On a particular day , the maximum frequency reflected from the ionosphere is ` 9 MHz`. On another day , it was found to increase by `1 MHz`. Calculate , the ratio of the maximum electron densities of the ionosphere on the two days .

Answer

To solve the problem, we need to use the relationship between the maximum frequency reflected from the ionosphere and the maximum electron density. The relationship can be expressed as follows:

\[ f = k \sqrt{N} \]

Where:
- \( f \) is the maximum frequency reflected (in Hz),
- \( N \) is the maximum electron density (in electrons per cubic meter),
- \( k \) is a constant.

For our calculations, we can ignore the constant \( k \) since we are interested in the ratio of electron densities on two different days.

### Step-by-step Solution:

1. **Identify the frequencies**:
- On the first day, the maximum frequency \( f_1 = 9 \, \text{MHz} \).
- On the second day, the maximum frequency \( f_2 = 9 \, \text{MHz} + 1 \, \text{MHz} = 10 \, \text{MHz} \).

2. **Express the electron densities**:
- Let the maximum electron density on the first day be \( N_1 \).
- Let the maximum electron density on the second day be \( N_2 \).

3. **Set up the equations using the relationship**:
- From the relationship \( f = k \sqrt{N} \), we can write:
\[ f_1 = k \sqrt{N_1} \implies N_1 = \left(\frac{f_1}{k}\right)^2 \]
\[ f_2 = k \sqrt{N_2} \implies N_2 = \left(\frac{f_2}{k}\right)^2 \]

4. **Calculate the ratio of the electron densities**:
- The ratio of the maximum electron densities can be expressed as:
\[ \frac{N_2}{N_1} = \frac{\left(\frac{f_2}{k}\right)^2}{\left(\frac{f_1}{k}\right)^2} = \left(\frac{f_2}{f_1}\right)^2 \]

5. **Substitute the frequencies**:
- Substitute \( f_1 = 9 \, \text{MHz} \) and \( f_2 = 10 \, \text{MHz} \):
\[ \frac{N_2}{N_1} = \left(\frac{10}{9}\right)^2 \]

6. **Calculate the numerical value**:
- Now calculate \( \left(\frac{10}{9}\right)^2 \):
\[ \frac{N_2}{N_1} = \left(\frac{10}{9}\right)^2 = \frac{100}{81} \approx 1.2346 \]

7. **Final ratio**:
- The ratio of the maximum electron densities \( N_2 : N_1 = 100 : 81 \).

### Final Answer:
The ratio of the maximum electron densities of the ionosphere on the two days is \( \frac{100}{81} \) or approximately \( 1.2346 \).