A man's annual income has increased by Rs. 2.6 lakhs but the tax on income that he has to pay has reduced from 17.1% to 13.3% in both the cases 35% of the income is tax free. What is his initial income (in Rs lakhs) if he paid equal taxes in both cases?

Options

A. `9.1`
B. 10
C. `8.4`
D. `6.8`

Answer

To solve the problem, we need to determine the initial income of the man based on the information provided about his tax situation before and after an increase in income.

### Step-by-Step Solution:

1. **Define Variables:**
Let the initial income of the man be \( x \) lakhs.

2. **Calculate the New Income:**
After an increase of Rs. 2.6 lakhs, the new income will be:
\[ \text{New Income} = x + 2.6 \text{ lakhs} \]

3. **Determine Tax-Free Income:**
Since 35% of the income is tax-free, the taxable income is 65% of the total income. Therefore, the taxable income before the increase is:
\[ \text{Taxable Income (initial)} = 0.65x \]
And after the increase, it is:
\[ \text{Taxable Income (new)} = 0.65(x + 2.6) \]

4. **Calculate Taxes Paid:**
The tax rate before the increase was 17.1%, so the tax paid initially is:
\[ \text{Tax Paid (initial)} = 0.171 \times 0.65x \]
The tax rate after the increase is 13.3%, so the tax paid now is:
\[ \text{Tax Paid (new)} = 0.133 \times 0.65(x + 2.6) \]

5. **Set Up the Equation:**
Since the problem states that the man paid equal taxes in both cases, we can set the two tax equations equal to each other:
\[ 0.171 \times 0.65x = 0.133 \times 0.65(x + 2.6) \]

6. **Simplify the Equation:**
We can cancel \( 0.65 \) from both sides:
\[ 0.171x = 0.133(x + 2.6) \]

7. **Distribute and Rearrange:**
Expanding the right side gives:
\[ 0.171x = 0.133x + 0.3468 \]
Rearranging this gives:
\[ 0.171x - 0.133x = 0.3468 \]
\[ 0.038x = 0.3468 \]

8. **Solve for \( x \):**
Dividing both sides by 0.038 gives:
\[ x = \frac{0.3468}{0.038} \approx 9.1 \]

9. **Conclusion:**
Therefore, the initial income of the man is approximately:
\[ \text{Initial Income} = 9.1 \text{ lakhs} \]