In the month of January, Aran's income and expenses were X 15000 and X 9000, respectively. His monthly expenses vary directly as the square of his monthly income. What is his income when it just equals his expenses?

Options

A. X 25000
B. X 2000
C. X 35000
D. X 15000

Answer

To solve the problem, we need to establish the relationship between Aran's income and expenses based on the information given.

### Step-by-Step Solution:

1. **Identify the variables**:
- Let \( I \) be Aran's income.
- Let \( E \) be Aran's expenses.

2. **Given values**:
- In January, \( I = 15000 \) and \( E = 9000 \).

3. **Understand the relationship**:
- The problem states that Aran's expenses vary directly as the square of his income. This can be expressed mathematically as:
\[ E = k \cdot I^2 \]
where \( k \) is a constant of proportionality.

4. **Find the constant \( k \)**:
- Using the values from January, we can substitute \( I \) and \( E \) into the equation:
\[ 9000 = k \cdot (15000)^2 \]
- Calculate \( (15000)^2 \):
\[ (15000)^2 = 225000000 \]
- Substitute this value back into the equation:
\[ 9000 = k \cdot 225000000 \]
- Solve for \( k \):
\[ k = \frac{9000}{225000000} = \frac{1}{25000} \]

5. **Set up the equation for when income equals expenses**:
- We need to find \( I \) when \( I = E \). Thus, we set:
\[ I = k \cdot I^2 \]
- Substitute \( k \):
\[ I = \frac{1}{25000} \cdot I^2 \]

6. **Rearranging the equation**:
- Multiply both sides by \( 25000 \) to eliminate the fraction:
\[ 25000I = I^2 \]
- Rearranging gives:
\[ I^2 - 25000I = 0 \]

7. **Factoring the equation**:
- Factor out \( I \):
\[ I(I - 25000) = 0 \]
- This gives us two solutions:
\[ I = 0 \quad \text{or} \quad I = 25000 \]

8. **Conclusion**:
- Since income cannot be zero in this context, the solution is:
\[ I = 25000 \]

### Final Answer:
Aran's income when it just equals his expenses is **X 25000**.