Math questions

A backpack, a bottle and forty six dollars

A backpack and a water bottle cost 46 dollars together. The backpack costs 40 dollars more than the bottle. What does the bottle cost?

Six dollars arrives almost immediately, and it is wrong. Check it: a 6 dollar bottle with a 46 dollar backpack comes to 52, which is over the stated total. Take the 40 dollar gap out of 46 first, and 6 is left for the two items to share equally. The bottle is 3 and the backpack is 43. A backpack and a bottle for $46 is the version you can answer here.

Shane Frederick built a three question test around exactly this shape, published in the Journal of Economic Perspectives in 2005. The design is the interesting part. Each question has an answer that arrives without effort, and that answer is wrong.

Why does the first answer arrive so fast?

Three mechanisms produce most of the wrong answers on this page.

An easier question gets answered instead. The bottle question quietly becomes a subtraction, because subtraction is what 46 and 40 suggest. The gap has to be removed before anything is halved, and once it is, the arithmetic is trivial.

Two numbers get averaged when the time behind them differs. Drive 60 miles out at 30 mph and the same 60 back at 60 mph. Averaging 30 and 60 gives 45. The trip took 3 hours for 120 miles, so the average speed is 40 mph. Averaging the speeds only works when both legs take the same time, and the slow leg here takes twice as long. Thirty miles out, sixty miles back is that question with the numbers in place.

Doubling gets read as a straight line. Algae covering a pond doubles every day and covers it fully on day 18. Day 9 is the middle of the calendar and nowhere near the middle of the growth: by day 9 the pond is under a thousandth covered. One step back from full is one halving, so day 17 is the answer. The pond that fills on day 18 is short enough to answer on a platform.

How do you get the number without paper?

Five moves cover most of what a question like this asks for.

Remove the gap, then split what is left. Any two quantities given by a total and a difference come apart this way. Subtract the difference from the total and halve the remainder for the smaller one.

Put total over total. Average speed is total distance divided by total time. Average price is total money divided by total units. Average score is total points divided by total people. The word average almost never means the average of the numbers in front of you.

Round to a friendly number, then subtract the extra. A cafe selling 24 croissants a day for four weeks sells over 28 days. Read 24 times 28 as 24 times 30 minus 24 times 2: 720 minus 48 gives 672. Four weeks of croissants also catches the people who stop at one week.

Halve one side and double the other. Forty eight copies at 12.50 becomes 24 at 25, then 12 at 50, then 6 at 100, which is 600. The product never changes and the decimals disappear on the first trade.

Square anything ending in five in two steps. Take the front digit, multiply it by the next number up, and write 25 after the result. For 25, that is 2 times 3 giving 6, then 25, so 625. For 35 it gives 1,225, and for 85 it gives 7,225. A patio 25 feet on each side is the one with a garden behind it.

There is a sixth worth knowing because it turns up in every conversation about savings. Divide 72 by an annual rate and you get the years to double. At 6 percent that is 12 years against an exact 11.9, which is close enough to say out loud. Dividing 100 by the rate gives 16.7 and is almost five years late, which is the sort of error that survives a whole meeting unchallenged. Six percent and the rule of 72 is the card that makes you commit to a number.

Where the answers usually go wrong

Two percent moves get treated as cancelling. A jacket up 20 percent then down 20 percent lands at 96 percent of where it started, because the rise is charged on the old price and the fall on the bigger one. Twenty percent up, twenty percent down is the cleanest case, and the whole family lives in Percent traps.

A unit gets skipped in the middle. Four weeks becomes 28 days before anything is multiplied. Kilometres become minutes before they become hours. Most factor of ten errors are a conversion that never happened.

A decimal point moves once too often. Halving and doubling exists to avoid this. So does taking a tenth first and doubling it, rather than doubling first.

The answer never gets checked against the question. Six dollars for the bottle takes four seconds to disprove. Plugging your answer back into the sentence is the cheapest step available, and it catches nearly everything.

Sequences and rough answers

Two other kinds of question turn up constantly, and both reward a first move rather than a formula.

For a sequence, take the differences before you look for a rule. The list 7, 10, 16, 28 has gaps of 3, 6 and 12, and those gaps double. The next gap is 24, so the next term is 52. Hunting for one fixed step fails here, and the differences hand you the pattern in one pass. Seven, ten, sixteen, twenty eight is the card version.

For a rough answer, name the two quantities you are multiplying and get each one roughly right. A concert floor 100 metres by 60 is 6,000 square metres, and at about 4 people per square metre that is 24,000 people. Area first, density second, one multiplication each. Guessing from an arena capacity mixes in the seated tiers and answers a different question. The concert floor and the crowd accepts a range, because that is what an estimate deserves.

Where these questions come from

Interviews use them, quizzes use them, and arguments in group chats use them most of all. The common thread is that each one can be settled by a person with no calculator and thirty seconds, which is exactly why they spread.

They also cover the arithmetic that turns up unannounced: a rate, an average, a doubling, a rough headcount, a price scaled from one part to a whole. None of it is advanced. All of it is easy to get wrong at speed.

Work through them above

The feed at the top of this page runs these as cards. One question, a box for your answer, and a line underneath naming the move rather than repeating the sum. Nothing is timed and nothing is scored.

For the same material at reading length, the blog has a set of harder puzzles and the argument over 8 divided by 2(2+2). For questions where the sentence is the obstacle, try Trick questions.