Matthew invested $5,300 in an account paying an interest rate of 2.1% compounded annually. Assuming no deposits or withdrawals are made, how long would it take, to the nearest year, for the value of the account to reach $6,020?

To find out how long it will take for Matthew's investment to grow from $5,300 to $6,020 with an interest rate of 2.1% compounded annually, 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 amount of money).
- $ r $ is the annual interest rate (in decimal).
- $ n $ is the number of times that interest is compounded per year.
- $ t $ is the time the money is invested or borrowed for, in years.

Given:
- $ A = 6,020 $
- $ P = 5,300 $
- $ r = 2.1\% = 0.021 $
- $ n = 1 $ (since the interest is compounded annually)

We need to solve for $ t $.

The formula simplifies to:

\[
6,020 = 5,300 \times \left(1 + 0.021\right)^{t}
\]

\[
\frac{6,020}{5,300} = \left(1.021\right)^{t}
\]

\[
1.13585 \approx 1.021^{t}
\]

To solve for $ t $, we take the natural logarithm (ln) of both sides:

\[
\ln(1.13585) = t \times \ln(1.021)
\]

\[
t = \frac{\ln(1.13585)}{\ln(1.021)}
\]

Let's calculate the value of $ t $.The calculation shows that it will take approximately $ 6.13 $ years for Matthew's investment to grow to $6,020. Therefore, to the nearest year, it will take **6 years**.

Would you like more details on this calculation, or do you have any other questions?

Here are 5 related questions you might find interesting:
1. How would the time change if the interest rate was increased to 3%?
2. What if Matthew wanted his investment to grow to $7,000 instead?
3. How would the calculation differ if the interest was compounded semi-annually?
4. What if Matthew makes an additional deposit every year? How would this affect the time?
5. How much interest will Matthew have earned at the end of the 6 years?

**Tip:** Understanding how compound interest works can help you make better financial decisions, as the frequency of compounding and the rate have significant impacts on your investment growth.

Explore how long it takes for an investment of $5,300 at 2.1% annual interest, compounded annually, to grow to $6,020. Includes step-by-step calculation using compound interest formula. Suitable for Grades 10-12.

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