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.