AS & A Level Maths

📘 Notes

Percentages and Fractions of an Amount (Year 7)

In this topic we find part of a whole using fractions or percentages.

1. Fractions of an Amount

To find a fraction $\dfrac{a}{b}$ of a number $N$:

Divide by the denominator $b$.
Multiply by the numerator $a$.

In symbols: \[ \dfrac{a}{b} \times N = a \times \dfrac{N}{b}. \]

Examples: Fractions of an amount

Example 1

Find $\dfrac{3}{4}$ of 20.

Step 1: $20 \div 4 = 5$.
Step 2: $5 \times 3 = 15$.
So $\dfrac{3}{4}$ of 20 is 15.

Example 2

Find $\dfrac{2}{5}$ of 45.

Step 1: $45 \div 5 = 9$.
Step 2: $9 \times 2 = 18$.
So $\dfrac{2}{5}$ of 45 is 18.

Example 3

Find $\dfrac{7}{10}$ of 60.

Step 1: $60 \div 10 = 6$.
Step 2: $6 \times 7 = 42$.
So $\dfrac{7}{10}$ of 60 is 42.

2. Percentages of an Amount

Percent means “out of 100”. For example:

$25\%$ means $\dfrac{25}{100}$.
$40\%$ means $\dfrac{40}{100}$.

You can find a percentage of an amount in two common ways:

Change the percentage to a fraction and use the fraction method.
Example: $25\% = \dfrac{25}{100} = \dfrac{1}{4}$.
Change the percentage to a decimal and multiply.
Example: $35\% = 0.35$.

Examples: Percentages of an amount

Example 4

Find $20\%$ of 50.

$20\% = 0.20$.
$0.20 \times 50 = 10$.
So $20\%$ of 50 is 10.

Example 5

Find $35\%$ of 80.

$10\%$ of 80 is $8$.
$30\%$ of 80 is $3 \times 8 = 24$.
$5\%$ of 80 is $\dfrac{8}{2} = 4$.
So $35\%$ of 80 is $24 + 4 = $ 28.

Example 6

Find $15\%$ of 120.

$10\%$ of 120 is $12$.
$5\%$ of 120 is $\dfrac{12}{2} = 6$.
So $15\%$ of 120 is $12 + 6 = $ 18.

3. Linking Fractions and Percentages

Some fractions and percentages are especially useful to remember:

Fraction	Percentage
$\dfrac{1}{2}$	$50\%$
$\dfrac{1}{4}$	$25\%$
$\dfrac{3}{4}$	$75\%$
$\dfrac{1}{5}$	$20\%$
$\dfrac{2}{5}$	$40\%$
$\dfrac{1}{10}$	$10\%$

You can use these to switch between fraction and percentage methods.

Examples: Using both fractions and percentages

Example 7

Find $\dfrac{3}{10}$ of 90.

Fraction method:
$90 \div 10 = 9$.
$9 \times 3 = 27$.
So $\dfrac{3}{10}$ of 90 is 27.

Example 8 (word problem)

There are 200 students in a school club.
$\dfrac{3}{5}$ of them play football. Find how many students play football.

$200 \div 5 = 40$.
$40 \times 3 = 120$.
So 120 students play football.

Example 9 (mixed: fraction and percentage)

A shop sells a T-shirt for \$40.
(a) Find $\dfrac{1}{4}$ of \$40.
(b) Find $25\%$ of \$40.
(c) What do you notice?

(a) $\dfrac{1}{4}$ of 40: $40 \div 4 = 10$.
(b) $25\%$ of 40: $25\% = \dfrac{1}{4}$, so answer is also 10.
(c) $\dfrac{1}{4}$ and $25\%$ give the same result because they are the same fraction of the amount.

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Fraction	Percentage
\(\dfrac{1}{2}\)	\(50\%\)
\(\dfrac{1}{4}\)	\(25\%\)
\(\dfrac{3}{4}\)	\(75\%\)
\(\dfrac{1}{5}\)	\(20\%\)
\(\dfrac{2}{5}\)	\(40\%\)
\(\dfrac{1}{10}\)	\(10\%\)

Percentages — Finding Percentages of Amounts