If the radii of two cyclinders are in the ratio 4: 3 and their heights are in the ratio 5: 6 find the ratio of their curved surface.

Answer

To find the ratio of the curved surface areas of two cylinders given the ratio of their radii and heights, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Ratios**:
- The ratio of the radii of the two cylinders is given as \( r_1 : r_2 = 4 : 3 \).
- The ratio of the heights of the two cylinders is given as \( h_1 : h_2 = 5 : 6 \).

2. **Write the Formula for Curved Surface Area**:
- The formula for the curved surface area (CSA) of a cylinder is given by:
\[ \text{CSA} = 2\pi rh \]
- Therefore, the curved surface area of the first cylinder (CSA1) is:
\[ \text{CSA}_1 = 2\pi r_1 h_1 \]
- And the curved surface area of the second cylinder (CSA2) is:
\[ \text{CSA}_2 = 2\pi r_2 h_2 \]

3. **Set Up the Ratio of the Curved Surface Areas**:
- We need to find the ratio of the curved surface areas:
\[ \frac{\text{CSA}_1}{\text{CSA}_2} = \frac{2\pi r_1 h_1}{2\pi r_2 h_2} \]
- The \( 2\pi \) cancels out:
\[ \frac{\text{CSA}_1}{\text{CSA}_2} = \frac{r_1 h_1}{r_2 h_2} \]

4. **Substitute the Ratios**:
- We can substitute the ratios of the radii and heights into the equation:
\[ \frac{\text{CSA}_1}{\text{CSA}_2} = \frac{r_1}{r_2} \cdot \frac{h_1}{h_2} = \frac{4/3}{5/6} \]

5. **Simplify the Expression**:
- To simplify \( \frac{4/3}{5/6} \), we multiply by the reciprocal:
\[ \frac{4}{3} \cdot \frac{6}{5} = \frac{4 \cdot 6}{3 \cdot 5} = \frac{24}{15} \]
- Now simplify \( \frac{24}{15} \):
\[ \frac{24 \div 3}{15 \div 3} = \frac{8}{5} \]

6. **Final Ratio**:
- Thus, the ratio of the curved surface areas of the two cylinders is:
\[ \frac{\text{CSA}_1}{\text{CSA}_2} = \frac{8}{5} \]

### Final Answer:
The ratio of the curved surface areas of the two cylinders is \( 8 : 5 \).