📑 On this page
⚡ The short answer
A proportion says two ratios are equal: a : b :: c : d, so a × d = b × c. Fourth proportional to a, b, c = bc/a. Third proportional to a and b = b²/a. Mean proportional between a and c = √(ac), not their average. Trap: check which term repeats before picking a formula.
🎬 The 30-second story
The poster that came out squashed
1
“12 × 18 cm photo, 30 cm wide poster. Width went up 18, so height is 18 + 18 = 36.”
🖼️
Aarav
Aarav is printing a poster for the fest stall.
2
“Photos stretch by multiplying. 12 became 30: that's × 2.5.”
📐
Zoya
On the proof, every face is wide and short.
3
“12 : 18 :: 30 : x. So 12x = 540 and x = 45.”
✏️
Meher
Meher writes it as a proportion.
4
“45, not 36? I'd have printed a pancake.”
🖨️
Aarav
12 : 18 is 2 : 3. So is 30 : 45. But 30 : 36 is 5 : 6.
Tara's maths: A proportion keeps the ratio, not the gap. Adding the same number to both sides changes the ratio, and the picture gets squashed.
What is a proportion, and why does a × d = b × c?
A proportion claims two ratios are the same: a : b :: c : d, read 'a is to b as c is to d'. The outer pair, a and d, are the extremes. The inner pair, b and c, are the means. Photo and poster form a proportion because both have the shape 2 : 3.
🧮 The formula
The cross rule
↔️
a × d
the extremes (outer pair)
=
🎯
b × c
the means (inner pair)
For 12 : 18 :: 30 : 45: extremes 12 × 45 = 540, means 18 × 30 = 540.
Why? Equal ratios mean the second pair is the first pair scaled by one number k, here 2.5. So a × d = 12 × (2.5 × 18) and b × c = 18 × (2.5 × 12): the same three numbers in a different order. The products have to match.
📌 One rule, three named answers
Every 'proportional' question is the cross rule with one slot empty. Fourth: the last term is missing. Third: a known term repeats, the last is missing. Mean: the repeated middle is missing. Name the slot and the formula writes itself.
How do you find the fourth proportional?
If a : b :: c : x, x is the fourth proportional to a, b and c. Cross rule: a × x = b × c, so x = bc ÷ a. Multiply the means, divide by the known extreme. The poster: 18 × 30 ÷ 12 = 45.
| Question says | Set-up | Answer |
|---|---|---|
| Fourth proportional to 2, 3, 4 | 2 : 3 :: 4 : x | 3 × 4 ÷ 2 = 6 |
| Fourth proportional to 3, 2, 4 | 3 : 2 :: 4 : x | 2 × 4 ÷ 3 = 8/3 |
| Fourth proportional to 12, 18, 30 | 12 : 18 :: 30 : x | 18 × 30 ÷ 12 = 45 |
→ scroll to see more
What is the mean proportional between two numbers?
Make the two means the same number: a : x :: x : c. That x is the mean proportional between a and c. Cross rule: x × x = a × c, so x = √(ac). Between 9 and 25: √225 = 15. Check: 9 : 15 and 15 : 25 are both 3 : 5. Same step up, twice.
📐 The figure
The mean proportional is a square hiding in a rectangle
Rectangle
Area = 9 × 25 = 225 cm²
Square, same area
Area = 15 × 15 = 225 cm²
15 is the mean proportional of 9 and 25
Figures are not to scale.
Why the mean proportional is not the average
⚔️ Side by side
Average vs mean proportional
➕Average
(a + c) ÷ 2
- •Same gap each side
- •9, 17, 25: +8, +8
✖️Mean proportional
√(ac)
- •Same multiplier each side
- •9, 15, 25: × 5/3, × 5/3
VS
| Test | Average | Mean proportional |
|---|---|---|
| Between 4 and 9 | 6.5 | 6 |
| Use it for | marks, totals | growth, scaling |
Unless the numbers are equal, the average is always bigger. That makes it CLAT's favourite wrong option.
What is a third proportional, and what is continued proportion?
Three numbers are in continued proportion when a : b :: b : c. One line, two names. b is the mean proportional between a and c. And c is the third proportional to a and b: b × b = a × c, so c = b² ÷ a.
📊 The data
Continued proportion: Meher's channel, × 1.5 a month
Month 1
400 subs
Month 2 · mean proportional of 400 and 900
600 subs
Month 3 · third proportional to 400 and 600
900 subs
Month 4
1,350 subs
Each bar is the last × 1.5. Any three in a row: 600² = 400 × 900 = 3,60,000.
⚠️ Third or mean? Look at what repeats
Third proportional to a and b: the known b repeats, a : b :: b : x, so x = b²/a. Mean proportional between a and c: the unknown repeats, a : x :: x : c, so x = √(ac). For 4 and 6, third = 9 but mean ≈ 4.9. Both will be in the options.
How do you solve proportion questions step by step?
✍️ Worked example
Warm-up
⏱ 40 sec
All three proportionals from one pair
(a) Mean proportional between 16 and 36. (b) Third proportional to 16 and 24. (c) Fourth proportional to 16, 24 and 36.
- 1(a) x² = 16 × 36 = 576, so x = 24.
- 2(b) 16 : 24 :: 24 : x, so x = 576 ÷ 16 = 36.
- 3(c) 16 : 24 :: 36 : x, so x = 24 × 36 ÷ 16 = 54.
Answer: 24, 36 and 54
⚡ Spot the multiplier
16 → 24 is × 1.5. The chain runs 16, 24, 36, 54: every answer is the next link.
✍️ Worked example
CLAT level
⏱ 30 sec
The town that grew at a steady rate
Rohan's town had 6,400 residents in 2020 and 8,100 in 2024, growing by the same percentage every year. How many in 2022?
- 1Same % a year means the same multiplier every 2 years, so 2020 : 2022 :: 2022 : 2024.
- 22022 is the mean proportional: √(6,400 × 8,100) = 80 × 90 = 7,200.
- 3Check: 6,400 × 9/8 = 7,200 and 7,200 × 9/8 = 8,100.
Answer: 7,200 residents
⚡ The exam-hall shortcut
Both are perfect squares: multiply the roots, 80 × 90. The average, 7,250, is the trap.
✍️ Worked example
Tricky
⏱ 60 sec
What do you add to make them proportional?
What number must be added to each of 6, 15, 20 and 43 so the results are in proportion? Options: 2, 3, 4, 5.
- 1(6 + x) : (15 + x) :: (20 + x) : (43 + x).
- 2Cross rule: 258 + 49x + x² = 300 + 35x + x².
- 3The x² cancels: 14x = 42, so x = 3. Check: 9 : 18 :: 23 : 46, both 1 : 2.
Answer: 3
⚡ Let the options work
Test options instead. x = 3: extremes 9 × 46 = 414, means 18 × 23 = 414. Done.
🎛️ Try it yourself — change any number
Try it: find the fourth proportional
Per unit
1.5
Answer
45
18 ÷ 12 × 30 = 45
12 : 18 :: 30 : x gives 45. 'Per unit' is the multiplier b ÷ a. Enter 16, 24, 24 for the third proportional (36). Swap a and c and watch the answer change.
How does proportion hide inside a CLAT caselet?
CLAT won't say 'mean proportional'. It writes: 'A study app's downloads grew by the same percentage every month: 2,500 in January, 3,600 in March. February's figure was lost.' 'Same percentage every month' is the signal: fixed growth over equal steps is a continued proportion.
📊 The data
The app caselet, rebuilt
January · given
2,500 downloads
February · √(2,500 × 3,600)
3,000 downloads
March · given
3,600 downloads
April · 3,600² ÷ 3,000
4,320 downloads
Multiplier = 3,000 ÷ 2,500 = 1.2: 20% a month. The average, 3,050, is the bait.
- ✓February? √(2,500 × 3,600) = 50 × 60 = 3,000.
- ✓April, if the trend holds? 3,600² ÷ 3,000 = 4,320.
- ✓Downloads from Delhi? Cannot be determined: no city split is given.
💡 Estimate before you calculate
The mean proportional sits below the average. Between 2,500 and 3,600 the average is 3,050, so any option at or above 3,050 is out at a glance.
What traps does CLAT set in proportion questions?
🧭 Decision flow
Which proportional are they asking for?
❓ Three numbers given, last term missing?
a : b :: c : x
Yes →
Fourth proportional: bc ÷ a
No ↓
❓ Is the unknown repeated in the middle?
a : x :: x : c
Yes →
Mean proportional: √(ac)
No ↓
❓ Does a known number repeat?
a : b :: b : x
Yes →
Third proportional: b² ÷ a
No →
Plain ratio question
Write both ratios, cross-multiply
- Averaging instead of rooting. Between 4 and 9: 6, not 6.5.
- Shuffled order. Fourth proportional to 2, 3, 4 is 6; to 3, 2, 4 it is 8/3.
- Decimals. Between 0.08 and 0.18: √0.0144 = 0.12, not 1.2.
- Negative roots. x² = 225 allows −15, but rupees and people are positive. Take 15.
How would CLAT ask about proportion?
No CLAT paper from 2018 to 2025 asks a direct 'mean proportional' question, so these two are CLAT-style, written like current caselets.
🧪 CLAT-style practice (not a past question)
Mean proportional in a caselet
A college moot court competition awards prize money to its top three teams. The organisers fix the amounts so that the first, second and third prizes are in continued proportion. The first prize is Rs 54,000 and the third prize is Rs 24,000.
What is the second prize?
ARs 30,000
BRs 36,000
CRs 39,000
DRs 42,000
Reveal answer + reasoning
Answer: B
First : second :: second : third, so the second prize is the mean proportional: √(54,000 × 24,000). Work in thousands: 54 × 24 = 1,296 = 36², so Rs 36,000. Check: 54 : 36 and 36 : 24 are both 3 : 2. Time: about 25 seconds.
Why the others fail
- A — 54 : 30 is 9 : 5 but 30 : 24 is 5 : 4. Ratios don't match.
- C — The average of 54,000 and 24,000: equal gaps, not equal ratios.
- D — 42² = 1,764, not 1,296.
🧪 CLAT-style practice (not a past question)
Fourth proportional, students to faculty
A law university runs a five-year programme and a three-year programme and keeps the student-to-faculty ratio exactly the same in both. The five-year programme has 840 students and 35 faculty members. The three-year programme has 22 faculty members.
How many students are in the three-year programme?
A480
B528
C560
D827
Reveal answer + reasoning
Answer: B
840 : 35 :: x : 22, so 35x = 840 × 22 = 18,480 and x = 528. Faster: 840 ÷ 35 = 24 students per teacher; 24 × 22 = 528. Time: about 20 seconds.
Why the others fail
- A — 24 × 20: uses 20 faculty, not 22.
- C — 840 × 2/3, borrowing 'three-year vs five-year', which isn't the ratio.
- D — 840 − 35 + 22: Aarav's poster mistake. It keeps the gap, not the ratio.
💥 Myth vs fact
Proportion myths that cost marks
🙅 Myth
The mean proportional is the mean of the two numbers.
Tap to bust it →
✅ Fact
It is √(ac). The plain average is bigger unless the numbers are equal.
🙅 Myth
Third and mean proportional are the same thing.
Tap to bust it →
✅ Fact
In a : b :: b : c, b is the mean and c is the third.
🙅 Myth
Adding the same number to every term keeps the proportion.
Tap to bust it →
✅ Fact
12 : 18 is 2 : 3; add 18 and 30 : 36 is 5 : 6. Only multiplying is safe.
Quick check: can you find the missing proportional?
🎯 Quick quiz
Five questions, three minutes
0 / 5
Q1. Find the fourth proportional to 5, 8 and 15.
x = 8 × 15 ÷ 5 = 24.Q2. Find the mean proportional between 0.08 and 0.18.
√0.0144 = 0.12. 0.13 is the average.Q3. Find the third proportional to 12 and 18.
18² ÷ 12 = 27. 14.7 is √(12 × 18), the mean proportional.Q4. What must be subtracted from each of 23, 30, 57 and 78 so that the results are in proportion?
Test 6: 17 × 72 = 1,224 and 24 × 51 = 1,224. So 6.Rohan's savings pot grows by the same percentage every year. It held ₹10,000 at the end of year 1 and ₹14,400 at the end of year 3.
Q5. How much did it hold at the end of year 2?
√(10,000 × 14,400) = 100 × 120 = ₹12,000. ₹12,200 is the average.🎯 Proportion in a nutshell
- a : b :: c : d means a × d = b × c.
- Fourth = bc ÷ a. Third = b² ÷ a. Mean = √(ac).
- The mean proportional is never above the average.
- Same % growth each period is a continued proportion.
📎 Formula card
- Proportion: a : b :: c : d ⇔ a × d = b × c
- Fourth proportional (a : b :: c : x): x = bc/a
- Third proportional (a : b :: b : x): x = b²/a
- Mean proportional (a : x :: x : c): x = √(ac) ≤ (a + c)/2
- Componendo-dividendo: (a + b) : (a − b) = p : q ⇒ a : b = (p + q) : (p − q)
Frequently asked questions
What is the mean proportional between two numbers?
The mean proportional between a and c is the number x that makes a : x :: x : c true. By the cross rule x × x = a × c, so x = √(ac). Between 9 and 25 it is 15, because 9 : 15 and 15 : 25 are both 3 : 5. It is the side of a square with the same area as an a by c rectangle.
What is the difference between third proportional and mean proportional?
Both come from a continued proportion a : b :: b : c. The mean proportional is the repeated middle term b, found as √(ac) when a and c are known. The third proportional is the last term c, found as b² ÷ a when a and b are known. For 4 and 6, the third proportional is 9 and the mean proportional is about 4.9.
How do you find the fourth proportional?
Write the three numbers in the order given as a : b :: c : x and cross-multiply, so x = bc ÷ a. The fourth proportional to 5, 8 and 15 is 8 × 15 ÷ 5 = 24. Keep the order exactly as the question names it: 2, 3, 4 gives 6, but 3, 2, 4 gives 8/3.
Is the mean proportional the same as the average?
No. The average of a and c is (a + c) ÷ 2 and keeps the gaps equal. The mean proportional is √(ac) and keeps the ratios equal. Between 4 and 9 the average is 6.5 but the mean proportional is 6. The average is always at least as big, so in CLAT it is usually the trap option.
Keep going
← Previous topic
Combining Ratios: A:B, B:C and Overall Ratios
Join A:B and B:C into A:B:C, and merge group ratios without traps
Next topic →
Direct and Inverse Proportion
Fixed ratio or fixed product? The one test that solves both
📚
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
Direct and Inverse Proportion
The cross rule in action: fixed ratios for direct, fixed products for inverse.
Ratio Basics
Simplifying and comparing ratios, the step before any proportion.
Population Growth and Depreciation
Same % every year is a continued proportion; the middle year is the mean proportional.
Compound Interest
CI amounts year after year are a continued proportion with factor (1 + r/100).
Scaling
Stretch a shape and every length moves in proportion, like Aarav's poster.
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 (you're here)
- 5.Direct and Inverse Proportion
- 6.The Unitary Method
- 7.Partnership: Sharing Profit by Capital × Time
- 8.Problems on Ages: The Constant-Gap Trick