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To solve the problem of determining the annual interest rate for Nanuk's savings account, we can use the compound interest formula:

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

Where:
- \( A \) is the amount of money accumulated after \( t \) years, including interest.
- \( P \) is the principal amount (the initial deposit).
- \( r \) is the annual interest rate (expressed as a decimal).
- \( n \) is the number of times that interest is compounded per year.
- \( t \) is the time the money is invested for in years.

Given:
- \( A = £2545.10 \)
- \( P = £2000 \)
- \( t = 4 \) years
- Interest is compounded annually, so \( n = 1 \).

Substituting the values into the formula:

\[
2545.10 = 2000 \left(1 + \frac{r}{1}\right)^{4 \times 1}
\]

Simplifying:

\[
2545.10 = 2000 \left(1 + r\right)^4
\]

Next, we divide both sides by 2000:

\[
\frac{2545.10}{2000} = \left(1 + r\right)^4
\]

\[
1.27255 = \left(1 + r\right)^4
\]

To solve for \( r \), take the fourth root of both sides:

\[
1 + r = \left(1.27255\right)^{\frac{1}{4}}
\]

\[
1 + r = 1.0623
\]

Subtract 1 from both sides:

\[
r = 0.0623
\]

Convert the decimal to a percentage:

\[
r = 6.23\%
\]

Rounded to one decimal place, the annual interest rate is **6.2%**.

Would you like any further details on this calculation or have any questions?

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Here are some related questions you might find interesting:

1. What is the difference between simple and compound interest?
2. How would the interest rate change if the interest were compounded quarterly instead of annually?
3. How does the compound interest formula change if additional deposits are made regularly?
4. What would the final amount be if the interest rate were instead 5%?
5. How long would it take to double the initial investment at the calculated interest rate?

**Tip:** Understanding how to manipulate the compound interest formula is key to solving many finance-related problems, especially those involving investments and loans.

Nanuk has a savings account which gathers compound interest each year. He opened the account with a deposit of £2000. After 4 years there is £2545.10 in the account. Work out the annual interest rate of the savings account. Give your answer to 1 d.p.

Learn how to calculate the annual interest rate for a compound interest savings account. Starting with £2000 and after 4 years, the total amount is £2545.10. This problem covers compound interest formula and exponential growth concepts. Suitable for Grades 9-11.

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