Calculate what $930 would grow to at 5% per year compounded daily for 5 years. (Use the Table 12.2.) Note: Round your final answer to the nearest cent.
Answer (1)
Identify the principal amount, interest rate, compounding frequency, and time period.
Apply the compound interest formula: F V = P ( 1 + n r ) n t .
Substitute the given values into the formula: F V = 930 ( 1 + 365 0.05 ) 365 × 5 .
Calculate the future value and round to the nearest cent: $1194.12 .
Explanation
Understanding the Problem We are asked to calculate the future value of an investment of $930 at an annual interest rate of 5% compounded daily for 5 years. We will use the compound interest formula to solve this problem.
Identifying the Formula The formula for compound interest is: F V = P ( 1 + n r ) n t where:
FV 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
Assigning Values In this problem, we have:
Principal amount, P = $930
Annual interest rate, r = 5% = 0.05
Number of times interest is compounded per year, n = 365 (daily)
Number of years, $t = 5
Plugging in the Values Now, we plug these values into the formula: F V = 930 ( 1 + 365 0.05 ) 365 × 5
Calculating the Result Calculating the future value: F V = 930 ( 1 + 365 0.05 ) 1825 F V = 930 ( 1 + 0.000136986 ) 1825 F V = 930 ( 1.000136986 ) 1825 F V = 930 × 1.283999131 F V = 1194.1191
Rounding the Answer Rounding the future value to the nearest cent, we get: F V = $1194.12
Final Answer Therefore, $930 would grow to $1194.12 at 5% per year compounded daily for 5 years.
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 estimate your future wealth. Similarly, when taking out a loan, understanding the compound interest can help you assess the total cost of borrowing and compare different loan options. This concept is also vital in investment strategies, where maximizing returns through compound growth is a primary goal.
