A sum of money is invested at $12 \%$ compounded quarterly. About how long will it take for the amount of money to double?
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. 5.9 years
B. 6.1 years
C. 23.4 years
D. 24.5 years
Answer (2)
Apply the compound interest formula: V ( t ) = P ( 1 + n r ) n t .
Set V ( t ) = 2 P to find the doubling time.
Substitute r = 0.12 and n = 4 into the formula and solve for t .
Calculate t = 4 l n ( 1.03 ) l n ( 2 ) ≈ 5.9 years. The final answer is 5.9 years .
Explanation
Understanding the Problem We want to find out how long it takes for an investment to double at a 12% interest rate, compounded quarterly. We'll use the compound interest formula to solve this problem.
Stating the Formula The compound interest formula is given by: V ( t ) = P ( 1 + n r ) n t Where: V ( t ) is the value of the investment after t years, P is the principal (initial) investment, r is the annual interest rate (as a decimal), n is the number of times the interest is compounded per year, t is the number of years.
Identifying the Values We want to find the time t when the investment doubles, so V ( t ) = 2 P . The annual interest rate is r = 12% = 0.12 , and the interest is compounded quarterly, so n = 4 .
Plugging in the Values Substitute these values into the formula: 2 P = P ( 1 + 4 0.12 ) 4 t Divide both sides by P :
2 = ( 1 + 4 0.12 ) 4 t Simplify the expression inside the parenthesis: 2 = ( 1 + 0.03 ) 4 t 2 = ( 1.03 ) 4 t
Using Logarithms Now, take the natural logarithm of both sides: ln ( 2 ) = ln (( 1.03 ) 4 t ) Use the logarithm power rule: ln ( 2 ) = 4 t ln ( 1.03 ) Solve for t :
t = 4 ln ( 1.03 ) ln ( 2 )
Calculating t Now we calculate the value of t :
t = 4 ln ( 1.03 ) ln ( 2 ) ≈ 4 × 0.02956 0.6931 ≈ 0.11824 0.6931 ≈ 5.86
Final Answer So, it will take approximately 5.86 years for the investment to double. Looking at the multiple-choice options, the closest answer is 5.9 years.
Examples
Understanding compound interest is crucial for making informed financial decisions. For example, if you invest money in a savings account or a certificate of deposit (CD), the interest earned is often compounded. Knowing how long it takes for your investment to double helps you plan for long-term financial goals, such as retirement or saving for a down payment on a house. The formula V ( t ) = P ( 1 + n r ) n t allows you to calculate the future value of your investment based on the initial investment ( P ), the annual interest rate ( r ), the number of times the interest is compounded per year ( n ), and the number of years ( t ). By understanding these factors, you can make strategic decisions to maximize your returns.
It will take approximately 5.9 years for an investment to double at a 12% interest rate compounded quarterly. This is calculated using the compound interest formula and involves logarithmic manipulation. Thus, the correct answer is 5.9 years.
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