Questions: Suppose that your income is 25,000 per year and you receive a cost-of-living raise each year. If inflation is constant at 8.3% annually, in how many years will you be making 30,900 per year? Round your answer to the nearest whole number.

Solution Steps

We start with the exponential growth equation relating the final salary \( S_f \), the initial salary \( S_i \), the growth rate \( r \), and the number of years \( t \):

\[ S_f = S_i \cdot (1 + r)^t \]

Given:

• \( S_i = 25000 \)
• \( S_f = 30900 \)
• \( r = 0.083 \)

Rearranging the equation to solve for \( t \):

\[ t = \frac{\log\left(\frac{S_f}{S_i}\right)}{\log(1 + r)} \]

Substituting the known values:

\[ t = \frac{\log\left(\frac{30900}{25000}\right)}{\log(1 + 0.083)} \]

Calculating the values:

1. Calculate \( \frac{30900}{25000} = 1.236 \).
2. Calculate \( \log(1.236) \approx 0.09691 \).
3. Calculate \( \log(1.083) \approx 0.03514 \).

Now substituting these values into the equation for \( t \):

\[ t \approx \frac{0.09691}{0.03514} \approx 2.757 \]

Rounding \( t \) to the nearest whole number gives:

\[ t \approx 3 \]

Final Answer