Direct and Inverse Variation
1. Direct Variation
Two quantities x and y are in DIRECT variation if they INCREASE or DECREASE together in such a way that their RATIO remains constant.
y ∝ x means y = kx, where k is the CONSTANT of variation.
Condition: y₁/x₁ = y₂/x₂ = y₃/x₃ = ... = k
'If x increases, y increases PROPORTIONALLY. The graph is a STRAIGHT LINE through the origin.'
Examples of Direct Variation
- Cost of articles ∝ Number of articles (more articles → more cost)
- Distance travelled ∝ Time (at constant speed)
- Work done ∝ Number of workers (each does equal work)
- Weight ∝ Mass (on Earth)
2. Solving Direct Variation Problems
Method: Set up the proportion: x₁/y₁ = x₂/y₂ or x₁ : y₁ = x₂ : y₂
Worked Example: 5 kg of rice costs Rs 225. Find the cost of 12 kg of rice.
Let cost of 12 kg = Rs y. 5/225 = 12/y 5y = 225 × 12 5y = 2700 y = Rs 540
'When solving direct variation, KEEP THE ORDER CONSISTENT — same quantity in the numerator on both sides.'
Worked Example: A car travels 180 km in 3 hours at constant speed. How far will it travel in 5 hours?
180/3 = y/5 3y = 180 × 5 3y = 900 y = 300 km
3. Inverse Variation
Two quantities x and y are in INVERSE variation if when one INCREASES, the other DECREASES such that their PRODUCT remains constant.
y ∝ 1/x means xy = k, where k is the CONSTANT of variation.
Condition: x₁y₁ = x₂y₂ = x₃y₃ = ... = k
'If x increases, y DECREASES proportionally. The graph is a RECTANGULAR HYPERBOLA.'
Examples of Inverse Variation
- Number of workers ∝ 1/Time (more workers → less time to finish a job)
- Speed ∝ 1/Time (faster speed → less time for same distance)
- Number of pipes ∝ 1/Time (more pipes → less time to fill a tank)
4. Solving Inverse Variation Problems
Method: Set up the equation: x₁y₁ = x₂y₂
Worked Example: 8 workers can build a wall in 15 days. How many days will 12 workers take?
Let number of days = y. 8 × 15 = 12 × y 120 = 12y y = 10 days
Worked Example: A car travels a certain distance at 60 km/h in 4 hours. How much time will it take at 80 km/h?
60 × 4 = 80 × t 240 = 80t t = 3 hours
5. Mixed Problems — Identifying Direct vs Inverse
| Situation | Type | Reason |
|---|---|---|
| More books → more cost | DIRECT | Ratio constant |
| More workers → less time | INVERSE | Product constant |
| More speed → more distance (same time) | DIRECT | Ratio constant |
| More speed → less time (same distance) | INVERSE | Product constant |
| More men → more food needed | DIRECT | Ratio constant |
| More men → fewer days (same food stock) | INVERSE | Product constant |
Worked Example: 15 men can dig a pond in 12 days. How many men are needed to dig it in 9 days?
This is INVERSE variation. 15 × 12 = m × 9 180 = 9m m = 20 men
6. Time and Work Problems
Worked Example: A can do a piece of work in 10 days. B can do the same work in 15 days. How long will they take working together?
A's 1 day work = 1/10 B's 1 day work = 1/15 Combined 1 day work = 1/10 + 1/15 = 3/30 + 2/30 = 5/30 = 1/6 Total time = 6 days
'Work done is INVERSELY proportional to the number of days. The more people, the fewer the days.'
Common Mistakes and Fixes
| Mistake | Fix |
|---|---|
| 'All proportions are direct' | Check: if one INCREASES and the other DECREASES, it is INVERSE |
| 'Using cross multiplication for inverse' | For inverse, use PRODUCT (x₁y₁ = x₂y₂), not cross multiplication |
| 'Confusing 'time for a job' with 'time per day'' | If doing MORE work per day, fewer TOTAL days needed — inverse |
| 'Forgetting to check reasonableness' | If direct: answer should be LARGER when starting value is larger |
ICSE Exam Focus (5–6 marks)
- 2-mark questions: Identify whether direct or inverse variation
- 3-mark questions: Simple word problems on direct or inverse variation
- 4-mark questions: Multi-step problems (combined variation)
- 6-mark questions: Time and work problems with efficiency differences
Self-Test
Q1. If x and y vary directly, and x = 6 when y = 15, find y when x = 10. A1. 6/15 = 10/y → 6y = 150 → y = 25.
Q2. If x and y vary inversely, and x = 12 when y = 8, find y when x = 16. A2. 12 × 8 = 16 × y → 96 = 16y → y = 6.
Q3. 6 taps can fill a tank in 40 minutes. How many taps are needed to fill it in 30 minutes? A3. Inverse: 6 × 40 = n × 30 → n = 240/30 = 8 taps.
Q4. A car covers a distance of 240 km in 4 hours. How much distance will it cover in 7 hours at the same speed? A4. Direct: 240/4 = y/7 → 4y = 1680 → y = 420 km.
Q5. A and B can do a piece of work in 12 days and 18 days respectively. In how many days can they complete it together? A5. A's 1 day = 1/12, B's 1 day = 1/18. Combined = 1/12 + 1/18 = 5/36. Days = 36/5 = 7.2 days.
Q6. If 5 men or 7 women can earn Rs 875 per day, how much will 10 men and 5 women earn per day? A6. 5 men = 7 women. 1 man = 7/5 women. 10 men = 14 women. Total = 14 + 5 = 19 women. 7 women earn Rs 875. 19 women earn (19/7) × 875 = Rs 2375.
