Year 7 Maths Percentages

Expressing Quantities as Percentages

Q: What is an exact percentage?

One with no rounding, such as 33 1/3% for 1/3.

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Theory

To write one quantity as a percentage of another, write it as a fraction (part over whole, in the same units), then multiply by $100\%$. $300$ mL of $1.2$ L is $\dfrac{300}{1200} \times 100\% = 25\%$. This Year 7 maths page shows exact and rounded answers, discounts, profit and loss, and comparing marks.

$\text{Percentage} = \dfrac{\text{part}}{\text{whole}} \times 100\%$. The part (the quantity) is the numerator; the whole (the total amount) is the denominator.

Step 1. Change both quantities to the same units. Step 2. Write the part over the whole and simplify (divide by the HCF). Step 3. Multiply by $100\%$: $\dfrac{12}{40} = \dfrac{3}{10}$ and $\dfrac{3}{10} \times 100\% = 30\%$.

Shortcut. Per cent means “out of $100$”, so if you can, make the denominator $100$: $\dfrac{9}{20} = \dfrac{45}{100} = 45\%$.

Exact or rounded. $\dfrac{2}{3} \times 100\% = \dfrac{200}{3}\% = 66\dfrac{2}{3}\%$ exactly. Rounded to one decimal place it is $66.7\%$.

Choose the right whole. A discount is compared with the old price. A profit or loss is compared with the cost price (or the selling price, if the question asks). To compare results out of different totals, change each one to a percentage.

A quiz out of $20$ marks. The whole ($20$ marks) lines up with $100\%$. $15$ out of $20$ is $\dfrac{3}{4}$ of the way along, so it is $75\%$.

The rule:

\[ \text{Percentage} = \dfrac{\text{part}}{\text{whole}} \times 100\% \]

Percentage = part / whole \times 100

\[ \text{Discount } \% = \dfrac{\text{discount}}{\text{old price}} \times 100\% \]

Discount = discount / old price \times 100

\[ \text{Profit } \% = \dfrac{\text{profit}}{\text{cost price}} \times 100\% \]

Profit = profit / cost price \times 100

          Same units first. \(30\) cm of \(2\) m is \(\dfrac{30}{200} \times 100\% = 15\%\), not \(1500\%\).
        

          Exact means exact. \(\dfrac{1}{12} = 8\dfrac{1}{3}\%\) exactly; \(8\%\) is only rounded. When you round, say how many decimal places.
        

How to write a quantity as a percentage of another

Change both quantities to the same units.
Write the part over the whole and simplify.
Multiply by $100\%$ (or make the denominator $100$).
Give an exact answer as a mixed numeral, or round if asked.

Example 1 — Out of

(i) Write $11$ out of $25$ as a percentage.
(ii) Jai scored $39$ out of $60$ on a test. What is his score as a percentage?

Solution

$\dfrac{11}{25}$	$=$	$\dfrac{44}{100}$
$\dfrac{39}{60}$	$=$	$\dfrac{13}{20}$
$\dfrac{13}{20} \times 100\%$	$=$	$65\%$

11 / 25 = 44 / 100 = 44, 39 / 60 = 13 / 20, 13 / 20 \times 100 = 65

(i) $44\%$
(ii) $65\%$. Check: $65 \times 60 \div 100 = 39$.

Example 2 — Millilitres and litres

Express $300$ mL of $1.2$ L as a percentage.

Solution

$1.2 \times 1000$	$=$	$1200$
$\dfrac{300}{1200} \times 100\%$	$=$	$\dfrac{1}{4} \times 100\%$
	$=$	$25\%$

1.2 L = 1200 mL, 300 / 1200 \times 100 = 1 / 4 \times 100 = 25

$25\%$

Example 3 — An exact percentage

Express $2700$ m of $8.1$ km as a percentage.

Solution

$8.1 \times 1000$	$=$	$8100$
$\dfrac{2700}{8100} \times 100\%$	$=$	$\dfrac{1}{3} \times 100\%$
	$=$	$33\dfrac{1}{3}\%$

8.1 km = 8100 m, 2700 / 8100 \times 100 = 1 / 3 \times 100 = 331 / 3

$33\dfrac{1}{3}\%$

Example 4 — Change to the same units

(i) Express $350$ m as a percentage of $2$ km.
(ii) Express $20$ min as an exact percentage of $1$ h $30$ min.

Solution

$\dfrac{350}{2000}$	$=$	$\dfrac{7}{40}$
$\dfrac{7}{40} \times 100\%$	$=$	$17.5\%$
$\dfrac{20}{90}$	$=$	$\dfrac{2}{9}$
$\dfrac{2}{9} \times 100\%$	$=$	$22\dfrac{2}{9}\%$

350 / 2000 = 7 / 40, 7 / 40 \times 100 = 17.5 20 / 90 = 2 / 9, 2 / 9 \times 100 = 222 / 9

(i) $17.5\%$ ($2$ km $= 2000$ m)
(ii) $22\dfrac{2}{9}\%$ ($1$ h $30$ min $= 90$ min)

Example 5 — Discount as a percentage

A jacket was $\$65$. In a sale it is $\$52$.
(i) What percentage of the old price do you pay?
(ii) What percentage is the discount?

Solution

$\dfrac{52}{65} \times 100\%$	$=$	$\dfrac{4}{5} \times 100\%$
	$=$	$80\%$
$65 - 52$	$=$	$13$
$\dfrac{13}{65} \times 100\%$	$=$	$20\%$

52 / 65 \times 100 = 4 / 5 \times 100 = 8065 - 52 = 13, 13 / 65 \times 100 = 20

(i) $80\%$
(ii) $20\%$ (compared with the old price)

Example 6 — Loss as a percentage

Ruby buys a bike for $\$150$ and sells it for $\$126$. Write her loss as a percentage of
(i) the cost price
(ii) the selling price, correct to one decimal place.

Solution

$150 - 126$	$=$	$24$
$\dfrac{24}{150} \times 100\%$	$=$	$16\%$
$\dfrac{24}{126} \times 100\%$	$=$	$\dfrac{400}{21}\%$

150 - 126 = 24, 24 / 150 \times 100 = 16, 24 / 126 \times 100 = 400 / 21 = 19.047

(i) $16\%$
(ii) $\dfrac{400}{21} = 19.047\ldots$, and the next digit after the $0$ is $4$, so round down: $19.0\%$. The same $\$24$ is a bigger percentage of a smaller whole.

Example 7 — Comparing results

Ava scored $27$ out of $30$ in spelling and $44$ out of $48$ in maths. Which result is better?

Solution

$\dfrac{27}{30} \times 100\%$	$=$	$90\%$
$\dfrac{44}{48} \times 100\%$	$=$	$91\dfrac{2}{3}\%$

27 / 30 \times 100 = 90, 44 / 48 \times 100 = 912 / 3

Maths is better: $91\dfrac{2}{3}\% > 90\%$.

Common pitfalls

Different units. Change $2$ m to $200$ cm before writing $30$ cm as a percentage of it.

The wrong whole. A price drops from $\$60$ to $\$45$. The $\$15$ discount is $\dfrac{15}{60} = 25\%$ of the old price, not $\dfrac{15}{45} = 33\dfrac{1}{3}\%$.

Rounding when exact is asked. Give $\dfrac{1}{12}$ as $8\dfrac{1}{3}\%$, not $8\%$.

Comparing raw marks. $44$ out of $48$ and $27$ out of $30$ have different totals: change both to percentages first.

Frequently asked questions

How do you write one quantity as a percentage of another?

Write both in the same units, put the part over the whole, then multiply by $100\%$.

What is 30 out of 40 as a percentage?

$\dfrac{30}{40} = \dfrac{3}{4} = 75\%$.

What is an exact percentage?

One with no rounding, such as $33\dfrac{1}{3}\%$ for $\dfrac{1}{3}$.

What is a discount compared with?

The old price.

What is a profit or loss compared with?

The cost price, unless the question asks for the selling price.

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\(\dfrac{11}{25}\)	\(=\)	\(\dfrac{44}{100}\)
\(\dfrac{39}{60}\)	\(=\)	\(\dfrac{13}{20}\)
\(\dfrac{13}{20} \times 100\%\)	\(=\)	\(65\%\)

\(1.2 \times 1000\)	\(=\)	\(1200\)
\(\dfrac{300}{1200} \times 100\%\)	\(=\)	\(\dfrac{1}{4} \times 100\%\)
\(\)	\(=\)	\(25\%\)

\(8.1 \times 1000\)	\(=\)	\(8100\)
\(\dfrac{2700}{8100} \times 100\%\)	\(=\)	\(\dfrac{1}{3} \times 100\%\)
\(\)	\(=\)	\(33\dfrac{1}{3}\%\)

\(\dfrac{350}{2000}\)	\(=\)	\(\dfrac{7}{40}\)
\(\dfrac{7}{40} \times 100\%\)	\(=\)	\(17.5\%\)
\(\dfrac{20}{90}\)	\(=\)	\(\dfrac{2}{9}\)
\(\dfrac{2}{9} \times 100\%\)	\(=\)	\(22\dfrac{2}{9}\%\)

\(\dfrac{52}{65} \times 100\%\)	\(=\)	\(\dfrac{4}{5} \times 100\%\)
\(\)	\(=\)	\(80\%\)
\(65 - 52\)	\(=\)	\(13\)
\(\dfrac{13}{65} \times 100\%\)	\(=\)	\(20\%\)

\(150 - 126\)	\(=\)	\(24\)
\(\dfrac{24}{150} \times 100\%\)	\(=\)	\(16\%\)
\(\dfrac{24}{126} \times 100\%\)	\(=\)	\(\dfrac{400}{21}\%\)

\(\dfrac{27}{30} \times 100\%\)	\(=\)	\(90\%\)
\(\dfrac{44}{48} \times 100\%\)	\(=\)	\(91\dfrac{2}{3}\%\)

Expressing Quantities as Percentages

📖 Theory

How to write a quantity as a percentage of another

Common pitfalls

Frequently asked questions

Theory