Two wires that are made up of two different materials whose specific resistance are in the ratio 2 : 3, length 3 : 4 and area 4 : 5. The ratio of their resistances is

Options

A. `6:5`
B. `6:8`
C. `5:8`
D. `1:2`

Answer

To find the ratio of the resistances of two wires made of different materials, we can use the formula for resistance:

\[ R = \frac{\rho L}{A} \]

Where:
- \( R \) is the resistance,
- \( \rho \) is the specific resistance (resistivity),
- \( L \) is the length of the wire,
- \( A \) is the cross-sectional area of the wire.

Given:
- The ratio of specific resistances (\( \rho_1 : \rho_2 \)) is \( 2 : 3 \).
- The ratio of lengths (\( L_1 : L_2 \)) is \( 3 : 4 \).
- The ratio of areas (\( A_1 : A_2 \)) is \( 4 : 5 \).

We need to find the ratio of the resistances \( R_1 : R_2 \).

### Step-by-Step Solution:

1. **Write the formula for the ratio of resistances:**
\[ \frac{R_1}{R_2} = \frac{\rho_1 L_1 / A_1}{\rho_2 L_2 / A_2} \]

2. **Rearrange the formula:**
\[ \frac{R_1}{R_2} = \frac{\rho_1}{\rho_2} \cdot \frac{L_1}{L_2} \cdot \frac{A_2}{A_1} \]

3. **Substitute the given ratios:**
- \( \frac{\rho_1}{\rho_2} = \frac{2}{3} \)
- \( \frac{L_1}{L_2} = \frac{3}{4} \)
- \( \frac{A_1}{A_2} = \frac{4}{5} \) implies \( \frac{A_2}{A_1} = \frac{5}{4} \)

So,
\[ \frac{R_1}{R_2} = \frac{2}{3} \cdot \frac{3}{4} \cdot \frac{5}{4} \]

4. **Calculate the product:**
\[ \frac{R_1}{R_2} = \frac{2 \cdot 3 \cdot 5}{3 \cdot 4 \cdot 4} \]

5. **Simplify the expression:**
- The \( 3 \) in the numerator and denominator cancels out:
\[ = \frac{2 \cdot 5}{4 \cdot 4} = \frac{10}{16} = \frac{5}{8} \]

6. **Final ratio of resistances:**
\[ R_1 : R_2 = 5 : 8 \]

### Conclusion:
The ratio of the resistances \( R_1 : R_2 \) is \( 5 : 8 \).