.
2
$\frac{1}{3} + \frac{1}{2} = \frac{2}{6} + \frac{3}{6}$ — Brackets first. 6 is the smallest shared bottom: $\frac{1}{3} = \frac{2}{6}$ and $\frac{1}{2} = \frac{3}{6}$ .
3
$= \frac{5}{6}$ — Same bottom, so add the tops: 2 + 3 = 5.
4
$\frac{5}{6} \div 2 = \frac{5}{12}$ — Halving cuts every piece in two, so double the bottom: 6 × 2 = 12. The top stays 5.

Common mistake

Working left to right: $\frac{7}{9} - \frac{2}{3} = \frac{1}{9}$ , then $\times \frac{1}{2}$ gives $\frac{1}{18}$ . That is wrong, because times must come first.

Which Method? Read the Question

Look at the sign between the fractions and at what the question asks for.

You see	You do
Same bottoms	add or take away the tops
Show that $\frac{a}{b} \pm \frac{c}{d}$	same bottom, write both new fractions
"The rest" or "do not"	take the fractions away from 1
Mixed numbers	top heavy first, then same bottom
“Simplest form”	divide top and bottom by the same number until nothing else divides both
$\times$ or $\div$ mixed with + or −	brackets first, then × and ÷, then + and −
“Halfway between”	add the two fractions, then divide by 2
Fraction of an amount, or $\times$ and $\div$	Multiplication and Division of Fractions
Changing $2 \frac{3}{4}$ to top heavy	Mixed Numbers and Improper Fractions

Key idea

Check the size. Adding two fractions gives more than either one. If your answer is smaller, you added the bottoms.

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