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11-CS-1 Engineering Economics · December 2017

Question 1 of 5: Simple, Compound, Continuous, and Gradient Interest

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

National Exams — December 2017 — 11-CS-1 Engineering Economics. Open book; non-communicating calculator permitted. Any four of the five questions constitute a complete paper; all questions are of equal value. Fully worked solutions to all five questions follow; standard compound-interest factors are used and minor rounding is immaterial.

Question 1: Simple, Compound, Continuous, and Gradient Interest (25 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

(a) Simple-Interest Return

$$I = P\,i\,n = 1{,}000(0.10)(3) = \boxed{\$300}\quad(\text{final amount } \$1{,}300)$$

(b) Effective Rate — 24% Compounded Daily

$$i_a = \left(1+\tfrac{0.24}{365}\right)^{365}-1 = \boxed{27.11\%}$$

(c) Effective Rate — 40% Compounded Continuously

$$i_a = e^{0.40}-1 = 1.49182-1 = \boxed{49.18\%}$$

(d) Present Worth of the Repair-Cost Gradient

Assumption (permitted by NOTE 1): repairs fall every six months, so four years contains eight repair events. The first is $500 now, at $t=0$, and the last is $4,000 at $t=3.5$ yr — eight payments $500, $1,000, …, $4,000 spanning the four-year window.

Because the payments are semi-annual and the quoted rate is monthly, first convert the rate to the payment period. The monthly rate is $0.12/12 = 1\%$, so the effective semi-annual rate is

$$i_{sa} = (1.01)^{6}-1 = 0.0615202 = 6.1520\%$$

The eight payments are a $500 base annuity plus a $500 arithmetic gradient. The standard $(P/A)$ and $(P/G)$ factors place the first payment one period after the valuation point, so they value the series at $t=-0.5$ yr; one compounding then carries it to the present:

$$P_{-0.5} = 500(P/A,6.1520\%,8) + 500(P/G,6.1520\%,8) = 500(6.172605)+500(19.676835)$$
$$P_{-0.5} = 3{,}086.30 + 9{,}838.42 = \$12{,}924.72$$
$$PW = 12{,}924.72\,(1.061520) = \boxed{\$13{,}719.85 \approx \$13{,}720}$$

Direct summation confirms it: $PW = 500\sum_{k=1}^{8} k\,(1.061520)^{-(k-1)} = 500(27.4397) = \$13{,}719.85$.

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