MATHEMATICS FOR INTERIOR DESIGN GSC1105
Ratio &
Proportion.
Prepared by:
General Studies Department.
Writing and using ratios
• A ratio compares two or more quantities.
• Ratios occur often in real life.
• A scale model of a car is labeled Scale 1: 18.
Part
Length on model
Length on actual car
Length of car door
7 cm
7 x 18 = 126 cm
Diameter of wheels
3 cm
3 x 18 = 54 cm
Length of car
26 cm
26 x 18 = 468 cm
EXAMPLE
To make 12 small cupcakes you need 150 g of flour
and 30 g of melted butter. Grace is making 36
cupcakes. How much of each ingredient does she
need?
Answer:
36 ÷12 = 3
so she will need three times as much of each of the
ingredients.
Flour 3 x 150 g= 450 g
Melted butter 3 x 30 g= 90 g
Simplifying ratios
• Simplifying a ratio is similar to simplifying
fractions.
• If we refer to example 1,the amounts of flour
and melted butter were 150g and 30 g.
• We write this as a ratio 150 : 30
• Dividing by 10 gives
15 : 3
• Dividing by 3 gives 5 : 1
• This means that you need 5 times as much
flour as melted butter.
EXAMPLE :
Write these ratios in their simplest form.
(a) 8 : 12
(b) 18 : 24 : 6
(c) 40 cm : 1 m
Answer:
(a)
8 :12 = 2 : 3 (dividing by 4)
(b)
18 : 24 : 6 = 3 : 4 : 1
(dividing by 6)
(c)
40 cm : 1 m
= 40 cm :100 cm Units not the same,
= 40 : 100 Change m to cm.
= 4 : 10
(dividing by 10)
=2:5
(dividing by 2)
Writing a ratio as a
fraction
•
A ratio can also be written as fractions, to help you
solve problems. When items are divided in a ratio 1 :
2, the fraction ½ gives you another method of
solving the problem.
EXAMPLE :
In a school the ratio of boys to girls is 5 : 7.
There are 265 boys. How many girls are there?
Answer: Method A (using multiplying)
Suppose there are x girls.
The ratio 5 : 7 has to be the same as 265 : x
265÷ 5 = 53 ,
so multiply both sides of the ratio by 53
Boys: 5 x 53 = 265 Girls: 7 x 53 = 371
There are 371 girls in the school.
Method B (using fractions)
Suppose there are x girls.
The ratio 5 : 7 has to be the same as 265 : x
Using fractions, = 5/7 = 265/x
It can also be written as = 7/5 = x / 265
Multiplying both sides by 265, x= (7 x 265)/5
This gives x = 371, therefore there are 371 girls
in the school.
Writing a ratio in the form 1: n or n:1
• In Example 1,
120 : 20 simplified to 6 : 1.
• Not all ratios simplify so that one value is 1.
For example, 6 : 4 simplifies to 3 : 2.
• If you are need to write this ratio in the form
n : 1 you cannot leave the answer as 3 : 2.
EXAMPLE :
Write these ratios in the form n : 1.
(a) 27 : 4
(b) 6 cm : 25 mm (e) $2.30 : 85c
ANSWER:
(a) 27 : 4 = 6.75 :1
(b) 6 cm : 25 mm = 60 mm : 25 mm
= 60 : 25
= 2.4 :1
(c) $2.38 : 85c
= 238 : 85
= 2.8:1
Dividing quantities in a given ratio
• Ali and Siti share $60 so that Ali receives twice as
much as Siti.
• You can think of this as, Ali : Siti = 2 : 1.
•
Ali will get this much
Siti will get this much
• where each bag of money contains the same
amount.
• To work out each man's share, you need to divide
the money into 3.
(2 + 1 = 3) equal parts (or shares).
1 part = $60 ÷ 3 = $20
Ali receives 2 X $20 = $40
2 parts
Siti receives 1 X $20 = $20
1 part
Check:
$20 + $40 = $60
Proportion
Two quantities are in direct proportion ,if their ratio stays the
same as they increase or decrease.
• Ratio method
For example, a mass of 5 kg attached to a spring stretches
it by 40 cm.
A mass of 15 kg attached to the same spring will stretch
it 120 cm.
The ratio mass : extension is
5 : 40
= 1 : 8 (dividing by 5)
15 : 120
= 1 : 8 (dividing by 15)
• The ratio mass : extension is the same so the two
quantities are in direct proportion.
Inverse proportions
When two quantities are in direct proportion
•
as one increases, so does the other
•
as one decreases, so does the other.
Or
If you travel at an average speed of 50 km/h it will
take you 2 hours.
If you only average 40 km/h it will take you 2+ hours
• As the speed decreases, the time increases.
• As the speed increases, the time decreases
EXAMPLE :
Two people take 6 hours to paint a fence. How
long will it take 3 people?
ANSWER:
2 people take 6 hours.
1 person takes 6 x 2 = 12 hours
3 people will take 12 ÷ 3 = 4 hours
Map Scales
• Map scales are written as ratios.
A scale of 1 : 40 000 means that 1 cm on the map
represents 40 000 cm on the ground.
When you answer questions involving map scales you
need to
• use the scale of the map
• convert between metric units of length so that your
answer is in sensible units.
EXAMPLE:
The scale of a map is 1: 40 000. The distance
between Gadong conter point and Empire
Country club on the map is 40 cm.
What is the actual distance between Gadong
conter point and Empire Country club? Give
your answer in kilometer
ANSWER:
1M = 100 CM and 1km = 1000 cm
Distance on map = 40 cm
Distance on the ground
= 40 x 40 000 cm = 1600 000 cm
= 1600 000 ÷ 100 m
= 16 000 m
= 16 000 ÷ 1000 km = 16 km
EXAMPLE :
The distance between two towns is 20 km. How far
apart will they be on a map of scale 1:180 000?
Distance on the ground = 20 km
= 20 x 1000 m
= 20 000 m
= 20 000 x 100 cm
= 2 000 000 cm
Distance on map = 2000 000 cm ÷ 180 000
=11.1 cm (to 3 s.f.)
Reference:
• Core Mathematics for IGCSE 2nd Edition, Ric
Pimentel and Terry Wall, University of Cambridge.