Find the requested value (to the nearest dollar) and tell what each of the other values represents. (a) Find An: An = 1,700  1 − (1 + 0.03)−20 0.03  An = $ Identify the following. (Assume the rate and time are using the same time period.) periodic payment $ rate time (b) Find R: 150,000 = R  1 − (1 + 0.005)−365 0.005  R = $ Identify the following. (Assume the rate and time are using the same time period.) present value $ rate time Find the requested value (to the nearest dollar) and tell what each of the other values represents. (a) Find An: An = 1,700
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The Correct Answer and Explanation is:
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(a) Find An
Given formula: An=1,700⋅1−(1+0.03)−200.03A_n = 1{,}700 \cdot \frac{1 – (1 + 0.03)^{-20}}{0.03}
This is the formula for the future value of an ordinary annuity with:
- Periodic payment = $1,700
- Rate per period (r) = 0.03
- Number of periods (n) = 20
Step-by-step Calculation:
(1+0.03)−20=(1.03)−20≈0.5536757(1 + 0.03)^{-20} = (1.03)^{-20} \approx 0.5536757 1−0.5536757=0.44632431 – 0.5536757 = 0.4463243 0.44632430.03≈14.87748\frac{0.4463243}{0.03} \approx 14.87748 An=1,700⋅14.87748≈25,292A_n = 1{,}700 \cdot 14.87748 \approx 25{,}292
✅ Answer: An≈$25,292A_n \approx \text{\$25,292}
Explanation of values:
- Periodic payment: $1,700 (the amount deposited or invested each period)
- Rate: 0.03 (3% per period)
- Time: 20 periods (could be years, months, etc., as long as consistent with rate)
(b) Find R
Given formula: 150,000=R⋅1−(1+0.005)−3650.005150{,}000 = R \cdot \frac{1 – (1 + 0.005)^{-365}}{0.005}
This is the present value of an ordinary annuity with:
- Present value = $150,000
- Rate per period (r) = 0.005
- Number of periods (n) = 365
Step-by-step Calculation:
(1+0.005)−365=(1.005)−365≈0.158204(1 + 0.005)^{-365} = (1.005)^{-365} \approx 0.158204 1−0.158204=0.8417961 – 0.158204 = 0.841796 0.8417960.005≈168.3592\frac{0.841796}{0.005} \approx 168.3592 R=150,000168.3592≈891R = \frac{150{,}000}{168.3592} \approx 891
✅ Answer: R≈$891R \approx \text{\$891}
Explanation of values:
- Present value: $150,000 (the current value of future payments)
- Rate: 0.005 (0.5% per period)
- Time: 365 periods
Overall Explanation
This problem involves two fundamental financial formulas used to calculate the present and future values of ordinary annuities, where payments are made at regular intervals.
In Part (a), we’re given a fixed periodic payment of $1,700, with a 3% interest rate per period, and 20 periods in total. The formula used is for the future value of an ordinary annuity: An=P⋅1−(1+r)−nrA_n = P \cdot \frac{1 – (1 + r)^{-n}}{r}
This formula calculates the total value after making consistent payments over time, accounting for compound interest. Here, we calculated the total amount accumulated after 20 periods, which came to $25,292.
In Part (b), the problem flips — it gives a present value and asks for the periodic payment. This uses the present value of an ordinary annuity formula: PV=R⋅1−(1+r)−nrPV = R \cdot \frac{1 – (1 + r)^{-n}}{r}
Here, the present value of $150,000 is equal to the sum of 365 payments, each discounted at a 0.5% rate. Solving for R, we find each payment is approximately $891.
Both cases demonstrate how time value of money works — money today is worth more than the same amount in the future. These tools are essential in financial planning, loans, retirement savings, and investment decisions.
