Rajesh’s salary increases by 50% whereas the tax increases by 75%. What is the percentage increase in tax rate?

To find the percentage increase in the tax rate, we need to analyze the problem step-by-step. Let’s define some variables and go through the calculations.

### Step 1: Define Variables

- Let \( S \) be the initial salary of Rajesh.
- Let \( T \) be the initial tax amount paid.
- The initial tax rate \( R \) is given by:
  \[
  R = \frac{T}{S}
  \]

### Step 2: Calculate the New Salary and Tax

- Rajesh’s salary increases by 50%, so the new salary \( S' \) is:
  \[
  S' = S + 0.5S = 1.5S
  \]

- The tax increases by 75%, so the new tax amount \( T' \) is:
  \[
  T' = T + 0.75T = 1.75T
  \]

### Step 3: Calculate the New Tax Rate

- The new tax rate \( R' \) based on the new salary and new tax amount is:
  \[
  R' = \frac{T'}{S'} = \frac{1.75T}{1.5S}
  \]

### Step 4: Find the Percentage Increase in Tax Rate

- The percentage increase in tax rate is given by:
  \[
  \text{Percentage increase in tax rate} = \left( \frac{R' - R}{R} \right) \times 100
  \]

Let's express \( R' - R \) in terms of \( R \):

\[
R' = \frac{1.75T}{1.5S} = \frac{1.75}{1.5} \cdot \frac{T}{S} = \frac{1.75}{1.5} \cdot R
\]

Calculating the factor:

\[
\frac{1.75}{1.5} = \frac{7}{6}
\]

So,

\[
R' = \frac{7}{6}R
\]

Now, calculate \( R' - R \):

\[
R' - R = \frac{7}{6}R - R = \left(\frac{7}{6} - 1\right)R = \frac{1}{6}R
\]

Therefore, the percentage increase in the tax rate is:

\[
\text{Percentage increase} = \left( \frac{\frac{1}{6}R}{R} \right) \times 100 = \frac{1}{6} \times 100 \approx 16.67\%
\]

### Conclusion

The percentage increase in the tax rate is **16.67%**.

Would you like more details, or do you have any questions about this solution?

Here are some related questions you might find interesting:

1. How does an increase in salary and tax affect the net income of an individual?
2. What would be the new tax rate if the salary doubled and the tax increased by 100%?
3. How can we calculate the tax rate increase if the tax is a fixed amount regardless of salary changes?
4. How does compounding affect the calculation of tax rates over multiple years?
5. How would the calculation change if there were deductions before the tax rate was applied?

**Tip:** Understanding how to work with percentage changes is crucial in financial calculations. Make sure to isolate the variable changes to clearly understand the effects on the final outcome.

Learn how to calculate the percentage increase in tax rate when Rajesh's salary increases by 50% and his tax increases by 75%. Follow a detailed step-by-step solution including formulas for percentage increase and proportional relationships. Suitable for high school students and above.

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