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?

Start with the thinking

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

Step-by-step solution

1. Balance

   $$pQ=F+vQ$$

2. Rearrange

   $$Q=F/(p-v)$$

3. Contribution margin — p − v = $33.00 − $25.50 = $7.50/unit

4. Substituting — Q = $41,000/$7.50 = 5,467 units/yr

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

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.

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

1. Formula

   $$0=-P+A(P/A,i,n)$$

2. Set up

   $$(P/A,i,5)=P/A=119000/25000=4.7600$$

3. Solve for i

   $$theratesatisfying[1-(1+i{)}^{-}5]/i=4.7600isi=1.66$$

4. After-tax cash flow — A_at = A(1 − t) = $25,000(1 − 0.35) = $16,250

5. After-tax (P/A) required — 7.3231

6. Solve again — i_at ≈ 0.01%

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

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?

Start with the thinking

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

Step-by-step solution

1. Balance

   $$pQ=F+vQ$$

2. Rearrange

   $$Q=F/(p-v)$$

3. Contribution margin — p − v = $23.00 − $10.00 = $13.00/unit

4. Substituting — Q = $119,000/$13.00 = 9,154 units/yr

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

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.

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

1. Formula

   $$0=-P+A(P/A,i,n)$$

2. Set up

   $$(P/A,i,8)=P/A=85000/18000=4.7222$$

3. Solve for i

   $$theratesatisfying[1-(1+i{)}^{-}8]/i=4.7222isi=13.47$$

4. After-tax cash flow — A_at = A(1 − t) = $18,000(1 − 0.27) = $13,140

5. After-tax (P/A) required — 6.4688

6. Solve again — i_at ≈ 4.98%

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

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?

Start with the thinking

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

Step-by-step solution

1. Balance

   $$pQ=F+vQ$$

2. Rearrange

   $$Q=F/(p-v)$$

3. Contribution margin — p − v = $32.50 − $17.00 = $15.50/unit

4. Substituting — Q = $90,000/$15.50 = 5,806 units/yr

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

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.

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

1. Formula

   $$0=-P+A(P/A,i,n)$$

2. Set up

   $$(P/A,i,10)=P/A=106000/22000=4.8182$$

3. Solve for i

   $$theratesatisfying[1-(1+i{)}^{-}10]/i=4.8182isi=16.08$$

4. After-tax cash flow — A_at = A(1 − t) = $22,000(1 − 0.32) = $14,960

5. After-tax (P/A) required — 7.0856

6. Solve again — i_at ≈ 6.81%

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

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?

Start with the thinking

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

Step-by-step solution

1. Balance

   $$pQ=F+vQ$$

2. Rearrange

   $$Q=F/(p-v)$$

3. Contribution margin — p − v = $20.00 − $9.00 = $11.00/unit

4. Substituting — Q = $135,000/$11.00 = 12,273 units/yr

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

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.

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

1. Formula

   $$0=-P+A(P/A,i,n)$$

2. Set up

   $$(P/A,i,10)=P/A=135000/33000=4.0909$$

3. Solve for i

   $$theratesatisfying[1-(1+i{)}^{-}10]/i=4.0909isi=20.73$$

4. After-tax cash flow — A_at = A(1 − t) = $33,000(1 − 0.31) = $22,770

5. After-tax (P/A) required — 5.9289

6. Solve again — i_at ≈ 10.84%

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

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?

Start with the thinking

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

Step-by-step solution

1. Balance

   $$pQ=F+vQ$$

2. Rearrange

   $$Q=F/(p-v)$$

3. Contribution margin — p − v = $43.00 − $24.50 = $18.50/unit

4. Substituting — Q = $158,000/$18.50 = 8,541 units/yr

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

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.

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

1. Formula

   $$0=-P+A(P/A,i,n)$$

2. Set up

   $$(P/A,i,8)=P/A=200000/33000=6.0606$$

3. Solve for i

   $$theratesatisfying[1-(1+i{)}^{-}8]/i=6.0606isi=6.62$$

4. After-tax cash flow — A_at = A(1 − t) = $33,000(1 − 0.25) = $24,750

5. After-tax (P/A) required — 8.0808

6. Solve again — i_at ≈ 0.01%

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