Determine the final amount after 3 years if $7000 is compounded monthly at 3 %. Show step by step working

We are asked to find the final amount of an investment of \$7000 after 3 years, compounded monthly at an interest rate of 3%.

We can use the compound interest formula:

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

Where:
- $A$ is the final amount.
- $P = 7000$ is the initial principal (the starting amount).
- $r = 0.03$ is the annual interest rate (3% expressed as a decimal).
- $n = 12$ is the number of times the interest is compounded per year (monthly compounding).
- $t = 3$ is the time the money is invested for (3 years).

### Step 1: Calculate the monthly interest rate
The monthly interest rate is given by:

\[
\frac{r}{n} = \frac{0.03}{12} = 0.0025
\]

### Step 2: Calculate the total number of compounding periods
The total number of compounding periods over 3 years is:

\[
nt = 12 \times 3 = 36
\]

### Step 3: Substitute values into the formula
Now, substitute the values into the compound interest formula:

\[
A = 7000 \left(1 + 0.0025\right)^{36}
\]

Simplifying the expression inside the parentheses:

\[
A = 7000 \left(1.0025\right)^{36}
\]

### Step 4: Calculate the power
Using a calculator, we can compute:

\[
1.0025^{36} \approx 1.093443
\]

### Step 5: Final Calculation
Now multiply by the principal amount:

\[
A = 7000 \times 1.093443 \approx 7654.10
\]

### Final Answer:
The final amount after 3 years is approximately **\$7654.10**.

Would you like more details on any of the steps?

Here are 5 related questions you might consider:
1. How would the final amount change if the interest were compounded annually?
2. What is the formula for continuous compounding, and how does it compare with monthly compounding?
3. How long would it take for the investment to double at this interest rate?
4. What would happen if the interest rate increased to 5% but was still compounded monthly?
5. What is the effect of the number of compounding periods on the final amount?

**Tip:** Increasing the frequency of compounding (e.g., from yearly to monthly) generally increases the final amount, even if the interest rate remains the same.

Learn how to calculate the final amount after 3 years for an investment of $7000 compounded monthly at 3%. This step-by-step guide uses the compound interest formula to show detailed calculations and explanations. Suitable for high school students learning about interest rates and exponential growth.

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