To determine if you're making a percentage more or less after a raise while accounting for inflation, you can follow these steps:
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Calculate the New Salary After the Raise: If your current salary is \( S \) and you receive a raise of \( n% \), your new salary \( S_{\text{new}} \) will be: \[ S_{\text{new}} = S \times (1 + \frac{n}{100}) \]
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Adjust for Inflation: If inflation is \( m% \), the value of money effectively decreases. To find out how much your salary needs to increase to keep up with inflation, calculate the inflation-adjusted salary \( S_{\text{adjusted}} \): \[ S_{\text{adjusted}} = S \times (1 + \frac{m}{100}) \]
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Compare the New Salary to the Adjusted Salary: To find the real change in your income considering inflation, you can calculate the difference in percentage terms: \[ \text{Real Change} = \left(\frac{S_{\text{new}} - S_{\text{adjusted}}}{S_{\text{adjusted}}}\right) \times 100 \]
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Substituting the Values: Replacing \( S_{\text{new}} \) and \( S_{\text{adjusted}} \) in the equation gives: \[ \text{Real Change} = \left(\frac{S \times (1 + \frac{n}{100}) - S \times (1 + \frac{m}{100})}{S \times (1 + \frac{m}{100})}\right) \times 100 \] Once simplified, it results in: \[ \text{Real Change} = \left(\frac{1 + \frac{n}{100} - (1 + \frac{m}{100})}{1 + \frac{m}{100}}\right) \times 100 \] \[ = \left(\frac{\frac{n}{100} - \frac{m}{100}}{1 + \frac{m}{100}}\right) \times 100 \] \[ = \frac{n - m}{1 + \frac{m}{100}} \]
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Outcome:
- If \( n > m \), you're effectively making more money in real terms.
- If \( n < m \), you're making less in real terms.
- If \( n = m \), your purchasing power remains unchanged.
This formula lets you determine the actual change in your purchasing power after accounting for both the raise and the inflation.