The ratio between two numbers is `5:3` and difference between their squares Is 144. Find the numbers-

Options

A. 15,9
B. 10,6
C. 5,3
D. 20,12

Answer

To solve the problem, we will follow these steps:

1. **Define the Numbers**:
Let the first number be \(5x\) and the second number be \(3x\). This is based on the given ratio of \(5:3\).

2. **Set Up the Equation**:
According to the problem, the difference between their squares is \(144\). Therefore, we can write the equation:
\[ (5x)^2 - (3x)^2 = 144 \]

3. **Expand the Squares**:
Expanding the squares in the equation gives us:
\[ 25x^2 - 9x^2 = 144 \]

4. **Simplify the Equation**:
Now, simplify the left side:
\[ 16x^2 = 144 \]

5. **Solve for \(x^2\)**:
Divide both sides by \(16\):
\[ x^2 = \frac{144}{16} \]
Simplifying this gives:
\[ x^2 = 9 \]

6. **Find \(x\)**:
Taking the square root of both sides, we find:
\[ x = 3 \]

7. **Calculate the Numbers**:
Now, substitute \(x\) back to find the two numbers:
- First number: \(5x = 5 \times 3 = 15\)
- Second number: \(3x = 3 \times 3 = 9\)

8. **Conclusion**:
The two numbers are \(15\) and \(9\).