MathsICSE Class 10 · Ratio and Proportion

Properties of Proportion and the k-Method: Proofs and Applications (ICSE Class 10)

Updated 15 October 2026 · Fact-checked

To prove a result when a/b = c/d, let each ratio equal k, so a = bk and c = dk. Substitute into both sides of the statement and simplify each to the same expression in k (or in b and d). The other properties, invertendo (b/a = d/c) and alternendo (a/c = b/d), follow from the same idea.

Replace a and c with bk and dk, simplify each side on its own, and watch b and d cancel so both sides become the same expression in k.

Illustration drawn for QuestionBuddy.

Understand Properties of Proportion and the k-Method

If $\frac{a}{b} = \frac{c}{d}$ , the CISCE syllabus expects you to know four named properties and their combinations:

Invertendo: $\frac{b}{a} = \frac{d}{c}$ (turn both ratios upside down).
Alternendo: $\frac{a}{c} = \frac{b}{d}$ (swap the means).
Componendo: $\frac{a + b}{b} = \frac{c + d}{d}$ .
Dividendo: $\frac{a - b}{b} = \frac{c - d}{d}$ .

Componendo and dividendo together are treated in their own topic. This topic covers the most reliable way to prove results about proportions: the k-method.

The k-method. Put each equal ratio equal to $k$ :

$\frac{a}{b} = \frac{c}{d} = k \Rightarrow a = bk, c = d k .$

Now every expression in $a, b, c, d$ can be rewritten in $b, d$ and $k$ . To prove that $LHS = RHS$ , simplify the LHS and the RHS separately, and show they reduce to the same thing. For example, to prove $\frac{3 a + 5 b}{3 a - 5 b} = \frac{3 c + 5 d}{3 c - 5 d}$ : LHS $= \frac{3 bk + 5 b}{3 bk - 5 b} = \frac{3 k + 5}{3 k - 5}$ , and RHS $= \frac{3 d k + 5 d}{3 d k - 5 d} = \frac{3 k + 5}{3 k - 5}$ . They are equal.

The same method works for three or more equal ratios: $\frac{x}{a} = \frac{y}{b} = \frac{z}{c} = k$ gives $x = ak$ , $y = bk$ , $z = c k$ . And for continued proportion $a : b = b : c = k$ it gives $b = c k$ , $a = c k^{2}$ .

Why it works. After substitution, the common factor ( $b$ or $d$ , or their powers) cancels, leaving only $k$ . If both sides of the claimed identity reduce to the same expression, the identity is true for every proportion.

Direct proportion. The syllabus also includes 'direct simple applications on proportions'. Two quantities are in direct proportion if their ratio is constant: $\frac{y}{x} = k$ . In 2024 Q1(vii), a train at uniform speed covered 60 m in 2 s, so distance ÷ time = 30 and 90 m took 3 s while 5 s gave 150 m; the examiners reported candidates who guessed instead of using the constant ratio.

Properties of Proportion and the k-Method: formulas to remember

k-substitution: $\frac{a}{b} = \frac{c}{d} = k \Rightarrow a = bk, c = d k$; Then express everything in $b$ , $d$ and $k$ .
Invertendo: $\frac{a}{b} = \frac{c}{d} \Rightarrow \frac{b}{a} = \frac{d}{c}$; Turn both ratios upside down.
Alternendo: $\frac{a}{b} = \frac{c}{d} \Rightarrow \frac{a}{c} = \frac{b}{d}$; Swap the middle terms.
Three equal ratios: $\frac{x}{a} = \frac{y}{b} = \frac{z}{c} = k \Rightarrow x = ak, y = bk, z = c k$; Works for any number of equal ratios.
Direct proportion: $\frac{y _{1}}{x _{1}} = \frac{y _{2}}{x _{2}} = k$; The ratio stays constant.

How to solve Properties of Proportion and the k-Method questions

For proofs, never work on both sides at once; simplify each side separately.

1Write 'Let $\frac{a}{b} = \frac{c}{d} = k$ , so $a = bk$ and $c = d k$ .'
2Substitute into the LHS and simplify fully (take out common factors, cancel).
3Substitute into the RHS and simplify fully.
4Show both equal the same expression; write 'LHS = RHS. Hence proved.'
5For a numerical ratio like $x : y = 3 : 4$ , put $x = 3 k$ , $y = 4 k$ and simplify the required ratio.
6For direct proportion questions, find the constant ratio first, then use it for each unknown.

Properties of Proportion and the k-Method 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.

Keeping a ratio fixed when scaling

Suppose a pancake recipe uses flour and milk in the ratio 3 : 2 by weight, and you need 450 g of flour for a larger batch.

How much milk do you need? Use the k-method.

Try it first, then show the solutionSolution

Flour $= 3 k$ , milk $= 2 k$ .
$3 k = 450 \Rightarrow k = 150$ ; milk $= 2 k = 300$ .

Answer: 300 g of milk.

Memory hook: A ratio is a recipe: find one k, and every ingredient follows.

Direct proportion on a map

Imagine a map on which 2 cm stands for 5 km on the ground.

What ground distance does 7 cm represent?

Try it first, then show the solutionSolution

Ground distance ÷ map distance $= \frac{5}{2} = 2.5$ km per cm.
$7 \times 2.5 = 17.5$ .

Answer: 17.5 km.

Memory hook: Direct proportion: same ratio, every time.

Tricks to remember Properties of Proportion and the k-Method

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

Let k do the work
Equal ratios? Name them $k$ . Every top becomes bottom × $k$ .
Use it when: Any 'if … prove that' question.
Alternendo swaps the middles
$a : b :: c : d$ → $a : c :: b : d$ : the two middle terms change places.
Use it when: Naming the property.
Invertendo turns upside down
Flip both fractions.
Use it when: Naming the property.
Sides separately
Simplify LHS on its own, RHS on its own, then compare. Never cross-multiply a 'prove that'.
Use it when: Writing proofs examiners accept.

Properties of Proportion and the k-Method: solved examples

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

Example 1: Naming a property · 1 mark · MCQBoard level

If $a : b = c : d$ , then by alternendo:
(a) $b : a = d : c$
(b) $a : c = b : d$
(c) $(a + b) : b = (c + d) : d$
(d) $(a - b) : b = (c - d) : d$

Try it first, then show the solutionSolution

Alternendo swaps the means $b$ and $c$ .

Answer: (b)

Example 2: A k-method proof · 3 marks · Short answerBoard level

If $\frac{a}{b} = \frac{c}{d}$ , prove that $\frac{3 a + 5 b}{3 a - 5 b} = \frac{3 c + 5 d}{3 c - 5 d}$ .

Try it first, then show the solutionSolution

Let $\frac{a}{b} = \frac{c}{d} = k$ : $a = bk$ , $c = d k$ .
LHS $= \frac{3 bk + 5 b}{3 bk - 5 b} = \frac{b ( 3 k + 5 )}{b ( 3 k - 5 )} = \frac{3 k + 5}{3 k - 5}$ .
RHS $= \frac{3 d k + 5 d}{3 d k - 5 d} = \frac{3 k + 5}{3 k - 5}$ . LHS = RHS.

Answer: Proved: both sides equal $\frac{3 k + 5}{3 k - 5}$ .

Example 3: Three equal ratios with cubes · 3 marks · Short answerHarder board level

If $\frac{x}{a} = \frac{y}{b} = \frac{z}{c}$ , prove that $\frac{x ^{3} + y ^{3} + z ^{3}}{a ^{3} + b ^{3} + c ^{3}} = \frac{x y z}{ab c}$ .

Try it first, then show the solutionSolution

Let each ratio be $k$ : $x = ak$ , $y = bk$ , $z = c k$ .
LHS $= \frac{k ^{3} ( a ^{3} + b ^{3} + c ^{3} )}{a ^{3} + b ^{3} + c ^{3}} = k^{3}$ .
RHS $= \frac{ab c k ^{3}}{ab c} = k^{3}$ . LHS = RHS.

Answer: Proved: both sides equal $k^{3}$ .

Example 4: Squares and products · 4 marks · Short answerHarder board level

If $a : b = c : d$ , prove that $\frac{a ^{2} + ab}{c ^{2} + c d} = \frac{b ^{2} - 2 ab}{d ^{2} - 2 c d}$ .

Try it first, then show the solutionSolution

$a = bk$ , $c = d k$ .
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 $= \frac{b ^{2} - 2 b ^{2} k}{d ^{2} - 2 d ^{2} k} = \frac{b ^{2} ( 1 - 2 k )}{d ^{2} ( 1 - 2 k )} = \frac{b ^{2}}{d ^{2}}$ .
LHS = RHS.

Answer: Proved: both sides equal $\frac{b ^{2}}{d ^{2}}$ .

Example 5: A square-root identity · 4 marks · Short answerHOTS / competency

If $\frac{a}{b} = \frac{c}{d} = \frac{e}{f}$ (all positive), prove that each ratio equals $\frac{a ^{2} + c ^{2} + e ^{2}}{b ^{2} + d ^{2} + f ^{2}}$ .

Try it first, then show the solutionSolution

Let each ratio be $k$ ( $k > 0$ ): $a = bk$ , $c = d k$ , $e = f k$ .
$\frac{a ^{2} + c ^{2} + e ^{2}}{b ^{2} + d ^{2} + f ^{2}} = \frac{k ^{2} ( b ^{2} + d ^{2} + f ^{2} )}{b ^{2} + d ^{2} + f ^{2}} = k^{2}$ .
$k^{2} = k$ , which is each ratio.

Answer: Proved: the square root equals $k$ .

Example 6: Case study: fuel and distance · 4 marks · Case-basedHOTS / competency

A car's fuel use is in direct proportion to the distance it travels. It uses 18 litres for 240 km.
(a) How much fuel does it need for 400 km?
(b) How far can it go on 45 litres?
(c) Show that the pairs (18, 240) and your answer to (a) with 400 are in proportion.

Try it first, then show the solutionSolution

Fuel per km $= \frac{18}{240} = 0.075$ L.
(a) $400 \times 0.075 = 30$ L.
(b) $\frac{45}{0.075} = 600$ km.
(c) $\frac{18}{240} = \frac{3}{40}$ and $\frac{30}{400} = \frac{3}{40}$ , so $18 : 240 :: 30 : 400$ .

Answer: (a) 30 L (b) 600 km (c) both ratios are $\frac{3}{40}$ .

Where students lose marks in Properties of Proportion and the k-Method

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

Mistake: Working on both sides together
Cross-multiplying a 'prove that' assumes what is to be proved.
Wrong working
$\frac{3 a + 5 b}{3 a - 5 b} = \frac{3 c + 5 d}{3 c - 5 d} \Rightarrow (3 a + 5 b) (3 c - 5 d) = \dots$
Right working
Simplify LHS to $\frac{3 k + 5}{3 k - 5}$ , RHS to $\frac{3 k + 5}{3 k - 5}$ .
Do this: Treat LHS and RHS separately.
Usually costs: Most of the proof marks.
Mistake: Forgetting to take the common factor out
The cancellation of $b$ or $d$ is the heart of the method.
Wrong working
$\frac{3 bk + 5 b}{3 bk - 5 b} = \frac{3 k + 5}{3 k - 5 b}$ .
Right working
$\frac{b ( 3 k + 5 )}{b ( 3 k - 5 )} = \frac{3 k + 5}{3 k - 5}$ .
Do this: Factor completely before cancelling.
Usually costs: The simplification marks.
Mistake: Guessing in proportion tables
Each pair must have the same ratio.
Wrong working
60 m in 2 s, so 90 m in 4 s.
Right working
Speed 30 m/s: 90 m in 3 s; 5 s gives 150 m.
Do this: Find the constant ratio first.
Usually costs: Reported for 2024 Q1(vii).

Exam tips

Begin every proof with 'Let $\frac{a}{b} = \frac{c}{d} = k$ '.
End with 'LHS = RHS. Hence proved.'
In 2024 Q1(vii) (distance and time at uniform speed) the answer was $x = 3$ , $y = 150$ ; the examiners reported candidates who guessed or used wrong proportion logic.
When a ratio is given as numbers ( $x : y = 3 : 4$ ), the k-method is quicker than dividing.

Practice questions on Properties of Proportion and the k-Method

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 $x : y = 2 : 3$ , then $(x + y) : y$ is:
(a) 5 : 2
(b) 5 : 3
(c) 3 : 5
(d) 2 : 5
Hint
Put $x = 2 k$ , $y = 3 k$ .
Answer
(b) 5 : 3
Q2 · 3 marks · Short answerBoard level
If $x : y = 3 : 4$ , find $(5 x - 2 y) : (3 x + y)$ .
Hint
$x = 3 k$ , $y = 4 k$ .
Answer
$\frac{15 k - 8 k}{9 k + 4 k} = \frac{7}{13}$ , so 7 : 13.
Q3 · 3 marks · Short answerHarder board level
If $\frac{a}{b} = \frac{c}{d}$ (with $b^{2} \neq = d^{2}$ ), prove that $\frac{a ^{2} - c ^{2}}{b ^{2} - d ^{2}} = \frac{a c}{b d}$ .
Hint
Both sides become $k^{2}$ .
Answer
LHS $= \frac{k ^{2} ( b ^{2} - d ^{2} )}{b ^{2} - d ^{2}} = k^{2}$ ; RHS $= \frac{bk \cdot d k}{b d} = k^{2}$ . Proved.
Q4 · 3 marks · Short answerHarder board level
If $a : b = c : d$ , prove that $(ma + nb) : b = (m c + n d) : d$ .
Hint
Both sides become $mk + n$ .
Answer
$\frac{mbk + nb}{b} = mk + n$ and $\frac{m d k + n d}{d} = mk + n$ . Proved.
Q5 · 1 mark · Assertion-reasonHarder board level
Assertion (A): If $\frac{a}{b} = \frac{c}{d}$ , then $\frac{a}{c} = \frac{b}{d}$ .
Reason (R): Alternendo: in a proportion the means may be interchanged.
(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
Which terms are the means?
Answer
(c)
Q6 · 4 marks · Short answerHOTS / competency
If $\frac{x}{b + c} = \frac{y}{c + a} = \frac{z}{a + b}$ , prove that $(b - c) x + (c - a) y + (a - b) z = 0$ .
Hint
Put $x = k (b + c)$ and so on, and use $(p - q) (p + q) = p^{2} - q^{2}$ .
Answer
LHS $= k [(b^{2} - c^{2}) + (c^{2} - a^{2}) + (a^{2} - b^{2})] = k \times 0 = 0$ . Proved.
Q7 · 3 marks · Short answerHarder board level
A cyclist moves at a uniform speed. Complete the table: distance 75 m in 3 s; $x$ m in 7 s; 200 m in $y$ s.
Hint
Distance ÷ time is constant.
Answer
Speed $= 25$ m/s, so $x = 175$ and $y = 8$ .

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 alternendo.Show answer
If $a : b = c : d$ , then $a : c = b : d$ .
2.State invertendo.Show answer
If $a : b = c : d$ , then $b : a = d : c$ .
3. $x : y = 5 : 2$ . Find $(x - y) : (x + y)$ .Show answer
3 : 7.
4. $\frac{a}{b} = \frac{c}{d} = k$ . Write $a$ and $c$ .Show answer
$a = bk$ , $c = d k$ .
5.12 pens cost ₹90. Cost of 20 pens?Show answer
₹150.

Properties of Proportion and the k-Method: frequently asked questions

What is the k-method in proportion?

You set each equal ratio equal to k, so a = bk and c = dk, then substitute into both sides of the statement and show they simplify to the same expression.

What are invertendo and alternendo?

If a : b = c : d, invertendo gives b : a = d : c and alternendo gives a : c = b : d. Both keep the proportion true.

Can I cross-multiply to prove a proportion identity?

It is better not to. Simplify each side separately with the k-method; examiners give method marks for that structure.