Income of a man is increased by Rs 1800 and payable tax is decreased from `18% t o 15%`. In both cases 29.95 income tax is free. Find initial income.

Answer

To find the initial income of the man, we will follow the steps outlined below:

### Step 1: Define the variables
Let the initial income of the man be Rs. \( x \).

### Step 2: Set up the tax equations
1. The initial tax payable at an 18% rate is:
\[ \text{Tax at 18%} = 0.18x \]
2. After the income is increased by Rs. 1800, the new income becomes \( x + 1800 \). The tax payable at a 15% rate is:
\[ \text{Tax at 15%} = 0.15(x + 1800) \]

### Step 3: Set the tax equations equal
Since the amount of income tax that is free remains the same in both cases, we can set up the equation:
\[ 0.18x = 0.15(x + 1800) \]

### Step 4: Simplify the equation
Expanding the right side:
\[ 0.18x = 0.15x + 0.15 \times 1800 \]
Calculating \( 0.15 \times 1800 \):
\[ 0.15 \times 1800 = 270 \]
So the equation becomes:
\[ 0.18x = 0.15x + 270 \]

### Step 5: Rearranging the equation
Subtract \( 0.15x \) from both sides:
\[ 0.18x - 0.15x = 270 \]
This simplifies to:
\[ 0.03x = 270 \]

### Step 6: Solve for \( x \)
Now, divide both sides by 0.03:
\[ x = \frac{270}{0.03} \]
Calculating the right side:
\[ x = 9000 \]

### Conclusion
The initial income of the man is Rs. 9000.

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