Inverse Proportion
Inverse proportion describes a relationship where one quantity increases as the other decreases, such that their product remains constant.
If more workers are put on a job, the time to finish decreases. If a vehicle moves faster, it takes less time to cover the same distance. These are examples of inverse proportion.
In Class 8 Mathematics (NCERT), inverse proportion is studied alongside direct proportion in the chapter Direct and Inverse Proportions. Understanding both types helps in solving a wide range of real-life problems.
Two quantities x and y are in inverse proportion if their product x × y always remains the same constant value.
What is Inverse Proportion?
Definition: Two quantities x and y are said to be in inverse proportion if:
- An increase in x causes a proportional decrease in y.
- A decrease in x causes a proportional increase in y.
- The product x × y always remains constant.
This constant product is denoted by k:
- x × y = k (constant) for all pairs of values.
- Equivalently, y = k/x.
- We write x ∝ 1/y (read as "x is inversely proportional to y").
Key conditions for inverse proportion:
- The two quantities change in opposite directions — one increases, the other decreases.
- The product of corresponding values is always the same.
- If x₁ × y₁ = x₂ × y₂, then x and y are in inverse proportion.
Inverse Proportion Formula
Formula:
x₁ × y₁ = x₂ × y₂
Where:
- x₁ and y₁ are the first pair of corresponding values.
- x₂ and y₂ are the second pair of corresponding values.
Alternate forms:
- y = k/x, where k = x × y is the constant.
- x₁/x₂ = y₂/y₁ (ratio form — note the reversal of y values).
To find a missing value:
y₂ = (x₁ × y₁) / x₂
Derivation and Proof
Understanding inverse proportion step by step:
- Suppose 6 workers can complete a task in 12 days.
- The total work = 6 × 12 = 72 worker-days. This is the constant k.
- If 9 workers do the same task: 9 × y₂ = 72, so y₂ = 72/9 = 8 days.
- If 4 workers do the same task: 4 × y₂ = 72, so y₂ = 72/4 = 18 days.
- More workers → fewer days. Fewer workers → more days. Product is always 72.
Why the product stays constant:
- The total amount of work (or any underlying quantity) does not change.
- If you double the number of workers (from 6 to 12), the time halves (from 12 to 6 days).
- If you halve the workers (from 6 to 3), the time doubles (from 12 to 24 days).
- The product workers × days = total work = constant.
Graphical understanding:
- The graph of inverse proportion is a curved line (rectangular hyperbola), NOT a straight line.
- The curve approaches both axes but never touches them.
- This curve is different from the straight-line graph of direct proportion.
Types and Properties
Methods to solve inverse proportion problems:
1. Product Method (using x₁ × y₁ = x₂ × y₂):
- Find the product from the given pair.
- Use the same product to find the unknown value.
- Example: 6 workers take 12 days. How many days for 9 workers? 6 × 12 = 9 × y₂. y₂ = 72/9 = 8 days.
2. Ratio Method (x₁/x₂ = y₂/y₁):
- Set up the ratio with y values reversed.
- Cross multiply to solve.
- Example: 6/9 = y₂/12. y₂ = (6 × 12)/9 = 8 days.
3. Unitary Method (adjusted for inverse):
- Find what happens for 1 unit, then adjust.
- Example: 6 workers → 12 days. 1 worker → 12 × 6 = 72 days. 9 workers → 72/9 = 8 days.
- Note: For inverse proportion, 1 unit gives a larger value (multiply), not smaller.
Identifying inverse proportion in word problems:
- More workers → less time to finish the same job.
- More speed → less time for the same distance.
- More pipes filling a tank → less time to fill.
- Wider road → fewer lanes needed for the same traffic.
Solved Examples
Example 1: Example 1: Workers and days
Problem: 8 workers can complete a task in 15 days. How many days will 12 workers take?
Solution:
Given:
- Workers₁ = 8, Days₁ = 15
- Workers₂ = 12, Days₂ = ?
Workers and days are in inverse proportion (more workers = fewer days):
- x₁ × y₁ = x₂ × y₂
- 8 × 15 = 12 × Days₂
- 120 = 12 × Days₂
- Days₂ = 120/12 = 10 days
Answer: 12 workers will complete the task in 10 days.
Example 2: Example 2: Speed and time
Problem: A car travelling at 60 km/h takes 5 hours to reach a destination. How long will it take at 75 km/h?
Solution:
Given:
- Speed₁ = 60 km/h, Time₁ = 5 hours
- Speed₂ = 75 km/h, Time₂ = ?
Speed and time are in inverse proportion (more speed = less time):
- 60 × 5 = 75 × Time₂
- 300 = 75 × Time₂
- Time₂ = 300/75 = 4 hours
Answer: At 75 km/h, the car will take 4 hours.
Example 3: Example 3: Pipes filling a tank
Problem: 6 pipes can fill a tank in 10 hours. How many hours will 15 pipes take?
Solution:
Given:
- Pipes₁ = 6, Time₁ = 10 hours
- Pipes₂ = 15, Time₂ = ?
More pipes = less time (inverse proportion):
- 6 × 10 = 15 × Time₂
- 60 = 15 × Time₂
- Time₂ = 60/15 = 4 hours
Answer: 15 pipes will fill the tank in 4 hours.
Example 4: Example 4: Food supply for soldiers
Problem: A camp has enough food for 120 soldiers for 30 days. If 40 more soldiers join, how long will the food last?
Solution:
Given:
- Soldiers₁ = 120, Days₁ = 30
- Soldiers₂ = 120 + 40 = 160, Days₂ = ?
More soldiers = food lasts fewer days (inverse proportion):
- 120 × 30 = 160 × Days₂
- 3600 = 160 × Days₂
- Days₂ = 3600/160 = 22.5 days
Answer: The food will last 22.5 days (or 22 days and 12 hours).
Example 5: Example 5: Checking inverse proportion
Problem: Check if the following values are in inverse proportion:
- x: 3, 6, 9, 12
- y: 24, 12, 8, 6
Solution:
Find x × y for each pair:
- 3 × 24 = 72
- 6 × 12 = 72
- 9 × 8 = 72
- 12 × 6 = 72
Since all products are equal (72), x and y are in inverse proportion with k = 72.
Answer: Yes, x and y are in inverse proportion.
Example 6: Example 6: Number of days and hours per day
Problem: Raman can finish a project working 6 hours a day in 20 days. If he works 8 hours a day, how many days will he take?
Solution:
Given:
- Hours/day₁ = 6, Days₁ = 20
- Hours/day₂ = 8, Days₂ = ?
Total work = 6 × 20 = 120 hours. This is constant.
- 8 × Days₂ = 120
- Days₂ = 120/8 = 15 days
Answer: Working 8 hours a day, Raman will finish in 15 days.
Example 7: Example 7: Distributing sweets
Problem: 420 sweets are to be distributed equally among children. If there are 20 children, each gets some sweets. How many sweets will each child get if there are 35 children?
Solution:
Given:
- Children₁ = 20, Sweets each₁ = 420/20 = 21
- Children₂ = 35, Sweets each₂ = ?
More children = fewer sweets each (inverse proportion):
- 20 × 21 = 35 × Sweets each₂
- 420 = 35 × Sweets each₂
- Sweets each₂ = 420/35 = 12
Answer: Each child will get 12 sweets.
Example 8: Example 8: Typing speed
Problem: A typist typing at 40 words per minute takes 30 minutes to type a document. How long will a typist typing at 50 words per minute take?
Solution:
Given:
- Speed₁ = 40 wpm, Time₁ = 30 minutes
- Speed₂ = 50 wpm, Time₂ = ?
Faster typing = less time (inverse proportion):
- 40 × 30 = 50 × Time₂
- 1200 = 50 × Time₂
- Time₂ = 1200/50 = 24 minutes
Answer: The faster typist will take 24 minutes.
Example 9: Example 9: Finding the constant
Problem: If x and y are in inverse proportion and x = 14 when y = 6, find (a) the constant k, and (b) the value of y when x = 21.
Solution:
(a) Finding k:
- k = x × y = 14 × 6 = 84
(b) Finding y when x = 21:
- x × y = k
- 21 × y = 84
- y = 84/21 = 4
Answer: k = 84, and y = 4 when x = 21.
Example 10: Example 10: Gear teeth and rotations
Problem: A gear with 20 teeth makes 15 rotations per minute. How many rotations per minute will a meshed gear with 30 teeth make?
Solution:
Given:
- Teeth₁ = 20, Rotations₁ = 15
- Teeth₂ = 30, Rotations₂ = ?
More teeth = fewer rotations (inverse proportion):
- 20 × 15 = 30 × Rotations₂
- 300 = 30 × Rotations₂
- Rotations₂ = 300/30 = 10
Answer: The gear with 30 teeth will make 10 rotations per minute.
Real-World Applications
Real-world applications of inverse proportion:
- Work and labour: More workers on a construction project means fewer days to complete it (assuming equal efficiency).
- Speed and travel time: A faster vehicle covers the same distance in less time.
- Pipes and tanks: More pipes filling a tank together reduces the time to fill it.
- Food and people: The same stock of food lasts fewer days if more people share it.
- Gears and machinery: A gear with more teeth rotates fewer times per minute when meshed with a smaller gear.
- Farming: More tractors ploughing a field reduces the total time needed.
- Money and cost: If the price per item increases, fewer items can be bought with the same budget.
- Road construction: More machines working on a road segment reduces completion time.
Key Points to Remember
- Two quantities are in inverse proportion if one increases while the other decreases at a constant product.
- The formula is: x₁ × y₁ = x₂ × y₂.
- The product x × y = k (constant) for all pairs.
- Equivalently, y = k/x or x ∝ 1/y.
- If x doubles, y halves. If x triples, y becomes one-third.
- The graph of inverse proportion is a curve (rectangular hyperbola), not a straight line.
- To check inverse proportion, verify that x × y gives the same value for all pairs.
- In the unitary method for inverse proportion, the value for 1 unit is larger (multiply, not divide).
- Do NOT confuse with direct proportion, where both quantities change in the same direction.
- The ratio form is x₁/x₂ = y₂/y₁ (note: y values are reversed).
Practice Problems
- 10 workers can dig a trench in 6 days. How many days will 15 workers take?
- A train travelling at 80 km/h takes 3 hours for a journey. How long will a train at 60 km/h take?
- A camp has food for 200 soldiers for 45 days. If 50 soldiers leave, how long will the food last?
- Check if x and y are in inverse proportion: x = 4, 8, 12, 16 and y = 36, 18, 12, 9.
- If y is inversely proportional to x, and y = 10 when x = 8, find y when x = 20.
- 12 taps can fill a swimming pool in 8 hours. How many taps are needed to fill it in 6 hours?
- A car at 50 km/h takes 6 hours to reach a city. At what speed should it travel to reach in 5 hours?
- 30 cows can graze a field in 16 days. In how many days will 40 cows graze the same field?
Frequently Asked Questions
Q1. What is inverse proportion?
Inverse proportion is a relationship between two quantities where one increases as the other decreases, such that their product remains constant. For example, more speed means less time for the same distance.
Q2. What is the formula for inverse proportion?
The formula is x₁ × y₁ = x₂ × y₂. Alternatively, y = k/x, where k is the constant product.
Q3. How is inverse proportion different from direct proportion?
In direct proportion, both quantities change in the same direction (x/y = constant). In inverse proportion, they change in opposite directions (x × y = constant).
Q4. What does the graph of inverse proportion look like?
The graph is a curved line (rectangular hyperbola) that approaches both axes but never touches them. It is NOT a straight line.
Q5. How do you check if values are in inverse proportion?
Multiply each x value by its corresponding y value. If all the products are equal, the values are in inverse proportion.
Q6. Can speed and time be inversely proportional?
Yes. For a fixed distance, speed and time are inversely proportional. If you double the speed, the time is halved. The product speed × time = distance (constant).
Q7. What is the constant of proportionality in inverse proportion?
The constant k = x × y. It represents the total work, total distance, or total quantity that stays unchanged while the two related quantities vary inversely.
Q8. How do you use the unitary method for inverse proportion?
For 1 unit, the value becomes larger (multiply by the original quantity). For example: 6 workers take 12 days. 1 worker takes 12 × 6 = 72 days. 9 workers take 72/9 = 8 days.
Q9. Give a real-life example of inverse proportion.
If 4 people take 6 hours to paint a room, 8 people would take 3 hours. More painters means less time. The product 4 × 6 = 8 × 3 = 24 (constant).
Q10. Are all real-life situations either direct or inverse proportion?
No. Many real-life situations involve combinations of direct and inverse relationships, or are not proportional at all. Direct and inverse proportion are simplified mathematical models that work well for many (but not all) cases.
