Exam math toolkit

How to attack any math question

1Read the last sentence first. Circle what is being asked: a price, a percentage, a tax, a balance?
2Write the formula before you touch the calculator.
3Change percentages to decimals: 6.2% = 0.062, 1.45% = 0.0145, 10% = 0.10.
4Plug in the numbers and calculate. Round only at the end, to the cent.
5Check that the answer makes sense. A margin must be smaller than the markup; a tax must be smaller than the wages.

Exam trap

Wrong answer choices are often the result of one common mistake: dividing by price instead of cost, forgetting a wage limit, or using the employee rate when the question asks for the total. Before choosing, ask which mistake each wrong choice represents.

1. Markup vs margin

Know this

Markup = profit ÷ cost. Margin = profit ÷ price. Margin = markup ÷ (1 + markup). Markup = margin ÷ (1 − margin). Price from a target margin = cost ÷ (1 − margin).

Markup on cost	Equals margin on price
10%	9.1%
15%	13.0%
20%	16.7%
25%	20%
33.3%	25%
50%	33.3%
100%	50%

Worked example

Worked example: price for a 20% margin

Total cost (direct + overhead): $80,000. You want a 20% margin.
Price = $80,000 ÷ (1 − 0.20) = $80,000 ÷ 0.80 = $100,000.
Profit = $100,000 − $80,000 = $20,000. Margin check: $20,000 ÷ $100,000 = 20%.
Markup: $20,000 ÷ $80,000 = 25%.

Price $100,000 (a 25% markup gives a 20% margin).

Worked example

Guided practice: fill the blanks

Cost: $30,000. Target margin: 25%.
Price = $30,000 ÷ (1 − ___) = $30,000 ÷ ___ = ___.
Profit = price − cost = ___.
Markup = profit ÷ ___ = ___.

0.25; 0.75; $40,000. Profit $10,000. Markup = $10,000 ÷ $30,000 (cost) = 33.3%.

Worked example

On your own

You priced a job at $54,000. Your cost is $45,000. What are the markup and the margin?
Hint: find the profit first.

Profit $9,000. Markup = $9,000 ÷ $45,000 = 20%. Margin = $9,000 ÷ $54,000 = 16.7%.

2. Overhead recovery

Know this

Overhead rate = annual overhead ÷ expected annual direct costs. Job overhead = job direct costs × overhead rate. Bid = (direct + overhead) + profit.

Worked example

Annual overhead $150,000; expected direct costs $1,000,000. Rate = 15%.
Job direct costs $40,000 × 15% = $6,000 overhead.
Total cost $46,000. Profit 10% × $46,000 = $4,600.

Bid $50,600.

Worked example

Guided practice: fill the blanks

Annual overhead $96,000; expected direct costs $800,000. Rate = $96,000 ÷ $800,000 = ___.
Job direct costs $25,000 × ___ = ___ overhead.
Total cost ___; add 10% profit = ___.

12%; 12% → $3,000; $28,000; $28,000 × 1.10 = $30,800 bid.

Worked example

On your own

Overhead is $210,000 a year and you expect $1,400,000 of direct costs. A job has $60,000 of direct costs. You add 8% profit on total cost. What is the bid?

Rate 15%. Overhead $9,000. Total cost $69,000. Profit $5,520. Bid $74,520.

3. Break-even

Know this

Break-even sales = fixed costs ÷ gross margin %. Sales for a profit goal = (fixed costs + target profit) ÷ gross margin %. Break-even units = fixed costs ÷ (price − variable cost per unit).

Worked example

Fixed overhead $240,000 a year. Gross margin 20%.
Break-even = $240,000 ÷ 0.20 = $1,200,000.
To earn $60,000 profit: ($240,000 + $60,000) ÷ 0.20 = $1,500,000.

$1,200,000 to break even; $1,500,000 to earn $60,000.

Worked example

Guided practice: fill the blanks

You install water heaters. Fixed costs $12,000 a month. Price $500 each; variable cost $300 each.
Contribution per unit = $500 − ___ = ___.
Break-even units = $12,000 ÷ ___ = ___.

$300; $200; $200; 60 units a month.

Worked example

On your own

Monthly overhead is $15,000 and your gross margin is 25%. What monthly sales volume breaks even?

$15,000 ÷ 0.25 = $60,000 a month.

4. Job costing: direct vs indirect

Know this

Job cost = only the costs caused by that job. Office and sales costs are G&A and are recovered through the overhead rate, not added line by line.

Worked example

Field labor $18,000; payroll taxes and insurance on that labor $4,500; materials $22,000; subcontractor $9,000; lift rental for the job $1,500.
Direct cost = $18,000 + $4,500 + $22,000 + $9,000 + $1,500 = $55,000.
Contract $68,000. Gross profit = $68,000 − $55,000 = $13,000.
Overhead at 12% of direct cost = $6,600. Net job profit = $13,000 − $6,600 = $6,400.

Direct cost $55,000; gross profit $13,000; net after overhead $6,400.

Worked example

Guided practice: fill the blanks

Costs: materials $12,000; field labor $8,000; office manager salary share $1,200; dump fees for the job $600; office utilities $300.
Direct: $12,000 + ___ + ___ = ___.
G&A: ___ + ___ = ___.

$8,000 + $600 = $20,600 direct. $1,200 + $300 = $1,500 G&A.

Worked example

On your own

Subcontractor $15,000; lumber $6,500; office rent $2,000; job-site trailer rental $900; company advertising $700. What is the direct job cost?

$15,000 + $6,500 + $900 = $22,400 direct. Rent and advertising ($2,700) are G&A.

5. Percent complete and earned revenue

Know this

Percent complete = cost to date ÷ estimated total cost. Earned revenue = contract price × percent complete. Billed − earned = overbilling (if positive) or underbilling (if negative).

Worked example

Contract $500,000; estimated total cost $400,000; cost to date $100,000; billed $150,000.
Percent complete = $100,000 ÷ $400,000 = 25%.
Earned revenue = $500,000 × 25% = $125,000.
Billed − earned = $150,000 − $125,000 = $25,000.

25% complete; earned $125,000; overbilled $25,000 (liability).

How to attack any math question

1Read the last sentence first. Circle what is being asked: a price, a percentage, a tax, a balance?
2Write the formula before you touch the calculator.
3Change percentages to decimals: 6.2% = 0.062, 1.45% = 0.0145, 10% = 0.10.
4Plug in the numbers and calculate. Round only at the end, to the cent.
5Check that the answer makes sense. A margin must be smaller than the markup; a tax must be smaller than the wages.

Exam trap

1. Markup vs margin

Know this

Markup = profit ÷ cost. Margin = profit ÷ price. Margin = markup ÷ (1 + markup). Markup = margin ÷ (1 − margin). Price from a target margin = cost ÷ (1 − margin).

Markup on cost	Equals margin on price
10%	9.1%
15%	13.0%
20%	16.7%
25%	20%
33.3%	25%
50%	33.3%
100%	50%

Worked example

Worked example: price for a 20% margin

Total cost (direct + overhead): $80,000. You want a 20% margin.
Price = $80,000 ÷ (1 − 0.20) = $80,000 ÷ 0.80 = $100,000.
Profit = $100,000 − $80,000 = $20,000. Margin check: $20,000 ÷ $100,000 = 20%.
Markup: $20,000 ÷ $80,000 = 25%.

Price $100,000 (a 25% markup gives a 20% margin).

Worked example

Guided practice: fill the blanks

Cost: $30,000. Target margin: 25%.
Price = $30,000 ÷ (1 − ___) = $30,000 ÷ ___ = ___.
Profit = price − cost = ___.
Markup = profit ÷ ___ = ___.

0.25; 0.75; $40,000. Profit $10,000. Markup = $10,000 ÷ $30,000 (cost) = 33.3%.

Worked example

On your own

You priced a job at $54,000. Your cost is $45,000. What are the markup and the margin?
Hint: find the profit first.

Profit $9,000. Markup = $9,000 ÷ $45,000 = 20%. Margin = $9,000 ÷ $54,000 = 16.7%.

2. Overhead recovery

Know this

Overhead rate = annual overhead ÷ expected annual direct costs. Job overhead = job direct costs × overhead rate. Bid = (direct + overhead) + profit.

Worked example

Annual overhead $150,000; expected direct costs $1,000,000. Rate = 15%.
Job direct costs $40,000 × 15% = $6,000 overhead.
Total cost $46,000. Profit 10% × $46,000 = $4,600.

Bid $50,600.

Worked example

Guided practice: fill the blanks

Annual overhead $96,000; expected direct costs $800,000. Rate = $96,000 ÷ $800,000 = ___.
Job direct costs $25,000 × ___ = ___ overhead.
Total cost ___; add 10% profit = ___.

12%; 12% → $3,000; $28,000; $28,000 × 1.10 = $30,800 bid.

Worked example

On your own

Overhead is $210,000 a year and you expect $1,400,000 of direct costs. A job has $60,000 of direct costs. You add 8% profit on total cost. What is the bid?

Rate 15%. Overhead $9,000. Total cost $69,000. Profit $5,520. Bid $74,520.

3. Break-even

Know this

Break-even sales = fixed costs ÷ gross margin %. Sales for a profit goal = (fixed costs + target profit) ÷ gross margin %. Break-even units = fixed costs ÷ (price − variable cost per unit).

Worked example

Fixed overhead $240,000 a year. Gross margin 20%.
Break-even = $240,000 ÷ 0.20 = $1,200,000.
To earn $60,000 profit: ($240,000 + $60,000) ÷ 0.20 = $1,500,000.

$1,200,000 to break even; $1,500,000 to earn $60,000.

Worked example

Guided practice: fill the blanks

You install water heaters. Fixed costs $12,000 a month. Price $500 each; variable cost $300 each.
Contribution per unit = $500 − ___ = ___.
Break-even units = $12,000 ÷ ___ = ___.

$300; $200; $200; 60 units a month.

Worked example

On your own

Monthly overhead is $15,000 and your gross margin is 25%. What monthly sales volume breaks even?

$15,000 ÷ 0.25 = $60,000 a month.

4. Job costing: direct vs indirect

Know this

Job cost = only the costs caused by that job. Office and sales costs are G&A and are recovered through the overhead rate, not added line by line.

Worked example

Field labor $18,000; payroll taxes and insurance on that labor $4,500; materials $22,000; subcontractor $9,000; lift rental for the job $1,500.
Direct cost = $18,000 + $4,500 + $22,000 + $9,000 + $1,500 = $55,000.
Contract $68,000. Gross profit = $68,000 − $55,000 = $13,000.
Overhead at 12% of direct cost = $6,600. Net job profit = $13,000 − $6,600 = $6,400.

Direct cost $55,000; gross profit $13,000; net after overhead $6,400.

Worked example

Guided practice: fill the blanks

Costs: materials $12,000; field labor $8,000; office manager salary share $1,200; dump fees for the job $600; office utilities $300.
Direct: $12,000 + ___ + ___ = ___.
G&A: ___ + ___ = ___.

$8,000 + $600 = $20,600 direct. $1,200 + $300 = $1,500 G&A.

Worked example

On your own

Subcontractor $15,000; lumber $6,500; office rent $2,000; job-site trailer rental $900; company advertising $700. What is the direct job cost?

$15,000 + $6,500 + $900 = $22,400 direct. Rent and advertising ($2,700) are G&A.

5. Percent complete and earned revenue

Know this

Percent complete = cost to date ÷ estimated total cost. Earned revenue = contract price × percent complete. Billed − earned = overbilling (if positive) or underbilling (if negative).

Worked example

Contract $500,000; estimated total cost $400,000; cost to date $100,000; billed $150,000.
Percent complete = $100,000 ÷ $400,000 = 25%.
Earned revenue = $500,000 × 25% = $125,000.
Billed − earned = $150,000 − $125,000 = $25,000.

25% complete; earned $125,000; overbilled $25,000 (liability).