Quantitative Aptitude · Ratio and Proportion, Indices and Logarithms

Proportion and Continued Proportion for CA Foundation

Updated 1 October 2026 · Fact-checked

A proportion says two ratios are equal: a : b = c : d, so ad = bc. The fourth proportional is d = bc ÷ a. In continued proportion a : b = b : c, so b² = ac. The mean proportional is √(ac) and the third proportional is b² ÷ a.

Understand Proportion and Continued Proportion

A ratio compares two quantities of the same kind. A proportion is a statement that two ratios are equal. If a : b = c : d, we write a : b :: c : d and say a, b, c, d are in proportion.

In a proportion, a and d are the extremes (first and last terms). b and c are the means (middle terms). The key rule is: product of extremes = product of means, so ad = bc. Almost every question uses this one rule.

If three of the four terms are known, you can find the fourth. That unknown term d is called the fourth proportional to a, b, c. The order matters: d = bc ÷ a.

A continued proportion is a chain where the same term repeats as consequent and then antecedent. Three quantities a, b, c are in continued proportion if a : b = b : c. Here b is the mean proportional between a and c, and c is the third proportional to a and b. Extending the chain gives a : b = b : c = c : d, and so on.

The difference is simple. A proportion needs four terms (they can all be different). A continued proportion has the middle terms equal, so three terms are enough to define it.

Key formulas to remember

Proportion: a : b = c : d ⇔ a ÷ b = c ÷ d ⇔ ad = bc; Extremes are a and d. Means are b and c. Terms are non-zero.
Fourth proportional: Fourth proportional to a, b, c = bc ÷ a; Order matters: a : b = c : x gives x = bc ÷ a.
Continued proportion (three terms): a : b = b : c ⇔ b² = ac; The middle term b is the mean proportional.
Mean proportional: Mean proportional between a and c = √(ac); For positive a and c, take the positive root.
Third proportional: Third proportional to a and b = b² ÷ a; From a : b = b : x. Put the given terms in order.
Continued proportion (four terms): a : b = b : c = c : d ⇒ b² = ac, c² = bd, and with common ratio k: b = ak, c = ak², d = ak³; Use k to express every term in one variable.
Duplicate, triplicate, sub-duplicate ratios: Duplicate of a : b = a² : b²; triplicate = a³ : b³; sub-duplicate = √a : √b; If a, b, c are in continued proportion, a : c = a² : b² (the duplicate ratio of a : b).

How to solve Proportion and Continued Proportion questions

Use this method for any question on proportion, mean, third or fourth proportional, and continued proportion.

1Read the wording and note the order of terms. 'Fourth proportional to a, b, c' means a : b = c : x.
2Write the proportion as a statement: first : second = third : fourth.
3Identify the type. If the two middle terms are the same, it is a continued proportion.
4Apply cross multiplication: product of extremes = product of means.
5Solve for the unknown. For mean proportional take a square root. For third proportional divide b² by a.
6For continued proportion with more than three terms, put a common ratio k and write terms as a, ak, ak², ak³.
7Check by substituting back: both ratios should simplify to the same value.
8Compare your answer with the four options. Pick the one that matches exactly.

Quickest way: Cross-multiply and eliminate options

When to use it: Use when the question gives four options and you have under a minute.

Memorise three results: fourth = bc ÷ a, third = b² ÷ a, mean = √(ac).
Write the given numbers in order before calculating. This avoids order errors.
For mean proportional, test options: the correct option squared must equal a × c.
For 'are a, b, c in continued proportion?' just check b² = ac.
If a number is not a perfect square or the arithmetic is heavy, cancel common factors first.
Skip questions with long chains of unknowns on the first pass and return later. Wrong answers cost 0.25 marks.

Common mistakes in Proportion and Continued Proportion

Mixing the order when finding the fourth or third proportional
Students assume the largest or given numbers can be placed in any order.
Fix: Write the proportion in the order the question gives. Fourth to a, b, c is a : b = c : x, so x = bc ÷ a.
Writing the mean proportional as (a + c) ÷ 2
It is confused with the arithmetic mean.
Fix: Mean proportional is the geometric type: √(ac). Use b² = ac.
Forgetting to take the square root for mean proportional
Students stop after finding ac.
Fix: Always finish with √(ac). Check that your answer squared gives ac.
Using b² ÷ c for the third proportional
Students mix up which term is divided.
Fix: For a : b = b : x, x = b² ÷ a. Divide by the first term.
Saying a, b, c are in continued proportion because a : b = c : d
Proportion and continued proportion are treated as the same.
Fix: Continued proportion needs the middle terms equal: a : b = b : c. Check b² = ac.
Cross-multiplying wrongly, such as ab = cd
Students multiply along the row instead of across.
Fix: Multiply extremes (first and last) and means (middle two): ad = bc.

Worked examples

Example 1

The fourth proportional to 4, 9 and 12 is: (A) 27 (B) 18 (C) 36 (D) 24

Show the solution

Write 4 : 9 = 12 : x.
Cross multiply: 4 × x = 9 × 12.
4x = 108.
x = 108 ÷ 4 = 27.
Check: 4 : 9 and 12 : 27 both reduce to 4 : 9.

Answer: (A) 27

Example 2

The mean proportional between 4 and 25 is: (A) 14.5 (B) 100 (C) 10 (D) 20

Show the solution

Mean proportional = √(a × c).
a × c = 4 × 25 = 100.
√100 = 10.
Check: 4 : 10 = 2 : 5 and 10 : 25 = 2 : 5. Both are equal.
Option (A) is the arithmetic mean, so it is a trap.

Answer: (C) 10

Example 3

If 3, x, 12 are in continued proportion and 12, 18, y are in continued proportion, then x + y is: (A) 33 (B) 24 (C) 30 (D) 42

Show the solution

For 3, x, 12: x² = 3 × 12 = 36, so x = 6.
For 12, 18, y: 18² = 12 × y.
324 = 12y, so y = 27.
x + y = 6 + 27 = 33.
Check: 12 : 18 = 2 : 3 and 18 : 27 = 2 : 3.

Answer: (A) 33

Exam tips

Most questions are direct: find the fourth, third or mean proportional. Practise these until they take under 30 seconds.
Watch for option traps like the arithmetic mean or a wrongly ordered answer.
Questions on 'a, b, c in continued proportion' often ask you to prove a relation like a : c = a² : b². Use b² = ac or substitute b = ak, c = ak².
Do not skip the order check. Many wrong answers come from reading the terms in the wrong sequence.
Use property tricks from componendo and dividendo when the question gives sums or differences of terms.

Practice questions from Ratio and Proportion, Indices and Logarithms

Proportion and Continued Proportion: frequently asked questions

What is the difference between proportion and continued proportion?

A proportion has four terms with a : b = c : d, and the product of extremes equals the product of means. A continued proportion has the middle terms equal, a : b = b : c, so b² = ac. Continued proportion is a special case of proportion.

How do I find the mean proportional?

Multiply the two given numbers and take the square root. For 4 and 25, the product is 100 and the mean proportional is 10. You can check that 4 : 10 equals 10 : 25.

What is the formula for the third proportional?

The third proportional to a and b is b² ÷ a. It comes from a : b = b : x. For 4 and 6, it is 36 ÷ 4 = 9.

Is the order of terms important in a proportion?

Yes. The fourth proportional to a, b, c is bc ÷ a, which differs from the answer if you swap the terms. Always write the proportion in the order given in the question.