One Quantity as a Fraction of Another

"Express $A$ as a fraction of $B$" simply means: write $A$ over $B$ and simplify.

The rule

$A$ as a fraction of $B$ $=\;\dfrac{A}{B}$

Which number goes on top?
The quantity that comes immediately after the word "express" goes on the top.
The quantity that comes after the word "of" goes on the bottom.
"Express $12$ as a fraction of $20$" $\Rightarrow \dfrac{12}{20}$.

Worked Example 1 — A straightforward case

Express $12$ as a fraction of $20$.

①Write the fraction: $\dfrac{12}{20}$

②Find the HCF of $12$ and $20$: it is $4$.

③Divide top and bottom by $4$: $\dfrac{12 \div 4}{20 \div 4} = \dfrac{3}{5}$

Answer: $\dfrac{3}{5}$

You cannot put centimetres over metres. Always convert both quantities into the same unit before writing the fraction.

Read both quantities and check their units.
Convert so that both are in the same unit (usually the smaller one).
Write the fraction $\dfrac{\text{first}}{\text{second}}$, dropping the units.
Simplify by dividing top and bottom by their HCF.

Worked Example 2 — Different units

Express $45$ cm as a fraction of $2$ m.

①Convert: $2$ m $= 200$ cm

②Fraction $= \dfrac{45}{200}$

③HCF of $45$ and $200$ is $5$: $\dfrac{45 \div 5}{200 \div 5} = \dfrac{9}{40}$

Answer: $\dfrac{9}{40}$

Worked Example 3 — Time

Express $50$ minutes as a fraction of $3$ hours.

①$3$ hours $= 3 \times 60 = 180$ minutes

②$\dfrac{50}{180}$

③HCF is $10$: $\dfrac{5}{18}$

Answer: $\dfrac{5}{18}$

There is nothing special to do when the first quantity is larger than the second — you still write $\dfrac{A}{B}$. The answer is then an improper fraction (top heavier than bottom), which you may leave as it is or write as a mixed number.

What a fraction greater than 1 tells you:
$\dfrac{7}{4} = 1.75$ means the first quantity is $1.75$ times — that is, $75\%$ bigger than — the second.

Worked Example 4 — An improper fraction

Express $70$ as a fraction of $40$.

①$\dfrac{70}{40}$

②HCF is $10$: $\dfrac{7}{4}$

③As a mixed number: $1\dfrac{3}{4}$; as a decimal: $1.75$

Answer: $\dfrac{7}{4}$ (or $1\dfrac{3}{4}$)

Worked Example 5 — Bigger, with units

Express $2.5$ kg as a fraction of $800$ g.

①$2.5$ kg $= 2500$ g

②$\dfrac{2500}{800}$

③Divide both by $100$: $\dfrac{25}{8}$ (HCF of 25 and 8 is 1, so this is simplest form)

Answer: $\dfrac{25}{8} = 3\dfrac{1}{8} = 3.125$

A very common exam twist is that you are given the parts and must first work out the total before writing the fraction.

Worked Example 6 — Finding the total first

A box contains $8$ red, $12$ blue and $5$ green counters. What fraction of the counters are blue?

①Total $= 8 + 12 + 5 = 25$

②Blue fraction $= \dfrac{12}{25}$

③$12$ and $25$ share no factors, so this is already simplest.

Answer: $\dfrac{12}{25}$

Careful: "what fraction of the red counters are blue?" would be a different question: $\dfrac{12}{8} = \dfrac{3}{2}$.

Worked Example 7 — Working backwards

$\dfrac{3}{8}$ of the students in a year group study Spanish. There are $96$ Spanish students. How many students are in the year group?

①$\dfrac{3}{8}$ of the total $= 96$

②So $\dfrac{1}{8}$ of the total $= 96 \div 3 = 32$

③The whole is $\dfrac{8}{8}$: $\;32 \times 8 = 256$

Answer: $256$ students

Once you have the fraction, you can always convert it. All three are answers to the same question: "how many times bigger?"

To go from…	…to	Method
Fraction	Decimal	Divide top by bottom
Decimal	Percentage	$\times 100$
Fraction	Percentage	$\dfrac{A}{B} \times 100$

Worked Example 8 — All three forms

In a test Priya scored $27$ out of $45$. Express her score as a fraction, a decimal and a percentage.

①Fraction: $\dfrac{27}{45}$. HCF is $9$: $\dfrac{3}{5}$

②Decimal: $3 \div 5 = 0.6$

③Percentage: $0.6 \times 100 = 60\%$

Comparing quantities: to decide which of two fractions is bigger, convert both to decimals. E.g. $\dfrac{7}{12} = 0.583$ and $\dfrac{5}{9} = 0.556$, so $\dfrac{7}{12}$ is bigger.

If $A : B = 3 : 5$, then:
• $A$ as a fraction of $B$ is $\dfrac{3}{5}$ (part : part)
• $A$ as a fraction of the total is $\dfrac{3}{8}$ (part : whole)
• $B$ as a fraction of $A$ is $\dfrac{5}{3}$

Worked Example 9 — Switching between them

In a car park the ratio of cars to vans is $9 : 4$.

①Vans as a fraction of cars: $\dfrac{4}{9}$

②Total parts $= 9 + 4 = 13$

③Vans as a fraction of all vehicles: $\dfrac{4}{13}$

④Cars as a fraction of vans: $\dfrac{9}{4} = 2.25$, so there are $2.25$ times as many cars as vans.

The rule

"$A$ as a fraction of $B$" $= \dfrac{A}{B}$. Whatever follows "of" goes on the bottom.

Units

Convert both to the same unit before writing the fraction.

Simplifying

Divide top and bottom by their highest common factor.

Bigger than 1

Perfectly allowed — leave as an improper fraction or convert to a mixed number.

Part or whole?

Read carefully: "of the blue" vs "of the total" give different denominators.

Converting

Fraction $\to$ decimal: divide. Decimal $\to$ percentage: $\times 100$.

Question 1

Express $18$ as a fraction of $30$, in its simplest form.

▶ Show solution

$\dfrac{18}{30}$. The HCF of $18$ and $30$ is $6$.

$\dfrac{18 \div 6}{30 \div 6} = \dfrac{3}{5}$

Question 2

Express $60$ cm as a fraction of $1.5$ m.

▶ Show solution

$1.5$ m $= 150$ cm.

$\dfrac{60}{150}$; HCF is $30$; $\dfrac{2}{5}$

Question 3

Express $75$ minutes as a fraction of $1$ day.

▶ Show solution

$1$ day $= 24 \times 60 = 1440$ minutes.

$\dfrac{75}{1440}$; HCF is $15$; $\dfrac{5}{96}$

Question 4

Express $84$ as a fraction of $56$. Give your answer as an improper fraction in its simplest form and as a mixed number.

▶ Show solution

$\dfrac{84}{56}$. The HCF of $84$ and $56$ is $28$.

$\dfrac{84 \div 28}{56 \div 28} = \dfrac{3}{2}$

As a mixed number: $1\dfrac{1}{2}$ (so $84$ is one and a half times $56$).

Question 5

A bag contains $14$ apples, $10$ pears and $16$ oranges. What fraction of the fruit are (a) pears, (b) not oranges?

▶ Show solution

Total $= 14 + 10 + 16 = 40$.

(a) $\dfrac{10}{40} = \dfrac{1}{4}$

(b) Not oranges $= 14 + 10 = 24$, so $\dfrac{24}{40} = \dfrac{3}{5}$

Question 6

A recipe uses $250$ g of flour and $600$ g of total ingredients. Express the mass of flour as a fraction of the total, and as a percentage to 1 d.p.

▶ Show solution

$\dfrac{250}{600}$; HCF is $50$; $\dfrac{5}{12}$

Percentage $= \dfrac{5}{12} \times 100 = 41.666\ldots = 41.7\%$ (1 d.p.)

Question 7

$\dfrac{5}{9}$ of a crowd at a match support the home team. There are $2700$ home supporters. How many people are in the crowd?

▶ Show solution

$\dfrac{5}{9}$ of the crowd $= 2700$.

$\dfrac{1}{9}$ of the crowd $= 2700 \div 5 = 540$.

Whole crowd $= 540 \times 9 = 4860$ people.

Question 8

In class A, $18$ out of $30$ students walk to school. In class B, $21$ out of $35$ do. Which class has the greater fraction of walkers?

▶ Show solution

Class A: $\dfrac{18}{30} = \dfrac{3}{5} = 0.6$

Class B: $\dfrac{21}{35} = \dfrac{3}{5} = 0.6$

Neither — the fractions are equal. Both are $60\%$.

Question 9

The ratio of boys to girls in a club is $7 : 5$. Express the number of girls as a fraction of (a) the number of boys, (b) the whole club.

▶ Show solution

(a) Girls : boys $= 5 : 7$, so girls are $\dfrac{5}{7}$ of the boys.

(b) Total parts $= 7 + 5 = 12$, so girls are $\dfrac{5}{12}$ of the club.

Question 10

A tank holds $1.2\text{ m}^3$ of water. A bucket holds $15$ litres. Express the bucket's capacity as a fraction of the tank's capacity, and state how many bucketfuls fill the tank.

▶ Show solution

$1\text{ m}^3 = 1000$ litres, so the tank holds $1.2 \times 1000 = 1200$ litres.

Fraction $= \dfrac{15}{1200}$; divide both by $15$: $\dfrac{1}{80}$

Since the bucket is $\dfrac{1}{80}$ of the tank, it takes $80$ bucketfuls to fill it.