To determine how much more money Sparky will have in 30 years if he earns 13% compounded annually instead of 11%, we need to calculate the future value of his annuity for both interest rates and then find the difference.

Step 1: Calculate the Future Value of the Annuity at 13% Interest

The future value $FV$ of an ordinary annuity is calculated using the formula: $FV = PMT \times \frac{(1 + r)^n - 1}{r}$

Where:

• $PMT$ is the annual payment ($7,500)
• $r$ is the annual interest rate (0.13 for 13%)
• $n$ is the total number of payments (30)

Plugging in the values: $FV_{13\%} = 7500 \times \frac{(1 + 0.13)^{30} - 1}{0.13}$

Calculating the values inside the parentheses first: $(1 + 0.13)^{30} \approx 52.0101$

Now: $FV_{13\%} = 7500 \times \frac{52.0101 - 1}{0.13}$ $FV_{13\%} = 7500 \times \frac{51.0101}{0.13}$ $FV_{13\%} = 7500 \times 392.3854$ $FV_{13\%} \approx 2,942,890.50$

Step 2: Calculate the Future Value of the Annuity at 11% Interest

Using the same formula, we calculate the future value at 11% interest: $FV_{11\%} = 7500 \times \frac{(1 + 0.11)^{30} - 1}{0.11}$

Calculating the values inside the parentheses first: $(1 + 0.11)^{30} \approx 22.8916$

Now: $FV_{11\%} = 7500 \times \frac{22.8916 - 1}{0.11}$ $FV_{11\%} = 7500 \times \frac{21.8916}{0.11}$ $FV_{11\%} = 7500 \times 199.015$ $FV_{11\%} \approx 1,492,612.50$

Step 3: Calculate the Difference

Now, we find the difference between the future values at 13% and 11% interest rates: $\text{Difference} = FV_{13\%} - FV_{11\%}$ $\text{Difference} = 2,942,890.50 - 1,492,612.50$ $\text{Difference} \approx 1,450,278$

Therefore, Sparky will have approximately $1,450,278 more in 30 years if he earns 13% compounded annually instead of 11%.

Would you like any further details or explanations on this solution? Here are five more questions you might find interesting:

1. How does the future value change if the number of years is increased or decreased?
2. What is the effect of making monthly contributions instead of annual ones?
3. How do you calculate the present value of an annuity with different interest rates?
4. What is the impact of different compounding periods (monthly, quarterly) on the future value?
5. How can you determine the interest rate required to reach a specific future value?

Tip: Always compare different interest rates and compounding periods to understand their long-term impact on investments.