Income of a person is increased by Rs. 4800. Tax rate decreases from `12% t o 10%` . Payable tax is same as before. If in both condition `20%` of income is tax free. Find his current income.

Answer

To solve the problem step by step, we will follow the reasoning laid out in the video transcript.

### Step 1: Define the Current Income
Let the current income of the person be \( X \).

### Step 2: Determine the Increased Income
The income is increased by Rs. 4800, so the new income after the increase will be:
\[ \text{New Income} = X + 4800 \]

### Step 3: Identify the Tax-Free Portion
20% of the income is tax-free. Therefore, the taxable portion of the income is 80%. Thus, the taxable income before the increase is:
\[ \text{Taxable Income (before)} = 0.8X \]
And the taxable income after the increase is:
\[ \text{Taxable Income (after)} = 0.8(X + 4800) \]

### Step 4: Calculate the Taxes
The tax rate before the increase was 12%, and the tax rate after the increase is 10%. Therefore, the tax amounts are:
- Tax before the increase:
\[ \text{Tax (before)} = 0.12 \times 0.8X = 0.096X \]
- Tax after the increase:
\[ \text{Tax (after)} = 0.10 \times 0.8(X + 4800) = 0.08(X + 4800) \]

### Step 5: Set Up the Equation
According to the problem, the tax payable remains the same before and after the increase. Thus, we can set up the equation:
\[ 0.096X = 0.08(X + 4800) \]

### Step 6: Simplify the Equation
Expanding the right side:
\[ 0.096X = 0.08X + 0.08 \times 4800 \]
Calculating \( 0.08 \times 4800 \):
\[ 0.08 \times 4800 = 384 \]
So the equation becomes:
\[ 0.096X = 0.08X + 384 \]

### Step 7: Rearrange the Equation
Now, we will move \( 0.08X \) to the left side:
\[ 0.096X - 0.08X = 384 \]
This simplifies to:
\[ 0.016X = 384 \]

### Step 8: Solve for \( X \)
Now, divide both sides by \( 0.016 \):
\[ X = \frac{384}{0.016} = 24000 \]

### Step 9: Find the Current Income
The current income after the increase is:
\[ \text{Current Income} = X + 4800 = 24000 + 4800 = 28800 \]

Thus, the current income of the person is **Rs. 28800**.

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