Consider the following investment. (Round your answers to the nearest cent.) $5,500 at $5 \frac{3}{4}\% compounded quarterly for $7 \frac{1}{2}$ years
(a) Find the future value of the given amount.
$
(b) Interpret the future value of the given amount. After $7 \frac{1}{2}$ years, the investment is worth $
Answer (2)
Using the compound interest formula, the future value of a $5,500 investment at a 5.75% interest rate compounded quarterly for 7.5 years is approximately $8,194.17. The investment grows due to compound interest, meaning the interest earned also earns interest over time. Therefore, the worth of the investment after 7.5 years is about $8,194.17.
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Use the compound interest formula: A = P ( 1 + n r ) n t .
Substitute the given values: P = 5500 , r = 0.0575 , n = 4 , and t = 7.5 .
Calculate the future value: A = 5500 ( 1 + 4 0.0575 ) ( 4 ) ( 7.5 ) ≈ 8439.53 .
After 7 2 1 years, the investment is worth approximately $8439.53 .
Explanation
Understanding the Problem We are given an investment of $5 , 500 at an annual interest rate of 5 4 3 % compounded quarterly for 7 2 1 years. We need to find the future value of this investment and interpret its meaning.
Stating the Compound Interest Formula The formula for compound interest is: A = P ( 1 + n r ) n t where:
A is the future value of the investment/loan, including interest
P is the principal investment amount (the initial deposit or loan amount)
r is the annual interest rate (as a decimal)
n is the number of times that interest is compounded per year
t is the number of years the money is invested or borrowed for
Identifying the Given Values We are given:
Principal amount: P = $5 , 500
Annual interest rate: r = 5 4 3 % = 5.75% = 0.0575
Compounding period: Quarterly, so n = 4 times per year
Investment time: t = 7 2 1 years = 7.5 years
Substituting the Values Substitute the given values into the formula: A = 5500 ( 1 + 4 0.0575 ) ( 4 ) ( 7.5 )
Calculating the Parentheses Calculate the value inside the parentheses: 1 + 4 0.0575 = 1 + 0.014375 = 1.014375
Calculating the Exponent Calculate the exponent: ( 4 ) ( 7.5 ) = 30
Calculating the Future Value Calculate the future value: A = 5500 ( 1.014375 ) 30
Calculating the Power Calculate ( 1.014375 ) 30 ≈ 1.53446
Calculating A Calculate A = 5500 × 1.53446 ≈ 8439.53
Rounding the Result Round the future value to the nearest cent: A ≈ $8439.53
Interpreting the Future Value After 7 2 1 years, the investment is worth approximately $8439.53 .
Examples
Understanding compound interest is crucial for making informed financial decisions. For instance, when planning for retirement, knowing how your savings will grow over time with compound interest helps you determine how much you need to save regularly. Similarly, when taking out a loan, understanding the compound interest can help you assess the total cost of borrowing and compare different loan options effectively. This knowledge empowers you to make sound financial choices and achieve your long-term financial goals.
