KS3 Maths · Key Stage 3 · Unit 7: Similarity, Rates and Graphs

Similar shapes, speed, density and pressure, and distance-time graphs

Years 8–9About 80 minutesCalculator allowed86 marks

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NameClassDate

Before you start

Recall

You will need these in every question below.

A1
Write the ratio $18 : 24$ in its simplest form.
[1 mark]
Answer

A2
Write 2 hours 45 minutes as a number of hours.
[1 mark]
Answer
A3
Change 3.6 km into metres.
[1 mark]
Answer
A4
Find the missing number: $9 : 15 = 12 : \dots$
[2 marks]
Answer
A5
A straight line passes through the points $(0, 3)$ and $(4, 15)$ . Work out its gradient.
[2 marks]
Answer

Examples, then your turn

Core

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

Example 1

Triangle $P QR$ is similar to triangle $A B C$ . $A$ matches $P$ , $B$ matches $Q$ and $C$ matches $R$ . $A B = 5$ cm, $B C = 8$ cm, $C A = 11$ cm and $P Q = 7.5$ cm. Work out the length $x$ .

Not drawn accurately

Hint Find the scale factor from two matching sides. Every length in the bigger triangle is the matching length times the scale factor.

B1
Triangle $P QR$ is similar to triangle $A B C$ . $A$ matches $P$ , $B$ matches $Q$ and $C$ matches $R$ . $A B = 6$ cm, $B C = 9$ cm, $C A = 10$ cm and $P Q = 8.4$ cm. Work out the length $x$ .
[2 marks]
Not drawn accurately
Answer

Example 2

A train travels 186 km in 1 hour 30 minutes. Work out its average speed in km/h.

Hint Write the time in hours first: 30 minutes is 0.5 of an hour. Then speed = distance ÷ time.

B2
A coach travels 147 km in 1 hour 45 minutes. Work out its average speed in km/h.
[2 marks]
Answer

Example 3

A block of metal has a volume of 35 cm³ and a density of 8.4 g/cm³. Work out its mass.

Hint Density = mass ÷ volume, so mass = density × volume.

B3
A stone has a mass of 702 g. Its density is 2.7 g/cm³. Work out its volume.
[2 marks]
Answer

Example 4

Tom cycles from home to a shop, waits, and cycles home. The graph shows his journey. Work out his speed on the way to the shop, in km/h.

Hint Read the distance and the time for the first straight section. Change the minutes into hours before you divide.

B4
Zara walks from home to a park, rests, and walks home. The graph shows her walk. Work out her speed on the way home, in km/h.
[2 marks]
Answer

Practice

Core

Each question asks for something different. Show your working.

C1
Each part describes two triangles. Write SSS, SAS, ASA or RHS to show why the two triangles are congruent.
(a)
Both triangles have sides of 6 cm, 11 cm and 13 cm.
[1 mark]
Answer
(b)
Both triangles have a right angle, a hypotenuse of 17 cm and another side of 8 cm.
[1 mark]
Answer
(c)
Both triangles have angles of $3 5^{\circ}$ and $8 0^{\circ}$ , and the side between those two angles is 12 cm.
[1 mark]
Answer
Total for C1: 3 marks

C2
Two rectangles are similar. The smaller one is 4 cm long and the larger one is 10 cm long. The area of the smaller rectangle is 12 cm². Work out the area of the larger rectangle.
[2 marks]
Answer
C3
Change 72 km/h into metres per second.
[2 marks]
Answer
C4
A box has a weight of 180 newtons (N). It rests on a face with an area of 0.6 m². Work out the pressure on the floor, in N/m².
[2 marks]
Answer
C5
A runner runs 6 km at 12 km/h, then a further 6 km at 8 km/h. Work out her average speed for the whole 12 km.
[3 marks]
Answer
C6
A post 1.5 m tall casts a shadow 2.4 m long. At the same time, a tree casts a shadow 9.6 m long. Work out the height $h$ of the tree.
[2 marks]
Not drawn accurately
Answer
C7
Silver has a density of 10.5 g/cm³. Copper has a density of 8.9 g/cm³.
(a)
A silver ring has a mass of 31.5 g. Work out its volume.
[2 marks]
Answer
(b)
A copper ring has the same volume as the silver ring. Work out its mass.
[1 mark]
Answer
Total for C7: 3 marks
C8
On a car's distance-time graph, one straight section goes from the point (1.5 hours, 20 km) to the point (4 hours, 120 km). Work out the speed of the car on this section.
[2 marks]
Answer
C9
A car travels 54 km at an average speed of 72 km/h. How many minutes does the journey take?
[2 marks]
Answer

Test-style questions

Stretch

Answer every question in the space given. Show your working: most of the marks are for method.

D1
Two picture frames are similar. The larger frame is twice as long and twice as wide as the smaller one. The smaller frame has an area of 40 cm². Sam says, "The larger frame has an area of 80 cm²."
Sam is wrong. Explain why, and work out the area of the larger frame.
[2 marks]
Answer

D2
In the diagram, $D$ is on $A B$ and $E$ is on $A C$ . $D E$ is parallel to $B C$ . $A D = 6$ cm, $D B = 4$ cm, $D E = 9$ cm and $E C = 3$ cm.
Not drawn accurately
(a)
Work out the length of $B C$ .
[2 marks]
Answer
(b)
Work out the length of $A E$ . You must show your working.
[3 marks]
Answer
Total for D2: 5 marks
D3
Ben leaves home at 9 am and cycles to a lake. He stops at the lake, then cycles home. The graph shows his journey.
(a)
Work out Ben's speed on the way to the lake, in km/h.
[2 marks]
Answer
(b)
Work out Ben's speed on the way home, in km/h.
[2 marks]
Answer
(c)
Work out Ben's average speed for the whole journey, including his stop. Give your answer to 1 decimal place.
[2 marks]
Answer
Total for D3: 6 marks
D4
Ali leaves a car park at 10:00 am and walks along a path at 4 km/h. Bea leaves the same car park at 10:45 am and cycles along the same path at 16 km/h.
At what time does Bea catch up with Ali? How far from the car park are they then? You must show your working.
[4 marks]
Answertime $\dots\dots$ distance $\dots\dots$ km
D5
A crate has a weight of 2400 N. It rests on a face that is 1.2 m long and 0.8 m wide.
(a)
Work out the pressure on the floor, in N/m².
[2 marks]
Answer
(b)
The crate is turned onto a face that is 0.8 m long and 0.5 m wide. Show that the pressure goes up by 3500 N/m².
[2 marks]
Total for D5: 4 marks
D6
Two water bottles are similar. The smaller bottle is 15 cm tall and the larger bottle is 20 cm tall. The smaller bottle holds 270 ml.
Work out how much the larger bottle holds. You must show your working.
[3 marks]
Answer
D7
A cheetah can run at 29 m/s. A car is travelling at 100 km/h. Which is faster? You must show your working.
[2 marks]
Answer
D8
In triangles $A B C$ and $D E F$ , angle $B$ = angle $E = 6 4^{\circ}$ , angle $C$ = angle $F = 4 9^{\circ}$ , and $B C = E F = 10$ cm.
Not drawn accurately
(a)
Write down the condition that shows the two triangles are congruent.
[1 mark]
Answer
(b)
Work out the size of angle $D$ .
[2 marks]
Answer
Total for D8: 3 marks
D9
A train travels at 90 km/h for 40 minutes. It then travels at 120 km/h for 1 hour 10 minutes.
Work out the average speed of the train for the whole journey. Give your answer to 1 decimal place.
[4 marks]
Answer
D10
Two triangles are similar. Their areas are 18 cm² and 50 cm². The smaller triangle has a base of 6 cm.
Work out the base of the larger triangle.
[3 marks]
Answer

Challenge

UKMT level

No route is given. Spend five minutes on a problem before you look at a hint. The hints are on the last page of the worksheet, and you can use one at a time.

E1
Two triangles are similar. Their perimeters are 30 cm and 45 cm. The area of the larger triangle is 40 cm² more than the area of the smaller triangle.
Work out the area of the smaller triangle.
[4 marks]
Hint 1 · What to try
The perimeters give you the length scale factor. What does that do to the areas?
Hint 2 · The first line
The length scale factor is $45 \div 30 = 1.5$ , so the larger area is $1. 5^{2} = 2.25$ times the smaller area.
Hint 3 · The full method
If the smaller area is $S$ , then $2.25 S - S = 40$ , so $1.25 S = 40$ and $S = 32$ .
Answer

E2
Two trains start 300 km apart and travel towards each other on parallel tracks. One train is 20 km/h faster than the other. They pass each other after 2 hours 30 minutes.
Work out the speed of the slower train.
[3 marks]
Hint 1 · What to try
Think about how fast the gap between the two trains closes.
Hint 2 · The first line
Together they close 300 km in 2.5 hours, so the gap closes at $300 \div 2.5 = 120$ km/h.
Hint 3 · The full method
The two speeds add to 120 and differ by 20, so they are 70 km/h and 50 km/h.
Answer
E3
If Jo walks to school at 5 km/h, she arrives 6 minutes late. If she walks at 6 km/h, she arrives 4 minutes early.
How far is it from Jo's home to school?
[4 marks]
Hint 1 · What to try
The two walks take different times. How many minutes apart are they?
Hint 2 · The first line
The slow walk takes 10 minutes longer, which is $\frac{1}{6}$ of an hour. For a distance $d$ km, $\frac{d}{5} - \frac{d}{6} = \frac{1}{6}$ .
Hint 3 · The full method
$\frac{d}{5} - \frac{d}{6} = \frac{d}{30}$ , so $\frac{d}{30} = \frac{1}{6}$ and $d = 5$ .
Answer
E4
Two solid cubes are made from the same metal. One has a mass of 250 g and the other has a mass of 2 kg. The smaller cube has a surface area of 150 cm².
Work out the surface area of the larger cube.
[3 marks]
Hint 1 · What to try
The same metal means the masses compare in the same way as the volumes.
Hint 2 · The first line
The volume scale factor is $2000 \div 250 = 8$ , so the length scale factor is 2.
Hint 3 · The full method
Areas are multiplied by $2^{2} = 4$ , so the surface area is $150 \times 4 = 600$ cm².
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can find a missing side of similar shapes from the scale factor, use area and volume scale factors, and say which condition makes two triangles congruent.
I can work with speed, density and pressure, change between units, and find an average speed for a journey.
I can read a distance-time graph, work out a speed from a straight section, and find where two journeys meet.

Answers and mark scheme

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