262 Chapter 5 Ratios and rates
Exercise 5D Understanding
1 a Convert 10 000 cm to:
i mm ii m iii km
b Convert 560 m to:
i km ii cm iii mm
Example 9 Comparing scale distance to real distance
A doll’s house has a toy lounge chair 2 cm wide. The real lounge chair is 1 m wide.
a Write both toy width and real chair width in cm.
b How much bigger is the real chair than the toy chair?
c What is the scale for toy chair : real chair?
d What is the scale factor?
Solution Explanation
a 2 cm, 100 cm 1 m × 100 = 100 cm
b 50 times bigger 2 cm × 50 = 100 cm
c 1 : 50 A scale always starts with 1.
The second number shows how many times larger the real
object is.
d 50 A scale is a ratio of two numbers but a scale factor is one
number.
2 Here are pictures of a real classic racing car and a model racing car.
model car length = 4 cm real car length = 4 m
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Number and Algebra 263
B Write the model car length in cm and real car length in cm.
C How many times longer than the toy car is the real car?
D What is the scale for the model car : real car?
E What is the scale factor?
A model ship is 60 cm long and the real ship is ×?
300 m long.
B Write model length in cm and real ship length Model Real
in cm. ship ship
C How many times longer is the real ship
compared to the model?
D What is the scale for the model ship : real ship?
Example 10 Calculating the real distance
A map has a scale ratio of 1 : 20 000.
B What is the scale factor?
Drilling
C What actual real distance in cm would 1 cm on the map represent?
for Gold D What actual real distance in mm would 5 mm on the map represent?
5D1
Solution Explanation
B 20 000 The scale factor is the second number in the scale ratio
C 20 000 cm 1 cm × 20 000 = 20 000 cm
D 100 000 mm 5 mm × 20 000 = 100 000 mm
A map has a scale ratio of 1 : 50 000.
×50 000
B What is the scale factor?
C What actual real distance in cm would 1 cm
on the map represent? Map Real
D What actual real distance in m would 1 m distance distance
on the map represent?
E What actual real distance in mm would ÷50 000
5 mm on the map represent?
F What actual real distance in cm would 5 cm
on the map represent?
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264 Chapter 5 Ratios and rates
Fluency
5D
Example 11 Converting scale distance to actual distance
A map has a scale of 1 : 20 000.
Find the actual distance in m for each scaled distance (map distance).
B 2 cm C 5 mm
Solution Explanation
B Actual distance = 2 cm × 20 000 Scale factor = 20 000
= 40 000 cm ×20 000
= 400 m
Map Real
÷100 to convert cm to m.
C Actual distance = 5 mm × 20 000 Multiply scaled distance by scale factor.
= 100 000 mm 5 mm times scale factor gives the answer
= 100 m in mm.
÷1000 to convert mm to m.
Find the actual distance for each of the following scaled distances. Give your answer
in the unit that is in brackets after each question.
t Multiply by the
B Scale 1 : 2 J 310 cm (cm) JJ 2.5 mm (mm)
scale factor.
C Scale 1 : 10 000 J 2 cm (m) JJ 4 mm (m) t Keep the units
D Scale 1 : 20 000 J 80 cm (km) JJ 1.25 m (km) the same as the
E Scale 1 : 400 J 16 mm (m) JJ 72 cm (m) question.
t Then convert to
F Scale 1 : 0.01 J 3 cm (mm) JJ 0.815 m (mm) the required units.
Example 12 Converting actual distance to scaled distance
A model boat has a scale of 1 : 500.
Find the scaled length in mm for each of these actual lengths.
B 50 m (mm) C 4550 mm (mm)
Solution Explanation
B Scaled length = 50 m ÷ 500 Divide by the scale factor, 500.
= 0.01 m 50 m ÷ 500 gives the answer in m.
= 10 cm × 100 to convert m to cm.
= 100 mm × 10 to convert cm to mm.
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Number and Algebra 265
Solution Explanation
C Scaled length = 4550 mm ÷ 500 Divide actual distance by scale factor.
= 45.5 mm ÷ 5 Shortcut: ÷100, then ÷5 (or vice versa)
= 9.1 mm The answer is in mm.
Model Real
÷500
Find the scaled length for each of these actual lengths.
t Divide by the
Give your final answer in the unit that is in brackets after scale factor.
each question. t Keep the units
B Scale 1 : 200 J 200 m (m) JJ 4 km (m) the same as
the question.
C Scale 1 : 500 J 10 000 m (m) JJ 1 km (m)
t Then convert
D Scale 1 : 10 000 J 1350 m (cm) JJ 736.5 m (cm) to the required
E Scale 1 : 250 000 J 5000 m (mm) JJ 1250 m (mm) units.
F Scale 1 : 0.05 J 7.5 cm (m) JJ 8.2 mm (m)
Change the two measurements provided in each scale into
t Convert
the same unit and then write the scale as a ratio of two larger unit to
numbers in simplest form. smaller unit.
B 2 cm : 200 m C 5 mm : 500 cm D 12 mm : 360 cm t Divide by HCF.
E 4 mm : 600 m F 4 cm : 5 m G 1 cm : 2 km
H 28 mm : 2800 m I 3 cm : 0.6 mm J 1.1 m : 0.11 mm
Example 13 Determining the scale and scale factor
State the scale factor in the following situations.
B 4 mm on a scale drawing represents an actual distance of 50 cm.
C An actual length of 0.1 mm is represented by 3 cm on a scaled drawing.
Solution Explanation
B Scale = 4 mm : 50 cm Write the ratio drawing length : actual length.
= 4 mm : 500 mm Convert to ‘like’ units.
Scale = 4 : 500 Write the scale without units.
= 1 : 125 Divide both numbers by 4. (HCF = 4)
Scale factor = 125 Ratio is now in the form 1 : scale factor.
The actual size is 125 times larger than the scaled
drawing.
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266 Chapter 5 Ratios and rates
5D Solution Explanation
C Scale = 3 cm : 0.1 mm Write the ratio drawing length : actual length.
= 30 mm : 0.1 mm Convert to ‘like’ units.
Scale = 30 : 0.1 Write the scale without units.
= 300 : 1 Multiply both numbers by 10 so both numbers
1 are whole.
= 1:
300 Divide both numbers by 300.
1 Ratio is now in the form 1 : scale factor.
Scale factor =
300 The actual size is 300 times smaller than the scaled
drawing.
Find the scale and the scale factor for each of the
t Same units
following.
t Scale ratios start
Skillsheet B 2 mm on a scale drawing represents an actual distance with 1
5C of 50 cm. t Write scale factors
C 4 cm on a scale drawing represents an actual distance as whole numbers
or fractions
of 2 km.
D 1.2 cm on a scale drawing represents an actual
distance of 0.6 km.
E 5 cm on a scale drawing represents an actual distance of 900 m.
F An actual length of 7 mm is represented by 4.9 cm on a scaled drawing.
G An actual length of 0.2 mm is represented by 12 cm on a scaled drawing.
Problem-solving and Reasoning
A model city has a scale ratio of 1 : 1000.
Scaled height means
B Find the actual height in m of a skyscraper that has a model height.
scaled height of 8 cm.
C Find the scaled length in cm of a train platform that is
45 m long in real life.
Blackbottle and Toowoola are 17 cm apart on a map with a scale of 1 : 50 000.
How many km apart are the towns in real life?
The house plan opposite has a scale of 1 : 150. For each
Length in mm × scale
room listed below do these two steps.
factor = real length
t Use a ruler to measure the internal length and width in mm
in mm.
t Use the scale to calculate the real dimensions in m to
1 decimal place.
B Bedroom 1 C Family room D Patio
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Number and Algebra 267
Scale 1 : 150
For each question below do
these two steps.
t Use a ruler to measure the
straight line map distance in
cm to the nearest millimetre.
t Use the scale to calculate
the real distance and
give each answer to the
nearest 100 km.
B Hobart to Cairns
C Perth to Sydney
D Darwin to Adelaide
E Brisbane to Melbourne
F Australia’s furthest point
west to furthest point east.
Scale 1 : 50 000 000
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268 Chapter 5 Ratios and rates
5D Enrichment: Design a bedroom
For this activity you will design and draw the floor plan of
How many times
a bedroom. Builders use millimetres for units so keep all
larger is the bedroom
units in millimetres for this activity. length than your
This bedroom has length = 3000 mm and width = 4000 mm. page length? Choose
a whole number for
The furniture in the bedroom is illustrated here. These the scale factor.
pictures are not shown to scale. The real dimensions are scale = 1 : scale factor
given for the ‘top view’. The ‘top view’ is how it is seen
looking down from above.
Top views with real dimensions
500 mm 500 mm
1400 mm 500 mm
400 mm
Top views with real dimensions
600 mm
900 mm
1200 mm
2000 mm
You are to design a scaled drawing of a bedroom including this furniture.
B Find a scale that will allow the drawing of this bedroom to fit on one page.
C Use your scale to change all the real dimensions to scaled lengths and widths in mm.
D Draw a scaled rectangle for the bedroom.
E Choose where each piece of furniture will be placed in the bedroom and draw
the scaled top view of each.
F Choose where a window and a door will be placed in the bedroom and draw the
scaled top view of each.
G Label dimensions with the real measurements in mm.
H Write the scale next to your bedroom plan.
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