DIRECT AND INVERSE PROPORTION
What is Proportion?
A proportion is a way of comparing two quantities. It tells us how one thing changes when another thing changes.
Example: “If 2 apples cost ₦4, then 4 apples cost ₦8.” The cost and the number of apples are related — there is a proportion between them.
Direct Proportion (Direct Variation)
Meaning
Two quantities are in direct proportion if when one increases, the other increases too in a fixed way; and when one decreases, the other decreases in the same fixed way.
Put simply: More of one → more of the other. Less of one → less of the other.
How to write it
y ∝ x (y varies directly as x)
y = kx (k is a constant)
Example (Direct):
If 5 pens cost ₦10, how much do 8 pens cost?
- Cost per pen = ₦10 ÷ 5 = ₦2
- Cost for 8 pens = 8 × ₦2 = ₦16
Answer: ₦16 — as pens increase, cost increases (direct proportion).
Inverse Proportion (Inverse Variation)
Meaning
Two quantities are in inverse proportion if when one increases, the other decreases so that their product stays the same.
Put simply: More of one → less of the other. Less of one → more of the other.
How to write it
y ∝ 1/x (y varies inversely as x)
y = k / x (k is a constant)
Example (Inverse):
If 5 workers can finish a job in 10 days, how many days will 10 workers take?
- Product (workers × days) = 5 × 10 = 50 (this is k)
- For 10 workers: days = 50 ÷ 10 = 5 days
Answer: 5 days — more workers means fewer days (inverse proportion).
How to Decide Which Kind (Direct or Inverse)
| If you see … | It’s |
|---|
| “As one goes up, the other goes up” | Direct proportion |
| “If you double x, y doubles” | Direct proportion |
| “As one goes up, the other goes down” | Inverse proportion |
| “If you double x, y halves” | Inverse proportion |
Formulas
Direct proportion: y = kx (k is constant)
Inverse proportion: y = k / x (k is constant)
Here x and y are the two quantities and k is the constant of proportionality (it does not change).
Examples
Example 1 (Direct)
If 3 books cost ₦150, how much do 7 books cost?
- Cost per book = ₦150 ÷ 3 = ₦50
- Cost for 7 books = 7 × ₦50 = ₦350
Answer: ₦350
Example 2 (Inverse)
If 4 people can paint a wall in 6 hours, how many hours will 8 people take?
- Product (people × hours) = 4 × 6 = 24
- For 8 people: hours = 24 ÷ 8 = 3 hours
Answer: 3 hours
Example 3 (Inverse)
Sam can do a job in 5 days working alone. If his friend helps (2 people), how many days will they take together?
- Product (people × days) = 1 × 5 = 5
- For 2 people: days = 5 ÷ 2 = 2.5 days
Answer: 2.5 days
APPLICATION OF DIRECT AND INVERSE PROPORTIONS
Meaning of Application
Application means how we use something in real life.
So, the application of direct and inverse proportions means how we use these ideas to solve real-life problems in areas such as shopping, business, work, construction, speed, and farming.
Application of Direct Proportion
Two quantities are said to be in direct proportion when one quantity increases and the other quantity also increases in the same way.
We use direct proportion when “more means more” or “less means less.”
Examples of places we use Direct Proportion:
- Shopping: The more items you buy, the more you pay.
- Cooking: The more people you want to serve, the more ingredients you need.
- Farming: The larger the farm, the more seeds required.
- Travelling: The longer the time, the farther the distance covered.
- Work and Pay: The longer you work, the more money you earn.
Examples on Direct Proportion
Example 1:
If 3 oranges cost ₦150, how much will 8 oranges cost?
- More oranges → more money → Direct proportion.
- Cost per orange = ₦150 ÷ 3 = ₦50.
- Cost of 8 oranges = 8 × ₦50 = ₦400.
Answer: ₦400
Example 2:
If 5 books cost ₦2,500, how much will 9 books cost?
- More books → more money → Direct proportion.
- Cost per book = ₦2,500 ÷ 5 = ₦500 per book.
- Cost of 9 books = ₦500 × 9 = ₦4,500.
Answer: ₦4,500
Example 3:
A car travels 120 km in 3 hours at a constant speed. How far will it travel in 8 hours?
- More time → more distance → Direct proportion.
- Speed = distance ÷ time = 120 ÷ 3 = 40 km/h.
- In 8 hours → distance = 40 × 8 = 320 km.
Answer: 320 km
Example 4:
If 4 bags of rice cost ₦12,000, how much will 10 bags cost?
- More bags → more cost → Direct proportion.
- Cost per bag = ₦12,000 ÷ 4 = ₦3,000 per bag.
- Cost for 10 bags = ₦3,000 × 10 = ₦30,000.
Answer: ₦30,000
Example 5:
A machine prints 3,000 copies in 5 hours. How many copies will it print in 12 hours at the same rate?
- More time → more copies → Direct proportion.
- Copies per hour = 3,000 ÷ 5 = 600 copies/hour.
- In 12 hours → 600 × 12 = 7,200 copies.
Answer: 7,200 copies
Application of Inverse Proportion
Two quantities are said to be in inverse proportion when one quantity increases and the other decreases in the same way.
We use inverse proportion when “more means less” or “less means more.”
Examples of places we use Inverse Proportion:
- Work and Time: The more workers there are, the less time it takes to finish the job.
- Speed and Time: The faster the speed, the less time the journey takes.
- Machines and Time: The more machines working, the less time needed to finish the work.
- Taps and Time: The more taps opened, the faster the tank fills up.
- Sharing Work: The more people share the work, the smaller each person’s share becomes.
Examples on Inverse Proportion
Example 1:
6 men can build a house in 15 days. How long will 10 men take to build the same house?
- More men → less time → Inverse proportion.
- Men × days = constant = 6 × 15 = 90.
- For 10 men → days = 90 ÷ 10 = 9 days.
Answer: 9 days
Example 2:
A car takes 9 hours to cover a certain distance at a speed of 60 km/h. How long will it take at 90 km/h?
- More speed → less time → Inverse proportion.
- Speed × time = constant = 60 × 9 = 540.
- For 90 km/h → time = 540 ÷ 90 = 6 hours.
Answer: 6 hours
Example 3:
8 taps can fill a tank in 12 hours. How long will 6 taps take to fill it?
- Fewer taps → more time → Inverse proportion.
- Taps × time = constant = 8 × 12 = 96.
- For 6 taps → time = 96 ÷ 6 = 16 hours.
Answer: 16 hours
Example 4:
4 pumps can empty a pond in 18 hours. How long will 6 pumps take?
- More pumps → less time → Inverse proportion.
- Pumps × time = constant = 4 × 18 = 72.
- For 6 pumps → time = 72 ÷ 6 = 12 hours.
Answer: 12 hours
Example 5:
12 men can complete a job in 15 days. After 5 days, 3 more men join them. How many more days will they need to finish the work?
- Same work → Inverse proportion.
- Total work = men × days = 12 × 15 = 180 man-days.
- Work done in 5 days = 12 × 5 = 60 man-days.
- Remaining work = 180 − 60 = 120 man-days.
- Now there are 12 + 3 = 15 men.
- Time to finish = 120 ÷ 15 = 8 days.
Answer: 8 more days
Summary
- Direct proportion: increase A → increase B; decrease A → decrease B.
- Inverse proportion: increase A → decrease B; decrease A → increase B.
- There is always a constant number (k) that links them.
- For direct: use y = kx. For inverse: use y = k / x.
