Break-even Analysis and Present Value of a New Machine: 22-Mec-B4 Integrated Manufacturing Systems, Undated paper, Q7

Question 7: Break-even Analysis and Present Value of a New Machine (20 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.

Part (a) — break-even at two selling prices

Given.

Quantity	Symbol	Value
Fixed cost over the relevant range	$F$	$25,000 per period
Variable cost per unit	$v$	$10.00 per unit
Selling price, original	$p_1$	$20.00 per unit
Selling price, increased	$p_2$	$25.00 per unit
Range over which the cost structure holds	—	1,500 to 2,500 units

Find. The break-even volume at each price, and the effect of the price increase on it.

Break-even chart. Raising the price steepens the revenue line, so it meets the unchanged total-cost line sooner; the break-even volume falls from 2,500 to 1,667 units.

Approach. Set revenue equal to total cost, solve for the volume, and recognise that the break-even point is the fixed cost divided by the contribution margin — so a price change acts on it only through that margin.

Set up the break-even condition. Profit is zero when revenue equals total cost, $$pQ = F + vQ \quad\Longrightarrow\quad Q_{BE} = \frac{F}{p - v},$$ the denominator $p-v$ being the contribution margin per unit — what each unit sold contributes toward covering the fixed cost.
Evaluate at the original price. The contribution margin is $20.00 - 10.00 = $ $10.00 per unit, so $$Q_{BE,1} = \frac{25{,}000}{10} = \boxed{2{,}500\ \text{units.}}$$ Note that this lands exactly on the upper end of the 1,500–2,500 unit range over which the stated cost structure is valid, so the answer is just admissible — the plant must run flat out at the top of the relevant range merely to break even, which is itself worth reporting to management.
Evaluate at the increased price. Raising the price to $25.00 lifts the contribution margin to $15.00 per unit, so $$Q_{BE,2} = \frac{25{,}000}{15} = 1{,}666.7 = \boxed{1{,}667\ \text{units (rounding up to a whole unit).}}$$
State the effect of the change. The break-even volume falls by $$2{,}500 - 1{,}667 = \boxed{833\ \text{units, a reduction of one third.}}$$ The proportion is no accident: the break-even volume varies inversely with the contribution margin, and the margin rose by half, so the volume fell to two-thirds of its former value. The price rose by 25 % but the break-even fell by 33 %, because the whole of the increase goes to contribution.
Rebuild the income statement as a check. At 1,667 units and $25.00, revenue is 41,675 and total cost is $25{,}000 + 10(1{,}667) = 41{,}670$, so profit is essentially zero as required; at exactly 1,666.67 units it is zero to the cent. The corresponding statement at the original price and 2,500 units gives revenue 50,000 against cost 50,000. It is also worth telling management what the price rise is worth at full volume: at 2,500 units the higher price yields $25(2{,}500) - [25{,}000 + 10(2{,}500)] = $ $12,500 of profit where the original price yielded nothing.

Check: the demand response is not given. The arithmetic above holds the volume-independent cost structure and asks only what happens to the break-even point. It says nothing about whether the market will still absorb 1,667 units at $25.00; unless demand is perfectly inelastic, some volume will be lost, and a complete recommendation would compare the new break-even against a revised sales forecast rather than against the old one. The examination question asks only for the effect on the break-even point.

Part (b) — present value of the expenditures on the new machine

Given.

Quantity	Symbol	Value
First cost of the machine	$P$	$24,000 at time zero
Economic life	$n$	8 years
Salvage value at the end of year 8	$S$	$4,000
Annual operating cost	$A$	$3,000 per year, years 1 to 8
Going rate of interest	$i$	10% per year

Find. The present value of the net expenditures associated with owning and operating the machine for its eight-year life.

Cash-flow diagram for the machine, drawn from the owner's point of view: downward arrows are expenditures, the single upward arrow at year 8 is the salvage receipt.

Approach. Discount each cash flow to time zero at 10 % — the first cost is already there, the operating costs form a uniform series handled by the series present-worth factor, and the salvage is a single future receipt handled by the single-payment present-worth factor and subtracted because it is a cash inflow.

Write the present-value expression. $$PV = P + A\,(P/A, i, n) - S\,(P/F, i, n),$$ the salvage entering with a minus sign because the question asks for the present value of expenditures and a receipt reduces them.
Evaluate the two interest factors at 10 per cent for eight years. $$(P/A, 10\%, 8) = \frac{1-(1.10)^{-8}}{0.10} = 5.3349,\qquad (P/F, 10\%, 8) = (1.10)^{-8} = 0.4665.$$
Discount the operating series and the salvage. $$A\,(P/A) = 3{,}000 \times 5.3349 = 16{,}004.78,\qquad S\,(P/F) = 4{,}000 \times 0.4665 = 1{,}866.03.$$ Only about 47 cents of each salvage dollar survives the eight years of discounting, which is why salvage rarely dominates a decision of this kind.
Add the three terms. $$PV = 24{,}000 + 16{,}004.78 - 1{,}866.03 = \boxed{PV = \textrm{CAD } 38{,}138.75.}$$
Convert to an annual equivalent as a cross-check. Spreading that present value uniformly over the eight years at 10 % gives $$EAC = PV\,(A/P, 10\%, 8) = 38{,}138.75 \times 0.187444 = \boxed{\textrm{CAD } 7{,}148.88 \text{ per year.}}$$ Rebuilding the same figure from its parts — capital recovery $24{,}000(0.187444) = 4{,}498.66$, plus operating 3,000.00, less sinking-fund salvage credit $4{,}000(0.087444) = 349.78$ — returns 7,148.88 exactly, using the identity $(A/P) - (A/F) = i$. The agreement confirms the present-value arithmetic.

The annual equivalent is the more useful number in practice, because it is the figure that must be compared against the annual cost of the alternative the machine is displacing. Note also that ignoring the salvage entirely would overstate the present value by $1,866 — under 5 % of the total — so a decision that turns on the salvage estimate is a decision that was close to begin with.

Final results.

Quantity	Result
(a) Contribution margin at $20.00 / $25.00	$10.00 / $15.00 per unit
(a) Break-even at $20.00	2,500 units
(a) Break-even at $25.00	1,667 units (1,666.7 exactly)
(a) Effect of the price increase	833 units lower, a reduction of one third
(a) Profit at 2,500 units under the higher price	$12,500
(b) $(P/A, 10\%, 8)$ / $(P/F, 10\%, 8)$	5.3349 / 0.4665
(b) Present value of operating costs	$16,004.78
(b) Present value of the salvage credit	$1,866.03
(b) Present value of net expenditures	$38,138.75
(b) Equivalent uniform annual cost	$7,148.88 per year

Topic. Cost-volume-profit (break-even) analysis, and present-worth comparison of an equipment expenditure.

Key relations. Break-even: $Q_{BE}=F/(p-v)$, and for a required profit $\pi$, $Q=(F+\pi)/(p-v)$. Present worth: $PV=P+A(P/A,i,n)-S(P/F,i,n)$ with $(P/A,i,n)=[1-(1+i)^{-n}]/i$ and $(P/F,i,n)=(1+i)^{-n}$. Annual equivalent: $EAC=PV(A/P,i,n)$, and $(A/P,i,n)-(A/F,i,n)=i$.

Why this works. Break-even analysis splits cost into a part that does not move with output and a part that moves proportionally, which turns the profit equation into a straight line and makes the crossing point a single division. Its whole value lies in the contribution margin: fixed cost tells you how deep the hole is and the margin tells you how fast each unit fills it, so a price increase is far more powerful than an equal-percentage cost reduction, because the whole of a price increase falls into the margin. The relevant range matters — fixed costs are only fixed over a stated band of output, which is why this question specifies 1,500 to 2,500 units. Present worth applies the same reasoning across time instead of across volume: money has a rental rate, so cash flows at different dates are not commensurable until each is brought to a common instant, and the two standard factors do exactly that for a single sum and for a uniform series.

Common pitfall. In (a), dividing the fixed cost by the price rather than by the contribution margin, which gives 1,250 units and understates the break-even by half; and failing to notice that the first break-even sits at the very top of the stated relevant range. In (b), adding the salvage instead of subtracting it, a sign error that moves the answer by 3,732 dollars; and applying the single-payment factor to the operating series or the series factor to the salvage.

Source. Chase & Jacobs, Operations and Supply Chain Management, 16th ed., ch. 5 (break-even analysis in capacity and process selection) and Appendix B (financial analysis, present-worth and annual-cost factors); Buffa & Sarin, Modern Production / Operations Management, 8th ed., ch. 3.

Question 7 of 7: Break-even Analysis and Present Value of a New Machine

Question 7: Break-even Analysis and Present Value of a New Machine (20 marks)

Part (a) — break-even at two selling prices

Part (b) — present value of the expenditures on the new machine