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Jacques deposited \$1,900 into an account that earns 4\% interest compounded semiannually. After $t$ years, Jacques has \$3,875.79 in the account. Assuming he made no additional deposits or withdrawals, how long was the money in the account?



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


A36 years
B2 years
C18 years

Verified Answer :

  • Use the compound interest formula \(V(t) = P(1 + \frac{r}{n})^{nt}\) with the given values \(P = 1,900\), \(V(t) = 3,875.79\), \(r = 0.04\), and \(n = 2\).
  • Simplify the equation to \(\frac{3,875.79}{1,900} = (1.02)^{2t}\).
  • Take the natural logarithm of both sides to isolate the variable \(t\): \(\ln(\frac{3,875.79}{1,900}) = 2t \ln(1.02)\).
  • Solve for \(t\) to find the final duration of \(\boxed{18 \text{ years}}\).

Explanation

  1. Problem Analysis and Setup

To determine the time \(t\) that Jacques's money was in the account, we start with the given compound interest formula:
\[V(t) = P\left(1 + \frac{r}{n}\right)^{nt}\]
From the problem description, we have the following data:

  • The principal amount \(P = \$1,900\).
  • The final amount after \(t\) years \(V(t) = \$3,875.79\).
  • The annual interest rate \(r = 4\% = 0.04\).
  • The compounding frequency per year \(n = 2\) (since it is compounded semiannually).

Our objective is to solve for the time \(t\) in years.

  1. Solving for Time t

Substituting the known values into the formula, we get:
\[3,875.79 = 1,900 \left(1 + \frac{0.04}{2}\right)^{2t}\]
First, simplify the term inside the parentheses:
\[1 + \frac{0.04}{2} = 1 + 0.02 = 1.02\]
The equation now becomes:
\[3,875.79 = 1,900 \times (1.02)^{2t}\]
Next, we isolate the exponential term by dividing both sides by 1,900:
\[\frac{3,875.79}{1,900} = (1.02)^{2t}\]
\[2.039889... = (1.02)^{2t}\]
To solve for the exponent \(2t\), we take the natural logarithm (\(\ln\)) of both sides:
\[\ln\left(\frac{3,875.79}{1,900}\right) = \ln\left(1.02^{2t}\right)\]
Using the logarithm power rule, \(\ln(x^a) = a \ln(x)\), we can write:
\[\ln\left(\frac{3,875.79}{1,900}\right) = 2t \times \ln(1.02)\]
Now, solve for \(t\):
\[t = \frac{\ln\left(\frac{3,875.79}{1,900}\right)}{2 \times \ln(1.02)}\]

  1. Calculation and Conclusion

Now we calculate the numerical value:
\[t = \frac{\ln(2.039889...)}{2 \times \ln(1.02)} \approx \frac{0.71286}{2 \times 0.019802} \approx \frac{0.71286}{0.039604} \approx 18\]
Rounding to the nearest whole number, we find that \(t = 18\) years. Comparing this result with the given options (36 years, 2 years, 18 years), it matches option (c). Thus, the money was in the account for 18 years.

Examples

Understanding compound interest is a vital skill for personal finance management. For example, if you decide to save for a major purchase or retirement, using the compound interest formula helps you estimate how much time you need to reach a specific financial goal. By adjusting variables like the principal investment, the interest rate, or the frequency of compounding, you can better plan your savings strategy and make informed decisions about where to keep your money to maximize growth over time.

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