Case interview math, shortcuts and worked problems

1. Percent change

   (new minus old) ÷ old

   Sales from $80 million to $92 million is 12 ÷ 80, up 15%. Always divide by the old number, and say whether you mean percent or percentage points.

2. Growth over years

   start × (1 + rate) to the power of years

   8% a year for 3 years is 1.26 times the start, 26% in total, a bit more than 8 × 3 = 24%. CAGR runs it backwards: the yearly rate that turns the start into the end.

3. Margins

   profit ÷ revenue

   Profit margin is profit over revenue. Contribution margin is price minus variable cost, per unit or as a share of price. Markup is profit over cost, which is a different, bigger number.

4. Break-even

   fixed costs ÷ contribution per unit

   $1,200,000 of fixed costs and $12 of contribution a unit means 100,000 units. Divide fixed costs by the contribution margin % instead and you get break-even revenue.

   More break-even examples

5. Market sizing

   buyers × units each × price

   A short chain of multiplications, each with a stated assumption. The math is easy and the zeros are where people slip.

   Market sizing, step by step

6. Weighted average

   each value × its share, added up

   Two products at 20% and 35% margin don't average 27.5%. Weight each by its share of revenue. This is how a mix shift moves a margin.

7. Payback

   upfront cost ÷ yearly cash it brings in

   A $1,500,000 machine that saves $300,000 a year pays back in 5 years. Quick to say, and it ignores everything after the payback year.

8. NPV basics

   today's value of future cash, minus the cost

   Each year's cash is divided by (1 + rate) once for every year you wait. At 10%, $1 a year from now is worth about $0.91 today. Cash that never ends is worth cash ÷ rate.

Each one saves a few seconds and, more to the point, a slipped zero.

1. Zeros

   Take them off, multiply, put them back

   300,000 × 50: 3 × 5 is 15, and there are 5 + 1 = 6 zeros, so 15,000,000.

2. Units

   Say thousands, millions and billions

   A thousand thousands is a million and a thousand millions is a billion. 30 million customers × $40 is 1,200 million, or $1.2 billion.

3. Rounding

   Round one up and the other down

   48 × 21 is close to 50 × 20, which is 1,000. The exact answer is 1,008. Say you rounded, and fix it at the end if the decision is close.

4. Break it apart

   Split a number into easy pieces

   35% of 240 is 10% three times (24 × 3 = 72) plus 5% (12), so 84. 35 × 12 is 35 × 10 plus 35 × 2, or 420.

5. Swap the percent

   x% of y is y% of x

   4% of 75 is hard. 75% of 4 is 3, and it is the same number.

6. Halve and double

   Halve one side, double the other

   25 × 16 is 50 × 8, which is 100 × 4, or 400. Keep going until one side is a round number.

7. Rule of 72

   Years to double ≈ 72 ÷ growth rate

   At 8% a year a market doubles in about 9 years (exactly 9.01). Backwards: a market that doubled in 6 years grew about 12% a year (exactly 12.2%).

8. Small growth rates

   Adding is close over a few years, and always low

   8% for 3 years is roughly 24% if you add, 26% if you compound. Fine for a first pass. Say "a bit over 24%" so the interviewer knows you know.

Know these by heart and division by awkward numbers turns into multiplication. They work backwards too: a 12.5% margin means profit is 1/8 of revenue.

1. Halves, quarters, eighths

   1/2 is 50%, 1/4 is 25%, 1/8 is 12.5%

   Also 3/8 is 37.5%, 5/8 is 62.5%, 7/8 is 87.5%.

2. Thirds and sixths

   1/3 is 33.3%, 2/3 is 66.7%, 1/6 is 16.7%

   Also 5/6 is 83.3%.

3. Fifths and tenths

   1/5 is 20%, 2/5 is 40%, 1/10 is 10%

   Also 3/10 is 30%.

4. Sevenths, ninths, elevenths

   1/7 is 14.3%, 1/9 is 11.1%, 1/11 is 9.1%

   Close enough to round: 1/7 is about 14%, 1/9 about 11%, 1/11 about 9%.

5. Twelfths and smaller

   1/12 is 8.3%, 1/15 is 6.7%, 1/20 is 5%

   Also 1/25 is 4%.

A client's revenue went from $80 million to $92 million and its costs from $60 million to $72 million. What happened?

1. Revenue

   Up 15%

   12 ÷ 80 is 3 ÷ 20, or 15%.

2. Costs

   Up 20%

   12 ÷ 60 is 1 ÷ 5, or 20%. Costs grew faster than revenue.

3. Profit

   Flat at $20 million

   $80 million minus $60 million was $20 million. $92 million minus $72 million is still $20 million.

4. Margin

   Down 3.3 points

   20 ÷ 80 is 25%. 20 ÷ 92 is about 21.7%. That is 3.3 percentage points lower, or about 13% lower in percent terms.

5. So what

   Growth is costing as much as it earns

   Say it as a finding: the company sold more, but every extra dollar of sales came with a dollar of cost. Next, ask which costs grew and whether they are tied to the new sales.

A meal kit sells for $40 and costs $28 to make and ship. Fixed costs are $1,200,000 a year. How many kits a year to break even?

1. Contribution

   $12 a kit

   $40 minus $28. As a share of price, 12 ÷ 40 is 30%.

2. Break-even

   100,000 kits a year

   $1,200,000 ÷ $12. Take off the zeros: 12 ÷ 12 is 1, and 1.2 million has 5 more zeros than 12, so 100,000.

3. Check

   $4 million of sales

   $1,200,000 ÷ 30% is $4,000,000, and 100,000 kits × $40 is the same.

4. So what

   About 1,900 kits a week

   100,000 ÷ 52 is about 1,923. A weekly number is easier to judge: ask how many kits a week the company sells today.

Product A is 60% of revenue at a 20% margin. Product B is 40% at 35%. What is the company's margin, and what happens if B grows to half of revenue?

1. Weight A

   12 points

   60% × 20%: 0.6 × 20 is 12.

2. Weight B

   14 points

   40% × 35%: 0.4 × 35 is 14.

3. Blended

   26%

   12 + 14. A plain average of 20% and 35% would say 27.5%, which is wrong because A is the bigger product.

4. Mix shift

   27.5%, up 1.5 points

   At half and half: 0.5 × 20 + 0.5 × 35 is 10 + 17.5. Neither product's margin changed. Only the mix did, which is worth saying out loud in a profitability case.

   The profitability framework

A $2 billion market grows 8% a year. How big is it in 3 years, and when does it double?

1. Year by year

   About $2.52 billion

   $2 billion × 1.08 is $2.16 billion. × 1.08 again is about $2.33 billion. Once more is about $2.52 billion.

2. The shortcut

   About $2.48 billion

   Adding the rates gives 24%, or $2.48 billion. That is about $39 million short, close enough for a first pass if you say so.

3. Doubling

   About 9 years

   72 ÷ 8 is 9. The exact answer is 9.01 years. The rule of 70 gives 8.75. Both are fine, and 72 divides more evenly.

4. So what

   Compare it with the client

   A growth number means something next to another one. If the client grows 4% in an 8% market, it is losing share even while sales rise.

How much do dog owners in a city of 3 million people spend on dog food a year?

1. Dogs

   300,000

   Assumption: one dog for every 10 people. 3,000,000 ÷ 10 drops one zero.

2. A month

   $15 million

   Assumption: $50 of food per dog a month. 3 × 5 is 15, with 6 zeros: $15,000,000.

3. A year

   $180 million

   $15 million × 12 is $180 million. Say it in millions as you go, so the zeros never pile up.

4. Check

   $600 per dog a year

   Shrink it to one dog: $50 × 12 is $600 a year, which sounds right for one dog. The total is that times 300,000 dogs.

   Market sizing, top-down and bottom-up

A $1,500,000 packing machine saves $250,000 a year in labor and $50,000 in waste. It lasts 5 years. Is it worth it at a 10% discount rate?

1. Yearly saving

   $300,000 a year

   $250,000 plus $50,000.

2. Payback

   5 years

   $1,500,000 ÷ $300,000: 15 ÷ 3 is 5. That is the whole life of the machine, which is already a warning.

3. Discount

   3.79 years of saving, in today's money

   At 10%, a dollar in years 1 to 5 is worth about 0.91, 0.83, 0.75, 0.68, 0.62 today. Those add up to 3.79.

4. NPV

   Negative, about $0.36 million

   $300,000 × 3.79 is about $1.14 million, less than the $1.5 million it costs. At 10% the machine loses value.

5. So what

   Worth it only if it lasts longer

   If it lasted 10 years, the savings would be worth about $1.84 million today, an NPV of about $0.34 million. So the question to ask is how long the machine really lasts.