Find the present value PV of the given investment. (Round your answer to the nearest cent.) An investment earns 8% per year and is worth $60,000 after 19 months.
Answer (2)
Convert the time period to years: t = 12 19 .
Use the future value formula: F V = P V ( 1 + r t ) .
Rearrange the formula to solve for present value: P V = 1 + r t F V .
Substitute the given values and calculate: P V = 1 + 0.08 \tims 12 19 60000 ≈ 53254.44 . The present value is 53254.44 .
Explanation
Problem Analysis We are given that an investment earns 8% per year and is worth $60,000 after 19 months. We need to find the present value (PV) of this investment.
Converting Time Period First, we need to convert the time period from months to years. Since there are 12 months in a year, 19 months is equal to 12 19 years.
Stating the Formula The formula for future value (FV) with simple interest is given by: F V = P V ( 1 + r t ) where:
FV is the future value of the investment
PV is the present value of the investment
r is the annual interest rate (as a decimal)
t is the time in years
Identifying Given Values We are given:
FV = $60,000
r = 8% = 0.08
t = 12 19 years We need to find PV.
Rearranging the Formula Rearrange the formula to solve for PV: P V = 1 + r t F V
Substituting Values and Calculating Substitute the given values into the formula: P V = 1 + 0.08 × 12 19 60000 Now, we calculate the value of PV: P V = 1 + 0.08 × 12 19 60000 = 1 + 0.126666... 60000 = 1.126666... 60000 ≈ 53254.4378698
Rounding the Result Round the value of PV to the nearest cent: PV \approx $53254.44
Final Answer Therefore, the present value of the investment is approximately $53254.44.
Examples
Understanding present value is crucial in financial planning. For instance, if you want to have $60,000 in 19 months with an investment that yields 8% annually, calculating the present value tells you how much you need to invest today. This concept is widely used in loan calculations, investment analysis, and retirement planning to determine the current worth of future sums of money. By discounting future values to their present value, individuals and businesses can make informed decisions about investments and expenditures.
The present value (PV) of an investment worth $60,000 after 19 months at an annual interest rate of 8% is approximately $53,254.44. This is found by converting months to years and using the future value formula to solve for PV. Calculating interest over the time period and reformulating yields the present value needed today to reach that future sum.
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