- **Given information**
The mass of the earth is 81 times that of the moon and the radius of the earth is 3.5 times that of the moon.
- **Calculating the acceleration due to gravity**
The acceleration due to gravity is given by the formula \(g = \frac{GM}{r^2}\), where G is the universal gravitational constant, M is the mass of the object, and r is the distance from the center of the object.
- **Ratio of acceleration due to gravity on the moon to that on earth**
Let's denote the mass of the moon as \(M_{\text{moon}}\) and the radius of the moon as \(r_{\text{moon}}\). Given that the mass of the earth is 81 times that of the moon, we have \(M_{\text{earth}} = 81M_{\text{moon}}\). Similarly, the radius of the earth is 3.5 times that of the moon, so \(r_{\text{earth}} = 3.5r_{\text{moon}}\).
- Substituting the values into the formula for acceleration due to gravity, we get:
\[g_{\text{moon}} = \frac{GM_{\text{moon}}}{r_{\text{moon}}^2}\]
\[g_{\text{earth}} = \frac{GM_{\text{earth}}}{r_{\text{earth}}^2}\]
- **Now, we can find the ratio of the accelerations due to gravity**
\[ \frac{g_{\text{moon}}}{g_{\text{earth}}} = \frac{\frac{GM_{\text{moon}}}{r_{\text{moon}}^2}}{\frac{GM_{\text{earth}}}{r_{\text{earth}}^2}} = \frac{M_{\text{moon}}}{M_{\text{earth}}} \times \frac{r_{\text{earth}}^2}{r_{\text{moon}}^2}\]
\[ \frac{g_{\text{moon}}}{g_{\text{earth}}} = \frac{1}{81} \times \frac{(3.5)^2}{1} = \frac{1}{81} \times 12.25 = \frac{12.25}{81} \approx 0.1512\]
- **Conclusion**
The ratio of the acceleration due to gravity at the surface of the moon to that at the surface of the earth is approximately 0.1512.