Present worth at two maturities (same rate): A bill due in 4 years is worth $575 now, but if it were due in 2 years 6 months it would be worth $620 now. Assuming simple interest, find the sum due on the bill (face value).
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A$ 695
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B$ 725
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C$ 713
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DNone of these
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E$ 740
Answer
Correct Answer: $ 713
Explanation
Introduction / Context:Two present-worth figures are quoted for the same future bill but with different times to maturity. Under simple interest, present worth PW and sum due S satisfy PW = S / (1 + r t). By forming a ratio of the two given PWs we eliminate S and solve for the rate r, then back-substitute to obtain S.
Given Data / Assumptions:
- PW₁ = 575 at t₁ = 4 years.
- PW₂ = 620 at t₂ = 2.5 years.
- Simple interest at a constant annual rate r.
Concept / Approach:Use PW = S / (1 + r t). Then 620/575 = (1 + r*4) / (1 + r*2.5). Solve for r, then compute S from either PW equation. This approach avoids guessing and uses only proportional reasoning plus one substitution.
Step-by-Step Solution:
620/575 = (1 + 4r) / (1 + 2.5r)Simplify left side: 620/575 = 124/115.Cross-multiply: 115(1 + 4r) = 124(1 + 2.5r).115 + 460r = 124 + 310r ⇒ 150r = 9 ⇒ r = 0.06 = 6%.Use PW₂ = 620 = S / (1 + 0.06*2.5) = S / 1.15 ⇒ S = 620 * 1.15 = 713.Verification / Alternative check:Check PW using t₁ = 4: PW = 713 / (1 + 0.24) = 713 / 1.24 = 575 (matches).
Why Other Options Are Wrong:
- $695, $725, others: Do not satisfy both present-worth equations at a single rate.
Common Pitfalls:Mistaking compound for simple interest or mixing the two PWs without eliminating S first.
Final Answer:$ 713