john invests £2000 for 3 years. he recievs interest of £415.90. What was the compound interest rate to the nearest 0.1%?

To find the compound interest rate, we use the formula for compound interest:

\[
A = P(1 + r)^t
\]

Where:
- \( A \) is the final amount (£2000 + £415.90 = £2415.90),
- \( P \) is the principal amount (£2000),
- \( r \) is the compound interest rate (expressed as a decimal),
- \( t \) is the time period (3 years).

We need to solve for \( r \). Rewriting the formula:

\[
2415.90 = 2000(1 + r)^3
\]

Divide both sides by 2000:

\[
1.20795 = (1 + r)^3
\]

Now, take the cube root of both sides:

\[
1 + r = \sqrt[3]{1.20795}
\]

\[
1 + r = 1.06486
\]

Subtract 1 from both sides:

\[
r = 0.06486
\]

Convert to a percentage:

\[
r = 6.5\%
\]

Therefore, the compound interest rate is approximately **6.5%**.

Would you like further details on any step?

Here are some related questions to expand understanding:
1. How would the formula change if interest were compounded monthly?
2. What is the difference between compound and simple interest in this context?
3. How would the final amount change if the interest rate were 5% instead of 6.5%?
4. Can we calculate the interest rate if the investment period were 5 years?
5. What happens if John had invested an additional £500 after the first year?

**Tip:** To estimate the compound interest rate, try using logarithms if you want to avoid approximations.

John invests £2000 for 3 years. He receives interest of £415.90. What was the compound interest rate to the nearest 0.1%?

Discover how to calculate the compound interest rate for a 3-year investment of £2000 with an interest earned of £415.90. This step-by-step solution uses the compound interest formula to find the rate, providing insights into algebraic manipulation for students in Grades 10-12.

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