Breakeven Analysis

Engineering Economics · FE Reference Handbook section

Engineering Economics

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10 exam-style examples

~45 min

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Handbook notes for this section

Definitions and conditions exactly as the handbook states them.

By altering the value of any one of the variables in a situation, holding all of the other values constant, it is possible to find a
value for that variable that makes the two alternatives equally economical. This value is the breakeven point.
Breakeven analysis is used to describe the percentage of capacity of operation for a manufacturing plant at which income will
The payback period is the period of time required for the profit or other benefits of an investment to equal the cost of the

Core formulas for this FE topic

Definitions, applicability, units, assumptions and worked examples for each relation.

This section is conceptual; there are no equations to memorise.

Worked exam-style examples

The four ways this section is written on the real exam — thoughts first, then equations, then substitution.

Example 1

Break-even production quantity — Breakeven Analysis

A precast plant has $41,000 fixed annual cost, $25.50 variable cost per unit, and sells units at $33.00. What annual output breaks even?

Given

Fixed = $41,000
v = $25.50/unit
p = $33.00/unit

Find

Break-even quantity Q

Start with the thinking

Break-even sets total revenue equal to total cost.
The contribution margin is p − v.

Step-by-step solution

Balance
$pQ = F + vQ$
Rearrange
$Q = F/(p - v)$
Contribution margin — p − v = $33.00 − $25.50 = $7.50/unit
Substituting — Q = $41,000/$7.50 = 5,467 units/yr

Answer:

Q ≈ 5,467 units per year

Why the other options are there

1,242 units (variable cost ignored)
1,608 units (price ignored)

Reference: FE Reference Handbook — Engineering Economics → Breakeven Analysis

Example 2

Rate of return on an equipment investment, before and after tax — Breakeven Analysis

A contractor invests $119,000 in equipment that returns $25,000 per year for 5 years with no salvage. Determine the rate of return, and estimate the after-tax return at a 35% tax rate on the net income.

Given

P = $119,000
A = $25,000/yr
$n = 5 yr$
$Tax rate = 35%$

Find

Before-tax rate of return and an after-tax estimate

Start with the thinking

The rate of return is the interest rate that makes present worth zero — solved by trial or by the calculator's IRR.
A quick after-tax screen scales the annual return by (1 − tax rate) and re-solves.

Step-by-step solution

Formula
$0 = -P + A(P/A, i, n)$
Set up
$(P/A, i, 5) = P/A = 119000/25000 = 4.7600$
Solve for i
$the rate satisfying [1 - (1+i)^-5]/i = 4.7600 is i = 1.66%$
After-tax cash flow — A_at = A(1 − t) = $25,000(1 − 0.35) = $16,250
After-tax (P/A) required — 7.3231
Solve again — i_at ≈ 0.01%

Answer:

Before-tax ROR ≈ 1.66% per year

Why the other options are there

5.0% (simple total return)
21.0% (ignored the time value of money)

Reference: FE Reference Handbook — Engineering Economics → Breakeven Analysis

Example 3

Break-even production quantity — Breakeven Analysis (2)

A precast plant has $119,000 fixed annual cost, $10.00 variable cost per unit, and sells units at $23.00. What annual output breaks even?

Given

Fixed = $119,000
v = $10.00/unit
p = $23.00/unit

Find

Break-even quantity Q

Start with the thinking

Break-even sets total revenue equal to total cost.
The contribution margin is p − v.

Step-by-step solution

Balance
$pQ = F + vQ$
Rearrange
$Q = F/(p - v)$
Contribution margin — p − v = $23.00 − $10.00 = $13.00/unit
Substituting — Q = $119,000/$13.00 = 9,154 units/yr

Answer:

Q ≈ 9,154 units per year

Why the other options are there

5,174 units (variable cost ignored)
11,900 units (price ignored)

Reference: FE Reference Handbook — Engineering Economics → Breakeven Analysis

Example 4

Rate of return on an equipment investment, before and after tax — Breakeven Analysis (2)

A contractor invests $85,000 in equipment that returns $18,000 per year for 8 years with no salvage. Determine the rate of return, and estimate the after-tax return at a 27% tax rate on the net income.

Given

P = $85,000
A = $18,000/yr
$n = 8 yr$
$Tax rate = 27%$

Find

Before-tax rate of return and an after-tax estimate

Start with the thinking

The rate of return is the interest rate that makes present worth zero — solved by trial or by the calculator's IRR.
A quick after-tax screen scales the annual return by (1 − tax rate) and re-solves.

Step-by-step solution

Formula
$0 = -P + A(P/A, i, n)$
Set up
$(P/A, i, 8) = P/A = 85000/18000 = 4.7222$
Solve for i
$the rate satisfying [1 - (1+i)^-8]/i = 4.7222 is i = 13.47%$
After-tax cash flow — A_at = A(1 − t) = $18,000(1 − 0.27) = $13,140
After-tax (P/A) required — 6.4688
Solve again — i_at ≈ 4.98%

Answer:

Before-tax ROR ≈ 13.47% per year

Why the other options are there

69.4% (simple total return)
21.2% (ignored the time value of money)

Reference: FE Reference Handbook — Engineering Economics → Breakeven Analysis

Example 5

Break-even production quantity — Breakeven Analysis (3)

A precast plant has $90,000 fixed annual cost, $17.00 variable cost per unit, and sells units at $32.50. What annual output breaks even?

Given

Fixed = $90,000
v = $17.00/unit
p = $32.50/unit

Find

Break-even quantity Q

Start with the thinking

Break-even sets total revenue equal to total cost.
The contribution margin is p − v.

Step-by-step solution

Balance
$pQ = F + vQ$
Rearrange
$Q = F/(p - v)$
Contribution margin — p − v = $32.50 − $17.00 = $15.50/unit
Substituting — Q = $90,000/$15.50 = 5,806 units/yr

Answer:

Q ≈ 5,806 units per year

Why the other options are there

2,769 units (variable cost ignored)
5,294 units (price ignored)

Reference: FE Reference Handbook — Engineering Economics → Breakeven Analysis

Example 6

Rate of return on an equipment investment, before and after tax — Breakeven Analysis (3)

A contractor invests $106,000 in equipment that returns $22,000 per year for 10 years with no salvage. Determine the rate of return, and estimate the after-tax return at a 32% tax rate on the net income.

Given

P = $106,000
A = $22,000/yr
$n = 10 yr$
$Tax rate = 32%$

Find

Before-tax rate of return and an after-tax estimate

Start with the thinking

The rate of return is the interest rate that makes present worth zero — solved by trial or by the calculator's IRR.
A quick after-tax screen scales the annual return by (1 − tax rate) and re-solves.

Step-by-step solution

Formula
$0 = -P + A(P/A, i, n)$
Set up
$(P/A, i, 10) = P/A = 106000/22000 = 4.8182$
Solve for i
$the rate satisfying [1 - (1+i)^-10]/i = 4.8182 is i = 16.08%$
After-tax cash flow — A_at = A(1 − t) = $22,000(1 − 0.32) = $14,960
After-tax (P/A) required — 7.0856
Solve again — i_at ≈ 6.81%

Answer:

Before-tax ROR ≈ 16.08% per year

Why the other options are there

107.5% (simple total return)
20.8% (ignored the time value of money)

Reference: FE Reference Handbook — Engineering Economics → Breakeven Analysis

Example 7

Break-even production quantity — Breakeven Analysis (4)

A precast plant has $135,000 fixed annual cost, $9.00 variable cost per unit, and sells units at $20.00. What annual output breaks even?

Given

Fixed = $135,000
v = $9.00/unit
p = $20.00/unit

Find

Break-even quantity Q

Start with the thinking

Break-even sets total revenue equal to total cost.
The contribution margin is p − v.

Step-by-step solution

Balance
$pQ = F + vQ$
Rearrange
$Q = F/(p - v)$
Contribution margin — p − v = $20.00 − $9.00 = $11.00/unit
Substituting — Q = $135,000/$11.00 = 12,273 units/yr

Answer:

Q ≈ 12,273 units per year

Why the other options are there

6,750 units (variable cost ignored)
15,000 units (price ignored)

Reference: FE Reference Handbook — Engineering Economics → Breakeven Analysis

Example 8

Rate of return on an equipment investment, before and after tax — Breakeven Analysis (4)

A contractor invests $135,000 in equipment that returns $33,000 per year for 10 years with no salvage. Determine the rate of return, and estimate the after-tax return at a 31% tax rate on the net income.

Given

P = $135,000
A = $33,000/yr
$n = 10 yr$
$Tax rate = 31%$

Find

Before-tax rate of return and an after-tax estimate

Start with the thinking

The rate of return is the interest rate that makes present worth zero — solved by trial or by the calculator's IRR.
A quick after-tax screen scales the annual return by (1 − tax rate) and re-solves.

Step-by-step solution

Formula
$0 = -P + A(P/A, i, n)$
Set up
$(P/A, i, 10) = P/A = 135000/33000 = 4.0909$
Solve for i
$the rate satisfying [1 - (1+i)^-10]/i = 4.0909 is i = 20.73%$
After-tax cash flow — A_at = A(1 − t) = $33,000(1 − 0.31) = $22,770
After-tax (P/A) required — 5.9289
Solve again — i_at ≈ 10.84%

Answer:

Before-tax ROR ≈ 20.73% per year

Why the other options are there

144.4% (simple total return)
24.4% (ignored the time value of money)

Reference: FE Reference Handbook — Engineering Economics → Breakeven Analysis

Example 9

Break-even production quantity — Breakeven Analysis (5)

A precast plant has $158,000 fixed annual cost, $24.50 variable cost per unit, and sells units at $43.00. What annual output breaks even?

Given

Fixed = $158,000
v = $24.50/unit
p = $43.00/unit

Find

Break-even quantity Q

Start with the thinking

Break-even sets total revenue equal to total cost.
The contribution margin is p − v.

Step-by-step solution

Balance
$pQ = F + vQ$
Rearrange
$Q = F/(p - v)$
Contribution margin — p − v = $43.00 − $24.50 = $18.50/unit
Substituting — Q = $158,000/$18.50 = 8,541 units/yr

Answer:

Q ≈ 8,541 units per year

Why the other options are there

3,674 units (variable cost ignored)
6,449 units (price ignored)

Reference: FE Reference Handbook — Engineering Economics → Breakeven Analysis

Example 10

Rate of return on an equipment investment, before and after tax — Breakeven Analysis (5)

A contractor invests $200,000 in equipment that returns $33,000 per year for 8 years with no salvage. Determine the rate of return, and estimate the after-tax return at a 25% tax rate on the net income.

Given

P = $200,000
A = $33,000/yr
$n = 8 yr$
$Tax rate = 25%$

Find

Before-tax rate of return and an after-tax estimate

Start with the thinking

The rate of return is the interest rate that makes present worth zero — solved by trial or by the calculator's IRR.
A quick after-tax screen scales the annual return by (1 − tax rate) and re-solves.

Step-by-step solution

Formula
$0 = -P + A(P/A, i, n)$
Set up
$(P/A, i, 8) = P/A = 200000/33000 = 6.0606$
Solve for i
$the rate satisfying [1 - (1+i)^-8]/i = 6.0606 is i = 6.62%$
After-tax cash flow — A_at = A(1 − t) = $33,000(1 − 0.25) = $24,750
After-tax (P/A) required — 8.0808
Solve again — i_at ≈ 0.01%

Answer:

Before-tax ROR ≈ 6.62% per year

Why the other options are there

32.0% (simple total return)
16.5% (ignored the time value of money)

Reference: FE Reference Handbook — Engineering Economics → Breakeven Analysis