SME
Engineering Economics · Problem type
Break-even quantity
Use this for “A product has $120,000 a year of fixed cost, $30 a unit of variable cost and sells for $45. Find its break-even quantity and the profit at 10,000 units.”
Chart
R(10,000) = 450,000
R(Q) = TC(Q)
The curves cross at Q = 8,000.
This is an example. Type your own values to start a new problem.
QBE (units)
=
$
$
−
$
$
Profit
=
$
p
×
Q (units)
−
($
FC
+
$
v
×
Q (units)
)
R
Revenue
calculated · $
TC
Total cost
calculated · $
Formulas
QBE
=
FC
p
−
v
8,000
=
120,000
45 − 30
R
=
p
×
Q
450,000 = 45 × 10,000
TC
=
FC
+
v
×
Q
420,000 = 120,000 + 30 × 10,000
Profit
=
R
−
TC
30,000 = 450,000 − 420,000
Assumptions
•
Costs, revenue and quantities are a year’s; price and variable cost per unit stay the same at any quantity.
•
The price p must exceed v, or every unit loses money and there is no break-even.
•
Above QBE the product makes a profit; below it, a loss.
Step-by-step
WE KNOW
FC = $120,000, v = $30, p = $45, Q = 10,000 units
FIND
break-even quantity (QBE), revenue (R), total cost (TC), profit (Profit)
Step 1 · Find break-even quantity (QBE)
Use QBE = FC ÷ (p − v)
Each
unit
sold
keeps
p
−
v
toward
the
fixed
cost:
divide
the
fixed
cost
by
it.
QBE
=
120,000
45 − 30
QBE
=
120,000
15
QBE = 8,000 units
Step 2 · Find revenue (R)
Use R = p × Q
Price times the quantity sold.
R
=
45
×
10,000
R
=
$450,000
Step 3 · Find total cost (TC)
Use TC = FC + v × Q
The fixed cost plus the variable cost of every unit.
TC = 120,000 + 30 × 10,000
TC = 120,000 + 300,000
TC = $420,000
Step 4 · Find profit (Profit)
Use Profit = R − TC
Revenue less total cost: at or above QBE it is not below 0.
Profit = 450,000 − 420,000
Profit = $30,000
Check
Put the numbers back in. Both sides should match.
8,000
=
120,000
45 − 30
✓
450,000 = 45 × 10,000 ✓
420,000 = 120,000 + 30 × 10,000 ✓
30,000 = 450,000 − 420,000 ✓
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