The arithmetic, from the ground up
Percentages, percentage points, growth over several years, margins, and breakeven.
Key takeaways
- Percent and percentage points are different: If a margin goes from 10 percent to 12 percent, it rose by 2 percentage points, but by 20 percent (2 is 20 percent of 10).
- Cases use a small set of arithmetic moves again and again.
- Growth over several years compounds: each year grows on top of the last.
Cases use a small set of arithmetic moves again and again. None of it is hard once it is laid out plainly. Freezing on simple math is a common way to lose points, so the goal is to understand each move and then make it automatic through practice.
Percentages and percentage change
A percentage is a part out of 100. Twenty percent of 1,500 is 1,500 times 0.20, which is 300. Percentage change compares the size of a change with the starting value: if sales go from 200 to 250, the change is 50, and 50 out of the starting 200 is 25 percent growth.
Worked case
A quick percentage-change calculation
The prompt
Revenue rose from EUR 800,000 to EUR 920,000 over a year. What is the growth rate?
The structure
- Growth rate = change divided by starting value
- Change = 920,000 minus 800,000
- Divide the change by the starting 800,000
Working it through
1. Find the change
Subtract the start from the end. Drop the thousands while you work and add them back at the end.
Change in revenue (EUR thousands):920 - 800 = 1202. Compare with the starting value
Divide the change by where you started, then read it as a percentage.
Growth rate (fraction):120 ÷ 800 = 0.15
The recommendation
Revenue grew 15 percent over the year.
If a margin goes from 10 percent to 12 percent, it rose by 2 percentage points, but by 20 percent (2 is 20 percent of 10). Say "points" when you compare two percentages. Interviewers notice this.
Growth over several years
Growth over several years compounds: each year grows on top of the last. If a market of 100 grows 10 percent a year for two years, it becomes 100 times 1.1 times 1.1, which is 121, not 120. The average yearly rate is called the CAGR (compound annual growth rate). A quick tool is the rule of 72: divide 72 by the yearly growth rate to estimate how many years it takes to double. At 8 percent a year, a market doubles in about 9 years.
A market grows 6 percent a year. Using the rule of 72, about how many years until it doubles?
Profit, margin, and markup
Profit is revenue minus cost. Margin is profit as a share of the price: sell for 100, spend 60, and your profit is 40, which is a 40 percent margin. Markup is the same 40 measured against the cost of 60 instead, which is about 67 percent. People mix these up often, so always ask yourself: a share of the price (margin) or a share of the cost (markup)?
Breakeven
Breakeven is the sales level where you exactly cover your costs, so profit is zero. Divide your fixed costs by the contribution per unit, which is the price of one unit minus its own variable cost. If fixed costs are SGD 10,000 a month and each unit contributes SGD 5, you break even at 2,000 units a month.
Worked case
A breakeven calculation
The prompt
Fixed costs are USD 60,000 a year. Each unit sells for USD 25 and costs USD 10 in materials. How many units to break even?
The structure
- Breakeven units = fixed cost divided by contribution per unit
- Contribution per unit = price minus variable cost
- Fixed costs to cover each year
Working it through
1. Contribution per unit
Each sale brings in 25 and uses 10 of materials, so it contributes the difference toward fixed costs.
Contribution per unit (USD):25 - 10 = 152. Breakeven units
Divide the fixed costs by the contribution each unit gives.
Breakeven units:60,000 ÷ 15 = 4,000
The recommendation
The answer is 4,000 units a year, because each unit contributes USD 15 (a USD 25 price minus USD 10 of materials), and USD 60,000 of fixed costs divided by USD 15 is 4,000. This means every unit above 4,000 adds USD 15 of profit, and every unit below it loses USD 15. The risk is that price or material cost moves, which shifts the breakeven. As a next step, compare 4,000 units with realistic demand to confirm the business can clear it with room to spare.
You sell an item for 50 that costs you 30. What is the margin?
Market share went from 20 percent to 25 percent. How should you describe the change?
Sources for this lesson (1)
- Recognized public explanations of case-interview concepts and frameworks
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