Fractions of an Amount

What a fraction means: shade a fraction of a shape, say what fraction is shaded, write one amount as a fraction of another in simplest form, and find a simple fraction of an amount.

In the exam for Edexcel IGCSE Maths 4MA1, Cambridge IGCSE Maths 0580, Cambridge O Level Maths 4024 · 32 real past-paper questions below

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What a Fraction Means

A fraction is part of a whole. Cut the whole into equal parts, then count the parts you take.

Part	It tells you
Bottom number	how many equal parts the whole is cut into
Top number	how many of those parts you take

The 1 mark question: "Shade $\frac{3}{5}$ of the shape." The shape has 15 squares.

Worked example

1
$15 \div 5 = 3$ — Divide the 15 squares by the bottom number, 5. One fifth is 3 squares, because 5 equal groups of 3 make 15.
2
$3 \times 3 = 9$ — Multiply by the top number, 3. You want 3 of those fifths, so 3 groups of 3 squares.
3
$shade 9 squares$ — Shade any 9 of the 15. They do not have to sit together; only the count matters.
4
$\frac{9}{15} = \frac{3}{5}$ — Check: divide 9 and 15 by 3 and you are back to three fifths.

Common mistake

Shading 3 squares. That is only right when the shape has exactly 5 squares. Divide by the bottom first.

Part Over Total, Then Simplify

The 1 or 2 mark question: "What fraction of the shape is shaded?" Count the shaded squares, then all the squares.

Worked example

1
$\frac{16}{20}$ — Shaded on top, all the squares on the bottom, because the bottom is always the whole.
2
$= \frac{4}{5}$ — Divide top and bottom by 4, the biggest number that goes into 16 and 20. The size of the fraction stays the same.
3
$4 and 5$ — Only 1 divides both 4 and 5, so $\frac{4}{5}$ is the simplest form.

Rule

Simplest form: divide the top and the bottom by the same number until nothing else divides both.

The 2 mark question: "Write 45 grams as a fraction of 0.6 kg." Make the units the same first.

Worked example

1
$0.6 kg = 600 g$ — Change kg to grams (× 1000). You can only compare two amounts in the same units.
2
$\frac{45}{600}$ — Part over total: 45 g is the part, 600 g is the whole.
3
$= \frac{15}{200}$ — 45 and 600 both divide by 3, so divide top and bottom by 3.
4
$= \frac{3}{40}$ — 15 and 200 both divide by 5. Now only 1 divides 3 and 40, so stop.

Key idea

Write the fraction down before you simplify, like $\frac{45}{600}$ . On a 2 mark question that line earns a mark for working, even if your simplifying slips.

Common mistake

Writing shaded over unshaded, like $\frac{16}{4}$ . The bottom is always the whole: every square.

When the Pieces Are Not Equal

Some shapes mix big and small pieces. A fraction only works with equal pieces, so cut the big ones up first.

A rectangle is made of 2 big squares and 8 small squares. Each big square is the same as 4 small squares. 1 big square and 2 small squares are shaded.

Worked example

1
$2 \times 4 = 8$ — Cut each big square into 4 small ones. The 2 big squares make 8 small squares.
2
$8 + 8 = 16$ — Add the 8 small squares already there. The whole is 16 equal small squares.
3
$4 + 2 = 6$ — The shaded big square is 4 small squares, plus the 2 shaded small ones: 6 shaded.
4
$\frac{6}{16}$ — Shaded over all. Now it is fair, because every piece is the same size.
5
$= \frac{3}{8}$ — Divide top and bottom by 2. Only 1 divides 3 and 8, so this is the simplest form.

Common mistake

Counting the pieces as they are: 3 shaded out of 10 gives $\frac{3}{10}$ . That is wrong, because the pieces are not the same size.

Key idea

A square with both diagonals drawn is cut into 4 equal triangles. To shade $\frac{1}{4}$ , shade one triangle.

Count the Total First

Some questions, like a bar chart or table, do not give you the total. Find it first, because the bottom of the fraction is always the whole.

A bar chart shows how a class gets to school: 12 by bus, 8 walk and 10 by car. What fraction of the class do not walk?

Worked example

1
$12 + 8 + 10 = 30$ — Add every bar to get the total. Each student is in exactly one bar, so the class is 30.
2
$30 - 8 = 22$ — “Do not walk” is everyone except the 8 walkers, so take 8 from the total: 22 students.
3
$\frac{22}{30}$ — Part over total: the students who do not walk, over the whole class.
4
$= \frac{11}{15}$ — Divide top and bottom by 2. Only 1 divides 11 and 15, so stop.

Common mistake

Putting 8 on top. The question asks about the students who do not walk, so the 8 walkers are left out.

Key idea

“Not shaded” works the same way: take the shaded squares away from the total, then put what is left over the total.

A Fraction of an Amount

The simple question: find $\frac{3}{5}$ of 35. The bottom number cuts, the top number counts.

Worked example

1
$35 \div 5 = 7$ — Divide by the bottom number, 5. This cuts 35 into 5 equal parts, so one fifth is 7.
2
$7 \times 3 = 21$ — Multiply by the top number, 3, because you want 3 of those fifths.
3
$\frac{3}{5} of 35 = 21$ — Check: 21 is less than 35, as it must be when the fraction is less than 1.

Common mistake

Dividing by the top number: 35 ÷ 3. Always divide by the bottom first.

Key idea

Questions with "the rest", or finding the whole from a part, are in the lesson Multiplication and Division of Fractions.

Which Method? Read the Question

The words in the question tell you what to do.

You see	You do
Shade $\frac{3}{5}$ of the shape	count squares, ÷ bottom, × top
What fraction is shaded?	shaded over all, then simplify
Big and small pieces	cut into equal pieces, then shaded over all
A bar chart or table, or “not …”	add up the total, then part over total
Write 45 as a fraction of 600	same units, part over total, simplify
$\frac{3}{5}$ of 35	÷ bottom, × top
"The rest" or find the whole	Multiplication and Division of Fractions
Top heavy fractions	Mixed Numbers and Improper Fractions

Key idea

Check your answer: a fraction less than one of an amount is always smaller than the amount.

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