Sec 1 Maths · Ratio, rate and speed

Simplifying ratios questions with answers

Practice questions on simplifying ratios, from the basics up to exam level, each with a full worked solution. Try them on paper first, then open the solution to check your working.

Worked example

Express $24 : 36$ in its simplest form.

Step 1 of 1

Divide every term by the HCF

\text{HCF of } 24 \text{ and } 36 = 12

24 : 36 = \dfrac{24}{12} : \dfrac{36}{12}

\phantom{24 : 36} = 2 : 3

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Questions

Question 1Two whole numbers
Express $60 : 24$ in its simplest form.
Show the worked solution
1. Step 1: Divide every term by the HCF $\text{HCF of } 60 \text{ and } 24 = 12$ $60 : 24 = \dfrac{60}{12} : \dfrac{24}{12}$ $\phantom{60 : 24} = 5 : 2$
Question 2Three whole numbers
Express $15 : 20 : 30$ in its simplest form.
Show the worked solution
1. Step 1: Divide every term by the HCF $\text{HCF of } 15, 20 \text{ and } 30 = 5$ $15 : 20 : 30 = \dfrac{15}{5} : \dfrac{20}{5} : \dfrac{30}{5}$ $\phantom{15 : 20 : 30} = 3 : 4 : 6$
Question 3Decimals and fractions
Express $\dfrac{2}{3} : \dfrac{3}{4}$ in its simplest form.
Show the worked solution
1. Step 1: Multiply every term by the LCM of the denominators $\text{LCM of } 3 \text{ and } 4 = 12$ $\dfrac{2}{3} \times 12 : \dfrac{3}{4} \times 12 = 8 : 9$
2. Step 2: The terms have no common factor: this is the simplest form $8 : 9$
Question 4Different units: change first
Express the ratio $500$ ml $:$ $3$ litres in its simplest form.
Show the worked solution
1. Step 1: Change both quantities to the same (smaller) unit first $3 \text{ litres} = 3 \times 1000 \text{ ml}$ $\phantom{3 \text{ litres}} = 3000 \text{ ml}$
2. Step 2: Divide every term by the HCF (a ratio has no units) $\text{HCF of } 500 \text{ and } 3000 = 500$ $500 : 3000 = \dfrac{500}{500} : \dfrac{3000}{500}$ $\phantom{500 : 3000} = 1 : 6$
Question 5Ratios from a context (part : whole)
At a concert there are 28 adults and 12 children. Find the ratio of the number of adults to the total number of people, in its simplest form.
Show the worked solution
1. Step 1: Find the total $\text{Total} = 28 + 12$ $\phantom{\text{Total}} = 40$
2. Step 2: Write the ratio in the order asked $\text{Adults} : \text{Total} = 28 : 40$
3. Step 3: Divide every term by the HCF $\text{HCF of } 28 \text{ and } 40 = 4$ $28 : 40 = \dfrac{28}{4} : \dfrac{40}{4}$ $\phantom{28 : 40} = 7 : 10$
Question 6Aspect ratios: a picture on a screen
The aspect ratio of a rectangle is width : height. A photo with an aspect ratio of $5 : 4$ is shown on a screen with an aspect ratio of $3 : 2$ . It is made as large as possible without being cut off, so it fills the full height of the screen, as shown in the diagram. Find the fraction of the screen that is not covered.
Show the worked solution
1. Step 1: Both have the same height: make the heights the same number of units (LCM of the second terms) $\text{picture } 5 : 4 = 5 : 4$ $\text{screen } 3 : 2 = 6 : 4$
2. Step 2: The picture covers its width out of the screen's width $\text{Covered} = \dfrac{5}{6}$ $\phantom{\text{Covered}} = \dfrac{5}{6}$
3. Step 3: Not covered: 1 take away the part covered $1 - \dfrac{5}{6} = \dfrac{1}{6}$ $\dfrac{1}{6}$ of the screen is not covered.

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Worked example

Express

24 : 36

in its simplest form.

Step 1 of 1

Divide every term by the HCF

\text{HCF of } 24 \text{ and } 36 = 12

24 : 36 = \dfrac{24}{12} : \dfrac{36}{12}

\phantom{24 : 36} = 2 : 3

Questions

Question 1Two whole numbers

Express $60 : 24$ in its simplest form.

Show the worked solution

Step 1: Divide every term by the HCF $\text{HCF of } 60 \text{ and } 24 = 12$ $60 : 24 = \dfrac{60}{12} : \dfrac{24}{12}$ $\phantom{60 : 24} = 5 : 2$

Question 2Three whole numbers

Express $15 : 20 : 30$ in its simplest form.

Show the worked solution

Step 1: Divide every term by the HCF $\text{HCF of } 15, 20 \text{ and } 30 = 5$ $15 : 20 : 30 = \dfrac{15}{5} : \dfrac{20}{5} : \dfrac{30}{5}$ $\phantom{15 : 20 : 30} = 3 : 4 : 6$

Question 3Decimals and fractions

Express $\dfrac{2}{3} : \dfrac{3}{4}$ in its simplest form.

Show the worked solution

Step 1: Multiply every term by the LCM of the denominators $\text{LCM of } 3 \text{ and } 4 = 12$ $\dfrac{2}{3} \times 12 : \dfrac{3}{4} \times 12 = 8 : 9$
Step 2: The terms have no common factor: this is the simplest form $8 : 9$

Question 4Different units: change first

Express the ratio $500$ ml $:$ $3$ litres in its simplest form.

Show the worked solution

Step 1: Change both quantities to the same (smaller) unit first $3 \text{ litres} = 3 \times 1000 \text{ ml}$ $\phantom{3 \text{ litres}} = 3000 \text{ ml}$
Step 2: Divide every term by the HCF (a ratio has no units) $\text{HCF of } 500 \text{ and } 3000 = 500$ $500 : 3000 = \dfrac{500}{500} : \dfrac{3000}{500}$ $\phantom{500 : 3000} = 1 : 6$

Question 5Ratios from a context (part : whole)

At a concert there are 28 adults and 12 children. Find the ratio of the number of adults to the total number of people, in its simplest form.

Show the worked solution

Step 1: Find the total $\text{Total} = 28 + 12$ $\phantom{\text{Total}} = 40$
Step 2: Write the ratio in the order asked $\text{Adults} : \text{Total} = 28 : 40$
Step 3: Divide every term by the HCF $\text{HCF of } 28 \text{ and } 40 = 4$ $28 : 40 = \dfrac{28}{4} : \dfrac{40}{4}$ $\phantom{28 : 40} = 7 : 10$

Question 6Aspect ratios: a picture on a screen

The aspect ratio of a rectangle is width : height. A photo with an aspect ratio of $5 : 4$ is shown on a screen with an aspect ratio of $3 : 2$ . It is made as large as possible without being cut off, so it fills the full height of the screen, as shown in the diagram. Find the fraction of the screen that is not covered.

Show the worked solution

Step 1: Both have the same height: make the heights the same number of units (LCM of the second terms) $\text{picture } 5 : 4 = 5 : 4$ $\text{screen } 3 : 2 = 6 : 4$
Step 2: The picture covers its width out of the screen's width $\text{Covered} = \dfrac{5}{6}$ $\phantom{\text{Covered}} = \dfrac{5}{6}$
Step 3: Not covered: 1 take away the part covered $1 - \dfrac{5}{6} = \dfrac{1}{6}$ $\dfrac{1}{6}$ of the screen is not covered.