A woman deposits ​$10 comma 000 at the end of each year for 9 years in an investment account with a guaranteed interest rate of 6​% compounded annually.
​(a) Find the value in the account at the end of the 9 years.
​(b) Her sister works for an investment firm that pays 5​% compounded annually. If the woman deposits money with this firm instead of the one in part​ (a), how much will she have in her account at the end of 9 ​years?
​(c) How much would she lose or gain over 9 years by investing in her​ sister's firm?

This problem involves calculating the future value of an annuity, where a fixed amount is deposited regularly and interest is compounded annually.

### (a) Value in the account at the end of 9 years (6% interest)

To calculate the future value of an annuity, we use the formula:

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

Where:
- $ FV $ = future value of the investment
- $ P $ = payment per period (in this case, \$10,000)
- $ r $ = interest rate per period (6% compounded annually, so $ r = 0.06 $)
- $ n $ = number of periods (9 years)

Substitute the values:

\[
FV = 10{,}000 \times \left( \frac{{(1 + 0.06)^9 - 1}}{0.06} \right)
\]
\[
FV = 10{,}000 \times \left( \frac{{(1.06)^9 - 1}}{0.06} \right)
\]

First, calculate $ (1.06)^9 $:
\[
(1.06)^9 \approx 1.68948
\]

Now substitute this value:
\[
FV = 10{,}000 \times \left( \frac{{1.68948 - 1}}{0.06} \right)
\]
\[
FV = 10{,}000 \times \left( \frac{{0.68948}}{0.06} \right)
\]
\[
FV \approx 10{,}000 \times 11.4913 = 114{,}913
\]

So, the value in the account at the end of 9 years is approximately **\$114,913**.

### (b) Value in the account at the end of 9 years (5% interest)

The same formula applies, but this time the interest rate is 5% compounded annually, so $ r = 0.05 $.

\[
FV = 10{,}000 \times \left( \frac{{(1 + 0.05)^9 - 1}}{0.05} \right)
\]
\[
FV = 10{,}000 \times \left( \frac{{(1.05)^9 - 1}}{0.05} \right)
\]

First, calculate $ (1.05)^9 $:
\[
(1.05)^9 \approx 1.55133
\]

Now substitute this value:
\[
FV = 10{,}000 \times \left( \frac{{1.55133 - 1}}{0.05} \right)
\]
\[
FV = 10{,}000 \times \left( \frac{{0.55133}}{0.05} \right)
\]
\[
FV \approx 10{,}000 \times 11.0266 = 110{,}266
\]

So, the value in the account at the end of 9 years is approximately **\$110,266**.

### (c) Loss or gain over 9 years by investing in her sister's firm

To find the difference between the two investments:

\[
\text{Difference} = 114{,}913 - 110{,}266 = 4{,}647
\]

By investing with her sister's firm, the woman would have **lost \$4,647** over 9 years compared to the 6% investment account.

### Summary of Results:
- (a) Account value after 9 years at 6% interest: **\$114,913**
- (b) Account value after 9 years at 5% interest: **\$110,266**
- (c) Loss from investing with sister’s firm: **\$4,647**

Would you like more details on the calculations or have any questions?

#### 5 related questions:
1. How would the results change if the deposit period was longer than 9 years?
2. What if the woman deposited \$20,000 annually instead of \$10,000?
3. How do different compounding periods (e.g., monthly) affect the future value?
4. How does the formula change for an account with continuous compounding interest?
5. What is the effect of inflation on the future value of such an investment?

#### Tip:
When comparing investments, it's essential to consider the impact of different interest rates over time, as small rate changes can lead to significant differences in the future value.

A woman deposits $10,000 at the end of each year for 9 years in an investment account with a guaranteed interest rate of 6% compounded annually. (a) Find the value in the account at the end of the 9 years. (b) Her sister works for an investment firm that pays 5% compounded annually. If the woman deposits money with this firm instead of the one in part (a), how much will she have in her account at the end of 9 years? (c) How much would she lose or gain over 9 years by investing in her sister's firm?

Learn how to calculate the future value of an annuity with $10,000 annual deposits at interest rates of 6% and 5% compounded annually over 9 years. This step-by-step guide covers key financial math concepts, including compound interest and annuities, suitable for students in Grades 10-12.

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