MathsICSE Class 10 · Ratio and Proportion

Componendo and Dividendo: Property, Method and Solved Questions (ICSE Class 10)

Updated 15 October 2026 · Fact-checked

Componendo and dividendo says: if a/b = c/d, then (a + b)/(a − b) = (c + d)/(c − d). It also works backwards. Use it when an equation has the form (sum)/(difference) = number, or when adding and subtracting numerator and denominator will cancel awkward terms, and always write 'by componendo and dividendo'.

Add and subtract top and bottom to turn a ratio into sum over difference, or reverse it to get x/y; write 'by componendo and dividendo'.

Illustration drawn for QuestionBuddy.

Understand Componendo and Dividendo

The properties of proportion let you rewrite $\frac{a}{b} = \frac{c}{d}$ in useful new forms. The most powerful one, and the most examined, is componendo and dividendo:

$\frac{a}{b} = \frac{c}{d} \Rightarrow \frac{a + b}{a - b} = \frac{c + d}{c - d} .$

Why it is true. From $\frac{a}{b} = \frac{c}{d}$ , adding 1 to both sides gives $\frac{a + b}{b} = \frac{c + d}{d}$ (componendo); subtracting 1 gives $\frac{a - b}{b} = \frac{c - d}{d}$ (dividendo). Dividing the first by the second cancels $b$ and $d$ and leaves the result. The converse also holds: if $\frac{a + b}{a - b} = \frac{c + d}{c - d}$ , then $\frac{a}{b} = \frac{c}{d}$ . So you can apply the property to an equation that is already in 'sum over difference' form to get back to a simple ratio.

When to use it.

1. Sum over difference given. If $\frac{x + y}{x - y} = \frac{5}{3}$ , then $\frac{x}{y} = \frac{5 + 3}{5 - 3} = 4$ .
2. Cubes. If $\frac{( a + b ) ^{3}}{( a - b ) ^{3}} = \frac{64}{27}$ (2024 Q4(iii)), take the cube root first: $\frac{a + b}{a - b} = \frac{4}{3}$ , then apply the property: $\frac{a}{b} = \frac{7}{1}$ . Expressions like $\frac{x ^{3} + 3 x}{3 x ^{2} + 1}$ turn into $(\frac{x + 1}{x - 1})^{3}$ because $(x^{3} + 3 x^{2} + 3 x + 1)$ and $(x^{3} - 3 x^{2} + 3 x - 1)$ are the cubes of $(x + 1)$ and $(x - 1)$ .
3. Square roots. For $\frac{p + q}{p - q} = k$ , the property gives $\frac{p}{q} = \frac{k + 1}{k - 1}$ ; square both sides and, if needed, apply the property again.
4. Repeated terms. In 2025 Q5(i), each fraction had the same terms on top and below plus an $x^{2}$ term; adding or subtracting the denominator isolated $6 x^{2}$ and $9 x^{2}$ and led to $x = 5$ .

Write the property's name. The question usually says 'using properties of proportion'. The examiners reported, for both 2024 Q4(iii) and 2025 Q5(i), candidates who used cross-multiplication instead; that loses the method marks even if the answer is right.

Final form. Give a ratio as $a : b = 7 : 1$ , not as $\frac{a}{b} = 7$ , and check any value of $x$ by substituting back.

Componendo and Dividendo: formulas to remember

Componendo: $\frac{a}{b} = \frac{c}{d} \Rightarrow \frac{a + b}{b} = \frac{c + d}{d}$; Add 1 to both sides.
Dividendo: $\frac{a}{b} = \frac{c}{d} \Rightarrow \frac{a - b}{b} = \frac{c - d}{d}$; Subtract 1 from both sides.
Componendo and dividendo: $\frac{a}{b} = \frac{c}{d} \Rightarrow \frac{a + b}{a - b} = \frac{c + d}{c - d}$; And conversely.
Shortcut for a single number: $\frac{x + y}{x - y} = k \Rightarrow \frac{x}{y} = \frac{k + 1}{k - 1}$; Write $k$ as $\frac{k}{1}$ and apply the property.
Cube pattern: $\frac{x ^{3} + 3 x}{3 x ^{2} + 1} = \frac{m}{n} \Rightarrow (\frac{x + 1}{x - 1})^{3} = \frac{m + n}{m - n}$; Numerator + denominator is $(x + 1)^{3}$ ; numerator − denominator is $(x - 1)^{3}$ .

How to solve Componendo and Dividendo questions

Look for a 'sum over difference' shape, or terms that cancel when numerator and denominator are added and subtracted.

1Write the given equation as one fraction equal to another (or to a number over 1).
2If there are cubes or squares of a whole fraction, take the cube root or square root first.
3Apply componendo and dividendo, writing the property's name.
4Simplify: common factors such as 2 cancel top and bottom.
5If square roots remain, square both sides and, if useful, apply the property again.
6Solve the resulting simple equation, or write the ratio in the form $a : b$ .
7Check by substituting back into the original equation.

Componendo and Dividendo in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Recovering a ratio from sum and difference

Suppose a class teacher notes that the ratio of (boys + girls) to (boys − girls) in her class of 36 is 9 : 1.

Find the ratio of boys to girls and the number of each.

Try it first, then show the solutionSolution

$\frac{b + g}{b - g} = \frac{9}{1}$ ; by componendo and dividendo, $\frac{b}{g} = \frac{9 + 1}{9 - 1} = \frac{10}{8} = \frac{5}{4}$ .
$b : g = 5 : 4$ ; out of 36: $b = 20$ , $g = 16$ .

Answer: 5 : 4, that is 20 boys and 16 girls.

Memory hook: Sum over difference? Add and subtract to get back the simple ratio.

Speeds in a stream

Imagine a boat whose speed downstream ( $u + v$ ) and upstream ( $u - v$ ) are in the ratio 5 : 3, where $u$ is its speed in still water and $v$ the speed of the stream.

Find $u : v$ , and $u$ if the stream flows at 3 km/h.

Try it first, then show the solutionSolution

$\frac{u + v}{u - v} = \frac{5}{3} \Rightarrow \frac{u}{v} = \frac{5 + 3}{5 - 3} = 4$ .
$u : v = 4 : 1$ ; with $v = 3$ , $u = 12$ km/h.

Answer: $u : v = 4 : 1$ ; the boat's still-water speed is 12 km/h.

Memory hook: Downstream over upstream is a ready-made sum over difference.

Tricks to remember Componendo and Dividendo

Short memory aids. Say each one aloud once, then use it on a question the same day.

Top plus bottom over top minus bottom
Apply the same move to both sides: (top + bottom) ÷ (top − bottom).
Use it when: Applying the property correctly.
(k + 1) over (k − 1)
If sum over difference is $k$ , the plain ratio is $\frac{k + 1}{k - 1}$ .
Use it when: Fast MCQs.
Spot the binomial cube
$x^{3} + 3 x$ with $3 x^{2} + 1$ (or $a^{3} + 3 a b^{2}$ with $3 a^{2} b + b^{3}$ ): add and subtract to make perfect cubes.
Use it when: Cube-type questions.
Name it to gain it
Write 'by componendo and dividendo' every time; cross-multiplying loses method marks.
Use it when: Any 'using properties of proportion' question.

Componendo and Dividendo: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Ratio from sum over difference · 1 mark · MCQBoard level

If $\frac{x + y}{x - y} = \frac{5}{3}$ , then $x : y$ is:
(a) 5 : 3
(b) 4 : 1
(c) 8 : 3
(d) 1 : 4

Try it first, then show the solutionSolution

By componendo and dividendo: $\frac{2 x}{2 y} = \frac{5 + 3}{5 - 3} = \frac{8}{2}$ .
$\frac{x}{y} = 4$ .

Answer: (b) 4 : 1

Example 2: Finding x · 3 marks · Short answerBoard level

Using properties of proportion, solve $\frac{x + 4}{x - 4} = \frac{5}{3}$ .

Try it first, then show the solutionSolution

By componendo and dividendo: $\frac{( x + 4 ) + ( x - 4 )}{( x + 4 ) - ( x - 4 )} = \frac{5 + 3}{5 - 3}$ .
$\frac{2 x}{8} = \frac{8}{2} = 4$ .
$x = 16$ . Check: $\frac{20}{12} = \frac{5}{3}$ ✓.

Answer: $x = 16$

Example 3: A cube pattern · 4 marks · Short answerHarder board level

Using properties of proportion, solve $\frac{x ^{3} + 3 x}{3 x ^{2} + 1} = \frac{35}{19}$ .

Try it first, then show the solutionSolution

By componendo and dividendo: $\frac{x ^{3} + 3 x ^{2} + 3 x + 1}{x ^{3} - 3 x ^{2} + 3 x - 1} = \frac{35 + 19}{35 - 19} = \frac{54}{16} = \frac{27}{8}$ .
$(\frac{x + 1}{x - 1})^{3} = \frac{27}{8} \Rightarrow \frac{x + 1}{x - 1} = \frac{3}{2}$ .
Again by componendo and dividendo: $\frac{2 x}{2} = \frac{3 + 2}{3 - 2} = 5$ , so $x = 5$ .
Check: $\frac{125 + 15}{75 + 1} = \frac{140}{76} = \frac{35}{19}$ ✓.

Answer: $x = 5$

Example 4: Square roots · 4 marks · Short answerHarder board level

Solve $\frac{x + 3 + x - 3}{x + 3 - x - 3} = 2$ .

Try it first, then show the solutionSolution

By componendo and dividendo: $\frac{2 x + 3}{2 x - 3} = \frac{2 + 1}{2 - 1} = 3$ .
Square: $\frac{x + 3}{x - 3} = 9$ .
By componendo and dividendo again: $\frac{2 x}{6} = \frac{9 + 1}{9 - 1} = \frac{5}{4} \Rightarrow x = \frac{15}{4}$ .
Check: $x + 3 = \frac{27}{4}$ , $x - 3 = \frac{3}{4}$ ; ratio of square roots $= 3$ , and $\frac{3 + 1}{3 - 1} = 2$ ✓.

Answer: $x = \frac{15}{4}$

Example 5: A classic identity · 4 marks · Short answerHOTS / competency

If $x = \frac{2 ab}{a + b}$ ( $a \neq = b$ ), find the value of $\frac{x + a}{x - a} + \frac{x + b}{x - b}$ .

Try it first, then show the solutionSolution

$\frac{x}{a} = \frac{2 b}{a + b}$ ; by componendo and dividendo, $\frac{x + a}{x - a} = \frac{2 b + a + b}{2 b - a - b} = \frac{a + 3 b}{b - a}$ .
Similarly $\frac{x}{b} = \frac{2 a}{a + b}$ gives $\frac{x + b}{x - b} = \frac{3 a + b}{a - b}$ .
Sum $= \frac{a + 3 b}{b - a} + \frac{3 a + b}{a - b} = \frac{- ( a + 3 b ) + 3 a + b}{a - b} = \frac{2 a - 2 b}{a - b}$ .
$= 2$ .

Answer: 2

Example 6: Dividendo on matching terms · 3 marks · Short answerHOTS / competency

Using properties of proportion, solve $\frac{2 x ^{2} + x + 4}{x + 4} = \frac{3 x ^{2} + 2 x + 3}{2 x + 3}$ , $x \neq = 0$ .

Try it first, then show the solutionSolution

By dividendo (subtract 1 from each side): $\frac{2 x ^{2}}{x + 4} = \frac{3 x ^{2}}{2 x + 3}$ .
Divide both sides by $x^{2}$ ( $x \neq = 0$ ): $\frac{2}{x + 4} = \frac{3}{2 x + 3}$ .
$2 (2 x + 3) = 3 (x + 4) \Rightarrow 4 x + 6 = 3 x + 12 \Rightarrow x = 6$ .
Check: $\frac{82}{10} = 8.2$ and $\frac{123}{15} = 8.2$ ✓.

Answer: $x = 6$

Where students lose marks in Componendo and Dividendo

The slips examiners see most often, and what to write instead.

Mistake: Cross-multiplying instead of using the property
The question asks for properties of proportion; the method marks are for them.
Wrong working
$\frac{x + 4}{x - 4} = \frac{5}{3} \Rightarrow 3 x + 12 = 5 x - 20$ .
Right working
By componendo and dividendo: $\frac{2 x}{8} = \frac{8}{2}$ .
Do this: Name the property and apply it as the first step.
Usually costs: Examiners named this for 2024 Q4(iii) and 2025 Q5(i).
Mistake: Wrong cube root
$364 = 4$ and $327 = 3$ .
Wrong working
$\frac{( a + b ) ^{3}}{( a - b ) ^{3}} = \frac{64}{27} \Rightarrow \frac{a + b}{a - b} = \frac{8}{3}$ .
Right working
$\frac{a + b}{a - b} = \frac{4}{3}$ .
Do this: Learn cubes up to $1 0^{3}$ .
Usually costs: Reported for 2024 Q4(iii).
Mistake: Sign errors when subtracting a bracket
$(x + 4) - (x - 4) = 8$ , not 0.
Wrong working
$(x + 4) - (x - 4) = x + 4 - x - 4 = 0$ .
Right working
$= x + 4 - x + 4 = 8$ .
Do this: Keep brackets while subtracting.
Usually costs: Reported for 2025 Q5(i).
Mistake: Leaving the answer as a fraction when a ratio is asked
The question asks for $a : b$ .
Wrong working
$\frac{a}{b} = 7$ .
Right working
$a : b = 7 : 1$ .
Do this: Finish in the form the question uses.
Usually costs: Reported for 2024 Q4(iii).

Exam tips

In 2024 Q4(iii) the expected steps were $\frac{a + b}{a - b} = \frac{4}{3}$ , then $\frac{2 a}{2 b} = \frac{7}{1}$ , so $a : b = 7 : 1$ .
In 2025 Q5(i) the expected route used the property to isolate $6 x^{2}$ and $9 x^{2}$ , reaching $x = 5$ .
When a variable appears as $x^{2}$ in both numerators, think 'dividendo'.
Always check $x$ by substitution; it catches sign errors.

Practice questions on Componendo and Dividendo

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
If $\frac{a}{b} = \frac{3}{2}$ , the value of $\frac{a + b}{a - b}$ is:
(a) 5
(b) $\frac{1}{5}$
(c) $\frac{5}{2}$
(d) 1
Hint
$\frac{3 + 2}{3 - 2}$ .
Answer
(a) 5
Q2 · 3 marks · Short answerBoard level
Using properties of proportion, solve $\frac{2 x + 3}{2 x - 3} = \frac{7}{5}$ .
Hint
Componendo and dividendo gives $\frac{4 x}{6}$ on the left.
Answer
$\frac{4 x}{6} = \frac{12}{2} = 6 \Rightarrow x = 9$ .
Q3 · 3 marks · Short answerHarder board level
If $\frac{( a + b ) ^{3}}{( a - b ) ^{3}} = \frac{343}{27}$ , find $a : b$ .
Hint
Cube root first.
Answer
$\frac{a + b}{a - b} = \frac{7}{3} \Rightarrow \frac{a}{b} = \frac{10}{4}$ , so $a : b = 5 : 2$ .
Q4 · 4 marks · Short answerHarder board level
Solve $\frac{2 + x + 2 - x}{2 + x - 2 - x} = 3$ .
Hint
After the property, square both sides.
Answer
$\frac{2 + x}{2 - x} = 2 \Rightarrow \frac{2 + x}{2 - x} = 4 \Rightarrow \frac{4}{2 x} = \frac{5}{3} \Rightarrow x = \frac{6}{5}$ .
Q5 · 1 mark · Assertion-reasonHarder board level
Assertion (A): If $\frac{x + 3}{x - 3} = 2$ , then $x = 9$ .
Reason (R): If $\frac{a}{b} = \frac{c}{d}$ , then $\frac{a + b}{a - b} = \frac{c + d}{c - d}$ .
(a) (A) is true and (R) is false.
(b) (A) is false and (R) is true.
(c) Both (A) and (R) are true, and (R) is the correct explanation for (A).
(d) Both (A) and (R) are true, but (R) is not the correct explanation for (A).
Hint
$\frac{2 x}{6} = \frac{2 + 1}{2 - 1}$ .
Answer
(c). $\frac{2 x}{6} = 3 \Rightarrow x = 9$ , using (R).
Q6 · 4 marks · Short answerHOTS / competency
If $x = \frac{a + 1 + a - 1}{a + 1 - a - 1}$ , prove that $x^{2} - 2 a x + 1 = 0$ .
Hint
Apply the property, square, then apply it again.
Answer
$\frac{x + 1}{x - 1} = \frac{a + 1}{a - 1}$ ; squaring, $\frac{x ^{2} + 2 x + 1}{x ^{2} - 2 x + 1} = \frac{a + 1}{a - 1}$ ; by componendo and dividendo, $\frac{2 ( x ^{2} + 1 )}{4 x} = \frac{2 a}{2}$ , so $x^{2} + 1 = 2 a x$ , that is $x^{2} - 2 a x + 1 = 0$ .
Q7 · 4 marks · Short answerHarder board level
Using properties of proportion, solve $\frac{x ^{3} + 12 x}{6 x ^{2} + 8} = \frac{14}{13}$ .
Hint
Numerator ± denominator are $(x + 2)^{3}$ and $(x - 2)^{3}$ .
Answer
By componendo and dividendo: $(\frac{x + 2}{x - 2})^{3} = \frac{14 + 13}{14 - 13} = \frac{27}{1} \Rightarrow \frac{x + 2}{x - 2} = 3 \Rightarrow \frac{2 x}{4} = \frac{4}{2} \Rightarrow x = 4$ .

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.State componendo and dividendo.Show answer
If $\frac{a}{b} = \frac{c}{d}$ , then $\frac{a + b}{a - b} = \frac{c + d}{c - d}$ .
2. $\frac{x + y}{x - y} = 3$ . Find $x : y$ .Show answer
$\frac{3 + 1}{3 - 1} = 2$ , so 2 : 1.
3. $\frac{a}{b} = \frac{5}{3}$ . Find $\frac{a + b}{a - b}$ .Show answer
4.
4.What are $(x^{3} + 3 x^{2} + 3 x + 1)$ and $(x^{3} - 3 x^{2} + 3 x - 1)$ ?Show answer
$(x + 1)^{3}$ and $(x - 1)^{3}$ .
5.Cube root of $\frac{125}{8}$ ?Show answer
$\frac{5}{2}$ .

Componendo and Dividendo: frequently asked questions

What is componendo and dividendo?

It is the property of proportion: if a/b = c/d, then (a + b)/(a − b) = (c + d)/(c − d). Componendo adds the denominator to the numerator; dividendo subtracts it.

Can I solve these questions by cross-multiplication?

You may get the right answer, but if the question says 'using properties of proportion', the examiners expect the property and give method marks for it.

How do I use componendo and dividendo with square roots?

Apply the property to get a ratio of two square roots, square both sides, and apply the property again if the result is a sum over difference.