How to Solve Continued Proportion Problems
Solve continued proportion aptitude problems using the mean proportional identity b² = ac, with a worked example and practice questions.
Expected Interview Answer
Three quantities a, b, c are in continued proportion when a:b = b:c, which means b is the mean proportional and b² = a×c, the single identity that solves almost every continued-proportion question.
Continued proportion extends simple ratio a:b into a chain a:b:c where the ratio between consecutive terms stays the same, so a/b = b/c. Cross-multiplying gives b² = ac, meaning the middle term is the geometric mean of the outer two, never their arithmetic average. This differs from a simple ratio problem because you are matching an entire chain of terms, not just two, and the shared middle term b must satisfy both ratios simultaneously. To extend a two-term ratio a:b into a three-term continued proportion a:b:c, multiply the second ratio so its first term equals b, keeping the chain consistent.
- One identity, b² = ac, unlocks most continued-proportion questions
- Clarifies the difference between mean proportional and simple average
- Extends cleanly to combining two ratios into one three-term chain
AI Mentor Explanation
A team’s run rate across three successive overs is in continued proportion when the ratio of over-1 runs to over-2 runs equals the ratio of over-2 runs to over-3 runs. If overs score 4, 8, and 16, then 4:8 equals 8:16, so 8 is the mean proportional and 8² = 4×16 = 64 checks out exactly. Continued proportion problems always reduce to this middle-term-squared identity, never to averaging the outer two scores.
Worked example (mean proportional)
Given
- a = 4, c = 9
Apply b² = ac
- b² = 4×9 = 36
- b = 6
Verify chain
- 4:6 = 6:9 = 2:3
Step-by-Step Explanation
Step 1
Identify the chain
Three terms a, b, c are continued proportion if a:b = b:c.
Step 2
Apply the mean-proportional identity
Cross-multiply to get b² = a×c.
Step 3
Solve for the unknown term
Take the square root of ac to find b, or solve for a or c if b is known.
Step 4
Verify the chain
Reduce both ratios a:b and b:c to confirm they match.
What Interviewer Expects
- Correct statement of the continued-proportion identity b² = ac
- Distinguishing mean proportional (geometric) from a simple average (arithmetic)
- Ability to extend two separate ratios into one continued-proportion chain
- Verification that the solved chain satisfies both ratio equalities
Common Mistakes
- Computing the arithmetic mean of a and c instead of the geometric mean
- Forgetting to take the square root after finding b²
- Misaligning the two ratios when combining them into a single chain
- Sign errors when a or c could have two square-root solutions
Best Answer (HR Friendly)
“Continued proportion means three numbers a, b, c form a chain where a is to b as b is to c. The key formula is b squared equals a times c, so the middle term is the geometric mean of the outer two, not their average. Whenever I see three linked quantities like that, I immediately reach for that squared-middle-term identity to solve for the unknown.”
Follow-up Questions
- How do you find the third proportional to two given numbers?
- How does continued proportion differ from a simple two-term ratio?
- How do you combine two separate ratios, a:b and b:c given differently, into one chain?
- What is the relationship between mean proportional and geometric mean more generally?
MCQ Practice
1. The mean proportional between 9 and 16 is?
b² = 9×16 = 144, so b = 12.
2. If a, b, c are in continued proportion with a = 2 and b = 6, then c is?
b² = a×c → 36 = 2c → c = 18.
3. Which relation defines a, b, c being in continued proportion?
Continued proportion means a:b = b:c, which cross-multiplies to b² = ac.
Flash Cards
Continued proportion condition? — a:b = b:c, meaning b is the mean proportional.
Mean proportional formula? — b² = a×c, so b = √(ac).
Mean proportional vs average? — Mean proportional is the geometric mean, not the arithmetic mean.
Third proportional to a and b? — The value c such that a:b = b:c, i.e. c = b²/a.