Jacques deposited $1,900 into an account that earns 4% interest compounded semiannually. After [tex]$t$[/tex] years, Jacques has [tex]$3,875.79[/tex] in the account. Assuming he made no additional deposits or withdrawals, how long was the money in the account?
Compound interest formula: [tex]V(t)=P\left(1+\frac{r}{n}\right)^{n t}[/tex]
[tex]$t=$[/tex] years since initial deposit
[tex]$n=$[/tex] number of times compounded per year
[tex]$r=$[/tex] annual interest rate (as a decimal)
[tex]$P=$[/tex] initial (principal) investment
[tex]$V(t)=$[/tex] value of investment after [tex]$t$[/tex] years
A. 2 years
B. 9 years
C. 18 years
D. 36 years
Answer (1)
Substitute the given values into the compound interest formula.
Simplify the equation and isolate the exponential term.
Take the natural logarithm of both sides and apply the power rule of logarithms.
Solve for t , which represents the number of years the money was in the account: 18 .
Explanation
Problem Analysis Let's analyze the problem. We are given the initial deposit, the interest rate, the compounding period, and the final value of the investment. We need to find the time it takes for the investment to reach the final value. We will use the compound interest formula to solve for t .
Understanding the Formula The compound interest formula is given by: V(t)=P\[1+\frac{r}{n}\]^{nt} where: V ( t ) = value of investment after t years P = initial (principal) investment r = annual interest rate (as a decimal) n = number of times compounded per year t = years since initial deposit
Identifying Given Values We are given: P = $1900 r = 4% = 0.04 n = 2 (compounded semiannually) V ( t ) = $3875.79 We need to find t .
Substituting Values into Formula Substitute the given values into the formula: 3875.79 = 1900\[1+\frac{0.04}{2}\]^{2t} Simplify the equation: 3875.79 = 1900 ( 1 + 0.02 ) 2 t 3875.79 = 1900 ( 1.02 ) 2 t
Simplifying the Equation Divide both sides by 1900: 1900 3875.79 = ( 1.02 ) 2 t 1900 3875.79 = 2.04 (approximately) So, we have: 2.04 = ( 1.02 ) 2 t
Applying Logarithms Take the natural logarithm of both sides: l n ( 2.04 ) = l n (( 1.02 ) 2 t ) Apply the power rule of logarithms: l n ( 2.04 ) = 2 t ⋅ l n ( 1.02 )
Solving for t Solve for t :
t = 2 ⋅ l n ( 1.02 ) l n ( 2.04 ) Using a calculator, we find: t ≈ 2 ⋅ 0.0198 0.713 ≈ 0.0396 0.713 ≈ 18
Final Answer Therefore, the money was in the account for approximately 18 years.
Examples
Compound interest is a powerful tool for growing wealth over time. For example, if you invest in a retirement account early in your career, the power of compounding can significantly increase your savings by the time you retire. Understanding compound interest can also help you make informed decisions about loans and other financial products, ensuring you choose options that minimize interest costs and maximize returns.
