Question

Three positive numbers are in the ratio 2 : 3 : 4. The sum of their squares is 2349. The average of the first two numbers is:

• A. 36
• B. 27.5
• C. 18
• D. 22.5

Hint:

You can also calculate the average in terms of \( x \) first: average of \( 2x \) and \( 3x \) is \( 2.5x \). Since \( x = 9 \), the average is simply \( 2.5 \times 9 = 22.5 \), saving calculation steps.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
Let the three numbers be \( 2x, 3x, \) and \( 4x \), where \( x \) is a common constant multiplier. We are given the sum of their squares is 2349 and need to find the average of the first two numbers, \( \frac{2x + 3x}{2} = 2.5x \).
Step 2: Key Formula or Approach:
The sum of squares equation is: \[ (2x)^2 + (3x)^2 + (4x)^2 = 2349 \]
Step 3: Detailed Explanation:
\[ 4x^2 + 9x^2 + 16x^2 = 2349 \]
\[ 29x^2 = 2349 \]
\[ x^2 = \frac{2349}{29} = 81 \]
\[ x = \sqrt{81} = 9 \]
The first two numbers are \( 2x = 2(9) = 18 \) and \( 3x = 3(9) = 27 \).
The average is:
\[ \text{Average} = \frac{18 + 27}{2} = \frac{45}{2} = 22.5 \]
Step 4: Final Answer:
The average of the first two numbers is 22.5.