1. Chapter at a Glance
In Proportional Reasoning-1 we compared quantities using ratios. Now we look at how two quantities change together. If you buy more pens, you pay more — that is direct proportion. If more workers share a job, each needs fewer days — that is inverse proportion. Knowing which kind of relationship you are looking at is the key to solving a huge number of everyday and exam problems.
- Direct proportion: x/y stays constant.
- Inverse proportion: x × y stays constant.
- How to recognise which type a situation is.
- Solving problems with tables, the unitary method and cross-multiplication.
- Combining ratios (compound situations) and sharing in a ratio.
- Graphs of direct and inverse proportion.
2. Direct Proportion
Two quantities x and y are in direct proportion if, when one increases (or decreases), the other increases (or decreases) by the same factor. Then the ratio stays fixed:
| Number of notebooks (x) | 1 | 2 | 5 | 8 |
|---|---|---|---|---|
| Cost in ₹ (y) | 40 | 80 | 200 | 320 |
| y / x | 40 | 40 | 40 | 40 |
Because y/x is always 40, cost is directly proportional to the number of notebooks. Here k = 40 is the price of one notebook.
If (x₁, y₁) and (x₂, y₂) are two pairs in direct proportion, then x₁ / y₁ = x₂ / y₂.
A car uses 6 litres of petrol to travel 84 km. How far will it go on 15 litres?
Solution: More petrol → more distance, so direct. 6/84 = 15/d → d = 84 × 15 / 6 = 210 km.
Examples of direct proportion: cost and quantity (fixed price), distance and time (fixed speed), wages and days worked (fixed daily wage), length on a map and real distance, ingredients and number of servings.
3. Inverse Proportion
Two quantities are in inverse proportion if, when one becomes k times, the other becomes 1/k times. Their product stays fixed:
| Number of workers (x) | 2 | 4 | 6 | 12 |
|---|---|---|---|---|
| Days to finish (y) | 24 | 12 | 8 | 4 |
| x × y | 48 | 48 | 48 | 48 |
The product 48 is the total "worker-days" of work — it does not change, however you split it.
If (x₁, y₁) and (x₂, y₂) are in inverse proportion, then x₁ × y₁ = x₂ × y₂.
At 60 km/h a train takes 5 hours for a journey. How long at 75 km/h?
Solution: Faster speed → less time, so inverse. 60 × 5 = 75 × t → t = 300 / 75 = 4 hours.
Examples of inverse proportion: speed and time (fixed distance), workers and days (fixed job), number of pipes and time to fill a tank, number of people and days food lasts, number of children and sweets each gets (fixed total).
4. Which Type Is It?
Ask: "If I double the first quantity, what happens to the second?"
| Situation | Double the first… | Type |
|---|---|---|
| Kg of rice and its cost | cost doubles | Direct |
| Pipes and time to fill a tank | time halves | Inverse |
| Height of a pole and its shadow (same time of day) | shadow doubles | Direct |
| Guests and days food stock lasts | days halve | Inverse |
| Age and height of a person | no fixed rule | Neither |
"Both increase" is not enough for direct proportion — they must increase by the same factor. A taxi fare with a fixed starting charge (₹50 + ₹15 per km) increases with distance but is not directly proportional, because fare/distance is not constant.
5. Graphs
6. Combining Proportions
Sometimes more than two quantities are linked. Handle them one at a time, deciding direct or inverse for each pair while keeping the others fixed.
8 workers working 6 hours a day finish a wall in 15 days. How many days will 12 workers working 5 hours a day take?
Solution: Total work in worker-hours is fixed: 8 × 6 × 15 = 720. New: 12 × 5 × d = 720 → d = 12 days.
A : B = 2 : 3 and B : C = 4 : 5. Find A : B : C.
Solution: Make B the same: A : B = 8 : 12, B : C = 12 : 15. So A : B : C = 8 : 12 : 15.
₹1,800 is shared among three friends in the ratio 2 : 3 : 4. Find each share.
Solution: Total parts = 9; one part = ₹200. Shares: ₹400, ₹600, ₹800.
7. Solved Problems (NCERT Style)
A hostel has food for 120 students for 25 days. If 30 more students join, how long will the food last?
Solution: Inverse. 120 × 25 = 150 × d → d = 20 days.
A 6 m pole casts a 4 m shadow. At the same time a tree casts a 10 m shadow. How tall is the tree?
Solution: Direct. 6/4 = h/10 → h = 15 m.
3 identical pipes fill a tank in 40 minutes. How long would 5 such pipes take?
Solution: Inverse. 3 × 40 = 5 × t → t = 24 minutes.
8. Master Formula Table
| Idea | Formula |
|---|---|
| Direct proportion | x/y = k, x₁/y₁ = x₂/y₂ |
| Inverse proportion | xy = k, x₁y₁ = x₂y₂ |
| Compound (work) | M₁ × H₁ × D₁ = M₂ × H₂ × D₂ (same job) |
| Combine ratios | Make the common term equal (LCM) |
| Share in ratio a : b : c | Share = (a / (a+b+c)) × total |
9. Master Summary Table
| Feature | Direct | Inverse |
|---|---|---|
| When x doubles | y doubles | y halves |
| Constant | Ratio y/x | Product xy |
| Graph | Straight line through origin | Falling curve |
| Classic example | Cost and quantity | Speed and time |
| Equation | y = kx | y = k/x |
Chart — days needed vs workers (job = 48 worker-days):
10. Mind Map
- y/x constant
- Line through origin
- xy constant
- Falling curve
- Double test
- Neither type
- Work = M × H × D
- Combine ratios
11. Tips and Tricks
- Always decide "direct or inverse?" before writing any equation.
- Check your answer with common sense: more workers must mean fewer days.
- For work problems, find the total work (worker-days or worker-hours) first.
- Cancel common factors before multiplying to keep numbers small.
- Using x₁/y₁ = x₂/y₂ for an inverse situation.
- Calling a relation with a fixed extra charge "direct proportion".
- Mixing units (hours and minutes) in the same equation.
12. Important One-liners
- Direct proportion: the ratio of the two quantities stays constant.
- Inverse proportion: the product of the two quantities stays constant.
- For a fixed distance, speed and time are inversely proportional.
- For a fixed speed, distance and time are directly proportional.
- The graph of direct proportion is a straight line through the origin.
- To combine A : B and B : C, make the B terms equal.
13. Exam-pattern Questions with Detailed Answers
If 5 pens cost ₹60, what do 12 pens cost?
Show answer
Direct. One pen = ₹12; 12 pens = ₹144.
Answer: ₹144
Which pair is in inverse proportion?
- Distance and petrol used
- Number of taps and time to fill a tank
- Length of cloth and its cost
- Number of books and their weight
Show answer
More taps fill the tank faster. The others are direct.
Answer: (B)
x and y are in direct proportion. When x = 4, y = 10. Find y when x = 10.
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k = y/x = 2.5; y = 2.5 × 10 = 25.
Answer: 25
x and y are in inverse proportion. When x = 6, y = 15. Find x when y = 9.
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xy = 90 → x = 90/9 = 10.
Answer: 10
A map has a scale of 1 cm : 5 km. Two towns are 7.4 cm apart on the map. Find the real distance.
Show answer
Direct. 7.4 × 5 = 37 km.
Answer: 37 km
18 workers can build a road in 20 days. How many workers are needed to finish it in 12 days?
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Inverse. 18 × 20 = n × 12 → n = 30.
Answer: 30 workers
If A : B = 3 : 4 and B : C = 6 : 7, find A : B : C.
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LCM of 4 and 6 is 12. A : B = 9 : 12, B : C = 12 : 14. So 9 : 12 : 14.
Answer: 9 : 12 : 14
An auto charges ₹30 fixed plus ₹12 per km. Is the fare directly proportional to distance? Give a reason.
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For 1 km fare = ₹42 (ratio 42); for 2 km fare = ₹54 (ratio 27). The ratio fare/distance is not constant, so it is not direct proportion.
Answer: No — the fixed charge breaks the constant ratio
10 machines working 8 hours a day produce 4,000 items in 5 days. How many items will 6 machines produce working 10 hours a day for 4 days?
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Items per machine-hour = 4000 / (10 × 8 × 5) = 10. New machine-hours = 6 × 10 × 4 = 240. Items = 2,400.
Answer: 2,400 items
A camp has food for 200 people for 30 days. After 10 days, 50 people leave. How long will the remaining food last?
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After 10 days, food left = 200 × 20 person-days = 4,000. Now 150 people: 4000 / 150 = 26⅔ days.
Answer: 26⅔ days (about 26 days and 16 hours)
A train covers a distance in 6 hours at 80 km/h. By how much must the speed increase to cover it in 5 hours?
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Distance = 480 km. New speed = 480/5 = 96 km/h. Increase = 16 km/h.
Answer: 16 km/h
Profit of ₹26,000 is shared by Asha, Bina and Chetan in the ratio ½ : ⅓ : ¼. Find each share.
Show answer
Clear the fractions by multiplying each term by 12 (LCM of 2, 3, 4): 6 : 4 : 3, total 13 parts. One part = 26,000 / 13 = ₹2,000. Shares: 6 × 2000, 4 × 2000, 3 × 2000.
Answer: Asha ₹12,000, Bina ₹8,000, Chetan ₹6,000
14. Quick Revision Summary
- Direct: y = kx; ratio constant; straight-line graph through origin.
- Inverse: xy = k; product constant; falling curve.
- Use the "double test" to identify the type.
- Work problems: total work = workers × hours × days stays fixed.
- Combine ratios by matching the common term; share totals by parts.