Question

Two numbers are such that the square of greater number is 504 less than 8 times the square of the other. If the numbers are in the ratio 3 : 4. Find the number.

• A. 15 and 20
• B. 6 and 8
• C. 12 and 16
• D. 9 and 12

Answer 1

The Correct Option is D

To solve this problem, we denote the two numbers by \(3x\) and \(4x\) based on their ratio being 3:4. Now, according to the problem, the square of the greater number, which is \((4x)^2\), is 504 less than 8 times the square of the other number, which is \((3x)^2\). We can set up the equation as follows:

\((4x)^2 = 8 \times (3x)^2 - 504\)

Let's simplify and solve the equation step by step:

• Calculate \( (4x)^2 \):
  \[(4x)^2 = 16x^2\]
• Calculate \( 8 \times (3x)^2 \):
  First, compute \((3x)^2\):
  \[(3x)^2 = 9x^2\]
  Then multiply by 8:
  \[8 \times 9x^2 = 72x^2\]
• Set the equation \(16x^2 = 72x^2 - 504\)
• Rearrange to solve for \(x\):
  \[ 16x^2 - 72x^2 = -504 \]
  \[ -56x^2 = -504 \]
  Divide by -56:
  \[ x^2 = \frac{504}{56} \]
• Calculate the division:
  \[ x^2 = 9 \]
  Taking the square root of both sides:
  \[ x = 3 \]
• Substitute back to get the numbers:
  The smaller number is \(3x\): \[3 \times 3 = 9\]
  The greater number is \(4x\): \[4 \times 3 = 12\]

Thus, the numbers are 9 and 12. This corresponds to the option "9 and 12".