Mental math under pressure
Rounding rules, error bounds, percent tricks, awkward divisions, and thirteen timed drills.
Key takeaways
- Round to go fast, but know how big your error is and whether it could change the decision.
- Rounding before a subtraction of close numbers can destroy the answer.
- In a product, round one input up and the other down, so the errors cancel.
- Percentage errors roughly add in a product, plus a little. If you round one input up 6 percent and another up 7.5 percent, the answer is about 14 percent too high (1.06 x 1.075 is about 1.14).
- Never round before subtracting two close numbers. Revenue of 10.4 and cost of 9.6 both round to 10, which gives a profit of 0 instead of 0.8, an error of 100 percent.
Key idea
Round to go fast, but know how big your error is and whether it could change the decision. That judgment is what separates quick math from careless math.
The lesson "Fast, clean, and checked" in Case math and quantitative reasoning covers the core habits: short forms for large numbers, units on every number, and sense checks. This lesson adds rules for rounding under pressure, simple error bounds, and tricks for awkward divisions.
Rounding: when it is safe and when it is not
Rounding rules
- In a product, round one input up and the other down, so the errors cancel. 47 x 2.14 is close to 50 x 2 = 100; the exact answer is 100.58.
- Percentage errors roughly add in a product, plus a little. If you round one input up 6 percent and another up 7.5 percent, the answer is about 14 percent too high (1.06 x 1.075 is about 1.14).
- Never round before subtracting two close numbers. Revenue of 10.4 and cost of 9.6 both round to 10, which gives a profit of 0 instead of 0.8, an error of 100 percent.
- Rounding is safe when the answer is far from the decision threshold. If a project returns three times its cost, a 10 percent error changes nothing. If it is within about 10 percent of break-even, compute exactly.
- Say the rounding out loud: "I will round 47 to 50, so my answer is about 6 percent high." The interviewer then sees that you know.
Profit is revenue of GBP 10.4 million minus cost of GBP 9.6 million a year. How do you calculate it fast?
Weaker answer
"Both are about 10 million, so profit is roughly zero."
Stronger answer
"I will not round these, because they are close: 10.4 minus 9.6 is GBP 0.8 million a year, a margin of about 8 percent."
Why the stronger answer wins: Rounding before a subtraction of close numbers can destroy the answer. Here a 4 percent rounding on each input became a 100 percent error on profit.
Error bounds in one line
If one input could be 20 percent too high or too low, the result carries the same 20 percent. If two inputs in a product could each be 10 percent too high, the result could be about 21 percent too high (1.1 x 1.1 = 1.21). Saying "this could be off by about 20 percent either way, and the answer holds unless it is off by more than half" is a strong sentence in any case.
Percent tricks
Five that save time
- Swap the order: x percent of y equals y percent of x. 8 percent of 25 is 25 percent of 8, which is 2.
- Build awkward percents: 15 percent is 10 percent plus half of that. 35 percent is a third plus a little.
- A rise then an equal fall is a net loss: up 10 percent then down 10 percent leaves 99 percent, a 1 percent fall.
- Percent change versus percentage points: a margin rising from 12 to 15 percent is up 3 percentage points, or 25 percent.
- Remove a percent by dividing, not subtracting: a price of AED 105 including 5 percent VAT is AED 100 before VAT (105 divided by 1.05), not AED 99.75.
Dividing by awkward numbers
Turn the division into something easy
- Divide by 7: multiply by about 0.143. A weekly figure of 2 million is about 286,000 a day.
- Divide by 12: divide by 3, then by 4. Or multiply by about 0.083. Annual to monthly is the most common use.
- Divide by 15: double the number, then divide by 30. 36 divided by 15 is 72 divided by 30, which is 2.4.
- Divide by 0.08: multiply by 12.5 (100 divided by 8). A fixed cost of EUR 2 million at an 8 percent contribution margin needs EUR 25 million of revenue to break even.
- Other easy swaps: divide by 0.25 means multiply by 4; by 0.125 means multiply by 8; by 0.05 means multiply by 20; by 0.04 means multiply by 25.
Timed drills: set the clock and answer before it ends
A Johannesburg store takes ZAR 2 million in a week. What is its average daily revenue, in ZAR thousands?
A Singapore gym chain earns SGD 5.4 million a year in memberships. What is that per month, in SGD thousands?
An Indian retailer splits a marketing budget of INR 36 lakh equally across 15 stores for the year. How much per store, in lakh?
A French brand has fixed costs of EUR 2 million a year and a contribution margin of 8 percent of revenue. What revenue does it need to break even, in EUR millions a year?
A market is worth AED 250 million a year. A company has 36 percent of it. What is its revenue, in AED millions a year?
Sales rise 10 percent one year and fall 10 percent the next. What is the net change over the two years, in percent? (Use a minus sign for a fall.)
Operating margin rises from 12 percent to 15 percent. By what percent did the margin increase (not percentage points)?
A US subscription app spends USD 360 to win a customer, who contributes USD 24 a month. How many months does it take to earn back the acquisition cost?
Rounding judgment drills: exact, rounded, and the size of the error
Exact: 4,870 x 0.198. You round to 5,000 x 0.2 = 1,000. By how many percent is the rounded answer too high? Answer to one decimal place.
Exact: a price of 47 times a volume of 1.86 million. You round both up to 50 x 2 million = 100 million. By how many percent is the rounded answer too high? Answer to one decimal place.
Exact: 47 x 2.14 million. You round to 50 x 2 million = 100 million, one input up and one down. By how many percent is the rounded answer too low? Answer to one decimal place.
Revenue is GBP 10.4 million and cost is GBP 9.6 million a year. Rounding both to 10 gives a profit of zero. What is the exact profit, in GBP millions a year?
You multiply two estimates. Each could be up to 10 percent too high. At most, how many percent too high could the product be?
When is rounding least safe?
What is the fastest way to divide by 0.08?
A margin rises from 10 percent to 12 percent. Which statement is correct?
Sources for this lesson (1)
- Recognized public explanations of case-interview concepts and terms
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