Case interview math practice
Six problems.
Six useful checks.
Practice the arithmetic, then explain what the number means. Work through percentage change, profitability, weighted averages, break-even, compound growth and capacity.
Questions, worked solutions and PDFs are open. No account needed.
01 / Percentage change
Sales rose by how much?
A shop sold 80 units last week and 92 this week. What is the percentage increase in units sold?
Assumptions: Compare equally long periods; every unit is counted once.
Check your reasoning for percentage change
15%.
The increase is 12 units. Divide by the starting value: 12 ÷ 80 = 0.15 = 15%.
Watch for: 12 is an absolute unit change. Percentage change uses the original 80 as its denominator.
02 / Profit and margin
More revenue. Less profit?
A retailer has $10 million in annual revenue and $8 million in annual costs. Next year, revenue grows 10% and costs grow 15%. Calculate next year's profit and profit margin. Did profit grow?
Assumptions: Profit means revenue minus the stated costs; use the same cost scope in both years.
Check your reasoning for profit and margin
$1.8 million profit; about 16.36% margin.
Revenue becomes $11 million and costs become $9.2 million. Profit is $11m − $9.2m = $1.8m. Margin is $1.8m ÷ $11m.
Profit falls 10% from $2m. Margin falls from 20% to about 16.36%: about 3.64 percentage points.
Watch for: Do not subtract the growth rates to calculate profit growth. Costs and revenue start at different values.
03 / Weighted average
The mix changed.
A store sells two products at $60 and $100. At first, 40% of units are the $60 product and 60% are the $100 product. The mix then changes to 60% and 40%. Find the average selling price before and after. What happens to revenue if total unit sales stay at 1,000?
Assumptions: Prices do not change. Shares refer to unit volume, not revenue.
Check your reasoning for weighted average
$84 before; $76 after.
First: 0.40 × $60 + 0.60 × $100 = $84. Then: 0.60 × $60 + 0.40 × $100 = $76.
At 1,000 units, revenue falls from $84,000 to $76,000: $8,000, or about 9.52%.
Watch for: The unweighted midpoint of $80 ignores the mix. A shift toward the lower-priced product reduces revenue even with unchanged volume.
04 / Break-even
How many units cover costs?
A service charges $12 per unit, has an $8 variable cost per unit, and incurs $360,000 in annual fixed costs. How many units must it sell each year to break even?
Assumptions: Price and unit cost stay constant; all stated fixed costs belong to the same year. Ignore taxes and capacity constraints.
Check your reasoning for break-even
90,000 units per year.
Each unit contributes $12 − $8 = $4 toward fixed costs. $360,000 ÷ $4 = 90,000 units.
Watch for: Dividing fixed costs by the $12 selling price ignores the variable cost. Check the answer: 90,000 × $4 exactly covers $360,000.
05 / Compound growth
Three years at 10%.
Annual revenue is $100 million today. It grows 10% each year for three years. What is revenue after year three, and what is the total percentage increase?
Assumptions: Each year's growth applies to the previous year's revenue.
Check your reasoning for compound growth
$133.1 million; a 33.1% total increase.
Year one is $110m, year two is $121m, and year three is $133.1m. Equivalently: $100m × 1.1³.
Watch for: Adding three annual rates gives 30%, which omits compounding. The annual growth rate is still 10%.
06 / Capacity
Can the team meet demand?
Four analysts each process 60 tickets per working day. The team works five days a week and receives 1,500 tickets a week. What is weekly capacity, the shortfall, and the minimum total number of analysts needed at the same productivity?
Assumptions: All analysts work all five days; tickets require comparable effort. Ignore leave, rework and scheduling variability.
Check your reasoning for capacity
1,200 tickets of capacity; 300 tickets short; five analysts in total.
4 × 60 × 5 = 1,200. Demand exceeds capacity by 1,500 − 1,200 = 300. Each analyst processes 300 tickets a week, so 1,500 ÷ 300 = 5.
Watch for: Five is the total required headcount, not five additional hires. Real staffing decisions also need variability and quality assumptions.