Worksheets · Foundation and Higher

Positive and fractional enlargements

8 exam-style questions, grades 3 to 6. Worked solutions and the marks are on the last page.

Question 1Non-calculator · 3 marks
This question is about enlargements.
(a) A rectangle measures 2 cm by 5 cm. It is enlarged by scale factor 3. Work out the perimeter of the enlarged rectangle. (2)
(b) A square with side 4 cm is enlarged to a square with side 10 cm. Write down the scale factor. (1)
Question 2Non-calculator · 4 marks
Shape $A$ is enlarged by scale factor $\frac{1}{4}$ to give shape $B$ .
(a) The longest side of $A$ is 18 cm. Work out the length of the longest side of $B$ . (2)
(b) Tom says, "An enlargement always makes a shape bigger." Explain why Tom is wrong. (1)
(c) A side of $B$ is 2 cm long. Work out the length of the matching side of $A$ . (1)
Question 3Non-calculator · 5 marks
An enlargement has centre C $(1,-2)$ and scale factor $1/2$ . Point A is $(7,1)$ .
(a) Find the coordinates of the image of A. (3)
(b) A segment of length 9 cm is enlarged in the same way. Find its image length. (2)
Question 4Non-calculator · 4 marks
A photograph 15 cm by 10 cm is enlarged by scale factor 1.6.
(a) Work out the length of the longer side of the enlargement. (2)
(b) Work out the area of the enlargement. (2)
Question 5Non-calculator · 3 marks
Rectangle R has vertices (2, 2), (6, 2), (6, 4) and (2, 4). It is enlarged by scale factor $\frac{1}{2}$ with centre (0, 0).
(a) Write down the coordinates of the image of (6, 4). (1)
(b) Work out the area of the image. (2)
Question 6Non-calculator · 4 marks
A model of a lorry is made using a scale factor of $\frac{1}{40}$ . The real lorry is 16 m long.
(a) Work out the length of the model in centimetres. (2)
(b) A model wheel has diameter 2.25 cm. Work out the diameter of the real wheel in metres. (2)
Question 7Non-calculator · 5 marks
Shape P is enlarged by scale factor k to give shape Q. A side of length 12 cm on P becomes 9 cm on Q.
(a) Find k. (2)
(b) Another side of Q is 6 cm long. Find the length of the corresponding side of P. (2)
(c) Explain how the value of k shows that Q is smaller than P. (1)
Question 8Non-calculator · 4 marks
The line segment $AB$ has $A(1, 2)$ and $B(3, 2)$ . It is enlarged to give $A'B'$ with $A'(-1, -1)$ and $B'(5, -1)$ .
(a) Write down the scale factor of the enlargement. (1)
(b) Find the coordinates of the centre of enlargement. (3)

Worked solutions and marks

Question 1

(a) 42 cm

Every length is multiplied by 3: 6 cm by 15 cm.
$2 \times (6 + 15) = 42$

M1 Multiplying both lengths by 3, or the original perimeter 14 by 3.
A1 Correct answer: 42 cm.

(b) 2.5

Scale factor $= \frac{\text{new length}}{\text{old length}} = \frac{10}{4} = 2.5$ .

B1 2.5 or $\frac{5}{2}$ .

Question 2

(a) 4.5 cm

Multiply by the scale factor: $18 \times \frac{1}{4} = 4.5$ .

M1 $18 \times \frac{1}{4}$ or $18 \div 4$ .
A1 Correct answer: 4.5 cm.

(b) A scale factor between 0 and 1 makes the image smaller.

$B$ is an enlargement of $A$ , but its lengths are a quarter of $A$ 's, so it is smaller.

C1 A scale factor between 0 and 1, such as $\frac{1}{4}$ , gives a smaller image.

(c) 8 cm

Going from $B$ back to $A$ multiplies by 4: $2 \times 4 = 8$ .

B1 Correct answer: 8 cm.

Question 3

(a) $(4,-1/2)$

Find the displacement from the centre to A.
$7-1=6$
Scale the displacement and add back the centre coordinates.
$x=1+\frac{1}{2}\times 6$
Therefore $(4,-1/2)$ .

M1 Find the displacement from the centre to A.
M1 Scale the displacement and add back the centre coordinates.
A1 Correct answer: $(4,-1/2)$

(b) $4.5$ cm

All lengths scale by the positive scale factor.
$9\times (1/2)$
Therefore $4.5$ cm.

M1 All lengths scale by the positive scale factor.
A1 Correct answer: $4.5$ cm

Question 4

(a) $24$ cm

Multiply the length by the scale factor.
$15\times 1.6$
Therefore $24$ cm.

M1 Multiply the length by the scale factor.
A1 Correct answer: $24$ cm

(b) $384$ cm²

Multiply the two enlarged side lengths.
$24\times 16$
Therefore $384$ cm².

M1 Multiply the two enlarged side lengths.
A1 Correct answer: $384$ cm²

Question 5

(a) $(3,2)$

Therefore $(3,2)$ .

B1 Correct answer: $(3,2)$

(b) $2$ square units

The image is 2 by 1; multiply the side lengths.
$2\times 1$
Therefore $2$ square units.

M1 The image is 2 by 1; multiply the side lengths.
A1 Correct answer: $2$ square units

Question 6

(a) $40$ cm

Convert to centimetres and divide by 40.
$1600/40$
Therefore $40$ cm.

M1 Convert to centimetres and divide by 40.
A1 Correct answer: $40$ cm

(b) $0.9$ m

Multiply by 40, then convert centimetres to metres.
$2.25\times 40/100$
Therefore $0.9$ m.

M1 Multiply by 40, then convert centimetres to metres.
A1 Correct answer: $0.9$ m

Question 7

(a) $\frac{3}{4}$

Divide the image length by the original length.
$9/12$
Therefore $\frac{3}{4}$ .

M1 Divide the image length by the original length.
A1 Correct answer: $\frac{3}{4}$

(b) $8$ cm

Reverse the enlargement by dividing by k.
$6/(3/4)$
Therefore $8$ cm.

M1 Reverse the enlargement by dividing by k.
A1 Correct answer: $8$ cm

(c) k = 3/4 is between 0 and 1, so every length is multiplied by a number less than 1.

Therefore k = 3/4 is between 0 and 1, so every length is multiplied by a number less than 1.

C1 Correct conclusion with supporting reasoning: k = 3/4 is between 0 and 1, so every length is multiplied by a number less than 1.

Question 8

(a) 3

$A'B' = 5 - (-1) = 6$ and $AB = 3 - 1 = 2$ , so the scale factor is $6 \div 2 = 3$ .

B1 Correct answer: Scale factor 3.

(b) $(2, 3.5)$

For centre $C$ , the image of $A$ is three times as far from $C$ :
$A' - C = 3(A - C)$
Rearrange.
$2C = 3A - A' = (3, 6) - (-1, -1) = (4, 7)$
So $C = (2, 3.5)$ . Check with $B$ : $\frac{3(3, 2) - (5, -1)}{2} = \frac{(4, 7)}{2}$ .

P1 A correct method for the centre: an equation such as $A' - C = 3(A - C)$ , or rays $A'A$ and $B'B$ drawn and extended to meet.
P1 Solving for one coordinate of the centre.
A1 Correct answer: $(2, 3.5)$ .

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Positive and fractional enlargements

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