The difference of the squares is of two numbers is 80% of the sum of their squares. The ratio of the larger number to the smaller number is

Options

A. `5:2`
B. `2:5`
C. `3:1`
D. `1:3`

Answer

To solve the problem, we need to find the ratio of the larger number to the smaller number given that the difference of the squares of two numbers is 80% of the sum of their squares.

### Step-by-Step Solution:

1. **Let the two numbers be \( x \) and \( y \)**:
- Here, we assume \( x \) is the larger number and \( y \) is the smaller number.

2. **Set up the equation based on the problem statement**:
- The difference of the squares of the two numbers is given by \( x^2 - y^2 \).
- The sum of the squares of the two numbers is given by \( x^2 + y^2 \).
- According to the problem, we have:
\[ x^2 - y^2 = 0.8 \times (x^2 + y^2) \]

3. **Convert 80% to a fraction**:
- \( 0.8 \) can be written as \( \frac{80}{100} = \frac{4}{5} \).
- Therefore, we can rewrite the equation as:
\[ x^2 - y^2 = \frac{4}{5} (x^2 + y^2) \]

4. **Clear the fraction by multiplying through by 5**:
- Multiply both sides by 5 to eliminate the fraction:
\[ 5(x^2 - y^2) = 4(x^2 + y^2) \]
- This simplifies to:
\[ 5x^2 - 5y^2 = 4x^2 + 4y^2 \]

5. **Rearrange the equation**:
- Bring all terms involving \( x^2 \) to one side and all terms involving \( y^2 \) to the other side:
\[ 5x^2 - 4x^2 = 4y^2 + 5y^2 \]
- This simplifies to:
\[ x^2 = 9y^2 \]

6. **Find the ratio \( \frac{x}{y} \)**:
- Taking the square root of both sides gives:
\[ \frac{x}{y} = \sqrt{\frac{x^2}{y^2}} = \sqrt{9} = 3 \]
- Therefore, the ratio of the larger number to the smaller number is:
\[ \frac{x}{y} = 3 \quad \text{or} \quad 3:1 \]

### Final Answer:
The ratio of the larger number to the smaller number is \( 3:1 \).

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