Amita wants to make monthly deposits of $200 into a savings account that offers 1.5% interest compounded monthly. Use the future balance of a periodic investment formula from Example 3 to determine how long will it take for the account balance to reach $5000. Round to the nearest tenth of a year.

To determine how long it will take for Amita's savings account to reach a balance of $5000, we will use the future balance of a periodic investment formula, which is:

\[
A = P \times \frac{(1 + r/n)^{nt} - 1}{r/n}
\]

Where:
- $ A $ is the future value of the account (in this case, $5000).
- $ P $ is the monthly deposit ($200).
- $ r $ is the annual interest rate (1.5%, or 0.015 as a decimal).
- $ n $ is the number of times interest is compounded per year (monthly, so $ n = 12 $).
- $ t $ is the time in years, which we are trying to find.

### Step 1: Plug in the known values into the formula

\[
5000 = 200 \times \frac{(1 + 0.015/12)^{12t} - 1}{0.015/12}
\]

### Step 2: Simplify the formula
First, calculate the monthly interest rate:

\[
r/n = \frac{0.015}{12} = 0.00125
\]

Substitute this back into the formula:

\[
5000 = 200 \times \frac{(1 + 0.00125)^{12t} - 1}{0.00125}
\]
\[
5000 = 200 \times \frac{(1.00125)^{12t} - 1}{0.00125}
\]

### Step 3: Solve for $ t $
Now, I will calculate $ t $ using logarithms, but I'll simplify further with actual numbers. Let's proceed to solve the equation.It will take approximately **2.1 years** for Amita's account balance to reach $5000, given monthly deposits of $200 and an interest rate of 1.5% compounded monthly.

Would you like further details or have any other questions?

Here are 5 related questions you might explore:
1. How would the time change if the interest rate were increased to 2%?
2. What if Amita wanted to deposit $300 instead of $200 monthly—how long would it take?
3. How much interest will Amita earn by the time the balance reaches $5000?
4. What would be the balance after 2.1 years if the interest were compounded annually instead of monthly?
5. How does the time required change if the final balance target is $10,000 instead of $5000?

**Tip:** Compounding frequency (monthly vs. annually) significantly affects the growth of an investment over time!

Learn how to calculate the time it will take for Amita to reach a $5000 balance with $200 monthly deposits and an interest rate of 1.5% compounded monthly. This problem explores the concept of compound interest and exponential growth, ideal for students in Grades 10-12.

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