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 Question:
We are given three positive numbers represented by a specific ratio, along with the sum of their squares.
We need to find the common multiplier to determine the exact values of the numbers, and then calculate the average (arithmetic mean) of the first two numbers.

Step 2: Key Formula or Approach:
Let the three numbers be \( 2x \), \( 3x \), and \( 4x \), where \( x > 0 \) is a common constant multiplier.
According to the problem, the equation is:
\[ (2x)^2 + (3x)^2 + (4x)^2 = 2349 \]

Step 3: Detailed Explanation:
1. Expand the square terms of the ratio variables:
\[ (2x)^2 = 4x^2 \]
\[ (3x)^2 = 9x^2 \]
\[ (4x)^2 = 16x^2 \] 2. Sum these squared terms together:
\[ 4x^2 + 9x^2 + 16x^2 = 2349 \]
\[ 29x^2 = 2349 \] 3. Solve for \( x^2 \) by dividing both sides by 29:
\[ x^2 = \frac{2349}{29} \] 4. Let us compute the division:
\[ 29 \times 80 = 2320 \]
\[ 2349 - 2320 = 29 \]
\[ 29 \times 81 = 2349 \]
So, \( x^2 = 81 \).
5. Since the numbers are positive, we take the positive square root of 81:
\[ x = \sqrt{81} = 9 \] 6. Now, calculate the actual values of the first two numbers:
First number = \( 2x = 2 \times 9 = 18 \)
Second number = \( 3x = 3 \times 9 = 27 \)
7. Find the average of the first two numbers:
\[ \text{Average} = \frac{\text{First Number} + \text{Second Number}}{2} \]
\[ \text{Average} = \frac{18 + 27}{2} = \frac{45}{2} = 22.5 \]

Step 4: Final Answer:
The average of the first two numbers is 22.5.