1. RATIO
A ratio is a way of comparing two or more quantities of the same kind.
For example, if there are 10 boys and 15 girls in a class, the ratio of boys to girls is:
10:15=2:3Important points:
- Always express ratios in their simplest form.
- Quantities being compared must be in the same units.
- The order of a ratio is important.
Question 1: Divide GH¢600 in the ratio 2:3.
Solution:
Step 1: Add the ratio parts.
Step 2: Find the value of one part.
600\div5=120Step 3: Calculate each share.
First share = 2\times120=\text{GH¢240}
Second share = 3\times120=\text{GH¢360}
Answer: GH¢240 and GH¢360.
Combining ratios
If A:B = 2:3 and B:C = 4:5, make the middle terms equal.
--A:B=8:12
--B:C=12:15
Therefore:
A:B:C=8:12:152. RATE
A rate compares two quantities measured in different units.
Examples include:
- Speed in kilometres per hour.
- Wages in cedis per day.
- Water flow in litres per minute.
- Fuel consumption in kilometres per litre.
Important formulas
-\text{Speed}=\frac{\text{Distance}}{\text{Time}}
-\text{Rate}=\frac{\text{Quantity}}{\text{Time}}
-\text{Unit rate}=\frac{\text{Total quantity}}{\text{Number of units}}
Question 2: A car travels 240 km in 4 hours. Calculate its average speed.
Solution:
-\text{Speed}=\frac{240}{4}
-=60\text{ km/h}
Question 3: A worker earns GH¢360 in 6 days. Find the amount earned in 10 days.
Solution:
Daily wage: 360\div6=\text{GH¢60}
Earnings for 10 days: 60\times10=\text{GH¢600}
3. PROPORTION
Proportion is an equality between two ratios. It is also used to describe how one quantity changes in relation to another.
There are three important types.
A. Direct proportion
Two quantities are directly proportional when an increase in one causes an increase in the other at a constant rate.
Similarly, a decrease in one causes a decrease in the other.
Examples:
- More exercise books bought means more money spent.
- More hours worked means more wages earned at a fixed hourly rate.
Formula:
--y\propto x
--y=kx
Where:
- y and x are the variables.
- kis the constant of proportionality.
Question 4: If y = 15 when x = 3, find y when x = 8.
Solution:
Since y=kx:
-15=3k
-k=5
When x=8:
y=5\times8=\boxed{40}
B. Inverse proportion
Two quantities are inversely proportional when an increase in one causes a decrease in the other.
Examples:
- More workers require fewer days to complete a fixed job.
- Higher speed means less time to cover a fixed distance.
Formula:
--y\propto\frac{1}{x}
--y=\frac{k}{x}
Or:
-xy=k
Question 5: Six workers complete a job in 12 days. How long will 8 workers take?
Solution:
-6\times12=8\times d
-72=8d
-d=\frac{72}{8}=\boxed{9\text{ days}}
C. Compound proportion
Compound proportion involves three or more quantities.
A quantity may depend directly on one variable and inversely on another.
Question 6: Four machines produce 200 items in 5 hours. How many items will 6 machines produce in 8 hours?
Solution:
Production increases with both machines and time.
-\text{Required output} =200\times\frac{6}{4}\times\frac{8}{5}
-=480\text{ items}
4. SQUARE AND CUBE PROPORTION
These are also important in WASSCE questions.
Direct square proportion
--y\propto x^2
--y=kx^2
If x doubles, y becomes four times its original value.
Inverse square proportion
-y\propto\frac{1}{x^2}
If x doubles, y becomes one-quarter of its original value.
Direct cube proportion
-y\propto x^3
If x doubles, y becomes eight times its original value.
5. MAP SCALES
A scale is a ratio comparing a distance on a map or drawing with the actual distance.
For example, a scale of 1:50,000 means:
1 cm on the map represents 50,000 cm on the ground.
Question 7: Two towns are 6 cm apart on a map with a scale of 1:100,000. Find the actual distance in kilometres.
Solution:
-6\times100,000=600,000\text{ cm}
Convert to kilometres:
-600,000\div100,000=6\text{ km}
Answer: 6 km.
6. IMPORTANT WASSCE EXAMINATION TIPS
| QUESTION CLUE | TYPE OF PROPORTION |
| More items, more cost | Direct |
| More workers, fewer days | Inverse |
| More speed, less time | Inverse |
| More hours, more wages | Direct |
| More machines, more production | Direct |
| Square of a quantity | Square proportion |
| Cube of a quantity | Cube proportion |
Essential formulas to memorise
1. Ratio sharing:
-\text{Share}=\frac{\text{Required parts}}{\text{Total parts}}\times\text{Total amount}
2. Rate:
-\text{Rate}=\frac{\text{Quantity}}{\text{Time}}
3. Direct proportion:
-y=kx
4. Inverse proportion:
-xy=k
5. Direct square proportion:
-y=kx^2
6. Inverse square proportion:
-y=\frac{k}{x^2}
Final examination advice: Before solving any proportion question, identify whether the relationship is direct or inverse. For compound proportion, determine separately how each variable affects the required quantity. This makes even lengthy WASSCE questions much easier to solve.