MathsICSE Class 10 · Chapter 6

Ratio and Proportion ICSE Class 10: Properties, Componendo-Dividendo and Solved Questions

Updated 14 October 2026 · Fact-checked

Four quantities a, b, c, d are in proportion when a : b = c : d, so ad = bc. ICSE Class 10 covers continued and mean proportion, the properties invertendo, alternendo, componendo and dividendo and their combinations, and simple direct applications. Most questions ask you to find an unknown or prove a result using these properties.

What Ratio and Proportion is about

A ratio compares two quantities of the same kind: $a : b = \frac{a}{b}$ . A proportion is a statement that two ratios are equal: $a : b = c : d$ , written $a : b :: c : d$ . Here $a$ and $d$ are the extremes, $b$ and $c$ the means, and the key fact is product of extremes = product of means, $a d = b c$ .

The CISCE 2027 syllabus has three parts: (a) proportion, continued proportion and mean proportion; (b) the properties componendo, dividendo, alternendo and invertendo, and their combinations; (c) direct simple applications on proportions only.

The combination componendo and dividendo is the workhorse of the chapter. It turns a messy fraction equation, especially one with square roots or cubes, into a simple one. The examiners' reports for 2024 and 2025 say the same thing twice: when a question says 'using properties of proportion', candidates who simply cross-multiply lose marks, and those who apply componendo-dividendo without brackets make sign errors. They also lost marks by leaving the answer as $\frac{a}{b} = 7$ instead of $a : b = 7 : 1$ .

Weightage: CISCE publishes no chapter or unit weightage for ICSE Mathematics. Ratio and Proportion is in the 2. Algebra (iii) Ratio and Proportion part of the 2027 syllabus and can be asked in both Section A and Section B of the 80-mark paper.Source: CISCE ICSE Mathematics syllabus, examination year 2027

Key concepts

Proportion and the fourth proportional

$a$ , $b$ , $c$ , $d$ are in proportion if $\frac{a}{b} = \frac{c}{d}$ , which is equivalent to $a d = b c$ . The order matters: '3, 5, 12, 20 are in proportion' means $3 : 5 = 12 : 20$ .

If $a$ , $b$ , $c$ are given, the number $d$ that completes the proportion is the fourth proportional: $d = \frac{b c}{a}$ .

Worked example

Find the fourth proportional to 3, 5 and 12.

Try it first, then show the solutionSolution

$3 : 5 = 12 : d \Rightarrow 3 d = 60$ .
$d = 20$ .

Answer: 20

Continued proportion, third proportional and mean proportional

$a$ , $b$ , $c$ are in continued proportion if $a : b = b : c$ , i.e. $b^{2} = a c$ . Then:

$c = \frac{b ^{2}}{a}$ is the third proportional to $a$ and $b$ ;
$b = a c$ is the mean proportional between $a$ and $c$ .

More generally $a, b, c, d, \dots$ are in continued proportion if $\frac{a}{b} = \frac{b}{c} = \frac{c}{d} = \dots$ . The terms then grow (or shrink) by the same factor each time, just like a geometric progression.

Worked example

Find (a) the third proportional to 4 and 10, (b) the mean proportional between 6 and 24.

Try it first, then show the solutionSolution

(a) $4 : 10 = 10 : c \Rightarrow c = \frac{100}{4} = 25$ .
(b) $b^{2} = 6 \times 24 = 144 \Rightarrow b = 12$ .

Answer: (a) 25 (b) 12

Invertendo, alternendo, componendo, dividendo

If $\frac{a}{b} = \frac{c}{d}$ , then:

Invertendo: $\frac{b}{a} = \frac{d}{c}$ (turn both upside down).
Alternendo: $\frac{a}{c} = \frac{b}{d}$ (swap the means).
Componendo: $\frac{a + b}{b} = \frac{c + d}{d}$ (add 1 to both sides).
Dividendo: $\frac{a - b}{b} = \frac{c - d}{d}$ (subtract 1 from both sides).

Each is just a short name for a step of algebra on equal fractions, so in an answer you write the name of the property next to the step.

Worked example

Starting from $\frac{3}{4} = \frac{9}{12}$ , write the result of each property.

Try it first, then show the solutionSolution

Invertendo: $\frac{4}{3} = \frac{12}{9}$ .
Alternendo: $\frac{3}{9} = \frac{4}{12}$ .
Componendo: $\frac{7}{4} = \frac{21}{12}$ .
Dividendo: $\frac{- 1}{4} = \frac{- 3}{12}$ .

Answer: All four new statements are true proportions.

Componendo and dividendo together

If $\frac{a}{b} = \frac{c}{d}$ , then
$\frac{a + b}{a - b} = \frac{c + d}{c - d} .$
The converse also holds: if $\frac{a + b}{a - b} = \frac{c + d}{c - d}$ , then $\frac{a}{b} = \frac{c}{d}$ .

In words: (numerator + denominator) over (numerator − denominator) on both sides. It is most useful when the numerator and denominator of a fraction differ only in one sign, such as $\frac{3 x + 4}{3 x - 4}$ : the sum and difference cancel one term each.

Always put brackets round the numerator and the denominator before adding or subtracting, to avoid sign errors.

Worked example

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

Try it first, then show the solutionSolution

Componendo and dividendo: $\frac{( 3 x + 4 ) + ( 3 x - 4 )}{( 3 x + 4 ) - ( 3 x - 4 )} = \frac{5 + 3}{5 - 3}$ .
$\frac{6 x}{8} = \frac{8}{2} = 4$ .
$x = \frac{32}{6} = \frac{16}{3}$ .

Answer: x = 16/3

Componendo-dividendo with square roots

Expressions like $\frac{p + q}{p - q}$ are made for componendo-dividendo: the sum gives $2 p$ and the difference gives $2 q$ , so the 2s cancel and you are left with $\frac{p}{q}$ . Square both sides, then finish with simple algebra (or componendo-dividendo again).

Worked example

Solve $\frac{x + 5 + x - 5}{x + 5 - x - 5} = 3$ .

Try it first, then show the solutionSolution

Write 3 as $\frac{3}{1}$ ; componendo and dividendo: $\frac{2 x + 5}{2 x - 5} = \frac{4}{2}$ .
$\frac{x + 5}{x - 5} = 2$ ; squaring, $\frac{x + 5}{x - 5} = 4$ .
$x + 5 = 4 x - 20 \Rightarrow x = \frac{25}{3}$ .

Answer: x = 25/3

The k-method for proving results

To prove a result that starts 'If $\frac{a}{b} = \frac{c}{d}$ , show that …', let $\frac{a}{b} = \frac{c}{d} = k$ . Then $a = bk$ and $c = d k$ . Substitute into both sides of what you have to prove and simplify each side separately until they are the same expression in $k$ (and $b$ , $d$ ).

For continued proportion $a : b = b : c = k$ , write $b = c k$ and $a = c k^{2}$ .

Worked example

If $\frac{a}{b} = \frac{c}{d}$ , show that $\frac{a ^{2} + c ^{2}}{b ^{2} + d ^{2}} = \frac{a c}{b d}$ .

Try it first, then show the solutionSolution

Let $a = bk$ , $c = d k$ .
LHS $= \frac{b ^{2} k ^{2} + d ^{2} k ^{2}}{b ^{2} + d ^{2}} = k^{2}$ .
RHS $= \frac{bk \cdot d k}{b d} = k^{2}$ . LHS = RHS.

Answer: Proved.

Direct simple applications

Two quantities are in direct proportion if their ratio stays the same: when one doubles, so does the other. Distance and time at a fixed speed, cost and quantity at a fixed price, and the amounts of ingredients in a recipe all behave this way.

Set up the two ratios in the same order and use product of extremes = product of means. A quick check: the bigger quantity must go with the bigger value.

Worked example

5 kg of rice costs ₹240. Find the cost of 8 kg.

Try it first, then show the solutionSolution

$5 : 240 = 8 : x$ .
$5 x = 1920 \Rightarrow x = 384$ .

Answer: ₹384

Maths in real life: Ratio and Proportion

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.

Direct proportion in a recipe

Suppose a recipe for 4 people uses 300 g of rice, and you are cooking for 10 people.

How much rice do you need?

Try it first, then show the solutionSolution

Rice is in direct proportion to the number of people: $4 : 300 = 10 : x$ .
$4 x = 3000 \Rightarrow x = 750$ .

Answer: 750 g

Memory hook: More people, more rice, same ratio.

Changing a ratio by adding one part

Imagine you have mixed white and blue paint in the ratio 5 : 2 to make 3.5 litres of light blue, and you want a darker shade in the ratio 5 : 3.

How much blue paint must you add?

Try it first, then show the solutionSolution

White $= \frac{5}{7} \times 3.5 = 2.5$ L; blue $= 1$ L.
Need white : blue $= 5 : 3$ with white unchanged: $2.5 : b = 5 : 3 \Rightarrow b = 1.5$ L.
Add $1.5 - 1 = 0.5$ L of blue.

Answer: 0.5 litre

Memory hook: Keep the unchanged part fixed and find the new partner.

Continued proportion

Suppose three siblings' ages are in continued proportion, the youngest being 4 and the middle one 6.

How old is the eldest?

Try it first, then show the solutionSolution

$4 : 6 = 6 : x$ .
$4 x = 36 \Rightarrow x = 9$ .

Answer: 9 years

Memory hook: Middle squared = first × last.

Topics in this chapter

Ratio and Proportion formula sheet

Proportion: $a : b = c : d ⟺ a d = b c$; Extremes a, d; means b, c.
Fourth proportional: $d = \frac{b c}{a}$; To a, b, c.
Continued proportion: $a : b = b : c ⟺ b^{2} = a c$; Third proportional c = b²/a.
Mean proportional: $b = a c$; Between a and c.
Invertendo and alternendo: $\frac{b}{a} = \frac{d}{c}; \frac{a}{c} = \frac{b}{d}$; From a/b = c/d.
Componendo and dividendo (separately): $\frac{a + b}{b} = \frac{c + d}{d}; \frac{a - b}{b} = \frac{c - d}{d}$; Add or subtract 1 on both sides.
Componendo and dividendo: $\frac{a + b}{a - b} = \frac{c + d}{c - d}$; Works both ways.
k-method: $\frac{a}{b} = \frac{c}{d} = k \Rightarrow a = bk, c = d k$; For proving identities.

How the board asks Ratio and Proportion

The question patterns to expect, and the method that works for each.

Question type	How it is asked	Method
MCQ on third, fourth or mean proportional	Section A, Question 1: three numbers in continued proportion with one missing, or a mean proportional.	Use $b^{2} = a c$ or $a d = b c$ ; check the order of the terms.
MCQ on direct proportion in a table	A table of distance and time (or cost and quantity) with two gaps.	Find the constant ratio from the complete column, then fill each gap.
Solve using properties of proportion (3-4 marks)	'Using properties of proportion, find x' with a fraction equation, often with surds or cubes.	Componendo-dividendo with brackets; name the property; do not cross-multiply.
Find a : b	An equation in a and b such as $\frac{( a + b ) ^{3}}{( a - b ) ^{3}} = \frac{64}{27}$ .	Take the cube root first, then componendo-dividendo; give the answer as a ratio.
Prove a result	'If a, b, c are in continued proportion, prove …' or 'If x = …, show that …'.	k-method, or componendo-dividendo then squaring.
Number to be added or subtracted	Find the number that must be added to each of four numbers to put them in proportion.	Write the proportion with x added, use product of extremes = product of means; the $x^{2}$ terms cancel.

Tricks to remember Ratio and Proportion

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

Extremes times extremes, means times means
In a : b = c : d, the outside pair (a, d) multiplies to the same as the inside pair (b, c).
Use it when: Every proportion check.
Sum over difference
Componendo-dividendo: '(top plus bottom) over (top minus bottom)' on both sides.
Use it when: Applying the property quickly and correctly.
Mean is in the middle
The mean proportional sits in the middle: a, √(ac), c.
Use it when: Mean and third proportional questions.
Bracket before you add
Put brackets round numerator and denominator before componendo-dividendo.
Use it when: Avoiding the sign errors examiners keep reporting.

Ratio and Proportion: board-level solved questions

Ordered from board level to HOTS. Cover the solution, write your own, then compare step by step.

Example 1: Mean proportional · 1 mark · MCQBoard level

The mean proportional between 9 and 16 is:
(a) 12.5
(b) 12
(c) 144
(d) 7

Try it first, then show the solutionSolution

$9 \times 16 = 144 = 12$ .

Answer: (b) 12

Example 2: Continued proportion check · 1 mark · Assertion-reasonBoard level

Assertion (A): 4, 6 and 9 are in continued proportion.
Reason (R): Three numbers $a$ , $b$ , $c$ are in continued proportion if $b^{2} = a c$ .
(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).

Try it first, then show the solutionSolution

$6^{2} = 36 = 4 \times 9$ , so (A) is true by the rule in (R).

Answer: (c)

Example 3: Cubes and componendo-dividendo twice · 4 marks · Short answerHarder board level

Using properties of proportion, solve $\frac{x ^{3} + 3 x}{3 x ^{2} + 1} = \frac{341}{91}$ .

Try it first, then show the solutionSolution

Componendo and dividendo: $\frac{x ^{3} + 3 x ^{2} + 3 x + 1}{x ^{3} - 3 x ^{2} + 3 x - 1} = \frac{341 + 91}{341 - 91} = \frac{432}{250}$ .
$\frac{( x + 1 ) ^{3}}{( x - 1 ) ^{3}} = \frac{216}{125}$ , so $\frac{x + 1}{x - 1} = \frac{6}{5}$ .
Componendo and dividendo again: $\frac{2 x}{2} = \frac{11}{1}$ .
$x = 11$ .

Answer: x = 11

Example 4: Show that a quadratic holds · 4 marks · Short answerHarder board level

If $x = \frac{a + 2 b + a - 2 b}{a + 2 b - a - 2 b}$ , use properties of proportion to show that $b x^{2} - a x + b = 0$ .

Try it first, then show the solutionSolution

Componendo and dividendo: $\frac{x + 1}{x - 1} = \frac{a + 2 b}{a - 2 b}$ .
Square: $\frac{x ^{2} + 2 x + 1}{x ^{2} - 2 x + 1} = \frac{a + 2 b}{a - 2 b}$ .
Componendo and dividendo: $\frac{2 ( x ^{2} + 1 )}{4 x} = \frac{2 a}{4 b}$ , so $\frac{x ^{2} + 1}{x} = \frac{a}{b}$ .
$b (x^{2} + 1) = a x \Rightarrow b x^{2} - a x + b = 0$ .

Answer: Shown.

Example 5: Dividendo on both sides · 4 marks · Short answerHarder board level

Using properties of proportion, find $x$ : $\frac{8 x ^{2} + 3 x - 2}{3 x - 2} = \frac{5 x ^{2} + 2 x + 4}{2 x + 4}$ , $x \neq = 0$ .

Try it first, then show the solutionSolution

Dividendo (subtract 1 from each side): $\frac{8 x ^{2}}{3 x - 2} = \frac{5 x ^{2}}{2 x + 4}$ .
Divide by $x^{2}$ ( $x \neq = 0$ ): $\frac{8}{3 x - 2} = \frac{5}{2 x + 4}$ .
$16 x + 32 = 15 x - 10$ .
$x = - 42$ .

Answer: x = −42

Example 6: Number to be added · 3 marks · Long answerBoard level

What number must be added to each of 6, 15, 20 and 43 so that the resulting numbers are in proportion?

Try it first, then show the solutionSolution

$(6 + x) (43 + x) = (15 + x) (20 + x)$ .
$258 + 49 x + x^{2} = 300 + 35 x + x^{2}$ .
$14 x = 42 \Rightarrow x = 3$ ; check $9 \times 46 = 18 \times 23 = 414$ .

Answer: 3

Example 7: A proof from continued proportion · 4 marks · Long answerHOTS / competency

If $a$ , $b$ , $c$ are in continued proportion, prove that $\frac{a ^{2} + b ^{2}}{b ^{2} + c ^{2}} = \frac{a}{c}$ . Check your result with 2, 6, 18. Is the converse true for any $a$ , $b$ , $c$ ? Give a reason.

Try it first, then show the solutionSolution

$b^{2} = a c$ , so LHS $= \frac{a ^{2} + a c}{a c + c ^{2}} = \frac{a ( a + c )}{c ( a + c )} = \frac{a}{c}$ .
Check: $\frac{4 + 36}{36 + 324} = \frac{40}{360} = \frac{1}{9} = \frac{2}{18}$ .
Converse: $\frac{a ^{2} + b ^{2}}{b ^{2} + c ^{2}} = \frac{a}{c} \Rightarrow a^{2} c + b^{2} c = a b^{2} + a c^{2} \Rightarrow a c (a - c) = b^{2} (a - c)$ .
If $a \neq = c$ , this gives $b^{2} = a c$ , so the converse holds; if $a = c$ , any $b$ works, so the converse fails in that case.

Answer: Proved; the converse holds only when a ≠ c.

Example 8: Ratio from componendo-dividendo · 4 marks · Long answerHarder board level

If $\frac{7 m - 5 n}{7 m + 5 n} = \frac{4}{9}$ , find $m : n$ .

Try it first, then show the solutionSolution

Componendo and dividendo: $\frac{( 7 m - 5 n ) + ( 7 m + 5 n )}{( 7 m - 5 n ) - ( 7 m + 5 n )} = \frac{4 + 9}{4 - 9}$ .
$\frac{14 m}{- 10 n} = \frac{13}{- 5}$ .
$\frac{m}{n} = \frac{13}{5} \times \frac{10}{14} = \frac{13}{7}$ .
$m : n = 13 : 7$ .

Answer: 13 : 7

Case-study practice (Section B style)

Read the situation, then answer every part. In the board paper this is a 10-mark question split into 3 + 3 + 4 marks.

Mixing juice at a school fete

At a school fete in Kochi, a stall makes mango drink by mixing concentrate and water in the ratio 2 : 7. A 2-litre bottle of concentrate costs ₹180.

(i) · 3 marks
How much concentrate and how much water are in 4.5 litres of the drink?
Answer
Answer: 1 L concentrate, 3.5 L water
Concentrate $= \frac{2}{9} \times 4.5 = 1$ L; water $= 3.5$ L.
(ii) · 3 marks
Customers find the drink too strong. How much water must be added to the 4.5 litres so that the ratio becomes 1 : 5?
Answer
Answer: 1.5 L
Concentrate stays 1 L. $1 : w = 1 : 5 \Rightarrow w = 5$ L. Add $5 - 3.5 = 1.5$ L.
(iii) · 4 marks
Using the original ratio 2 : 7, find the cost of the concentrate needed for 27 litres of drink. Explain why the cost is in direct proportion to the volume of drink.
Answer
Answer: ₹540
Concentrate $= \frac{2}{9} \times 27 = 6$ L. Cost per litre $= \frac{180}{2} = ₹90$ , so cost $= 6 \times 90 = ₹540$ . With a fixed ratio, the concentrate is a fixed fraction ( $\frac{2}{9}$ ) of the drink and costs a fixed amount per litre, so doubling the drink doubles the cost.

Photo prints in continued proportion

A photo studio prints a picture in widths of 8 cm, 12 cm and $x$ cm, chosen so that the three widths are in continued proportion. A fourth size is added later.

(i) · 3 marks
Find $x$ .
Answer
Answer: 18 cm
$8 : 12 = 12 : x \Rightarrow 8 x = 144 \Rightarrow x = 18$ .
(ii) · 3 marks
Show that 12 is the mean proportional between 8 and 18, and state the common ratio of the widths.
Answer
Answer: √(8 × 18) = 12; ratio 3 : 2
$8 \times 18 = 144 = 12$ . $\frac{12}{8} = \frac{18}{12} = \frac{3}{2}$ .
(iii) · 4 marks
The fourth width $w$ is chosen so that $8 : 12 = 18 : w$ . Find $w$ and show that 8, 12, 18, $w$ are in continued proportion.
Answer
Answer: w = 27
$8 w = 216 \Rightarrow w = 27$ . Ratios: $\frac{12}{8} = \frac{18}{12} = \frac{27}{18} = \frac{3}{2}$ , all equal, so the four widths are in continued proportion.

Where the marks are

How a typical answer is split into step marks. Show every step: a right final answer alone rarely earns full marks.

Find the third proportional to $(a^{2} - b^{2})$ and $(a - b)$ . (3 marks)

Step	Marks
$(a^{2} - b^{2}) : (a - b) = (a - b) : x$	1
$x = \frac{( a - b ) ^{2}}{a ^{2} - b ^{2}}$	1
$x = \frac{( a - b ) ^{2}}{( a - b ) ( a + b )} = \frac{a - b}{a + b}$	1

Using properties of proportion, solve $\frac{x ^{3} + 12 x}{6 x ^{2} + 8} = \frac{63}{62}$ . (4 marks)

Step	Marks
Componendo and dividendo: $\frac{x ^{3} + 6 x ^{2} + 12 x + 8}{x ^{3} - 6 x ^{2} + 12 x - 8} = \frac{125}{1}$	1
$\frac{( x + 2 ) ^{3}}{( x - 2 ) ^{3}} = \frac{125}{1} \Rightarrow \frac{x + 2}{x - 2} = \frac{5}{1}$	1
Componendo and dividendo: $\frac{2 x}{4} = \frac{6}{4}$	1
$x = 3$	1

If $x = \frac{a + 1 + a - 1}{a + 1 - a - 1}$ , using properties of proportion show that $x^{2} - 2 a x + 1 = 0$ . (5 marks)

Step	Marks
Componendo and dividendo: $\frac{x + 1}{x - 1} = \frac{2 a + 1}{2 a - 1}$	1
Squaring: $\frac{x ^{2} + 2 x + 1}{x ^{2} - 2 x + 1} = \frac{a + 1}{a - 1}$	1
Componendo and dividendo: $\frac{2 ( x ^{2} + 1 )}{4 x} = \frac{2 a}{2}$	1
$\frac{x ^{2} + 1}{2 x} = a$	1
$x^{2} + 1 = 2 a x \Rightarrow x^{2} - 2 a x + 1 = 0$	1

Where students lose marks in Ratio and Proportion

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

Mistake: Cross-multiplying when the question says 'using properties of proportion'
CISCE examiners' comments, 2024 and 2025: candidates solved by cross-multiplication instead of the properties, and did not apply componendo and dividendo.
Do this: Name the property ('componendo and dividendo') on the line where you use it. Cross-multiply only at the very end, on a simple fraction.
Usually costs: The method marks, even if the answer is right.
Mistake: Sign errors in componendo-dividendo
CISCE examiners' comments, 2025: candidates applied componendo-dividendo with sign errors; the examiners advise using brackets.
Wrong working
$\frac{6 x ^{2} + 3 x - 5 - 3 x - 5}{3 x - 5}$
Right working
$\frac{( 6 x ^{2} + 3 x - 5 ) - ( 3 x - 5 )}{3 x - 5} = \frac{6 x ^{2}}{3 x - 5}$
Do this: Bracket the numerator and the denominator before subtracting.
Usually costs: 1 to 2 marks.
Mistake: Wrong cube root or square root of a fraction
CISCE examiners' comments, 2024: candidates took the cube root of $\frac{64}{27}$ as $\frac{8}{3}$ instead of $\frac{4}{3}$ .
Do this: Take the root of the numerator and the denominator separately, and check by cubing: $(\frac{4}{3})^{3} = \frac{64}{27}$ .
Usually costs: Every step after it.
Mistake: Final answer not written as a ratio
CISCE examiners' comments, 2024: candidates left the answer as $\frac{a}{b} = 7$ instead of $a : b = 7 : 1$ .
Do this: If the question asks for $a : b$ , the last line must be ' $a : b = \dots : \dots$ '.
Usually costs: The final mark.
Mistake: Guessing a direct-proportion table
CISCE examiners' comments, 2024: in a distance-time table, candidates guessed or used wrong proportion logic.
Do this: Find the constant ratio from the complete column (e.g. 60 m in 2 s → 30 m per second) and use it for each gap.
Usually costs: 1 mark.

Practice questions on Ratio and Proportion

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
The third proportional to 5 and 15 is:
(a) 10
(b) 25
(c) 45
(d) 75
Hint
c = b²/a.
Answer
(c) 45
Q2 · 1 mark · MCQBoard level
The fourth proportional to 2, 7 and 10 is:
(a) 35
(b) 20
(c) 14
(d) 70
Hint
d = bc/a.
Answer
(a) 35
Q3 · 1 mark · MCQHarder board level
If $x : y = 5 : 2$ , then $\frac{x + y}{x - y}$ is:
(a) $\frac{3}{7}$
(b) $\frac{7}{3}$
(c) $\frac{5}{2}$
(d) $\frac{10}{3}$
Hint
Componendo and dividendo.
Answer
(b) $\frac{5 + 2}{5 - 2} = \frac{7}{3}$
Q4 · 1 mark · Assertion-reasonBoard level
Assertion (A): The mean proportional between 4 and 16 is 10.
Reason (R): The mean proportional between $a$ and $c$ is $a c$ .
(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
Is 10 the average of 4 and 16, or the square root of their product?
Answer
(b). By (R), the mean proportional is $4 \times 16 = 8$ , not 10 (10 is the arithmetic mean). So (A) is false and (R) is true.
Q5 · 4 marks · Short answerHarder board level
Using properties of proportion, solve $\frac{2 x + 3 + 2 x - 3}{2 x + 3 - 2 x - 3} = 2$ .
Hint
Componendo-dividendo, then square.
Answer
$\frac{2 x + 3}{2 x - 3} = \frac{3}{1}$ ; $\frac{2 x + 3}{2 x - 3} = 9 \Rightarrow 2 x + 3 = 18 x - 27 \Rightarrow x = \frac{15}{8}$ .
Q6 · 4 marks · Short answerHarder board level
If $\frac{a}{b} = \frac{c}{d}$ , prove that $\frac{a ^{2} + ab}{c ^{2} + c d} = \frac{b ^{2} - 2 ab}{d ^{2} - 2 c d}$ .
Hint
Let a = bk, c = dk.
Answer
LHS $= \frac{b ^{2} k ^{2} + b ^{2} k}{d ^{2} k ^{2} + d ^{2} k} = \frac{b ^{2} k ( k + 1 )}{d ^{2} k ( k + 1 )} = \frac{b ^{2}}{d ^{2}}$ . RHS $= \frac{b ^{2} - 2 b ^{2} k}{d ^{2} - 2 d ^{2} k} = \frac{b ^{2} ( 1 - 2 k )}{d ^{2} ( 1 - 2 k )} = \frac{b ^{2}}{d ^{2}}$ . Equal.
Q7 · 3 marks · Long answerBoard level
Find the mean proportional between $(x - y)$ and $(x^{3} - x^{2} y)$ .
Hint
Factorise $x^{3} - x^{2} y$ .
Answer
$(x - y) \cdot x^{2} (x - y) = x^{2} (x - y)^{2} = x (x - y)$ .
Q8 · 4 marks · Long answerHarder board level
Using properties of proportion, solve $\frac{x ^{4} + 1}{2 x ^{2}} = \frac{17}{8}$ .
Hint
Componendo and dividendo gives $\frac{( x ^{2} + 1 ) ^{2}}{( x ^{2} - 1 ) ^{2}} = \frac{25}{9}$ .
Answer
$\frac{x ^{2} + 1}{x ^{2} - 1} = \pm \frac{5}{3}$ . With $+ \frac{5}{3}$ : componendo-dividendo gives $x^{2} = 4$ , $x = \pm 2$ . With $- \frac{5}{3}$ : $x^{2} = \frac{1}{4}$ , $x = \pm \frac{1}{2}$ . So $x = \pm 2, \pm \frac{1}{2}$ .
Q9 · 4 marks · Long answerHOTS / competency
If $b$ is the mean proportional between $a$ and $c$ , prove that $\frac{a ^{2} - b ^{2} + c ^{2}}{a ^{- 2} - b ^{- 2} + c ^{- 2}} = b^{4}$ .
Hint
Multiply the denominator by $a^{2} b^{2} c^{2}$ and use $b^{2} = a c$ .
Answer
Denominator $= \frac{b ^{2} c ^{2} - a ^{2} c ^{2} + a ^{2} b ^{2}}{a ^{2} b ^{2} c ^{2}}$ . With $b^{2} = a c$ : numerator of this $= a c \cdot c^{2} - a^{2} c^{2} + a^{2} \cdot a c = a c (c^{2} - a c + a^{2}) = b^{2} (a^{2} - b^{2} + c^{2})$ , and $a^{2} b^{2} c^{2} = b^{2} \cdot b^{4} = b^{6}$ . So LHS $= (a^{2} - b^{2} + c^{2}) \times \frac{b ^{6}}{b ^{2} ( a ^{2} - b ^{2} + c ^{2} )} = b^{4}$ .
Q10 · 4 marks · Long answerHarder board level
If $\frac{x}{a} = \frac{y}{b} = \frac{z}{c}$ , show that $\frac{x ^{3}}{a ^{2}} + \frac{y ^{3}}{b ^{2}} + \frac{z ^{3}}{c ^{2}} = \frac{( x + y + z ) ^{3}}{( a + b + c ) ^{2}}$ .
Hint
x = ak, y = bk, z = ck.
Answer
LHS $= k^{3} (a + b + c)$ . RHS $= \frac{k ^{3} ( a + b + c ) ^{3}}{( a + b + c ) ^{2}} = k^{3} (a + b + c)$ . Equal.
Q11 · 1 mark · MCQHOTS / competency
A shop sells pens at a fixed price per pen. 4 pens cost ₹60, 7 pens cost ₹y and x pens cost ₹180. Then:
(a) x = 12, y = 105
(b) x = 12, y = 100
(c) x = 10, y = 105
(d) x = 9, y = 105
Hint
Price per pen is constant.
Answer
(a): ₹15 per pen, so $y = 105$ and $x = \frac{180}{15} = 12$ .
Q12 · 4 marks · Short answerHOTS / competency
If $(a + 3 b + 2 c + 6 d) (a - 3 b - 2 c + 6 d) = (a + 3 b - 2 c - 6 d) (a - 3 b + 2 c - 6 d)$ , prove that $a : b = c : d$ .
Hint
Rewrite as a proportion and apply componendo-dividendo twice.
Answer
$\frac{a + 3 b + 2 c + 6 d}{a + 3 b - 2 c - 6 d} = \frac{a - 3 b + 2 c - 6 d}{a - 3 b - 2 c + 6 d}$ . Componendo-dividendo: $\frac{2 ( a + 3 b )}{2 ( 2 c + 6 d )} = \frac{2 ( a - 3 b )}{2 ( 2 c - 6 d )}$ , so by alternendo $\frac{a + 3 b}{a - 3 b} = \frac{2 c + 6 d}{2 c - 6 d}$ . Componendo-dividendo again: $\frac{2 a}{6 b} = \frac{4 c}{12 d}$ , so $\frac{a}{b} = \frac{c}{d}$ .

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.When are a, b, c, d in proportion?Show answer
When a : b = c : d, i.e. ad = bc.
2.Define the mean proportional between a and c.Show answer
√(ac)
3.What is the third proportional to a and b?Show answer
b²/a
4.State componendo and dividendo.Show answer
If a/b = c/d then (a + b)/(a − b) = (c + d)/(c − d).
5.What is alternendo?Show answer
If a/b = c/d then a/c = b/d.
6.In the k-method, how do you write a and c?Show answer
a = bk, c = dk.
7.Mean proportional between 4 and 25?Show answer
10
8.Why use brackets in componendo-dividendo?Show answer
To subtract whole expressions correctly and avoid sign errors.

Best order to study it

Learn proportion, fourth, third and mean proportional with numbers.
Learn the four properties and say each one in words.
Practise componendo-dividendo on linear, square-root and cube expressions.
Practise proofs by the k-method.
Finish with direct-proportion applications and mixed questions.

Quick revision checklist

The night before the exam: can you do each of these without looking?

Can you find the fourth, third and mean proportional?
Can you state invertendo, alternendo, componendo and dividendo?
Can you apply componendo-dividendo with brackets and name it?
Can you deal with square roots and cubes using componendo-dividendo?
Can you prove a result with the k-method?
Do you give a : b in ratio form at the end?
Can you fill a direct-proportion table using the constant ratio?

Ratio and Proportion: frequently asked questions

Can I cross-multiply instead of using componendo and dividendo?

Only if the question does not ask you to use properties of proportion. When it does, examiners expect to see the property used and named.

What is the difference between mean proportional and third proportional?

For a, b, c in continued proportion, b is the mean proportional between a and c, and c is the third proportional to a and b.

Are inverse proportion problems in the ICSE Class 10 syllabus?

The 2027 syllabus says 'direct simple applications on proportions only', so practise direct proportion.

When is componendo-dividendo most useful?

When the numerator and denominator differ by the sign of one term, such as (√p + √q)/(√p − √q) or (x³ + 3x)/(3x² + 1), because adding and subtracting removes terms.

Is a : b the same as a/b?

They have the same value, but if a question asks for a ratio, write it as a : b in simplest whole numbers.

Based on: CISCE ICSE 2027 Mathematics syllabus; CISCE ICSE 2027 Mathematics specimen question paper and marking scheme; CISCE Analysis of Pupil Performance, ICSE Mathematics, 2025; CISCE Analysis of Pupil Performance, ICSE Mathematics, 2024