Quantitative Aptitude · Ratio and Proportion, Indices and Logarithms

Properties of Proportion: Componendo and Dividendo

Updated 1 October 2026

Componendo and dividendo are rules that simplify a proportion a/b = c/d by adding or subtracting the denominator to or from the numerator. Combined, they give (a+b)/(a−b) = (c+d)/(c−d). Use them when a ratio has the form (x+y)/(x−y) and you must find x/y or x.

Understand Properties of Proportion: Componendo and Dividendo

A proportion says two ratios are equal: a/b = c/d. Because the two sides are equal, you can change both sides in the same way and the equality stays true. The properties of proportion are standard ways of doing this.

The four basic properties are invertendo (flip both ratios), alternendo (swap the middle terms), componendo (add the denominator to the numerator) and dividendo (subtract the denominator from the numerator). Each one is just algebra on the equation a/b = c/d.

The real power comes when you use componendo and dividendo together. Dividing the componendo result by the dividendo result gives (a+b)/(a−b) = (c+d)/(c−d). This works in reverse as well. If a question gives you an expression like (x+y)/(x−y) = 5/3, you can read off x/y quickly without cross-multiplying.

The typical exam question gives a messy fraction with sums and differences, often with square roots or powers. You apply the rule to get a simple ratio, then solve. It saves time in an MCQ paper where each wrong answer costs 0.25 marks.

All the rules need the denominators to be non-zero. For dividendo and the combined rule, you also need a ≠ b and c ≠ d so you never divide by zero.

Key formulas to remember

Invertendo: If a/b = c/d, then b/a = d/c; Flip both sides. Needs a, c non-zero.
Alternendo: If a/b = c/d, then a/c = b/d; Swap the means (b and c). Needs c and d non-zero.
Componendo: If a/b = c/d, then (a + b)/b = (c + d)/d; Add 1 to both sides. The denominator stays as it is.
Dividendo: If a/b = c/d, then (a − b)/b = (c − d)/d; Subtract 1 from both sides. The denominator stays as it is.
Componendo and dividendo: If a/b = c/d, then (a + b)/(a − b) = (c + d)/(c − d); Needs a ≠ b and c ≠ d. Works in reverse: from (a+b)/(a−b) = p/q, get a/b = (p+q)/(p−q).
Reverse form: If (x + y)/(x − y) = p/q, then x/y = (p + q)/(p − q); The most used form in exam questions. Needs p ≠ q.

How to solve Properties of Proportion: Componendo and Dividendo questions

Use this method when a question gives an equation or ratio built from sums and differences and asks for a ratio or a value.

1Write the given relation as a single fraction equal to a single fraction, such as (A + B)/(A − B) = p/q.
2Check that the numerator and denominator are a sum and a difference of the same two quantities A and B.
3Apply componendo and dividendo in reverse: A/B = (p + q)/(p − q).
4Simplify the numbers. Reduce the ratio to its lowest terms.
5If A and B are powers or roots (such as √x or x²), remove them by squaring, cubing or taking roots on both sides.
6Substitute the ratio into the expression asked in the question, or solve for the variable.
7Check by picking easy numbers that fit the ratio and testing them in the original equation.

Quickest way: Read off the ratio, then test with numbers

When to use it: Use it when the question has the form (x + y)/(x − y) = p/q, or when the given fraction has the same pair of terms added on top and subtracted below.

Spot the pattern: same two terms added above and subtracted below.
Write x/y = (p + q)/(p − q) straight away. Top is the sum of the numbers, bottom is their difference.
Set x = (p + q)k and y = (p − q)k, reduced to lowest terms. For example, for p/q = 7/3 take x = 5, y = 2.
Plug these into each option or into the asked expression and eliminate wrong options.
If the pattern is not there, do not force it. Skip to cross-multiplication or move on if it is taking too long.

Common mistakes in Properties of Proportion: Componendo and Dividendo

Writing x/y = (p − q)/(p + q) instead of (p + q)/(p − q).
Students flip the signs from memory without checking which way the rule works.
Fix: Remember that the sum of the numbers goes on top. Test with p/q = 3/1: (x+y)/(x−y) = 3 gives x = 2y, so x/y = 2, which equals (3+1)/(3−1).
Applying componendo to the numerator but changing the denominator too.
Students think both terms of the ratio must change.
Fix: In componendo, (a + b)/b has the original denominator b. Only the numerator changes.
Forgetting to square or cube back after reaching a ratio of roots.
The ratio found is for √x/√y, not x/y, and students stop early.
Fix: Write down what A and B are before you start. At the end, raise both sides to the needed power.
Using the rule when the pattern is not a pure sum over difference.
Students see a plus and minus and apply the trick blindly, for example to (x + 2y)/(x − 3y).
Fix: The same two terms must appear in sum and difference. Otherwise treat 2y and 3y as separate and use cross-multiplication.
Dividing by zero when p = q.
If p = q, then p − q = 0, which means y = 0 in the original relation.
Fix: Check p ≠ q before you apply the reverse form. If they are equal, look at the original equation separately.

Worked examples

Example 1

If (x + y)/(x − y) = 7/3, then x : y is: (a) 5 : 2 (b) 2 : 5 (c) 7 : 3 (d) 3 : 7

Show the solution

The pattern is a sum over a difference, so use the reverse form.
x/y = (7 + 3)/(7 − 3) = 10/4.
Reduce 10/4 to lowest terms: 5/2.
Check with x = 5, y = 2: (5 + 2)/(5 − 2) = 7/3. This is correct.

Answer: x : y = 5 : 2, which is option (a).

Example 2

If a/b = c/d, what is (3a + 2b)/(3a − 2b) equal to? (a) (3c + 2d)/(3c − 2d) (b) (a + c)/(b + d) (c) (3c − 2d)/(3c + 2d) (d) (2c + 3d)/(2c − 3d)

Show the solution

Given a/b = c/d, multiply both sides by 3/2 to get 3a/2b = 3c/2d.
Apply componendo and dividendo to this: (3a + 2b)/(3a − 2b) = (3c + 2d)/(3c − 2d).
This matches option (a).
Check with a = 2, b = 1, c = 4, d = 2: left side = (6 + 2)/(6 − 2) = 2. Right side = (12 + 4)/(12 − 4) = 2. Both are equal.

Answer: Option (a): (3c + 2d)/(3c − 2d).

Example 3

If (√x + √y)/(√x − √y) = 3, then the value of x/y is: (a) 2 (b) 4 (c) 3 (d) 9

Show the solution

Write 3 as 3/1.
Apply the reverse form with A = √x and B = √y: √x/√y = (3 + 1)/(3 − 1) = 4/2 = 2.
Square both sides: x/y = 2² = 4.
Check with x = 4, y = 1: (2 + 1)/(2 − 1) = 3. This is correct.

Answer: x/y = 4, which is option (b).

Exam tips

Look for the pattern first. If you see the same two terms added above and subtracted below, use the reverse form and move on in under a minute.
Always check what A and B are. If they are roots or powers, the final step is to square or take roots, and one wrong option is usually the unsquared answer.
Use small-number checks to eliminate options. Pick numbers that fit your ratio and test them in the original expression.
If the problem does not fit the pattern after ten seconds, skip it. A wrong answer costs 0.25 marks, and time saved helps with other questions.
Practise writing the ratio as k multiples so you can quickly set x = (p + q)k and y = (p − q)k when the question asks for a numeric value.

Practice questions from Ratio and Proportion, Indices and Logarithms

Properties of Proportion: Componendo and Dividendo: frequently asked questions

What is the difference between componendo and dividendo?

Componendo adds the denominator to the numerator: (a + b)/b = (c + d)/d. Dividendo subtracts it: (a − b)/b = (c − d)/d. Both start from a/b = c/d.

When should I use componendo and dividendo in CA Foundation?

Use them when an equation has the same two terms added on top and subtracted below, such as (x + y)/(x − y) = p/q. They are very useful when the terms are roots or powers. If the pattern is missing, use cross-multiplication.

Does alternendo change the value of the ratio?

Alternendo swaps the middle terms, so a/b = c/d becomes a/c = b/d. The new ratios are different, but the proportion remains true. It helps compare like quantities.

Can I use componendo and dividendo when a = b?

No. If a = b, then a − b = 0 and the combined form would divide by zero. Check that the quantities are not equal before you apply the rule.