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23-CS-1 Engineering Economics · May 2018

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

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

Notes on this paper

National Exams — May 2018 — 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 follow; standard compound-interest factors are used and minor rounding is immaterial.

Question 1: Simple, Compound, Effective, and Continuous 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

$$I = P\,i\,n = 500(0.12)(3) = \boxed{\$180}$$

(b) Compound Amount — 10% Annual, 10 Years

$$F = 1{,}000(1.10)^{10} = 1{,}000(2.593742) = \boxed{\$2{,}593.74}$$

(c) Effective Rate — 24% Compounded Daily

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

(d) Doubling Time — 11% Continuous

With continuous compounding $F = Pe^{rt}$; setting $F/P = 2$:

$$e^{0.11t} = 2 \;\Rightarrow\; t = \frac{\ln 2}{0.11} = \frac{0.69315}{0.11} = \boxed{6.30\ \text{years}}$$

Two cross-checks are worth writing down because they are the marks the examiner is looking for. In (a) the same $500 at 12% compound interest would reach $500(1.12)^3 = \$702.46$, i.e. $202.46 of interest rather than $180: the $22.46 gap is the interest-on-interest that simple interest ignores, and it is the whole reason lenders quote one and borrowers feel the other. In (c) the daily compounding of a nominal 24% lifts the true annual cost to 27.11%, a 3.11-point premium; had the card compounded monthly the effective rate would be $(1+0.24/12)^{12}-1 = 26.82\%$ and continuously $e^{0.24}-1 = 27.12\%$, so 365-day compounding is already within 0.01 points of the continuous limit. Part (d) is the mirror image of (b): where (b) fixes the horizon and asks for the amount, (d) fixes the ratio $F/P=2$ and asks for the horizon, and because the doubling time $\ln 2/r$ depends on the rate alone it is independent of the sum invested—which is exactly why the question can say “any invested amount”.

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