A boy walked along two adjacent sides of a rectangular field. If he had walked along the diagonal, then he would have saved a distance equal to one-fourth of the larger side. The ratio of the larger to the smaller side is:

Options

A. `7:24`
B. `11 : 18`
C. `24 : 7`
D. `18 : 11 `

Answer

To solve the problem, we need to find the ratio of the larger side to the smaller side of a rectangular field based on the information given.

### Step-by-Step Solution:

1. **Define Variables**:
Let the length of the rectangle be \( l \) (larger side) and the width be \( b \) (smaller side).

2. **Distance Walked**:
When the boy walks along the two adjacent sides, he covers a distance of \( l + b \).

3. **Distance Saved**:
If he had walked along the diagonal, he would have saved a distance equal to one-fourth of the larger side, which is \( \frac{l}{4} \).

4. **Diagonal Distance**:
The distance along the diagonal \( d \) can be calculated using the Pythagorean theorem:
\[ d = \sqrt{l^2 + b^2} \]

5. **Setting Up the Equation**:
According to the problem, the distance saved can be expressed as:
\[ (l + b) - d = \frac{l}{4} \]
Substituting \( d \) into the equation gives:
\[ l + b - \sqrt{l^2 + b^2} = \frac{l}{4} \]

6. **Rearranging the Equation**:
Rearranging the equation, we have:
\[ \sqrt{l^2 + b^2} = l + b - \frac{l}{4} \]
Simplifying the right side:
\[ \sqrt{l^2 + b^2} = l + b - \frac{l}{4} = \frac{3l}{4} + b \]

7. **Squaring Both Sides**:
Now, square both sides to eliminate the square root:
\[ l^2 + b^2 = \left(\frac{3l}{4} + b\right)^2 \]

8. **Expanding the Right Side**:
Expanding the right side:
\[ l^2 + b^2 = \left(\frac{3l}{4}\right)^2 + 2 \cdot \frac{3l}{4} \cdot b + b^2 \]
This simplifies to:
\[ l^2 + b^2 = \frac{9l^2}{16} + \frac{3lb}{2} + b^2 \]

9. **Cancelling \( b^2 \)**:
Cancel \( b^2 \) from both sides:
\[ l^2 = \frac{9l^2}{16} + \frac{3lb}{2} \]

10. **Rearranging**:
Rearranging gives:
\[ l^2 - \frac{9l^2}{16} = \frac{3lb}{2} \]
This simplifies to:
\[ \frac{7l^2}{16} = \frac{3lb}{2} \]

11. **Cross-Multiplying**:
Cross-multiplying gives:
\[ 7l^2 = 24lb \]

12. **Finding the Ratio**:
Dividing both sides by \( l \) (assuming \( l \neq 0 \)):
\[ 7l = 24b \implies \frac{l}{b} = \frac{24}{7} \]

13. **Final Ratio**:
Therefore, the ratio of the larger side to the smaller side is:
\[ \text{Ratio of } l \text{ to } b = 24:7 \]