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

$t=$ 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$6,360.00
B$6,370.78
C$7,180.89
D$10,892.13

Verified Answer :

  • Identify the variables from the problem: \(P = 6000\), \(r = 0.015\), \(n = 12\), and \(t = 4\).
  • Set up the compound interest formula: \(V(4) = 6000\left(1 + \frac{0.015}{12}\right)^{12 \times 4}\).
  • Simplify the expression to \(V(4) = 6000(1.00125)^{48}\).
  • Calculate the final value, which results in approximately \(\$6,370.78\).
  • Final answer: \(\boxed{\$6,370.78}\).

Explanation

  1. Problem Data Analysis

To determine the value of the investment after 4 years, we identify the following variables given in the problem:

  • The principal amount, \(P = \$6,000\).
  • The annual interest rate, \(r = 1.5\% = 0.015\).
  • The compounding frequency, \(n = 12\) (compounded monthly).
  • The time period, \(t = 4\) years.

We use the compound interest formula:
\[V(t) = P\left(1 + \frac{r}{n}\right)^{nt}\]

  1. Substitution and Setup

Substitute the known values into the formula to find the value after 4 years:
\[V(4) = 6000 \times \left(1 + \frac{0.015}{12}\right)^{12 \times 4}\]

First, calculate the monthly interest rate:
\[\frac{r}{n} = \frac{0.015}{12} = 0.00125\]

Next, calculate the total number of compounding periods:
\[nt = 12 \times 4 = 48\]

Now, substitute these back into the equation:
\[V(4) = 6000 \times (1 + 0.00125)^{48}\]
\[V(4) = 6000 \times (1.00125)^{48}\]

  1. Calculation and Conclusion

Calculate the value of \((1.00125)^{48}\) and then multiply it by the principal:
\[(1.00125)^{48} \approx 1.06179676\]
\[V(4) \approx 6000 \times 1.06179676 = 6370.78056...\]

Rounding this to the nearest cent, we get:
\[V(4) = \boxed{\$6,370.78}\]
Comparing this result to the given options, we find it matches option b.

Examples

Understanding compound interest is crucial for personal finance, as it illustrates how your money grows over time. For example, if you save for a future goal, like a college fund or a dream vacation, knowing how your interest accumulates helps you plan better and make informed decisions about your savings account or investment choices. Even small differences in interest rates or frequency of compounding can significantly impact your total balance in the long run.

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