The interior angle of a regular polygon with n sides is 6 times that of an exterior angle of a regular polygon with `(3)/(2)` n sides. Then n equals

Options

A. 12
B. 20
C. 10
D. 18

Answer

To solve the problem, we need to find the value of \( n \) such that the interior angle of a regular polygon with \( n \) sides is 6 times the exterior angle of a regular polygon with \( \frac{3}{2}n \) sides.

### Step-by-Step Solution:

1. **Find the Interior Angle of a Regular Polygon with \( n \) Sides:**
The formula for the interior angle \( \alpha \) of a regular polygon with \( n \) sides is given by:
\[ \alpha = \frac{(n-2) \times 180}{n} \]

2. **Find the Exterior Angle of a Regular Polygon with \( \frac{3}{2}n \) Sides:**
The formula for the exterior angle \( \beta \) of a regular polygon with \( m \) sides is given by:
\[ \beta = \frac{360}{m} \]
For our case, \( m = \frac{3}{2}n \), so:
\[ \beta = \frac{360}{\frac{3}{2}n} = \frac{360 \times 2}{3n} = \frac{720}{3n} = \frac{240}{n} \]

3. **Set Up the Equation Based on the Given Condition:**
According to the problem, the interior angle is 6 times the exterior angle:
\[ \alpha = 6 \beta \]
Substituting the expressions for \( \alpha \) and \( \beta \):
\[ \frac{(n-2) \times 180}{n} = 6 \left(\frac{240}{n}\right) \]

4. **Simplify the Equation:**
Multiply both sides by \( n \) to eliminate the denominator:
\[ (n-2) \times 180 = 6 \times 240 \]
Calculate \( 6 \times 240 \):
\[ 6 \times 240 = 1440 \]
So the equation becomes:
\[ (n-2) \times 180 = 1440 \]

5. **Divide Both Sides by 180:**
\[ n - 2 = \frac{1440}{180} \]
Simplifying \( \frac{1440}{180} \):
\[ \frac{1440 \div 180}{180 \div 180} = \frac{8}{1} = 8 \]
Thus:
\[ n - 2 = 8 \]

6. **Solve for \( n \):**
Add 2 to both sides:
\[ n = 8 + 2 = 10 \]

### Final Answer:
Thus, the value of \( n \) is \( 10 \).