Ratio and Proportion: Mean, Third and Fourth Proportional

A ratio compares two quantities of the same kind by division. A : B = 3 : 5 means that for every 3 units of A there are 5 units of B, so A is 3/5 of B. The actual amounts can be 3 and 5, 30 and 50, or 3k and 5k for any k. Writing the quantities as 3k and 5k is the starting point for almost every ratio problem.

A proportion is a statement that two ratios are equal, written a : b :: c : d. The outer terms a and d are the extremes and the inner terms b and c are the means. In any proportion, the product of the extremes equals the product of the means: ad = bc. Every proportional formula on this page comes from that one rule.

When you know A : B and B : C, make the common term B equal in both, then read off A : B : C. Multiply each ratio by whatever makes the two B values match, usually their LCM.

Worked example 1: A : B = 2 : 3 and B : C = 4 : 5

The B values are 3 and 4, with LCM 12. Multiply the first ratio by 4 and the second by 3: A : B = 8 : 12 and B : C = 12 : 15. So A : B : C = 8 : 12 : 15. If ₹7,000 is shared this way, A gets 8/35 × 7,000 = ₹1,600.

A quick method for three ratios is to cross-multiply in a pattern, but making the shared term equal is easier to remember and works for any number of ratios.

Name	Of a : b
Duplicate ratio	a² : b²
Triplicate ratio	a³ : b³
Sub-duplicate ratio	√a : √b
Sub-triplicate ratio	∛a : ∛b
Inverse ratio	b : a, or 1/a : 1/b
Compounded ratio of a : b and c : d	ac : bd

The duplicate ratio of 3 : 4 is 9 : 16, and its sub-duplicate ratio is √3 : 2. The compounded ratio multiplies the first terms together and the second terms together.

Worked example 2: compounding three ratios

The compounded ratio of 2 : 3, 4 : 5 and 5 : 8 is (2 × 4 × 5) : (3 × 5 × 8) = 40 : 120 = 1 : 3.

Term	Proportion	Formula
Fourth proportional to a, b, c	a : b :: c : d	d = bc ÷ a
Third proportional to a, b	a : b :: b : c	c = b² ÷ a
Mean proportional between a, b	a : x :: x : b	x = √(ab)

Worked example 3: fourth proportional to 4, 6 and 10

d = 6 × 10 ÷ 4 = 15. Check: 4 : 6 = 10 : 15 = 2 : 3.

Worked example 4: third proportional to 9 and 12

c = 12² ÷ 9 = 144 ÷ 9 = 16.

Worked example 5: mean proportional between 4 and 25

x = √(4 × 25) = √100 = 10. Check: 4 : 10 = 10 : 25 = 2 : 5.

To share an amount in a ratio of fractions, clear the fractions first by multiplying every term by the LCM of the denominators.

Worked example 6: shares of 1/2 : 1/3 : 1/4

Multiply by 12 to get 6 : 4 : 3, a total of 13 parts. Out of ₹1,300, the first share is 6/13 × 1,300 = ₹600.

When the same number is added to both terms to change the ratio, write the new ratio as an equation and cross-multiply.

Worked example 7: adding to both terms

What must be added to each term of 3 : 5 to make it 5 : 6? (3 + x) ÷ (5 + x) = 5 ÷ 6, so 18 + 6x = 25 + 5x and x = 7. Check: 10 : 12 = 5 : 6.

Worked example 8: incomes and expenses

Two people's incomes are in the ratio 3 : 4 and their expenses in the ratio 5 : 7. Each saves ₹3,000. With incomes 3x and 4x and expenses 5y and 7y: 3x − 5y = 3,000 and 4x − 7y = 3,000. Solving gives y = 3,000 and x = 6,000, so the first income is ₹18,000.

✘ Combining A : B and B : C without first making B equal.
✘ Confusing the third proportional (b² ÷ a) with the fourth (bc ÷ a).
✘ Sharing by fractional ratios without clearing the fractions.
✘ Comparing quantities in different units before converting them.

Ratios drive partnership problems, problems on ages and alligation and mixture. The formula list is in the Quant cheat code.

Q1. If A : B = 3 : 4 and B : C = 6 : 7, what is A : C? easy

(a) 1 : 2 (b) 9 : 14 (c) 3 : 7 (d) 18 : 21

✅ Answer: (b) 9 : 14

💡 Explanation: Make B the same in both ratios. LCM of 4 and 6 is 12, so A : B = 9 : 12 and B : C = 12 : 14. Then A : C = 9 : 14.

Q2. What is the fourth proportional to 5, 8 and 15? easy

(a) 20 (b) 18 (c) 30 (d) 24

✅ Answer: (d) 24

💡 Explanation: If 5 : 8 :: 15 : d, the product of the extremes equals the product of the means: 5d = 8 × 15 = 120, so d = 24.

Q3. What is the mean proportional between 9 and 16? easy

(a) 12 (b) 12.5 (c) 25 (d) 144

✅ Answer: (a) 12

💡 Explanation: The mean proportional x satisfies 9 : x :: x : 16, so x² = 9 × 16 = 144 and x = 12.

Q4. What is the third proportional to 16 and 24? medium

(a) 32 (b) 40 (c) 36 (d) 20

✅ Answer: (c) 36

💡 Explanation: The third proportional c satisfies 16 : 24 :: 24 : c, so 16c = 24² = 576 and c = 36.

Q5. ₹1,080 is divided in the ratio 1/3 : 1/4 : 1/6. What is the largest share? medium

(a) ₹480 (b) ₹360 (c) ₹540 (d) ₹432

✅ Answer: (a) ₹480

💡 Explanation: Multiply each fraction by the LCM 12 to get 4 : 3 : 2, which is 9 parts. The largest share is 4/9 × 1,080 = ₹480.

Q6. What number must be subtracted from each of 23, 30, 57 and 78 so that the results are in proportion? hard

(a) 4 (b) 5 (c) 7 (d) 6

✅ Answer: (d) 6

💡 Explanation: Set (23 − x)(78 − x) = (30 − x)(57 − x). The x² terms cancel, leaving 1,794 − 101x = 1,710 − 87x, so 14x = 84 and x = 6. Check: 17 : 24 :: 51 : 72.

What is the difference between the third and the fourth proportional?

The fourth proportional to a, b and c is the d in a : b :: c : d, so d = bc ÷ a. The third proportional to a and b is the c in a : b :: b : c, so the middle term repeats and c = b² ÷ a.

Why do we multiply fractional ratios by the LCM?

Multiplying every term of a ratio by the same number does not change the ratio. Using the LCM of the denominators turns the fractions into whole numbers, which are much easier to add and divide.

All problems and solutions on this page are original. No outside facts are relied on; every numeric answer is recomputed by software before publishing.

How this page was prepared. Drafted with AI assistance (Claude (Anthropic AI model)) from the sources listed above. 14 calculations on this page were recomputed by software and matched. Automated checks also covered structure, internal links and duplicate text. Software checks cannot confirm facts. Reviewed by praveenkumar on 2026-10-23 against the sources listed above.

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