- Identify the variables: principal \(P = 1900\), future value \(V = 3875.79\), annual interest rate \(r = 0.04\), and compounding frequency \(n = 2\).
- Set up the compound interest formula: \(3875.79 = 1900(1 + 0.02)^{2t}\).
- Simplify and isolate the exponential term: \((1.02)^{2t} = 3875.79 / 1900\).
- Use natural logarithms to solve for time: \(t = \frac{\ln(3875.79 / 1900)}{2 \ln(1.02)} \approx 18\).
- Final Answer: \(\boxed{18 \text{ years}}\)
Explanation
- Analysis of the Problem Data
We start by identifying the variables given in the problem and the appropriate formula for compound interest. The principal amount \(P\) deposited is \(\$1,900\), the annual interest rate \(r\) is \(4\%\) or \(0.04\), the compounding frequency \(n\) is semiannually, which means \(n = 2\), and the final value \(V(t)\) after \(t\) years is \(\$3,875.79\). We use the compound interest formula to relate these values:
\[V(t) = P\left(1 + \frac{r}{n}\right)^{nt}.\]
- Setting up the Equation
Substituting the known values into the formula gives:
\[3875.79 = 1900\left(1 + \frac{0.04}{2}\right)^{2t}.\]
Simplifying the expression within the parentheses, we get:
\[3875.79 = 1900(1 + 0.02)^{2t} = 1900(1.02)^{2t}.\]
Next, we isolate the exponential term by dividing both sides by \(\$1,900\):
\[\frac{3875.79}{1900} = (1.02)^{2t}.\]
Performing the division, we find:
\[2.039889 \approx (1.02)^{2t}.\]
- Solving for Time \(t\)
To solve for the time \(t\), we take the natural logarithm (\(\ln\)) of both sides of the equation:
\[\ln(2.039889) = \ln((1.02)^{2t}).\]
Using the logarithm power rule \(\ln(a^b) = b \cdot \ln(a)\), we can write the right side as:
\[\ln(2.039889) = 2t \cdot \ln(1.02).\]
Now, solve for \(t\):
\[t = \frac{\ln(2.039889)}{2 \cdot \ln(1.02)}.\]
Using calculation, we find that:
\[t \approx \frac{0.71285}{2 \times 0.0198026} \approx \frac{0.71285}{0.039605} \approx 18.\]
Thus, the time the money was in the account is approximately 18 years.
Examples
Compound interest is fundamental to understanding how money grows over time in savings accounts, investments, and retirement funds. For instance, if you are planning to save for a major purchase like a house, a car, or a college education, using the compound interest formula helps you estimate exactly how long your savings will take to reach your target goal, enabling better financial planning and decision-making. Knowing how interest compounds helps individuals make more informed choices about where to keep their hard-earned money.