The number of sides in two regular polygons are in the ratio 5:4 and the difference between each interior angle of the polygons is `6^@`. Then the number of sides are.

Options

A. 15, 12
B. 5, 4
C. 10, 8
D. 20, 16

Answer

To solve the problem, we need to find the number of sides in two regular polygons given that their sides are in the ratio 5:4 and the difference between their interior angles is 6 degrees.

### Step-by-Step Solution:

1. **Define the Number of Sides:**
Let the number of sides of the first polygon be \( n_1 = 5x \) and the number of sides of the second polygon be \( n_2 = 4x \), where \( x \) is a common multiplier.

**Hint:** Start by expressing the number of sides in terms of a variable to represent the ratio.

2. **Formula for Interior Angle:**
The formula for the interior angle of a regular polygon with \( n \) sides is given by:
\[ \text{Interior angle} = \frac{180(n - 2)}{n} \]

3. **Calculate Interior Angles:**
For the first polygon (with \( n_1 = 5x \)):
\[ \text{Interior angle of first polygon} = \frac{180(5x - 2)}{5x} \]
For the second polygon (with \( n_2 = 4x \)):
\[ \text{Interior angle of second polygon} = \frac{180(4x - 2)}{4x} \]

4. **Set Up the Equation:**
According to the problem, the difference between the interior angles is 6 degrees:
\[ \frac{180(5x - 2)}{5x} - \frac{180(4x - 2)}{4x} = 6 \]

5. **Simplify the Equation:**
To simplify, we can multiply through by \( 20x \) (the least common multiple of \( 5x \) and \( 4x \)):
\[ 20x \left( \frac{180(5x - 2)}{5x} \right) - 20x \left( \frac{180(4x - 2)}{4x} \right) = 20x \cdot 6 \]
This simplifies to:
\[ 7200(5x - 2) - 9000(4x - 2) = 120x \]

6. **Expand and Combine Like Terms:**
Expanding both sides:
\[ 36000x - 14400 - 36000x + 18000 = 120x \]
Combine like terms:
\[ 36000x - 36000x + 18000 - 14400 = 120x \]
This simplifies to:
\[ 3600 = 120x \]

7. **Solve for \( x \):**
Divide both sides by 120:
\[ x = \frac{3600}{120} = 30 \]

8. **Find the Number of Sides:**
Now substitute \( x \) back to find \( n_1 \) and \( n_2 \):
\[ n_1 = 5x = 5 \cdot 30 = 150 \]
\[ n_2 = 4x = 4 \cdot 30 = 120 \]

9. **Final Answer:**
The number of sides in the two polygons are 150 and 120.

### Summary of Results:
- The first polygon has 150 sides.
- The second polygon has 120 sides.