Income of a person decreased by 3000 Rs. and rate of Income tax increased from `15% t o 18% ` . The ratio of paid tax is` 3 : 2. 26.5%` income is tax free. Find initial income.

Answer

To solve the problem step by step, we will denote the initial income as \( x \).

### Step 1: Determine the taxable income
Since 26.5% of the income is tax-free, the taxable income can be calculated as:
\[ \text{Taxable Income} = x - (26.5\% \text{ of } x) = x - 0.265x = 0.735x \]

**Hint:** To find the taxable income, subtract the tax-free percentage from the total income.

### Step 2: Calculate the tax before the decrease in income
The initial income tax rate is 15%, so the tax paid before the decrease in income is:
\[ \text{Tax Paid} = 15\% \text{ of } \text{Taxable Income} = 0.15 \times 0.735x = 0.11025x \]

**Hint:** Use the formula for calculating tax: Tax = Tax Rate × Taxable Income.

### Step 3: Determine the new income after the decrease
The income decreased by 3000 Rs, so the new income is:
\[ \text{New Income} = x - 3000 \]

### Step 4: Calculate the new taxable income
The new taxable income after the decrease is:
\[ \text{New Taxable Income} = (x - 3000) - (26.5\% \text{ of } (x - 3000)) = (x - 3000) - 0.265(x - 3000) = 0.735(x - 3000) \]

**Hint:** Remember to apply the tax-free percentage to the new income.

### Step 5: Calculate the tax paid after the decrease
The new income tax rate is 18%, so the tax paid after the decrease is:
\[ \text{New Tax Paid} = 18\% \text{ of New Taxable Income} = 0.18 \times 0.735(x - 3000) = 0.1313(x - 3000) \]

**Hint:** Again, use the tax formula with the new tax rate.

### Step 6: Set up the ratio of taxes paid
According to the problem, the ratio of the tax paid before and after the decrease is \( 3:2 \):
\[ \frac{0.11025x}{0.1313(x - 3000)} = \frac{3}{2} \]

### Step 7: Cross-multiply to solve for \( x \)
Cross-multiplying gives:
\[ 2 \times 0.11025x = 3 \times 0.1313(x - 3000) \]
\[ 0.2205x = 0.3939x - 393.9 \]

### Step 8: Rearranging the equation
Rearranging gives:
\[ 0.3939x - 0.2205x = 393.9 \]
\[ 0.1734x = 393.9 \]

### Step 9: Solve for \( x \)
Now, divide both sides by 0.1734 to find \( x \):
\[ x = \frac{393.9}{0.1734} \approx 2273.68 \]

### Step 10: Rounding to the nearest whole number
Since income is typically in whole numbers, we round \( x \) to:
\[ x \approx 2274 \text{ Rs} \]

### Final Answer
The initial income is approximately **2274 Rs**.

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