To calculate how much extra interest is earned with a 10% return compared to a 5% return over 7 years, we can use the compound interest formula:

$A = P \left(1 + \frac{r}{100}\right)^t$

Where:

• $A$ is the final amount
• $P$ is the initial investment ($X$ in this case)
• $r$ is the annual interest rate (5% or 10%)
• $t$ is the number of years (7 years)

We'll calculate the future value for both the 5% and 10% interest rates over 7 years, then find the difference between the two.

Step 1: Future Value at 10% Interest Rate

$A_{10} = X \left(1 + \frac{10}{100}\right)^7 = X \cdot (1.1)^7$

Step 2: Future Value at 5% Interest Rate

$A_{5} = X \left(1 + \frac{5}{100}\right)^7 = X \cdot (1.05)^7$

Step 3: Extra Interest Earned

The extra interest is simply the difference between these two future values: $\text{Extra Interest} = A_{10} - A_{5}$ $\text{Extra Interest} = X \cdot (1.1^7 - 1.05^7)$

Now, let's compute the values of $1.1^7$ and $1.05^7$.

• $1.1^7 \approx 1.9487$
• $1.05^7 \approx 1.4071$

So: $\text{Extra Interest} = X \cdot (1.9487 - 1.4071) = X \cdot 0.5416$

Thus, the extra interest earned is approximately $0.5416X$, or 54.16% of the original investment amount $X$.

Summary:

The extra interest you would earn by investing at 10% per year instead of 5% per year over 7 years is approximately 54.16% of the initial investment $X$.

---

Would you like further clarification or details? Here are some related questions:

1. How does compound interest differ from simple interest?
2. How would the extra interest change if the investment period were longer?
3. What would the outcome be with different interest rates?
4. How does inflation impact the value of returns in such investments?
5. What other factors should you consider when choosing between different investment options?

Tip: The longer the investment period, the more significant the impact of small differences in interest rates due to compounding.