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 need to find two numbers based on the given ratio and the difference between their squares. Here’s a step-by-step solution:

### Step 1: Define the Numbers
Let the first number be \( 5x \) and the second number be \( 3x \), where \( x \) is a common multiplier.

### Step 2: Write the Equation for the Difference of Squares
The difference between the squares of the two numbers can be expressed as:
\[ (5x)^2 - (3x)^2 = 144 \]

### Step 3: Expand the Squares
Now, we will expand the squares:
\[ 25x^2 - 9x^2 = 144 \]

### Step 4: Simplify the Equation
Combine like terms:
\[ 16x^2 = 144 \]

### Step 5: Solve for \( x^2 \)
To isolate \( x^2 \), divide both sides by 16:
\[ x^2 = \frac{144}{16} \]
\[ x^2 = 9 \]

### Step 6: Solve for \( x \)
Now, take the square root of both sides:
\[ x = 3 \]

### Step 7: Find the Numbers
Substituting \( x \) back into the expressions for the numbers:
- First number: \( 5x = 5 \times 3 = 15 \)
- Second number: \( 3x = 3 \times 3 = 9 \)

### Conclusion
The two numbers are **15** and **9**.

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