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Mean Proportion / Geometric Mean

Introduction

In some aptitude questions, you are asked to find a mean proportion between two numbers. This is also called the geometric mean.

It is frequently used in proportion problems, especially when two ratios are connected. If a : b = b : c, then b is called the mean proportional between a and c.

Pattern: Mean Proportion / Geometric Mean

Pattern

The mean proportion between two numbers a and b is √(a × b).

If a : x = x : b, then x² = a × b → x = √(a × b).

This is the key formula to solve such problems quickly.

Step-by-Step Example

Question

Find the mean proportion between 9 and 16.

Solution

  1. Step 1: Write the formula for mean proportion.

    If a : x = x : b, then x = √(a × b).

  2. Step 2: Substitute values.

    a = 9, b = 16 → x = √(9 × 16).

  3. Step 3: Simplify.

    x = √144 = 12.

  4. Step 4: Final Answer.

    The mean proportion between 9 and 16 is 12.

  5. Step 5: Quick Check.

    Ratio check: 9 : 12 = 3 : 4 and 12 : 16 = 3 : 4 ✅. So, the result is correct.

Question

If a number is the mean proportion between 25 and x, and the number is 35, find the value of x.

Solution

  1. Step 1: Write the relation for mean proportion.

    If 35 is the mean proportion between 25 and x, then 35² = 25 × x.

  2. Step 2: Simplify the equation.

    1225 = 25 × x.

  3. Step 3: Solve for x.

    x = 1225 ÷ 25 = 49.

  4. Step 4: Final Answer.

    The value of x is 49.

  5. Step 5: Quick Check.

    √(25 × 49) = √1225 = 35 ✅. This matches the given mean proportion.

Quick Variations

If one number and mean proportion are known: Use x = (mean²) ÷ known number. Example: Mean = 15, one number = 9 → other number = 225 ÷ 9 = 25.

If two ratios are given: Sometimes mean proportion connects them. Example: If a : b = b : c, then b is mean proportion between a and c.

Trick to Always Use

  • Formula: Mean proportion between a and b = √(a × b).
  • Check consistency: Ensure the two resulting ratios are equal.
  • Reverse use: If mean and one term are given, the other = (mean²) ÷ known term.

Summary

Summary

The Mean Proportion (Geometric Mean) connects two numbers in a proportion.

  • If a : x = x : b → x² = a × b → x = √(a × b).
  • Always verify by checking both ratios are equal.
  • Can be reversed if the mean and one number are given.

This concept is simple but powerful, especially in proportion-related aptitude questions.

Practice

(1/5)
1. Find the mean proportion between 4 and 9.
easy
A. 5
B. 6
C. 7
D. 8

Solution

  1. Step 1: Apply the geometric-mean formula

    Use mean proportion = √(a × b).

  2. Step 2: Multiply and take square root

    a = 4, b = 9 → 4 × 9 = 36 → √36 = 6.

  3. Final Answer:

    6 → Option B

  4. Quick Check:

    6² = 36 = 4 × 9 ✅
Hint: Take the square root of the product directly.
Common Mistakes: Adding or averaging instead of taking square root of product.
2. Find the mean proportion between 16 and 25.
easy
A. 19
B. 20
C. 21
D. 22

Solution

  1. Step 1: Apply the geometric-mean formula

    Mean proportion = √(a × b).

  2. Step 2: Multiply and take square root

    16 × 25 = 400 → √400 = 20.

  3. Final Answer:

    20 → Option B

  4. Quick Check:

    20² = 400 = 16 × 25 ✅
Hint: Perfect squares make the calculation easy.
Common Mistakes: Taking arithmetic mean (20.5) instead of geometric mean.
3. Find the mean proportion between 7 and 63.
easy
A. 18
B. 20
C. 21
D. 22

Solution

  1. Step 1: Apply the geometric-mean formula

    Mean proportion = √(a × b).

  2. Step 2: Multiply and take square root

    7 × 63 = 441 → √441 = 21.

  3. Final Answer:

    21 → Option C

  4. Quick Check:

    21² = 441 = 7 × 63 ✅
Hint: When one number is a multiple of the other, result is their geometric mean.
Common Mistakes: Using half of the sum (35) instead of square root of product.
4. Find the mean proportion between 12 and 27.
medium
A. 16
B. 17
C. 18
D. 19

Solution

  1. Step 1: Apply the geometric-mean formula

    Mean proportion = √(a × b).

  2. Step 2: Multiply and take square root

    12 × 27 = 324 → √324 = 18.

  3. Final Answer:

    18 → Option C

  4. Quick Check:

    18² = 324 = 12 × 27 ✅
Hint: Multiply first, then find square root quickly.
Common Mistakes: Using √(a + b) instead of √(a × b).
5. Find the mean proportion between 8 and 18.
medium
A. 11.5
B. 12
C. 12.5
D. 13

Solution

  1. Step 1: Apply the geometric-mean formula

    Mean proportion = √(a × b).

  2. Step 2: Multiply and take square root

    8 × 18 = 144 → √144 = 12.

  3. Final Answer:

    12 → Option B

  4. Quick Check:

    12² = 144 = 8 × 18 ✅
Hint: Look for perfect square products for exact answers.
Common Mistakes: Taking average (13) instead of geometric mean.

Mock Test

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