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Isabel deposits \$6,000 into an account that earns 1.5% interest compounded monthly. Assuming no more deposits and no withdrawals are made, how much money is in the account after 4 years?



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


A\$6,370.78
B\$6,360.00
C\$10,892.13
D\$7,180.89

Verified Answer :

  • Identify variables: \(P = 6000\), \(r = 0.015\), \(n = 12\), \(t = 4\).
  • Substitute into the compound interest formula: \(V(4) = 6000(1 + \frac{0.015}{12})^{12 \times 4}\).
  • Simplify and compute: \(V(4) = 6000(1.00125)^{48} \approx 6370.78\).
  • Select the matching option: \(\boxed{\$6,370.78}\) (Option a).

Explanation

  1. Problem Analysis

To determine the amount of money in the account after 4 years, we use the compound interest formula:
\[V(t) = P\left(1 + \frac{r}{n}\right)^{nt}\]
Given data:

  • Principal investment \(P = \$6,000\)
  • Annual interest rate \(r = 1.5\% = 0.015\)
  • Compounding frequency \(n = 12\) (monthly)
  • Time period \(t = 4\) years

These values represent our initial conditions for calculating the future value of the investment.

  1. Setting up the Equation

We now substitute the given values into the formula:
\[V(4) = 6000\left(1 + \frac{0.015}{12}\right)^{12 \times 4}\]
First, we calculate the periodic interest rate, \(\frac{r}{n}\):
\[\frac{0.015}{12} = 0.00125\]
Next, we determine the total number of compounding periods, \(nt\):
\[12 \times 4 = 48\]
Now, substitute these back into the expression:
\[V(4) = 6000(1 + 0.00125)^{48} = 6000(1.00125)^{48}\]

  1. Calculation and Conclusion

Now we calculate the value of the investment:
\[V(4) = 6000 \times (1.00125)^{48} \approx 6000 \times 1.06179676\]
\[V(4) \approx 6370.78057\]
Rounding to the nearest cent, the final amount in the account is \(\$6,370.78\). Comparing this to the given options:

A\(\$6,370.78\)
B\(\$6,360.00\)
C\(\$10,892.13\)
D\(\$7,180.89\)

The correct option is a.

Examples

Understanding compound interest is crucial for personal finance, such as savings accounts, retirement funds, or loans. By knowing how to calculate the growth of your money over time, you can make informed decisions about where to invest and how long to leave your money in an account. For example, if you deposit \(\$6,000\) into an account that earns \(1.5\%\) interest compounded monthly, you can see how much it grows over \(4\) years, helping you plan for future financial goals.

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