Ratios and scale

Divide quantities in a given ratio; interpret and use scale drawings, maps and plans; apply the scale factor relationships

k : k^2 : k^3

for lengths, areas and volumes of similar figures.

Worked examples

Dividing in a ratio

Straightforward

Problem

Three business partners share a profit of $42\,000 in the ratio

2:5:7

. How much does each partner receive?

Multiply each partner's number of parts by the value of 1 part.

Partner	Parts	Amount
A	2	$2 \times \$3000 = \$6000$
B	5	$5 \times \$3000 = \$15\,000$
C	7	$7 \times \$3000 = \$21\,000$

Map scales

Moderate

Problem

A map has a scale of

1:200\,000

. (a) Two towns are 3.5 cm apart on the map. Find the actual distance in km. (b) A road is 45 km long in real life. How long is it on the map, in cm?

Similar figures — length, area and volume

Challenging

Problem

Two similar prisms have corresponding lengths of 3 cm and 6 cm (scale factor

k = 2

). The smaller prism has surface area

= 54

cm² and volume

= 27

cm³. Find the surface area and volume of the larger prism.

Recall how length, area and volume scale for similar figures, and apply the scale factor to each measure.

Measure	Scales by	Calculation
Length	$k = 2$	$3 \times 2 = 6$ cm
Area	$k^2 = 4$	$54 \times 4 = 216$ cm²
Volume	$k^3 = 8$	$27 \times 8 = 216$ cm³

Practise

Q1·Straightforward

Divide $240 between Ali and Ben in the ratio

3:5

. How much does Ali receive?

Q2·Straightforward

A map has a scale of

1:50\,000

. A river is 4 cm long on the map. What is the actual length of the river in kilometres?

Q3·Straightforward

A scale drawing uses a scale of

1:20

. A room wall is 8 m long in real life. How long is the wall on the drawing, in centimetres?

Q4·Moderate

A concrete mix uses cement, sand and gravel in the ratio

1:2:4

. If 70 kg of gravel is used, how many kilograms of cement are needed?

Q5·Moderate

A map has a scale of

1:250\,000

. Two towns are 7.5 cm apart on the map. Find the actual distance between the towns in kilometres.

Q6·Moderate

Two similar triangles have bases of 6 cm and 9 cm. The smaller triangle has an area of 18 cm². Find the area of the larger triangle in cm².

Q7·Moderate

Divide 180 kg in the ratio

2:3:4

. What is the largest share, in kg?

Q8·Moderate

Two similar boxes have lengths of 5 cm and 20 cm. The smaller box has a volume of 50 cm³. Find the volume of the larger box in cm³.

Q9·Challenging

A map has a scale of

1:500\,000

. The driving distance between two cities is 120 km. What is this distance on the map, in centimetres?

Q10·Challenging

A scale model of a car is built to a scale of

1:25

. The model is 6 cm high. What is the actual height of the car in metres?

Q11·Challenging

Three people share profits in the ratio

5:3:2

. The largest share is $4500 more than the smallest share. Find the total profit in dollars.

Q12·Challenging

Two similar cylinders have heights of 2 cm and 5 cm. The smaller cylinder has a volume of 80 cm³. Find the volume of the larger cylinder in cm³.