Quick questions on Break-even analysis and linear cost or revenue models for HSC Maths Standard 2

For a linear model with $n$ units produced or sold:

Break-even is the quantity at which profit is zero, that is $C = R$:

<svg viewBox="0 0 480 320" role="img" aria-labelledby="be-t be-d" style="max-width: 100%; height: auto; display: block; margin: 1em auto;"> <title id="be-t">Break-even chart with cost and revenue lines crossing</title> <desc id="be-d">Cost line starts at the fixed-cost y-intercept and slopes upward. Revenue line passes through the origin with a steeper slope. The two lines intersect at the break-even quantity. To the left is a loss region; to the right is a profit region.</desc> <g font-family="system-ui, sans-serif" font-size="13" fill="currentColor"> <line x1="60" y1="280" x2="450" y2="280" stroke="currentColor" stroke-width="1.5"/> <polygon points="450,280 442,276 442,284" fill="currentColor"/> <line x1="60" y1="30" x2="60" y2="290" stroke="currentColor" stroke-width="1.5"/> <polygon points="60,30 56,38 64,38" fill="currentColor"/> <text x="445" y="298">quantity n</text> <text x="44" y="32">$</text> <line x1="60" y1="230" x2="440" y2="80" stroke="currentColor" stroke-width="2"/> <text x="380" y="78" font-style="italic">Cost C = F + vn</text> <line x1="60" y1="280" x2="440" y2="30" stroke="currentColor" stroke-width="2" stroke-dasharray="6 4"/> <text x="370" y="40" font-style="italic">Revenue R = pn</text> <circle cx="250" cy="155" r="5" fill="currentColor"/> <text x="258" y="150" font-weight="bold">break-even</text> <line x1="250" y1="155" x2="250" y2="280" stroke="currentColor" stroke-width="1" stroke-dasharray="3 3" opacity="0.5"/> <text x="248" y="295" font-size="11" text-anchor="middle">n_BE</text> <text x="50" y="245" font-size="11" font-style="italic">F</text> <text x="140" y="265" font-size="12" opacity="0.7">loss</text> <text x="350" y="200" font-size="12" opacity="0.7">profit</text> </g> </svg>

Standard phrasings: "find the number of units needed to break even", "find the profit if $x$ are sold", "by how much does the profit increase per extra unit sold" (this is just $p - v$).

The owner considers raising the price to \6$. New contribution margin:$\$4.80$. New break-even: $\frac{3200}{4.80} \approx 667$ coffees, a drop of $78$ coffees per month at break-even.

Without the $F$ term, break-even is meaningless. Read the question to identify what is fixed.

The break-even formula often gives a fractional answer. Round up for break-even quantities, since the business is still in loss at the rounded-down value.

Revenue is total sales. Profit is revenue minus all costs. The HSC often tests whether you know the difference.

Profit at $n$ units is $(p - v) n - F$. Once you have the contribution margin, it is a one-step calculation. :::