KS3 Maths · Year 7 · Chapter 11: Shape and ratio · Lesson 1

Ratio of lengths, areas and volumes

Year 7About 75 minutesNo calculator45 marks

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Date

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Quick questions to start.

A1
Needed today
Write $18 : 30$ in its simplest form.
[1 mark]
Answer
A2
Needed today
How many millimetres are there in $6.5$ cm?
[1 mark]
Answer
A3
Last lesson
Work out the value of $2^{5}$ .
[1 mark]
Answer
A4
From chapter 6
Work out the volume of a cuboid $5$ cm by $3$ cm by $2$ cm.
[1 mark]
Answer

Example, then your turn

Year 7

Your teacher works through each example with you. Then try the one beside it.

Example 1

Work out the ratio of the length of $A B$ to the length of $C D$ . Give it in its simplest form.

Not drawn accurately

Hint Change both lengths to the smaller unit first. A ratio has no units.

B1
Work out the ratio of the length of $P Q$ to the length of $R S$ . Give it in its simplest form.
[2 marks]
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Answer

Example 2

Work out the ratio of the volume of the cube to the volume of the cuboid, in its simplest form.

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Hint Volume of a cuboid: length

\times

width

\times

height.

B3
Work out the ratio of the volume of the smaller cuboid to the volume of the larger cuboid, in its simplest form.
[2 marks]
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Answer

Example 3

For squares $A$ and $B$ , work out the ratio of (a) their sides, (b) their perimeters, (c) their areas. (d) Write the area ratio in the form $1 : n$ .

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Hint Work each one out and simplify. For the form

1 : n

, divide both sides by the first number.

B4
For the two cubes, work out the ratio of (a) their edges, (b) their surface areas, (c) their volumes. Give each in its simplest form.
[3 marks]
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Answer

Practice

Year 7

C1
Three ropes are $40$ cm, $1.2$ m and $2$ m long. Write the ratio of their lengths in its simplest form.
[2 marks]
Answer
C2
A poster is pinned in one corner of a rectangular noticeboard. What fraction of the noticeboard is covered by the poster?
[3 marks]
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Answer

C3
Work out the ratio of the area of rectangle $A$ to the area of rectangle $B$ , in its simplest form.
[3 marks]
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Answer
C4
Work out the ratio of the area of triangle $P$ to the area of triangle $Q$ , in its simplest form.
[3 marks]
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Answer

Exam-style questions

GCSE Foundation

D1
Ben says, "A square with sides of $8$ cm has twice the area of a square with sides of $4$ cm."
Ben is wrong. Work out the ratio of the two areas, and explain what doubling the sides does to the area.
[4 marks]
Answer

D2
The ratio of the volume of the small box to the volume of the large box is $1 : 5$ . Both boxes are cuboids. Work out the height, $l$ , of the large box.
[4 marks]
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Answer $l = \dots\dots$ cm
D3
A rectangular garden has a rectangular lawn. The rest of the garden is flower beds.
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(a)
Work out the area of the lawn.
[1 mark]
Answer
(b)
Work out the area of the flower beds.
[1 mark]
Answer
(c)
Write the ratio of the area of the lawn to the area of the flower beds in its simplest form.
[2 marks]
Answer
(d)
Write the ratio in part (c) in the form $1 : n$ .
[1 mark]
Answer
Total for D3: 5 marks
D4
The ratio of the volumes of two cubes is $1 : 27$ . The smaller cube has edges of $4$ cm.
(a)
Work out the length of an edge of the larger cube.
[2 marks]
Answer
(b)
Work out the ratio of the surface areas of the two cubes, in its simplest form.
[3 marks]
Answer
Total for D4: 5 marks

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
The ratio of the perimeter of square $A$ to the perimeter of square $B$ is $2 : 5$ . Square $B$ has an area of $100$ cm². Work out the area of square $A$ .
[5 marks]
Hint 1 · What to try
Work out the side of square $B$ , then its perimeter.
Hint 2 · The first line
Square $B$ has sides of $10$ cm and a perimeter of $40$ cm, so square $A$ has a perimeter of $16$ cm.
Hint 3 · The full method
Square $A$ has sides of $16 \div 4 = 4$ cm.
Answer $\dots\dots$ cm²

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can write the ratio of two lengths in different units in its simplest form.
I can write the ratio of two areas or two volumes, and write a ratio in the form $1 : n$ .
I can say what happens to the area and the volume when every length is multiplied.
I can find a missing length from a ratio of volumes.

Answers and mark scheme

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