MathsICSE Class 10 · Ratio and Proportion

Continued Proportion, Mean Proportional and Third Proportional (ICSE Class 10)

Updated 15 October 2026 · Fact-checked

Three quantities a, b, c are in continued proportion when a : b = b : c, that is b² = ac. Then b is the mean proportional between a and c (b = √(ac)), and c is the third proportional to a and b (c = b²/a). The fourth proportional to a, b, c is d = bc/a.

Each term is the one before times the same number (here 2), so the middle term squared equals first times last: 6² = 3 × 12.

Illustration drawn for QuestionBuddy.

Understand Continued Proportion and Mean Proportional

Four quantities $a, b, c, d$ are in proportion, written $a : b :: c : d$ , when $\frac{a}{b} = \frac{c}{d}$ , that is when $a d = b c$ (product of extremes = product of means). Here $d$ is called the fourth proportional to $a$ , $b$ and $c$ :

$d = \frac{b c}{a} .$

Continued proportion. When the second and third terms are the same, $a : b :: b : c$ , the three quantities $a, b, c$ are said to be in continued proportion. Then

$\frac{a}{b} = \frac{b}{c} ⟺ b^{2} = a c .$

$b$ is the mean proportional between $a$ and $c$ : $b = a c$ (for positive quantities).
$c$ is the third proportional to $a$ and $b$ : $c = \frac{b ^{2}}{a}$ .

Each term is obtained from the one before by multiplying by the same number $k = \frac{b}{a}$ , so $a, b, c$ in continued proportion are $a, ak, a k^{2}$ , the first three terms of a G.P. For example, 3, 6, 12 ( $k = 2$ ), or 9, 12, 16 ( $k = \frac{4}{3}$ ).

Order matters. 'The third proportional to 6 and 9' means $6 : 9 :: 9 : c$ , so $c = \frac{81}{6} = 13.5$ . 'The mean proportional between 6 and 9' is $54 = 36$ . Do not mix the two.

Typical questions.

1. Find a mean, third or fourth proportional (often with decimals or algebra).
2. Find $x$ so that given expressions are in continued proportion: write (middle) $^{2}$ = (first)(last) and solve.
3. Prove an identity for quantities in continued proportion. Two routes: replace $b^{2}$ by $a c$ wherever it appears, or put $b = ak$ , $c = a k^{2}$ (the $k$ -method) and show both sides equal the same expression in $a$ and $k$ .

For four quantities in continued proportion, $a : b = b : c = c : d$ , each is $k$ times the previous: $a, ak, a k^{2}, a k^{3}$ .

Continued Proportion and Mean Proportional: formulas to remember

Proportion: $a : b :: c : d ⟺ a d = b c$; Extremes × extremes = means × means.
Fourth proportional: $d = \frac{b c}{a}$; To $a$ , $b$ , $c$ in that order.
Continued proportion: $a : b :: b : c ⟺ b^{2} = a c$; Middle term squared equals the product of the ends.
Mean proportional: $b = a c$; Between $a$ and $c$ .
Third proportional: $c = \frac{b ^{2}}{a}$; To $a$ and $b$ .
k-form: $a, b = ak, c = a k^{2} (, d = a k^{3})$; Useful for proofs.

How to solve Continued Proportion and Mean Proportional questions

Translate the words into one equation, then solve or substitute.

1Identify what is asked: fourth proportional, third proportional, mean proportional, or an unknown making terms proportional.
2Write the proportion in order: $a : b :: c : d$ or $a : b :: b : c$ .
3Use product of extremes = product of means ( $a d = b c$ or $b^{2} = a c$ ).
4Solve for the unknown; for a mean proportional take the positive square root unless told otherwise.
5For proofs, substitute $b^{2} = a c$ , or $b = ak$ , $c = a k^{2}$ , and simplify both sides.
6Check that the ratios are equal with your answer.

Continued Proportion and Mean Proportional 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.

Mean proportional as the middle of a growth

Suppose a lab culture has 200 cells at the start and 800 cells two hours later, and it multiplies by the same factor each hour.

How many cells were there after one hour?

Try it first, then show the solutionSolution

200, $x$ , 800 are in continued proportion: $x^{2} = 200 \times 800 = 160000$ .
$x = 400$ .

Answer: 400 cells (the culture doubles every hour).

Memory hook: The mean proportional is the 'middle' of steady multiplying.

Third proportional

Imagine three nesting boxes whose heights are in continued proportion. The smallest is 8 cm and the middle one is 12 cm.

Find the height of the largest box.

Try it first, then show the solutionSolution

$8 : 12 :: 12 : h$ .
$h = \frac{1 2 ^{2}}{8} = 18$ .

Answer: 18 cm.

Memory hook: Third proportional: square the middle, divide by the first.

Tricks to remember Continued Proportion and Mean Proportional

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

Middle squared = ends multiplied
In $a, b, c$ continued proportion, $b^{2} = a c$ .
Use it when: Every continued-proportion question.
Fourth: last two multiplied, over the first
$d = \frac{b \times c}{a}$ .
Use it when: Fourth proportional.
Same multiplier each step
Continued proportion is a mini G.P.: $a, ak, a k^{2}$ .
Use it when: Proofs and spotting examples.

Continued Proportion and Mean Proportional: solved examples

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

Example 1: Mean proportional · 1 mark · MCQBoard level

The mean proportional between 5 and 45 is:
(a) 225
(b) 15
(c) 25
(d) 405

Try it first, then show the solutionSolution

$b = 5 \times 45 = 225 = 15$ .

Answer: (b) 15

Example 2: Third and fourth proportionals · 3 marks · Short answerBoard level

Find (a) the third proportional to 6 and 9, (b) the fourth proportional to 4, 7 and 12.

Try it first, then show the solutionSolution

(a) $6 : 9 :: 9 : c \Rightarrow c = \frac{81}{6} = 13.5$ .
(b) $4 : 7 :: 12 : d \Rightarrow d = \frac{7 \times 12}{4} = 21$ .

Answer: (a) 13.5 (b) 21

Example 3: An unknown making a continued proportion · 3 marks · Short answerHarder board level

Find $x$ if $x + 1$ , $x + 4$ and $x + 10$ are in continued proportion. Write the three numbers.

Try it first, then show the solutionSolution

$(x + 4)^{2} = (x + 1) (x + 10)$ .
$x^{2} + 8 x + 16 = x^{2} + 11 x + 10 \Rightarrow 6 = 3 x \Rightarrow x = 2$ .
Numbers: 3, 6, 12 (each twice the one before).

Answer: $x = 2$ ; 3, 6, 12.

Example 4: Algebraic mean proportional · 3 marks · Short answerHarder board level

Find the mean proportional between $x - y$ and $x^{3} - x^{2} y$ (where $x > y > 0$ ).

Try it first, then show the solutionSolution

$x^{3} - x^{2} y = x^{2} (x - y)$ .
Mean proportional $= (x - y) \cdot x^{2} (x - y) = x^{2} (x - y)^{2}$ .
$= x (x - y)$ .

Answer: $x (x - y)$

Example 5: Proving an identity · 3 marks · Short answerHOTS / competency

If $a, b, c$ are in continued proportion, prove that $(a^{2} + b^{2}) (b^{2} + c^{2}) = (ab + b c)^{2}$ .

Try it first, then show the solutionSolution

Continued proportion gives $b^{2} = a c$ .
LHS $= (a^{2} + a c) (a c + c^{2}) = a (a + c) \cdot c (a + c) = a c (a + c)^{2}$ .
RHS $= [b (a + c)]^{2} = b^{2} (a + c)^{2} = a c (a + c)^{2}$ .
LHS = RHS.

Answer: Proved: both sides equal $a c (a + c)^{2}$ , using $b^{2} = a c$ .

Example 6: Case study: three square prints · 4 marks · Case-basedHOTS / competency

A photographer makes three square prints whose sides are in continued proportion. The smallest has side 9 cm and the largest 16 cm.
(a) Find the side of the middle print.
(b) Show that the areas of the three prints are also in continued proportion.
(c) By what factor does the side grow from one print to the next?

Try it first, then show the solutionSolution

(a) $s^{2} = 9 \times 16 = 144 \Rightarrow s = 12$ cm.
(b) Areas: 81, 144, 256. $14 4^{2} = 20736 = 81 \times 256$ ✓.
(c) $\frac{12}{9} = \frac{16}{12} = \frac{4}{3}$ .

Answer: (a) 12 cm (b) $14 4^{2} = 81 \times 256$ (c) $\frac{4}{3}$

Where students lose marks in Continued Proportion and Mean Proportional

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

Mistake: Confusing the mean and third proportionals
They are different terms of the continued proportion.
Wrong working
Third proportional to 6 and 9: $54$ .
Right working
$\frac{9 ^{2}}{6} = 13.5$ ; $54$ is the mean proportional.
Do this: Write the proportion $a : b :: b : c$ and see which term is missing.
Usually costs: The whole question.
Mistake: Using the arithmetic mean
The mean proportional is the geometric mean $a c$ , not $\frac{a + c}{2}$ .
Wrong working
Mean proportional of 5 and 45 $= 25$ .
Right working
$5 \times 45 = 225 = 15$ .
Do this: Remember 'middle squared = ends multiplied'.
Usually costs: The MCQ.
Mistake: Wrong order for the fourth proportional
$a : b :: c : d$ gives $d = \frac{b c}{a}$ , not $\frac{ab}{c}$ .
Wrong working
Fourth proportional to 4, 7, 12: $\frac{4 \times 7}{12}$ .
Right working
$\frac{7 \times 12}{4} = 21$ .
Do this: Write the proportion in the given order first.
Usually costs: The answer mark.

Exam tips

Write the definition line (' $a, b, c$ in continued proportion ⇒ $b^{2} = a c$ ') in every proof; it is the key method mark.
For decimals, convert to whole numbers where possible (0.08 × 0.18 = 144 × 10⁻⁴).
For 'find x' questions, the $x^{2}$ terms usually cancel, leaving a linear equation.
Check your answer by showing the two ratios are equal.

Practice questions on Continued Proportion and Mean Proportional

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 2 and 6 is:
(a) 12
(b) 18
(c) 3
(d) $23$
Hint
$\frac{6 ^{2}}{2}$ .
Answer
(b) 18
Q2 · 3 marks · Short answerBoard level
Find the mean proportional between 0.08 and 0.18.
Hint
$0.08 \times 0.18 = 0.0144$ .
Answer
$0.0144 = 0.12$ .
Q3 · 3 marks · Short answerHarder board level
What number must be added to each of 3, 7 and 15 so that the results are in continued proportion?
Hint
$(7 + x)^{2} = (3 + x) (15 + x)$ .
Answer
$49 + 14 x = 45 + 18 x \Rightarrow x = 1$ ; the numbers become 4, 8, 16.
Q4 · 3 marks · Short answerHarder board level
Find the fourth proportional to $2 a$ , $3 b$ and $4 ab$ .
Hint
$d = \frac{3 b \times 4 ab}{2 a}$ .
Answer
$6 b^{2}$
Q5 · 1 mark · MCQHarder board level
If $b$ is the mean proportional between $a$ and $c$ , then:
(a) $b = \frac{a + c}{2}$
(b) $b^{2} = a c$
(c) $a^{2} = b c$
(d) $c^{2} = ab$
Hint
$a : b :: b : c$ .
Answer
(b)
Q6 · 3 marks · Short answerHarder board level
If $a, b, c$ are in continued proportion, prove that $(a + b + c) (a - b + c) = a^{2} + b^{2} + c^{2}$ .
Hint
Write the LHS as $(a + c)^{2} - b^{2}$ .
Answer
LHS $= (a + c)^{2} - b^{2} = a^{2} + 2 a c + c^{2} - b^{2} = a^{2} + 2 b^{2} + c^{2} - b^{2} = a^{2} + b^{2} + c^{2}$ , using $a c = b^{2}$ .
Q7 · 4 marks · Short answerHOTS / competency
The mean proportional between two positive numbers is 6 and the third proportional to them is 48. Find the numbers.
Hint
$ab = 36$ and $\frac{b ^{2}}{a} = 48$ .
Answer
$b = \frac{36}{a}$ , so $\frac{1296}{a ^{3}} = 48 \Rightarrow a^{3} = 27 \Rightarrow a = 3$ , $b = 12$ .

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.Mean proportional between 2 and 18?Show answer
6.
2.Third proportional to 4 and 10?Show answer
25.
3.Fourth proportional to 2, 3, 8?Show answer
12.
4.Are 5, 10, 20 in continued proportion?Show answer
Yes: $1 0^{2} = 100 = 5 \times 20$ .
5.Write $a, b, c$ in continued proportion in terms of $a$ and $k$ .Show answer
$a, ak, a k^{2}$ .

Continued Proportion and Mean Proportional: frequently asked questions

What does continued proportion mean?

Three quantities a, b, c are in continued proportion if a : b = b : c. Equivalently b² = ac, so each term is the same multiple of the one before.

How do I find the mean proportional between two numbers?

Multiply the two numbers and take the square root. For 4 and 9 it is √36 = 6.

What is the difference between third and fourth proportional?

The third proportional to a and b is c with a : b = b : c, so c = b²/a. The fourth proportional to a, b, c is d with a : b = c : d, so d = bc/a.