Fundamentals of Business Mathematics and Statistics · Ratios, Variations and Proportions

Proportion and Continued Proportion: Rules and Solved Problems

Updated 10 October 2026 · Fact-checked

A proportion is an equality of two ratios, a : b = c : d, which means ad = bc. Fourth proportional is d, third proportional to a and b is b²÷a, and mean proportional between a and b is √(ab). For ratio-manipulation problems, use componendo and dividendo.

Understand Proportion and Continued Proportion

A ratio compares two quantities of the same kind, like 6 : 9. A proportion says 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.

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

The difference between ratio and proportion is simple. A ratio compares two quantities and has two terms. A proportion is a statement that two ratios are equal and has four terms (or three in a continued proportion). A ratio can be simplified. A proportion can be true or false.

In a continued proportion, a : b = b : c. Here b is the mean proportional between a and c, and c is the third proportional to a and b. From b² = ac, you get b = √(ac) and c = b² ÷ a. A longer chain a : b = b : c = c : d is also a continued proportion, where b² = ac and c² = bd.

The properties (invertendo, alternendo, componendo, dividendo) let you change a proportion into a simpler form without redoing the algebra. They are valid when no denominator becomes zero.

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.
Fourth proportional: Fourth proportional to a, b, c = bc ÷ a; Taken in the order a : b = c : x.
Third proportional: Third proportional to a and b = b² ÷ a; From a : b = b : x.
Mean proportional: Mean proportional between a and b = √(ab); For positive a and b, from a : x = x : b.
Continued proportion: a : b = b : c = c : d ⇒ b² = ac, c² = bd; Common ratio k gives b = ak, c = ak², d = ak³.
Invertendo: If a/b = c/d, then b/a = d/c; Flip both ratios.
Alternendo: If a/b = c/d, then a/c = b/d; Swap the means.
Componendo: If a/b = c/d, then (a + b)/b = (c + d)/d; Add 1 to both sides.
Dividendo: If a/b = c/d, then (a − b)/b = (c − d)/d; Subtract 1 from both sides.
Componendo and dividendo: If a/b = c/d, then (a + b)/(a − b) = (c + d)/(c − d); Valid when a ≠ b and c ≠ d.

How to solve Proportion and Continued Proportion questions

Use this order for most questions on proportion and continued proportion.

1Identify what is asked: fourth, third or mean proportional, or a proof or value problem using properties.
2Write the proportion in the correct order. Order matters: fourth proportional to a, b, c means a : b = c : x.
3Convert to a cross product: extremes product = means product.
4For continued proportion, use b² = ac. For a longer chain, put the common ratio as k and write the terms as ak, ak², ak³.
5If the question has a ratio like (x + y)/(x − y), try componendo and dividendo before expanding.
6Solve for the unknown and check that the answer is positive where quantities must be positive.
7Verify by substituting back: both ratios should be equal.

Quickest way: Cross-multiply and use the shortcut formulas

When to use it: Use this when the MCQ asks for a proportional or when a ratio of the form (a + b)/(a − b) is given.

Fourth proportional: multiply the two terms that are not first, divide by the first. For 2, 5, 6 it is 5 × 6 ÷ 2 = 15.
Third proportional: square the second term, divide by the first.
Mean proportional: multiply the two numbers and take the square root. Check whether the product is a perfect square.
If x/y = p/q and the question asks for (x + y)/(x − y), write (p + q)/(p − q) directly.
If answer options are numbers, plug each option back into the cross product to eliminate wrong ones.

Common mistakes in Proportion and Continued Proportion

Taking the third proportional to a and b as a²÷b.
Students mix up which term is squared.
Fix: Remember a : b = b : x. The repeated term b is squared, so x = b² ÷ a.
Using the average (a + b)/2 as the mean proportional.
The word 'mean' suggests the arithmetic mean.
Fix: Mean proportional is √(ab), the geometric mean of the two numbers.
Writing the fourth proportional in the wrong order.
Students cross-multiply without fixing the order a : b = c : x.
Fix: Always write the proportion in the given order of terms first, then solve for x.
Applying componendo and dividendo as (a + b)/(a − b) = (c − d)/(c + d).
The signs get swapped when writing quickly.
Fix: Plus over minus on the left must match plus over minus on the right.
Forgetting that the property needs a ≠ b, or a non-zero denominator.
Students treat the formula as always valid.
Fix: Check that the denominator you create is not zero before using the result.
Treating a : b = c : d as a + b = c + d.
Ratios are confused with differences.
Fix: Equal ratios mean ad = bc. Sums and differences are not equal in general.

Worked examples

Example 1

Find the mean proportional between 4 and 25, and the third proportional to 4 and 6.

Show the solution

Mean proportional = √(4 × 25) = √100 = 10.
Third proportional to 4 and 6 = 6² ÷ 4.
6² = 36, and 36 ÷ 4 = 9.
Check: 4 : 6 = 6 : 9. Both ratios equal 2/3.

Answer: Mean proportional = 10 and third proportional = 9.

Example 2

If x/y = 5/3, find the value of (x + y)/(x − y).

Show the solution

Apply componendo and dividendo: (x + y)/(x − y) = (5 + 3)/(5 − 3).
This equals 8/2.
8/2 = 4.
Check with x = 5, y = 3: (5 + 3)/(5 − 3) = 8/2 = 4.

Answer: 4

Exam tips

Questions are usually direct: find the fourth, third or mean proportional. Practise the three formulas until they are automatic.
When a ratio such as (a + b)/(a − b) appears, componendo and dividendo usually gives the answer in one line.
In continued proportion, put the common ratio as k and write terms as ak, ak², ak³ to solve proofs quickly.
There is no negative marking, so always mark an option. Eliminate those that fail the cross-product check.

Practice questions from Ratios, Variations and Proportions

Proportion and Continued Proportion in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

CA FoundationProportion and Continued Proportion

Proportion and Continued Proportion: frequently asked questions

What is the difference between ratio and proportion?

A ratio compares two quantities of the same kind, such as 3 : 5. A proportion states that two ratios are equal, such as 3 : 5 = 6 : 10. A ratio has two terms and a proportion has four.

How do I find the mean proportional between two numbers?

Multiply the two numbers and take the square root. For 9 and 16, the product is 144 and the mean proportional is 12. It comes from the condition a : x = x : b, so x² = ab.

When can I use componendo and dividendo?

Use it when a/b = c/d and you need a ratio like (a + b)/(a − b). It is valid only when the denominators involved are non-zero, so a ≠ b and c ≠ d.

What is the third proportional to a and b?

It is the number x such that a : b = b : x. So x = b² ÷ a. For 4 and 6 it is 9.