The equation for the future value of a deposit earning compound interest is
[tex]V(t)=P\left(1+\frac{r}{n}\right)[/tex]
where
[tex]P =[/tex] the initial deposit
[tex]t =[/tex] years invested
[tex]r =[/tex] rate at which interest is compounded annually
[tex]n =[/tex] number of times the interest is compounded per year
After 10 years, a $2,000-dollar investment compounded annually has grown to $3600. What is the interest rate to the nearest whole-number percent?
Answer (2)
The interest rate for a $2,000 investment that grows to $3,600 over 10 years compounded annually is approximately 6%. This is calculated using the compound interest formula and isolating the interest rate. After solving, we find that the interest rate to the nearest whole number is 6%.
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Substitute the given values into the compound interest formula.
Simplify the equation to isolate the term ( 1 + r ) 10 .
Take the 10th root of both sides to solve for 1 + r .
Isolate r , convert it to a percentage, and round to the nearest whole number, resulting in 6% .
Explanation
Understanding the Problem We are given the formula for the future value of an investment with compound interest: V ( t ) = P ( n t 1 + ( n r ) ) , where: V ( t ) is the future value of the investment P is the initial deposit r is the annual interest rate n is the number of times the interest is compounded per year t is the number of years the money is invested for. We are given the following information: P = $2000 V ( 10 ) = $3600 t = 10 years n = 1 (compounded annually) We need to find the interest rate r .
Substituting the Values Substitute the given values into the formula: 3600 = 2000 ( 1 × 10 1 + ( 1 r ) ) Simplify the equation: 3600 = 2000 ( 1 + r ) 10
Simplifying the Equation Divide both sides by 2000: 2000 3600 = ( 1 + r ) 10 Simplify the fraction: 1.8 = ( 1 + r ) 10
Taking the 10th Root Take the 10th root of both sides: 10 1.8 = 1 + r
Isolating r Isolate r :
r = 10 1.8 − 1
Calculating r Calculate the value of r :
r = 1. 8 0.1 − 1 r ≈ 1.0604 − 1 r ≈ 0.0604
Converting to Percentage Multiply r by 100 to express it as a percentage: r ≈ 0.0604 × 100 = 6.04% Round to the nearest whole number: r ≈ 6%
Final Answer Therefore, the interest rate to the nearest whole-number percent is 6%.
Examples
Understanding compound interest is crucial for making informed financial decisions. For instance, when planning for retirement, knowing the interest rate on your investments helps you project the future value of your savings. Similarly, when taking out a loan, understanding the interest rate allows you to calculate the total cost of borrowing and compare different loan options. This concept is also applicable in business, where companies use compound interest calculations to evaluate the profitability of long-term investments and projects. By mastering compound interest, you can effectively manage your finances and make sound investment choices.
