Two wires A and B of the same material have their lengths in the ratio `1 : 5` and diameters in the ratio ` 3 : 2`. If the resistance of the wire B is `180 Omega`, find the resistance of the wire A.

Answer

To find the resistance of wire A given the resistance of wire B and the ratios of their lengths and diameters, we can follow these steps:

### Step 1: Understand the relationship between resistance, length, and area
The resistance \( R \) of a wire can be expressed using the formula:
\[ R = \frac{\rho L}{A} \]
where:
- \( R \) is the resistance,
- \( \rho \) is the resistivity of the material,
- \( L \) is the length of the wire,
- \( A \) is the cross-sectional area of the wire.

### Step 2: Set up the ratios
Given:
- The lengths of wires A and B are in the ratio \( 1:5 \), so:
\[ \frac{L_A}{L_B} = \frac{1}{5} \]
- The diameters of wires A and B are in the ratio \( 3:2 \), so:
\[ \frac{D_A}{D_B} = \frac{3}{2} \]

### Step 3: Calculate the areas
The cross-sectional area \( A \) of a wire is related to its diameter \( D \) by the formula:
\[ A = \frac{\pi D^2}{4} \]
Thus, the areas of wires A and B can be expressed as:
\[ A_A = \frac{\pi D_A^2}{4}, \quad A_B = \frac{\pi D_B^2}{4} \]
Taking the ratio of the areas:
\[ \frac{A_A}{A_B} = \frac{\frac{\pi D_A^2}{4}}{\frac{\pi D_B^2}{4}} = \frac{D_A^2}{D_B^2} \]
Substituting the ratio of diameters:
\[ \frac{A_A}{A_B} = \left(\frac{D_A}{D_B}\right)^2 = \left(\frac{3}{2}\right)^2 = \frac{9}{4} \]

### Step 4: Set up the resistance ratio
Using the resistance formula and substituting the ratios:
\[ \frac{R_A}{R_B} = \frac{\rho L_A / A_A}{\rho L_B / A_B} = \frac{L_A}{L_B} \cdot \frac{A_B}{A_A} \]
Since the resistivity \( \rho \) cancels out:
\[ \frac{R_A}{R_B} = \frac{L_A}{L_B} \cdot \frac{A_B}{A_A} = \frac{1/5}{9/4} = \frac{1}{5} \cdot \frac{4}{9} = \frac{4}{45} \]

### Step 5: Calculate resistance of wire A
Given that the resistance of wire B \( R_B = 180 \, \Omega \):
\[ R_A = R_B \cdot \frac{4}{45} = 180 \cdot \frac{4}{45} \]
Calculating this gives:
\[ R_A = \frac{720}{45} = 16 \, \Omega \]

### Final Answer
The resistance of wire A is:
\[ \boxed{16 \, \Omega} \]