Answered

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



Compound interest formula: $V(t)=P\left(1+\frac{r}{n}\right)^{n t}$

$t=$ years since initial deposit

$n=$ number of times compounded per year

$r=$ annual interest rate (as a decimal)

$P=$ initial (principal) investment

$V(t)=$ value of investment after $t$ years


A5\%
B31\%
C26\%
D11\%

Verified Answer :

  • Identify the variables: principal \(P = \$2500\), future value \(V(2) = \$2762.35\), time \(t = 2\) years, and compounding frequency \(n = 12\).
  • Apply the compound interest formula \(V(t) = P(1 + r/n)^{nt}\) to solve for \(r\).
  • Simplify the expression to find \((1 + r/12)^{24} = 1.10494\).
  • Solve for \(r\) to obtain approximately \(0.05\), which is equivalent to \(5\%\), matching option \(\boxed{a}\).

Explanation

  1. Analyze the Problem and Data

To find the annual interest rate \(r\), we start with the compound interest formula:
\[V(t) = P\left(1 + \frac{r}{n}\right)^{nt}\]
Given data:

  • Principal investment (\(P\)) = \(\$2,500\)
  • Future value after 2 years (\(V(2)\)) = \(\$2,762.35\)
  • Time period (\(t\)) = \(2\) years
  • Compounding frequency (\(n\)) = \(12\) (monthly)

We substitute these values into the formula:
\[2762.35 = 2500\left(1 + \frac{r}{12}\right)^{12 \times 2}\]
\[2762.35 = 2500\left(1 + \frac{r}{12}\right)^{24}\]

  1. Solve for the Interest Rate r

To isolate the interest rate \(r\), we first divide both sides of the equation by the principal \(P = 2500\):
\[\frac{2762.35}{2500} = \left(1 + \frac{r}{12}\right)^{24}\]
\[1.10494 = \left(1 + \frac{r}{12}\right)^{24}\]
Next, we take the \(24^{th}\) root of both sides to solve for the term inside the parentheses:
\[\left(1 + \frac{r}{12}\right) = (1.10494)^{1/24}\]
Calculating the \(24^{th}\) root of \(1.10494\):
\[1 + \frac{r}{12} \approx 1.0041666\]
Subtracting \(1\) from both sides:
\[\frac{r}{12} \approx 0.0041666\]
Multiplying by \(12\):
\[r \approx 12 \times 0.0041666 \approx 0.049999\]

  1. Conclusion and Selection

The result for \(r\) is approximately \(0.049999\), which represents the decimal form of the annual interest rate. To express this as a percentage, we multiply by \(100\):
\[0.049999 \times 100 \approx 5\%\]
Comparing this result to the given options:

A\(5\%\)
B\(31\%\)
C\(26\%\)
D\(11\%\)

The calculated annual interest rate of \(5\%\) matches option a. Thus, the final answer is \(\boxed{a}\).

Examples

The concept of compound interest is essential for understanding how savings accounts, investments, and loans work over time. For example, if you deposit money into a high-yield savings account for college, the interest you earn is reinvested, allowing your balance to grow exponentially rather than linearly. By calculating the annual interest rate, you can compare different bank offers and choose the most effective way to grow your money, demonstrating the power of mathematical planning in your personal finances.

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