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Chapter 17: Problem 74

The annual amount of a series of payments to be made at the end of each of the next 12 years is \(\$ 500\). What is the present worth of the payments at 8 percent interest compounded annually? a) \(\$ 500\) c) \(\$ 6,000\) b) \(\$ 3,768\) d) \(\$ 6,480\)

Short Answer

Expert verified
Answer: The present worth is approximately \(\$ 3,768\).

Step by step solution

01

Understanding the Present Worth formula for annuities

The Present Worth (PW) of an annuity can be calculated using the formula: PW = A * [(1-(1+i)^{-n}) / i] Here, A is the amount of the annuity, i is the interest rate, and n is the number of payments. For this problem, A = $500, i = 8% or 0.08, and n = 12.
02

Plug the values in the formula

Now we plug the values in the formula and solve for PW. PW = 500 * [(1-(1+0.08)^{-12}) / 0.08]
03

Calculate the Present Worth

Let's solve this expression step-by-step: 1. Calculate (1+0.08): 1.08 2. Raise the result to the power of -12: (1.08)^{-12} ≈ 0.397 3. Subtract the result from step 2 from 1: 1 - 0.397 ≈ 0.603 4. Divide the result from step 3 by the interest rate: 0.603 / 0.08 ≈ 7.536 5. Multiply the result from step 4 by the amount of the annuity: 7.536 * 500 ≈ 3768 PW ≈ \(\$ 3,768\) The correct answer is (b) \(\$ 3,768\).

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