Ratios - Practice

Year 7 core

By the end of this topic you should be able to:

A ratio compares quantities of the same kind. $3 : 5$ is read “three to five”.

If there are $3$ apples for every $5$ oranges, the ratio of apples to oranges is $3 : 5$ . The total in each “group” is $3 + 5 = 8$ pieces of fruit.

Divide every part of the ratio by their greatest common factor.

Worked example 1 Simplify 18 : 24

$\gcd(18, 24) = 6$ . Divide both parts by $6$ :

18 : 24 \;=\; 3 : 4.

Multiplying every part of a ratio by the same number gives an equivalent ratio. $2 : 3 \;=\; 4 : 6 \;=\; 10 : 15$ .

Finding a missing part

If $a : b \;=\; c : \square$ , then $\square = \dfrac{b \times c}{a}$ .

Dividing a quantity in a ratio

If you split a quantity $Q$ in the ratio $a : b$ , the parts sum to $a + b$ .

\text{first share} = \frac{a}{a+b} \times Q, \qquad \text{second share} = \frac{b}{a+b} \times Q.

Worked example 2 Dividing an amount

Divide $60 in the ratio $2 : 3$ .

Check: $24 + 36 = 60$ , i.e. $60. OK.

Ratios turn up whenever two measurements of the same kind are compared - ingredients in a recipe, sides of similar shapes, parts of a mixture.

Worked example 3 Scaling a recipe

A recipe for $4$ people uses $800$ g of pasta. How much is needed for $7$ people?

Set up a ratio of people-to-pasta: $4 : 800$ . For $7$ people the ratio becomes $7 : \square$ .

\square = \dfrac{800 \times 7}{4} = 1400 \text{ g}.

Worked example 4 Sharing in a ratio

Concrete is mixed using cement, sand, and gravel in the ratio $1 : 2 : 4$ . How much gravel is in a $35$ kg batch?

Fluency

Fluency

Divide $40 in the ratio $3 : 5$ .
Divide $72 in the ratio $2 : 7$ .
Divide $48$ sweets in the ratio $1 : 2 : 3$ .
A recipe uses flour and sugar in the ratio $5 : 2$ . If there are $350$ g of flour, how much sugar is used?
Two numbers are in the ratio $4 : 5$ and their sum is $90$ . Find the numbers.
A $3 : 2$ ratio of boys to girls in a class of $30$ gives how many of each?

Reasoning

Ben says “the ratio $4 : 6$ is the same as $\dfrac{4}{6}$ , which is the same as the percentage $66.67\%$ of boys”. Explain what is right and what is confused in Ben’s statement.
A drink is made from concentrate and water in the ratio $1 : 4$ . Jen says ” $\dfrac{1}{4}$ of the drink is concentrate, which is $25\%$ ”. What has Jen mixed up, and what is the correct percentage?
Explain why the ratio $6 : 9$ is equivalent to $2 : 3$ , but is not equivalent to $2 : 5$ .
Two gears have $24$ and $36$ teeth. Write the gear ratio in simplest form and explain what the ratio means in plain words.

Problem solving

A cake recipe makes $12$ cupcakes and uses $300$ g flour, $180$ g sugar and $4$ eggs. How much of each is needed for $30$ cupcakes?
A map has scale $1 : 25\,000$ . Two towns are $8$ cm apart on the map. How many kilometres apart are they in reality?
Two friends share a $150 phone bill in the ratio of their usage. Anna used the phone for $180$ minutes, Ben for $120$ minutes. How much should each pay?
A rectangular garden has length and width in the ratio $5 : 3$ . If its perimeter is $48$ m, find its length and width.
A school of $480$ students is split into three houses in the ratio $3 : 4 : 5$ . How many students are in each house?
Paint is mixed from white and red in the ratio $7 : 3$ . How much red paint is needed to make $5$ litres of mixed paint?