Claire deposited [tex][tex]$2,500[/tex] into an account that accrues interest monthly. She made no additional deposits or withdrawals. After 2 years, Claire had [tex]$[/tex]2,762.35[/tex] in the account. What is the annual interest rate of the account?

Compound interest formula: [tex]V(t)=P\left(1+\frac{r}{n}\right)^{nt}[/tex]

- [tex]t[/tex] = years since initial deposit
- [tex]n[/tex] = number of times compounded per year
- [tex]r[/tex] = annual interest rate (as a decimal)
- [tex]P[/tex] = initial (principal) investment
- [tex]V(t)[/tex] = value of investment after [tex]t[/tex] years

Options:
A. [tex]5\%[/tex]
B. [tex]11\%[/tex]
C. [tex]26\%[/tex]
D. [tex]31\%[/tex]



Answer :

Absolutely, let's solve this step-by-step using the compound interest formula to find out the annual interest rate.

The compound interest formula is:
[tex]\[ V(t) = P \left(1 + \frac{r}{n} \right)^{nt} \][/tex]
where:
- [tex]\( P = 2500 \)[/tex] (initial principal investment, in dollars)
- [tex]\( V(t) = 2762.35 \)[/tex] (value of the investment after [tex]\( t \)[/tex] years, in dollars)
- [tex]\( t = 2 \)[/tex] (years since initial deposit)
- [tex]\( n = 12 \)[/tex] (number of times interest is compounded per year)
- [tex]\( r \)[/tex] is the annual interest rate (as a decimal)

To find the annual interest rate ([tex]\( r \)[/tex]) we need to rearrange the formula. Start by isolating [tex]\( \frac{r}{n} \)[/tex]:

[tex]\[ 2762.35 = 2500 \left(1 + \frac{r}{12} \right)^{12 \cdot 2} \][/tex]

Divide both sides by 2500:

[tex]\[ \frac{2762.35}{2500} = \left(1 + \frac{r}{12} \right)^{24} \][/tex]

Calculate the left-hand side:

[tex]\[ 1.10494 = \left(1 + \frac{r}{12} \right)^{24} \][/tex]

To solve for [tex]\( r \)[/tex], we take the 24th root of both sides:

[tex]\[ \left(1.10494 \right)^{\frac{1}{24}} = 1 + \frac{r}{12} \][/tex]

Calculate the 24th root:

[tex]\[ 1.004166 \approx 1 + \frac{r}{12} \][/tex]

Subtract 1 from both sides to isolate [tex]\( \frac{r}{12} \)[/tex]:

[tex]\[ \frac{r}{12} \approx 0.004166 \][/tex]

Multiply both sides by 12 to solve for [tex]\( r \)[/tex]:

[tex]\[ r \approx 0.004166 \times 12 \][/tex]
[tex]\[ r \approx 0.049999 \][/tex]

Convert [tex]\( r \)[/tex] to a percentage by multiplying by 100:

[tex]\[ r \approx 4.9999\% \][/tex]

Therefore, the annual interest rate for the account is approximately [tex]\( 5\% \)[/tex]. The correct answer from the given options is:

[tex]\[ \boxed{5\%} \][/tex]