A detailed description of the concepts of Ratio, Rates and Proportion, along with worked examples.

1. RATIOS, RATES AND PROPORTIONS
A ratio is a comparison of two quantities by
division. The ratio of 𝑎 𝑡𝑜 𝑏 can be written 𝑎: 𝑏 or
𝑎
𝑏
, where 𝑏 .
Ratios that name the same comparison are said
to be equivalent.
A statement that two ratios are equivalent,
such as
3
4
=
12
16
, is called a proportion.

2. RATIOS, RATES AND PROPORTIONS
In the proportion
𝑎
𝑏
=
𝑐
𝑑
, the products 𝑎 × 𝑑
and 𝑏 × 𝑐 are called cross products. These
cross products are equal and this property
is used to solve a proportion to find a
missing value.

3. RATIOS, RATES AND PROPORTIONS
Example 1: Solving Proportions
Solve each proportion.
a)
2
9
=
6
𝑝
b)
4
𝑞−3
=
2
7
2 6
= 2 × (𝑞 − 3) = 4 × 7
9 𝑝
2 × 𝑝 = 6 × 9 2𝑞 − 6 = 28
2𝑝 = 54 2𝑞 = 34
𝑝 = = 𝟐𝟕 𝑞 =𝟏𝟕

4. RATIOS, RATES AND PROPORTIONS
Example 2: Using Ratios
The ratio of the number of bones in a human’s ears
to the number of bones in the skull is 3:11.
If there are 22 bones in the skull.
How many bones are in the ears?
Step 1: Write a ratio comparing bones in ears to bones in skull.
𝑏𝑜𝑛𝑒𝑠 𝑖𝑛 𝑒𝑎𝑟𝑠 𝑡𝑜 𝑏𝑜𝑛𝑒𝑠 𝑖𝑛 𝑠𝑘𝑢𝑙𝑙 = 3: 11 𝑜𝑟
Step 2: Write a proportion. Let 𝑥 be the number of bones in the ears.
3 = 𝑥
11 22

5. RATIOS, RATES AND PROPORTIONS
Step 3: Cross multiply.
11 × 𝑥 = 3 × 22
11𝑥 = 66
𝑥 =
66
11
= 𝟔
There are 6 bones in the ears.

6. RATIOS, RATES AND PROPORTIONS
Example 3: Using Ratios
The ratio of matches lost to matches won for a cricket
team is 1:4. If the team has won 16 matches, how many
games did they lose?
Step 1: Write the ratio of matches lost to matches won.
1: 4
Step 2: Write a proportion. Let 𝑥 be the number of matches lost.
1 𝑥
=
4 16
Step 3: Cross multiply
4 × 𝑥 = 1 × 16
4𝑥 = 16
𝑥 = = 4 The team lost 4 matches.

7. RATIOS, RATES AND PROPORTIONS
A rate is a ratio with two quantities that have
different units, such as
60 𝑘𝑖𝑙𝑜𝑚𝑒𝑡𝑟𝑒𝑠
2 ℎ𝑜𝑢𝑟𝑠
.
Rates are usually written as unit rates.
In a unit rate, the second quantity is a unit.
For example, 1 hour, 1 day, 1 week.
Any rate can be converted to a unit rate.

8. RATIOS, RATES AND PROPORTIONS
Example 4: Finding Unit Rates
Raulf Laue of Germany flipped a pancake 416 times
in 120 seconds to set the world record. Find the
unit rate. Express your answer correct to the
nearest hundredth.
Step 1: Write a proportion to find an equivalent ratio with a second
quantity of 1.
416 𝑥
=
120 1
Step 2: Divide on the left side to find 𝑥.
𝑥 = = 3.47 The unit rate is 3.47 flips per second.

9. RATIOS, RATES AND PROPORTIONS
Example 5: Finding Unit Rates
Genevieve is paid $700 for 8 hours of work. Find
the unit rate.
Step 1: Write a proportion to find an equivalent ratio with a second
quantity of 1.
700 𝒙
=
8 1
Step 2: Divide on the left side to find 𝑥.
𝑥 = = $𝟖𝟕. 𝟓𝟎
The unit rate is $87.50.

10. RATIOS, RATES AND PROPORTIONS
A scale is a ratio between two sets of
measurements, such as 1 cm:50 km. A scale
drawing or scale model uses a scale to
represent an object as smaller or larger than
the actual object. A map is an example of a
scale drawing.

11. RATIOS, RATES AND PROPORTIONS
Example 6: Scale
A scale model of a human heart is 1.6m long. The scale
is 32:1. How many centimetres long is the actual heart
it represents?
Step 1: Convert 1.6m to cm.
1.6𝑚 = 1.6 × 100 𝑐𝑚 = 160 𝑐𝑚
Step 2: Write a proportion. Let 𝑥 be the actual length.
32 = 160
1 𝒙
Step 3: Use the cross products to solve.
32 × 𝑥 = 160 × 1
𝑥 = = 𝟓 The actual length of the heart is 5 cm.

12. RATIOS, RATES AND PROPORTIONS
Example 7: Scales and Scale Drawings
a) A contractor has a blueprint for a house drawn to
the scale 1 cm: 3 m.
A wall on the blueprint is 6.5 cm long. How long is the
actual wall?
Step 1: Convert 3m to cm.
3 𝑚 = 300 𝑐𝑚
Step 2: Write a proportion. Let 𝑥 be the actual length.
.5
𝑥
Step 3: Use the cross products to solve.
1 × 𝒙 = 6.5 × 300
𝒙 = 6.5 × 300 = 1950 The actual wall is 1950 cm long.
= 6

13. RATIOS, RATES AND PROPORTIONS
b) For the same building, the contractor has already
completed one wall that was 1200 cm long.
What was the length of this wall on the blueprint?
Step 1: Write a proportion. Use 𝑥 as the length on the
blueprint.
1 𝑥
=
300 1200
Step 2: Use the cross products to solve.
𝑥 × 300 = 1 × 1200
𝑥 = = 4 The length of the wall on the blueprint is 4cm.

14. RATIOS, RATES AND PROPORTIONS
Exercise:
1. In a school, the ratio of boys to girls is 3:2.
There are 312 boys. How many girls are there?
Answer: 3 𝑝𝑎𝑟𝑡𝑠 = 312
1𝑝𝑎𝑟𝑡 = = 104
2𝑝𝑎𝑟𝑡𝑠 = 104 × 2 = 208
There are 208 girls.

15. RATIOS, RATES AND PROPORTIONS
Find each unit rate. Round to 2 decimal
places where necessary.
Nuts cost $10.75 for 3g. 10.75 𝑥
Answer:
10.75
3
=
𝑥
1
𝑥 × 3 = 10.75
𝑥 =
10.75
3
= 3.58
Therefore, nuts cost $3.58/g.

16. RATIOS, RATES AND PROPORTIONS
2. Irene washes 30 cars in 5 hours.
Answer:
30
5
=
𝑥
1
𝑥 × 5 = 30
𝑥 =
30
5
= 6
Therefore, Irene’s car wash rate is 6 cars/hr.

17. RATIOS, RATES AND PROPORTIONS
3. A car travels 180 kilometres in 4 hours. What
is the car’s speed in metres per second?
Answer:
Convert kilometres to metres and hours to seconds.
180𝑘𝑚 = 180 × 1000 = 180000 𝑚
4ℎ𝑟 = 4 × 3600 = 14400 𝑠
Write a proportion to find the unit rate.
180000
14400
=
𝑥
1
𝑥 =
180000
14400
= 12.5
The speed is 12.5 m/s.

18. RATIOS, RATES AND PROPORTIONS
5) A scale model of a car is 9 centimetres long.
The scale is 1:24. How many centimetres long is
the car it represents?
Answer: Write a proportion. Use 𝑥 as the actual length.
9
𝑥
=
1
24
Use the cross products to solve.
𝑥 × 1 = 9 × 24
𝑥 = 9 × 24 = 216
The actual length of the car is 216 cm.