Suppose ​$289 is deposited at the end of every six months into an account earning 7.1 % per annum compounded annually. If the balance in the account is to be $ 11, 300, how many deposits are​ needed? 

To solve this problem, we will use the formula for the future value of an annuity:

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

Where:
- $ FV $ is the future value (total amount accumulated),
- $ P $ is the amount of each deposit,
- $ r $ is the annual interest rate,
- $ n $ is the number of compounding periods per year,
- $ t $ is the time in years.

Given:
- $ P = 289 $ (each deposit),
- $ r = 7.1\% = 0.071 $ (annual interest rate),
- $ FV = 11,300 $ (future value),
- Since the interest is compounded annually, $ n = 1 $.

Since deposits are made every 6 months, the number of deposits in one year is 2. Let's calculate how many deposits (denoted as $ N $) are needed to reach the future value of $11,300.

First, let's rewrite the formula in terms of $ t $, and solve for the time period in years.

\[
11,300 = 289 \times \frac{(1 + 0.071/1)^{1 \times t} - 1}{0.071/1}
\]

Let’s solve this step-by-step.

---

I will calculate the number of periods for you now.The solution for the time period is approximately $ t = 19.37 $ years.

Since deposits are made every 6 months (twice a year), the total number of deposits is:

\[
N = 2 \times 19.37 = 38.74
\]

Since the number of deposits must be a whole number, we round up to the next whole number. Thus, **39 deposits** are needed to reach the future value of $11,300.

Would you like more details or have any questions?

Here are 5 related questions:
1. How does changing the interest rate affect the number of deposits required?
2. What if the deposits were made quarterly instead of every six months?
3. How would the solution change if the interest was compounded semi-annually instead of annually?
4. What if the future value goal was $15,000 instead of $11,300?
5. How can the future value formula be applied to other financial goals?

**Tip**: When working with compounded interest, always make sure to align the frequency of deposits and compounding periods.

Suppose $289 is deposited at the end of every six months into an account earning 7.1% per annum compounded annually. If the balance in the account is to be $11,300, how many deposits are needed?

Learn how to calculate the number of deposits required to reach a future value of $11,300 with a $289 deposit every six months at an annual interest rate of 7.1%, compounded annually. Explore the future value formula and key financial concepts related to compound interest and annuities. Suitable for high school students (Grades 10-12).

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