23-CS-1 Engineering Economics · December 2019
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
National Exams — December 2019 — 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.
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.
Each investment is a single outlay at $t=0$, an expense series that starts at $A_1$ and rises by an arithmetic gradient $G$ over years 1–7, and one receipt $F$ at $t=7$. Its net present worth at a rate $i$ is
The part asks for a rate of return method, so the rate that makes $PW(i)=0$ is what must be found — first for each alternative on its own (the screening test against do-nothing), then for the increment between the survivors (the selection test).
Step 1 — standalone IRR of each investment. Solving $PW(i)=0$ for each:
| Investment | Standalone IRR | Passes MARR = 8%? | Present Worth at 8% |
|---|---|---|---|
| 1 | 9.01% | Yes | +$13,790 |
| 2 | 9.90% | Yes | +$74,950 |
| 3 | 7.17% | No | −$17,460 |
Investment 3 earns only 7.17%, below the 8% MARR, so it is rejected outright. (Equivalently, its present worth at 8% is negative: using $(P/A,8\%,7)=5.20637$, $(P/G,8\%,7)=14.0242$ and $(P/F,8\%,7)=0.583490$, $PW_3 = -185{,}000 - 55{,}000(5.20637) - 3{,}000(14.0242) + 850{,}000(0.583490) = -\$17{,}456$.)
Step 2 — incremental IRR between the survivors. Order the survivors by first cost and test the extra capital. The increment 2 − 1 is
Its first cash flow is an outflow, so this increment is an investment and the ordinary accept test applies (accept if its rate exceeds the MARR). Solving $\Delta PW(i)=0$:
At the MARR the same increment is worth $\Delta PW(8\%) = 74{,}950 - 13{,}790 = +\$61{,}160$, confirming the accept decision. Select Investment 2.
No. A standalone IRR is a percentage on whatever capital the alternative happens to require, and percentages cannot be compared across different investment sizes; the only valid rate-of-return test for mutually exclusive alternatives is whether each increment of capital earns at least the MARR — equivalently, maximise present worth.
On this paper the two criteria happen to agree: Investment 2 has both the highest standalone IRR (9.90%) and the highest present worth, so this exam does not itself supply a counter-example. One is easy to construct from its own data. Raise Investment 1's year-7 return from $600,000 to $700,000 and nothing else changes; Investment 1's standalone IRR rises to 12.86%, far above Investment 2's 9.90%, yet its present worth at 8% is only $+\$72{,}140$, still below Investment 2's $+\$74{,}950$. The incremental test would take Investment 2 anyway, and it would be right to — the higher-percentage alternative adds less total value.
No. $FW = PW(F/P,8\%,7)$ scales every alternative by the same positive constant, so the ranking is unchanged—Future Worth also selects Investment 2. No calculation is needed.
