https://file.aiquickdraw.com/mathbot/file/1728342625975b2iwshyd.jpg

To solve the problem, we use the formula for compound interest:

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

Where:
- \( A \) is the amount of money at the end of the investment (target £8000),
- \( P \) is the principal or initial amount (which we need to find),
- \( r \) is the annual interest rate (5%),
- \( t \) is the time in years (3 years).

We are given:
\[
A = 8000, \quad r = 5\%, \quad t = 3 \text{ years}.
\]

We need to solve for \( P \).

\[
8000 = P \left(1 + \frac{5}{100}\right)^3
\]
\[
8000 = P \left(1.05\right)^3
\]
\[
8000 = P \times 1.157625
\]
\[
P = \frac{8000}{1.157625}
\]
\[
P \approx 6910.85
\]

To the nearest pound, Cynthia needs to invest **£6911**.

Would you like more details or have any further questions?

Here are some related questions to consider:
1. What would the investment be if the interest rate was 6% instead of 5%?
2. How much would the investment grow to in 4 years instead of 3?
3. How would the required initial investment change if the final target was £10,000?
4. What would happen if interest was compounded monthly rather than annually?
5. What is the formula for continuous compound interest?

**Tip:** When calculating compound interest, the frequency of compounding (monthly, quarterly, yearly) can significantly impact the final amount.

Cynthia invests some money in a bank which pays 5% compound interest per year. She wants it to be worth over £8000 at the end of 3 years. What is the smallest amount, to the nearest pound, she can invest?

Discover how to calculate the initial investment required to reach £8000 in 3 years at 5% annual compound interest. This problem explores the compound interest formula and is suitable for high school students (Grades 10-12).

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