Amira is going to invest in an account paying an interest rate of 2.1% compounded quarterly. How much would Amira need to invest, *to the nearest ten dollars*, for the value of the account to reach $880 in 9 years?

To solve this problem, we need to use the formula for compound interest when interest is compounded periodically (e.g., quarterly, annually, etc.):

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

where:
- $ A $ is the amount of money in the account after time $ t $,
- $ P $ is the principal amount (the initial investment),
- $ r $ is the annual interest rate (as a decimal),
- $ n $ is the number of times the interest is compounded per year,
- $ t $ is the time in years.

Given:
- $ A = 880 $ (the final amount),
- $ r = 2.1\% = 0.021 $ (the interest rate),
- $ n = 4 $ (interest is compounded quarterly),
- $ t = 9 $ (time in years).

We need to find $ P $, the initial investment amount.

### Step-by-step Solution

1. **Set up the equation:**

\[
880 = P \left(1 + \frac{0.021}{4}\right)^{4 \times 9}
\]

2. **Simplify the expression inside the parentheses:**

\[
1 + \frac{0.021}{4} = 1 + 0.00525 = 1.00525
\]

3. **Calculate the exponent $ 4 \times 9 = 36 $:**

\[
880 = P \left(1.00525\right)^{36}
\]

4. **Compute $ \left(1.00525\right)^{36} $:**

\[
(1.00525)^{36} \approx 1.2034
\]

5. **Substitute this value back into the equation:**

\[
880 = P \times 1.2034
\]

6. **Solve for $ P $:**

\[
P = \frac{880}{1.2034}
\]

\[
P \approx 731.19
\]

7. **Round to the nearest ten dollars:**

\[
P \approx 730
\]

### Conclusion

Amira would need to invest **$730** to the nearest ten dollars for the value of the account to reach $880 in 9 years.

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

### Related Questions:

1. How much would Amira need to invest if the interest rate was 3% instead of 2.1%?
2. What would be the final amount if Amira invested $730 at an interest rate of 2.1% compounded quarterly for 12 years?
3. How does the frequency of compounding (quarterly vs. annually) affect the amount needed to reach a specific future value?
4. What would be the future value of an investment of $730 at 2.1% compounded quarterly after 15 years?
5. How would the required initial investment change if the target amount was $1,000 instead of $880?

**Tip:** When calculating compound interest for different compounding frequencies, remember that more frequent compounding leads to slightly higher returns due to interest being calculated on previously earned interest more often.

Learn how to calculate compound interest with a 2.1% interest rate compounded quarterly. Understand the formula A = P (1 + r/n)^(nt) and solve for the initial investment needed for an account to reach $880 in 9 years.

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