https://file.aiquickdraw.com/mathbot/file/1725722249444esglo0tk.jpg

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

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

where:
- \(A\) is the final amount in the account,
- \(P\) is the initial principal (the amount initially invested),
- \(r\) is the annual interest rate,
- \(n\) is the number of years.

Given:
- \(P = 2580\) (initial investment),
- \(r = 1\%\) (annual compound interest rate),
- \(A = 2684.76\) (final amount after \(n\) years).

We need to find the value of \(n\).

### Step-by-step solution:

1. Substitute the given values into the compound interest formula:

\[
2684.76 = 2580 \times \left(1 + \frac{1}{100}\right)^n
\]

2. Simplify the expression:

\[
2684.76 = 2580 \times \left(1.01\right)^n
\]

3. Divide both sides by 2580:

\[
\frac{2684.76}{2580} = 1.01^n
\]

4. Calculate the left-hand side:

\[
1.04065 \approx 1.01^n
\]

5. To solve for \(n\), take the logarithm of both sides:

\[
\log(1.04065) = \log(1.01^n)
\]

6. Use the logarithm rule \(\log(a^b) = b \log(a)\):

\[
\log(1.04065) = n \cdot \log(1.01)
\]

7. Solve for \(n\):

\[
n = \frac{\log(1.04065)}{\log(1.01)}
\]

Let's calculate the value of \(n\).The value of \(n\) is approximately 4 years.

So, it will take around 4 years for Amelia's investment to grow from £2580 to £2684.76 at a 1% per annum compound interest rate.

Would you like more details or have any questions? Here are some related questions that might interest you:

1. How do you calculate compound interest for different compounding periods (e.g., monthly, quarterly)?
2. What is the difference between simple interest and compound interest?
3. How does the interest rate affect the time required to reach a certain amount?
4. What happens if the interest rate is compounded continuously?
5. How can you calculate the future value of an investment with varying interest rates?

**Tip:** When solving problems involving compound interest, remember to carefully distinguish between simple and compound interest formulas to avoid mistakes.

Amelia invests £2580 for n years into a savings account. Amelia was paid 1% per annum compound interest. At the end of the n years, Amelia had £2684.76 in the savings account. Work out the value of n.

Learn how to calculate the number of years for an investment of £2580 to grow to £2684.76 at 1% per annum compound interest. This problem uses key algebraic concepts like the compound interest formula and logarithmic rules. Suitable for Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation