📑 On this page
⚡ The short answer
Two quantities are in direct proportion when their ratio is fixed (y = k × x): double one, the other doubles. They are in inverse proportion when their product is fixed (x × y = k): double one, the other halves. Ask: 'more of this means more or less of that?' Trap: +100% on one side is only −50% on the other.
🎬 The 30-second story
The fest-stall maths
1
“40 samosas for your stall? ₹600.”
🥟
Kabir
Aarav is running the snacks stall at the college fest.
2
“So 80 is ₹1,200. Six of us set up the stage in 4 hours, so with 12 people I'll block 8 hours.”
🎪
Aarav
One answer is right.
3
“More hands, fewer hours. 6 × 4 = 24 person-hours. Twelve people need 2.”
🛠️
Zoya
Zoya spots the fixed thing: the job.
4
“And if only 3 of us turn up?”
⏰
Rohan
24 ÷ 3 = 8 hours. Aarav answered for the wrong crew.
Tara's maths: Samosas and cost rise together at a fixed ₹15 each: direct, a fixed ratio. People and hours move opposite ways and always multiply to 24 person-hours: inverse, a fixed product. Find what is fixed, then solve.
What does direct proportion actually mean?
Two quantities are in direct proportion when one is always the same multiple of the other. Samosas cost ₹15 each, so cost = 15 × samosas. That 15 is the constant of proportionality: the price of one samosa. Divide cost by samosas in any row and you get 15.
📊 The data
Direct: the bill grows in step with the samosas
20 samosas
₹300
40 samosas · Kabir's quote
₹600
60 samosas
₹900
80 samosas
₹1,200
Each bar is ₹15 × samosas. Cost ÷ samosas = 15 in every row: that fixed ratio is what makes it direct.
Two quiet conditions hide inside 'direct'. Zero gives zero: no samosas, no bill. And the rate never changes: the 80th samosa costs what the 1st did. Add a delivery fee or a bulk discount and the two still rise together, but they are no longer in proportion.
What is inverse proportion, and why does the product stay fixed?
Inverse proportion is a fixed cake shared out. The stage needs 24 person-hours of work, whoever does it. Share it among more people and each person's time shrinks. Since people × hours must always rebuild 24, one going up forces the other down by the same factor.
📊 The data
Inverse: hours shrink as the hands grow
2 people
12 h
3 people
8 h
4 people
6 h
6 people · the real crew
4 h
8 people
3 h
12 people
2 h
Every row multiplies to 24 person-hours. Double 3 people to 6 and the time halves from 8 hours to 4.
📌 Every inverse question hides a fixed total
Find it and the question solves itself: distance for speed and time, total work for workers and days, food stock for people and days, budget for price and quantity. Write the total first, then divide by the new number.
How do you tell direct from inverse proportion?
⚔️ Side by side
Direct vs inverse at a glance
📈Direct
Ratio fixed: y ÷ x = k
- •Both rise or fall together
- •Double one, the other doubles
- •Zero gives zero
📉Inverse
Product fixed: x × y = k
- •One rises, the other falls
- •Double one, the other halves
- •Neither can be zero
VS
| Test | Direct | Inverse |
|---|---|---|
| Set-up | x₁/y₁ = x₂/y₂ | x₁ × y₁ = x₂ × y₂ |
| Hidden constant | a rate (₹ per item) | a total (distance, work, budget) |
| CLAT pair | quantity and cost | speed and time, fixed distance |
Pairs can switch sides: speed and distance are direct when time is fixed.
🧭 Decision flow
Which one is it? The two-question test
❓ If the first goes up, does the second go up too?
Yes →
❓ Does doubling the first exactly double the second?
Watch for a fixed fee
Yes →
Direct: keep the ratio fixed
No →
Related, but not proportional
Taxi: ₹50 base + ₹12 per km
No →
❓ Does their product stay the same?
Is a fixed total being shared?
Yes →
Inverse: keep the product fixed
No →
Opposite, but not inverse
Battery: 100% minus 10% an hour
What are the direct and inverse proportion formulas?
Direct: x₁/y₁ = x₂/y₂, so y₂ = y₁ × (x₂/x₁). Inverse: x₁ × y₁ = x₂ × y₂, so y₂ = y₁ × (x₁/x₂). The only difference is that the fraction is flipped. That flip is the whole topic.
🧮 The formula
The inverse engine: workers and days
👷
Workers
goes up
×
📅
Days
must go down
=
🧱
Fixed job
person-days never change
The percentage version: why +100% is only −50%
If x rises by a%, x is multiplied by (100 + a)/100. To keep the product fixed, y is multiplied by the flipped fraction, so y falls by a/(100 + a) × 100%. The base trap: divide by the new value (100 + a), not by 100.
| x changes by | y is multiplied by | y changes by |
|---|---|---|
| +10% | 10/11 | −9.09% |
| +20% | 5/6 | −16.67% |
| +25% | 4/5 | −20% |
| +50% | 2/3 | −33.33% |
| +100% | 1/2 | −50% |
| −20% | 5/4 | +25% |
| −50% | 2 | +100% |
→ scroll to see more
How do you solve direct and inverse proportion questions?
✍️ Worked example
Warm-up
⏱ 20 sec
Download time
Meher's phone downloads a 3.6 GB lecture pack in 8 minutes at a steady speed. How long will a 5.4 GB pack take?
- 1More data at the same speed means more time: direct.
- 2Ratio set-up: 3.6/8 = 5.4/t.
- 3t = 8 × (5.4/3.6) = 8 × 1.5 = 12.
Answer: 12 minutes
⚡ The exam-hall shortcut
5.4 is one and a half times 3.6, so the time is one and a half times 8. No equation needed.
✍️ Worked example
CLAT level
⏱ 45 sec
The mess food, with a twist
A hostel mess has food for 80 students for 15 days. After 5 days, 20 more students join. For how many more days will the food last?
- 1More eaters, fewer days: inverse. The fixed total is food, in student-days.
- 2Food left after 5 days = 80 × 10 = 800 student-days. (Using all 15 days is the trap.)
- 3100 students now share it: 800 ÷ 100 = 8.
Answer: 8 more days (13 days from the start)
⚡ The exam-hall shortcut
Remaining 10 days × (80/100) = 8. More people, so use the fraction below 1.
✍️ Worked example
Tricky
⏱ 60 sec
Bigger backdrop, bigger crew
12 volunteers, working 6 hours a day, paint the fest backdrop in 10 days. How many days will 15 volunteers take, working 8 hours a day, to paint a backdrop twice as large?
- 1Work = volunteers × hours a day × days. One backdrop = 12 × 6 × 10 = 720 volunteer-hours.
- 2Twice the backdrop = 1,440 volunteer-hours.
- 3New crew: 15 × 8 = 120 volunteer-hours a day, so 1,440 ÷ 120 = 12.
Answer: 12 days
⚡ The exam-hall shortcut
Start from 10 and ask of each change: more or fewer days? 10 × (12/15) × (6/8) × 2 = 12.
✍️ Worked example
Tricky
⏱ 40 sec
The 12 minutes saved
Rohan rides to coaching at a steady speed. If he rides 25% faster, he reaches 12 minutes earlier. How long does his usual ride take?
- 1Fixed distance, so speed and time are inverse.
- 2Speed × 5/4 means time × 4/5, so he saves 1/5 of the usual time.
- 3That 1/5 is 12 minutes, so the usual time = 12 × 5 = 60 minutes. Check: 60 − 48 = 12.
Answer: 60 minutes
⚡ Use the pair table
+25% speed means −20% time. 20% of the time is 12 minutes, so the whole is 60.
The 30-second route: multiply by the right fraction
- 1
Start with the old answerWrite the old value of what is asked: 15 days, 8 minutes, ₹600.
- 2
Ask more or less, one change at a timeFor each change, decide whether it pushes the answer up or down.
- 3
Multiply by the matching fractionShould grow? Use big over small. Should shrink? Small over big.
🎛️ Try it yourself — change any number
Try it: how long does the stage set-up take?
New days (or time)
3
Product stays
24
6 × 4 = 8 × y → y = 3
8 people give 3 hours (24 ÷ 8). Try 12 people (2 hours), 3 people (8 hours), then 5 people: 4.8 hours. The product never changes.
How does proportion hide inside a CLAT caselet?
CLAT rarely says 'inversely proportional'. It writes: 'Four teams travel 120 km to a moot court venue, all leaving at 7 a.m. at constant speeds: car 60 km/h, bus 40 km/h, scooter 30 km/h, cycle 20 km/h.' The fixed total is 120 km. Table it first.
📊 The data
Time to cover the same 120 km
Car, 60 km/h
2 h
Bus, 40 km/h · arrives 10 a.m.
3 h
Scooter, 30 km/h
4 h
Cycle, 20 km/h
6 h
Speed × time = 120 for every team. The car is 3 times as fast as the cycle and takes a third of the time.
- ✓'The bus must arrive by 9:30 a.m.' By what % must it speed up? Time falls from 3 h to 2.5 h, a cut of 1/6, so speed rises by 1/5: 120 ÷ 2.5 = 48 km/h, 20% faster.
- ✓'The cyclists ride 25% faster.' When do they arrive? 6 × 4/5 = 4.8 h = 4 h 48 min, so at 11:48 a.m.
- ✓'How much petrol does the car use?' Cannot be determined: the passage never gives the mileage.
💡 Estimate before you calculate
Decide the direction first and cross out options. If the cyclists speed up, the time must be under 6 hours. Fewer workers? The days must go up. CLAT's distractors are usually the wrong-direction answer.
What traps does CLAT set in proportion questions?
⚠️ +100% on one side is not −100% on the other
If a doubles, b halves: a 50% fall. A 100% fall would make b zero, which no inverse pair allows. The same slip turns +25% into '−25%' when the truth is −20%. Convert to a fraction and flip it.
- Wrong direction. More workers giving more days is a red flag.
- The fixed fee. A cab at ₹50 plus ₹12 a km costs ₹170 for 10 km and ₹290 for 20 km, not ₹340.
- Forgetting what is used. 'After 5 days, 20 more join': share only the remaining food.
- Mixed units. Hours a day vs total days, minutes vs hours. Match units first.
How has CLAT asked direct and inverse proportion?
📝 Asked in CLAT 2018
· Q144
Inverse proportion, % change
Stand-alone question (no passage).
a and b are inversely proportional to each other and are positive. If a increases by 100%, then b decreases by
A50%
B75%
C100%
D200%
Reveal answer + reasoning
Answer: A
Inverse means a × b stays fixed. A 100% increase doubles a, so b must become b/2: a fall of 50%. By the rule: 100/(100 + 100) × 100 = 50%. Time: about 15 seconds.
Why the others fail
- B — A 75% fall leaves b/4, which needs a to quadruple.
- C — Assumes the percentage just flips sign. b would be zero, impossible when b is positive.
- D — Nothing positive can fall by more than 100%.
📝 Asked in CLAT 2019
· Q53
Direct proportion, same meal
Stand-alone question (no passage).
A mess contractor can either serve 450 students with the meal that he prepares or can cater to 270 cops with the same meal. If 300 students have already eaten in the mess, how many cops can be fed with the remaining meal?
A20
B45
C90
D180
Reveal answer + reasoning
Answer: C
Our solution (2019 has no official key). The meal = 450 student-portions = 270 cop-portions. After 300 students eat, 150 student-portions are left: 150/450 = 1/3 of the meal. Cops fed are direct with the meal left: 1/3 × 270 = 90. Time: about 30 seconds.
Why the others fail
- A — No consistent route gives 20; a filler option.
- B — Half the right answer: one halving too many.
- D — 180 = 300 × 270/450, the cop-equivalent of the food already eaten. Wrong base.
💥 Myth vs fact
Proportion myths that cost marks
🙅 Myth
If one goes up while the other goes down, it is inverse.
Tap to bust it →
✅ Fact
Only if the product is fixed. A battery falls 100%, 90%, 80% per hour of use, but 100 × 0 ≠ 90 × 1. It is a straight-line fall, not inverse.
🙅 Myth
Anything that rises together is direct proportion.
Tap to bust it →
✅ Fact
Only if the ratio is fixed and zero gives zero. A base fare breaks it.
🙅 Myth
Speed and time are always inverse.
Tap to bust it →
✅ Fact
Only when the distance is fixed. Fix the time, and speed and distance are direct.
🙅 Myth
Double one and its inverse partner falls 100%.
Tap to bust it →
✅ Fact
It halves: a 50% fall.
Quick check: can you spot direct vs inverse?
🎯 Quick quiz
Five questions, two minutes
0 / 5
Q1. 9 identical pens cost ₹117. What do 14 such pens cost?
Direct. One pen = 117 ÷ 9 = ₹13, so 14 pens = 14 × 13 = ₹182.Q2. 10 identical pumps empty a tank in 6 hours. How long will 4 such pumps take?
Inverse. 10 × 6 = 60 pump-hours; 60 ÷ 4 = 15 hours.Q3. x and y are inversely proportional. If x decreases by 20%, what happens to y?
x × 4/5 means y × 5/4, a rise of 25%.Q4. Which pair is NOT in direct proportion?
The ₹50 base means fare ÷ distance keeps changing, so the ratio is not fixed.8 riders deliver 240 orders in 5 hours. All riders work at the same pace.
Q5. How many orders will 12 riders deliver in 4 hours?
Orders are direct with both riders and hours: 240 × (12/8) × (4/5) = 288.🎯 Direct and inverse proportion in a nutshell
- Direct: ratio fixed, y = k × x. Double x, y doubles.
- Inverse: product fixed, x × y = k. Double x, y halves.
- Find the fixed thing: a rate means direct, a total means inverse.
- Several changes? Multiply the old answer by one fraction per change.
- Inverse percentages flip: +25% → −20%, +100% → −50%.
📎 Formula card
- Direct: x₁/y₁ = x₂/y₂ → y₂ = y₁ × x₂/x₁
- Inverse: x₁ × y₁ = x₂ × y₂ → y₂ = y₁ × x₁/x₂
- Compound work: M₁H₁D₁/W₁ = M₂H₂D₂/W₂ (workers, hours a day, days, work)
- Inverse %: x up a% → y down a/(100 + a) × 100%
- Pairs: +25% ↔ −20% · +50% ↔ −33.33% · +100% ↔ −50% · −20% ↔ +25%
Frequently asked questions
What is the difference between direct and inverse proportion?
In direct proportion the ratio of two quantities stays fixed, so both rise or fall together by the same factor: double the samosas, double the bill. In inverse proportion their product stays fixed, so one rises as the other falls: double the workers, halve the days. Direct uses x1/y1 = x2/y2; inverse uses x1 × y1 = x2 × y2.
How do you know if a question is inverse proportion?
Ask whether more of the first quantity means less of the second, then look for a fixed total being shared: a fixed distance, job, food stock or budget. If that total stays the same in every case, the two quantities multiply to it and the pair is inverse. Moving in opposite directions alone is not enough.
If one quantity increases by 25%, how much does an inversely proportional quantity decrease?
By 20%. A 25% rise multiplies the first quantity by 5/4, so the other must be multiplied by 4/5 to keep the product fixed, which is a 20% fall. The general rule is a/(100 + a) × 100%, so +50% gives −33.33% and +100% gives −50%, never −100%.
Are speed and time directly or inversely proportional?
Inversely proportional, but only when the distance is fixed. Covering 120 km at 60 km/h takes 2 hours and at 30 km/h takes 4 hours; speed × time stays 120. If the time is fixed instead, speed and distance are directly proportional: drive faster for the same two hours and you go further.
Keep going
← Previous topic
Proportion, Third and Mean Proportional
Fourth, third and mean proportional from one cross rule
Next topic →
The Unitary Method
Find the value of one, then anything. Plus the fixed-charge trap
📚
Part of the chapter
Ratio & Proportion for CLAT Quantitative Techniques
See how this topic fits the whole of Ratio & Proportion →
Connected topics in other chapters
The Unitary Method
The 'find one unit first' route that solves the same questions the slow, safe way.
Time and Work: the LCM Method
Workers and days are the classic inverse pair; this page takes the idea to full CLAT questions.
Speed, Distance and Time
Speed and time are inverse only when the distance is fixed. See the whole triangle.
Proportion, Third and Mean Proportional
The cross rule a × d = b × c that sits under every direct-proportion set-up.
Successive Percentage Change
Why +25% on one side is −20% on the other: the same fraction maths, from the percentage side.
Every topic in Ratio & Proportion
- 1.Ratio Basics: Simplifying and Comparing Ratios
- 2.Dividing a Quantity in a Given Ratio
- 3.Combining Ratios: A:B, B:C and Overall Ratios
- 4.Proportion, Third and Mean Proportional
- 5.Direct and Inverse Proportion (you're here)
- 6.The Unitary Method
- 7.Partnership: Sharing Profit by Capital × Time
- 8.Problems on Ages: The Constant-Gap Trick