Subject: Mathematics | Level: JHS (Basic 7 – 9) | Topic: Ratio, Proportion, and Rates | Difficulty: Intermediate
Introduction
Ratio, proportion, and rates appear all the time in daily Ghanaian life. When a market woman shares her profit with a partner, when a carpenter mixes cement and sand, when a driver calculates fuel consumption for a trip, they are all using ratio and proportion. These concepts are also among the most frequently examined topics in the BECE Mathematics paper.
1. Ratio
A ratio compares two or more quantities of the same kind. It tells us how much of one thing there is compared to another.
Writing Ratios
Ratios can be written using the colon (:) or as a fraction.
Example: A class has 18 boys and 12 girls. The ratio of boys to girls = 18:12 = 3:2 (simplified by dividing both by 6).
Simplifying Ratios
Divide all parts of the ratio by their HCF.
Example: Simplify 24:36. HCF = 12. → 24÷12 : 36÷12 = 2:3
Example: Simplify 15:25:35. HCF = 5. → 3:5:7
Equivalent Ratios
Multiply or divide both parts of a ratio by the same number to get an equivalent ratio.
Example: 3:4 = 6:8 = 9:12 = 15:20
2. Dividing Quantities in a Given Ratio
This is one of the most common ratio questions in BECE. The method is always the same:
- Find the total number of parts
- Find the value of one part
- Multiply by the number of parts for each share
Example 1: Divide GHS 360 in the ratio 3:5.
Total parts = 3 + 5 = 8
One part = 360 ÷ 8 = GHS 45
First share = 3 × 45 = GHS 135
Second share = 5 × 45 = GHS 225
Check: 135 + 225 = 360 ✓
Example 2: Three friends Ama, Kojo, and Yaw share a profit of GHS 1,200 in the ratio 2:3:7. How much does each person receive?
Total parts = 2 + 3 + 7 = 12
One part = 1,200 ÷ 12 = GHS 100
Ama = 2 × 100 = GHS 200
Kojo = 3 × 100 = GHS 300
Yaw = 7 × 100 = GHS 700
Check: 200 + 300 + 700 = 1,200 ✓
Example 3 (Finding the original total): Kofi and Ama share some money in the ratio 3:5. Kofi receives GHS 270. What is the total amount shared?
Kofi’s share = 3 parts = GHS 270, so 1 part = 270 ÷ 3 = GHS 90
Total = 8 parts = 8 × 90 = GHS 720
3. Proportion
Proportion is about how two ratios relate to each other. There are two types:
Direct Proportion
Two quantities are in direct proportion when as one increases, the other increases at the same rate. If you double one, you double the other.
Example: 5 kg of tomatoes costs GHS 30. How much do 8 kg cost?
Method (Unitary Method):
1 kg costs 30 ÷ 5 = GHS 6
8 kg costs 8 × 6 = GHS 48
Inverse Proportion
Two quantities are in inverse proportion when as one increases, the other decreases at the same rate. If you double one, you halve the other.
Example: 4 workers take 15 days to build a wall. How long will 10 workers take?
More workers → fewer days (inverse proportion)
4 workers × 15 days = 60 worker-days (constant)
10 workers → 60 ÷ 10 = 6 days
4. Rates
A rate compares two quantities of different kinds. The most common rates in everyday life involve speed, unit cost, and pay per hour.
Speed, Distance, and Time
The relationship between speed, distance, and time is one of the most tested topics in BECE Mathematics:
Speed = Distance ÷ Time
Distance = Speed × Time
Time = Distance ÷ Speed
Example 1: A bus travels 240 km in 3 hours. Find its average speed.
Speed = 240 ÷ 3 = 80 km/h
Example 2: A car travels at 90 km/h for 2.5 hours. Find the distance covered.
Distance = 90 × 2.5 = 225 km
Example 3: A lorry travels 350 km at an average speed of 70 km/h. How long does the journey take?
Time = 350 ÷ 70 = 5 hours
Unit Price
Unit price = Total cost ÷ Quantity
Example: 6 notebooks cost GHS 18. What is the cost of 1 notebook?
Unit price = 18 ÷ 6 = GHS 3
Exchange Rates
Example: If GHS 1 = USD 0.07, how many US dollars will GHS 500 give?
USD = 500 × 0.07 = USD 35
Summary
- A ratio compares two quantities of the same kind. Always simplify by dividing by the HCF.
- To divide a quantity in a ratio: find total parts → find one part → multiply by each share.
- Direct proportion: both quantities increase or decrease together.
- Inverse proportion: one increases while the other decreases.
- Speed = Distance ÷ Time; Distance = Speed × Time; Time = Distance ÷ Speed.
Practice Questions
- Simplify the ratio 48:60.
- A school buys pens and pencils in the ratio 5:3. If they buy 120 pens, how many pencils do they buy?
- Divide GHS 2,100 among three people in the ratio 2:3:2.
- If 3 metres of fabric costs GHS 54, how much will 7 metres cost?
- A car travelling at 80 km/h takes 3.5 hours to complete a journey. How far is the journey?
Answers: (1) 4:5 (2) Ratio 5:3; 5 parts = 120 pens → 1 part = 24 → pencils = 3 × 24 = 72 pencils (3) Total = 7 parts; 1 part = 2,100÷7 = 300; Shares = GHS 600, GHS 900, GHS 600 (4) 1m = 54÷3 = 18; 7m = 7×18 = GHS 126 (5) Distance = 80 × 3.5 = 280 km
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