Ratios - Year 7 Mathematics

Year 7 core

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

write and simplify a ratio,
recognise equivalent ratios and find missing terms,
divide a quantity in a given ratio,
use ratios to model real situations involving lengths, areas and money.

1. What a ratio is

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.

2. Simplifying a ratio

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.

3. Equivalent ratios

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 $\square = \dfrac{b \times c}{a}$ .

4. Dividing a quantity in a given ratio

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 $2 : 3$ .

The parts sum to $2 + 3 = 5$ .
One “unit” is worth $60 \div 5 = 12$ , so $12.
The shares are $2 \times 12 = 24$ (so $24) and $3 \times 12 = 36$ (so $36).

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

5. Ratios of lengths, areas and volumes

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?

The parts sum to $1 + 2 + 4 = 7$ .
One part $= 35 \div 7 = 5$ kg.
Gravel is $4$ parts $= 4 \times 5 = 20$ kg.

Practice: Year 7 core

Fluency

Simplify and find missing parts

Simplify the ratio $12 : 18$ .
Simplify the ratio $35 : 14$ .
Simplify the ratio $40 : 60 : 100$ .
Write $250$ g : $1$ kg as a simplified ratio.
Write $45$ minutes : $2$ hours as a simplified ratio.
Find the missing number: $3 : 5 = 12 : \square$ .
Find the missing number: $\square : 8 = 15 : 10$ .
Find the missing number: $2 : 3 = \square : 18$ .

Fluency

Dividing a quantity in a ratio

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

Explain and spot the mistake

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

Real-world problems

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?

Answers

Answer key

Attempt the practice first. When you're ready to check, expand the answers below.

Show the full answer key

Year 7 core - answers

Fluency

Simplify and find missing parts

$2 : 3$
$5 : 2$
$2 : 3 : 5$
$1 : 4$ . Method: convert both to grams, $250 : 1000$ ; divide by $250$ .
$3 : 8$ . Method: $45 : 120$ ; divide by $15$ .
$20$
$12$
$12$ . Method: $2 \times 6 = 12$ , since $3 \times 6 = 18$ .

Fluency

Dividing a quantity in a ratio

$15 : $25
$16 : $56
$8,\ 16,\ 24$ sweets. Method: $1 + 2 + 3 = 6$ parts; each part $= 8$ .
$140$ g. Method: flour is $5$ parts, so $1$ part $= 70$ g; sugar $= 2 \times 70$ .
$40$ and $50$ . Method: $4 + 5 = 9$ parts; each part $= 90 \div 9 = 10$ .
$18$ boys, $12$ girls. Method: $3 + 2 = 5$ parts; each $= 6$ .

Explain and spot the mistake - answers

Reasoning

Explain and spot the mistake

A ratio of $4 : 6$ compares one group to the other, not to the whole. To find “what fraction of the total is boys” you need boys over total: $\dfrac{4}{4 + 6} = \dfrac{4}{10} = 40\%$ . Ben wrote $\dfrac{4}{6} \approx 66.67\%$ , which is the ratio of boys to girls, not boys to total.
In a $1 : 4$ ratio the parts are $1$ concentrate plus $4$ water, giving $5$ parts total. So concentrate is $\dfrac{1}{5} = 20\%$ of the drink, not $\dfrac{1}{4} = 25\%$ . Jen forgot to add the parts to find the total.
Dividing both $6$ and $9$ by $3$ gives $2 : 3$ , so the parts scale down by the same factor. For $2 : 5$ the same multiplier would need to take $2 \to 6$ and $5 \to 9$ , but $2 \times 3 = 6$ while $5 \times 3 = 15 \neq 9$ . The proportion doesn’t match, so $6 : 9 \neq 2 : 5$ .
$24 : 36 = 2 : 3$ . In plain words: for every $2$ turns of the first gear, the second gear makes $3$ turns (or equivalently, the first gear turns $1.5$ times faster than the second).

Real-world problems - answers

Problem solving

Real-world problems

Flour $750$ g, sugar $450$ g, eggs $10$ . Method: scale factor $= 30 \div 12 = 2.5$ .
$2$ km. Method: $8 \times 25\,000 = 200\,000$ cm $= 2$ km.
Anna $90, Ben $60. Method: ratio $180 : 120 = 3 : 2$ ; $5$ parts = $150; each part = $30.
Length $15$ m, width $9$ m. Method: $5 + 3 = 8$ parts; total length-plus-width (half of perimeter) $= 24$ m, so $1$ part $= 3$ m.
$120,\ 160,\ 200$ . Method: $3 + 4 + 5 = 12$ parts; each part $= 480 \div 12 = 40$ .
$1.5$ L red. Method: $7 + 3 = 10$ parts; each part $= 0.5$ L; red $= 3$ parts.

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