Ratio and Proportion • Topic 1 of 3

Continued Proportion

What is continued proportion? Three quantities a, b, c are said to be in continued proportion if the ratio of a to b is equal to the ratio of b to c. That is:

a : b = b : c or a/b = b/c

This means \(b^{2}\) = a × c. Here, b is called the mean proportional (or geometric mean) between a and c. The quantity c is called the third proportional to a and b.

Key terms:

  • Mean proportional between a and c = \(\sqrt{a × c}\)
  • Third proportional to a and b = \(b^{2}\)/a
  • Continued proportion can be extended to more terms: a, b, c, d are in continued proportion if a:b = b:c = c:d

Real-life analogy: Think of a photograph being enlarged. If the original photo has width 4 cm and height 6 cm (ratio 2:3), and the enlarged version has width 8 cm, then for the same ratio, the height should be 12 cm. Here, 4, 8, 16 would be in continued proportion? No — 4:8 = 1:2, 8:16 = 1:2, so 4, 8, 16 are in continued proportion! The mean proportional between 4 and 16 is 8.

Continued proportion a : b = b : c with cross-product b squared equals a times cContinued proportion: a : b = b : ca : bb : c=product of means = product of extremesb × b = a × c → b² = a×cMean proportional b equals square root of a c, and third proportional examples on a number trackMean & third proportionalMean proportional9b16b = √(9×16) = 12Third proportional46cc = 6²/4 = 9
Solved Examples
1
Worked Example
Check whether 6, 12, 24 are in continued proportion.
Solution
  1. Step 1: Check if a:b = b:c, i.e., 6:12 = 12:24
  2. Step 2: 6:12 = 6/12 = 1/2
  3. Step 3: 12:24 = 12/24 = 1/2
  4. Step 4: Both ratios are equal (1:2)

Answer: Yes, 6, 12, 24 are in continued proportion

2
Worked Example
Find the mean proportional between 16 and 25.
Solution
  1. Step 1: Mean proportional b = \(\sqrt{a × c}\)
  2. Step 2: b = \(\sqrt{16 × 25}\) = \(\sqrt{400}\)
  3. Step 3: b = 20

Answer: 20

3
Worked Example
If 3, x, 12 are in continued proportion, find the value of x. Also find the third proportional to 3 and x.
Solution
  1. Step 1: For continued proportion: 3/x = x/12
  2. Step 2: Cross multiply: \(x^{2}\) = 3 × 12 = 36
  3. Step 3: x = 6 (positive value)
  4. Step 4: Third proportional to 3 and 6 = \(6^{2}\)/3 = 36/3 = 12

Answer: x = 6; third proportional = 12

Key Points

  • Three numbers a, b, c are in continued proportion if a:b = b:c
  • Condition: \(b^{2}\) = a × c
  • b is the mean proportional between a and c: b = \(\sqrt{ac}\)
  • c is the third proportional to a and b: c = \(b^{2}\)/a
  • Continued proportion can be extended to more than three terms
  • In a continued proportion, the ratio between consecutive terms is constant
Quick Check — 4 MCQs
Tap an option to check your answer0 / 4
Q1.If $a:b::b:c$, then:
Explanation: The middle term squared $=$ product of extremes.
Q2.$b$ is the mean proportional between $a$ and $c$ if:
Explanation: $b=\sqrt{ac}$.
Q3.The mean proportional between $4$ and $9$ is:
Explanation: $\sqrt{36}=6$.
Q4.In a continued proportion $a,b,c$, $c$ is the:
Explanation: Third proportional.
Practice more MCQs → 📝 Assignment · 35 marks