Mental Math for Trading Interviews

The arithmetic screens at trading firms are blunt instruments. The most famous version, Optiver's 80 in 8, asks 80 questions in 8 minutes, scored $+1$ for a correct answer and $-2$ for a wrong one, and most prop shops run something structurally similar. That is six seconds per question, no calculator, with a penalty that makes guessing toxic. Nobody clears it through raw talent. The candidates who pass know a short list of techniques so well that the right one selects itself, and they have drilled the misses out of their hands. This is that list. Every technique gets the one-line reason it works and a worked example with every intermediate digit shown, because in mental math the digits are the lesson.

The toolkit

1. Difference of squares

Why it works: $(a - d)(a + d) = a^2 - d^2$, so any product whose factors sit symmetrically around a round number collapses into one easy square minus one tiny one.

Take $47 \times 53$. Both factors straddle 50 at distance 3, so the answer is $50^2 - 3^2 = 2500 - 9 = 2491$. It scales cleanly: $96 \times 104 = 100^2 - 4^2 = 10000 - 16 = 9984$. The habit to build is checking the midpoint of any two-factor product before doing anything else. Whenever the factors average to a multiple of ten, this is the move.

2. Squaring near a base

Why it works: $(a + d)^2 = a^2 + 2ad + d^2$, and when $a$ is a round base the cross term costs nothing.

For $62^2$, work from base 60 with $d = 2$: $60^2 + 2 \times 60 \times 2 + 2^2 = 3600 + 240 + 4 = 3844$. Traders usually run the rearranged form $a^2 = (a + d)(a - d) + d^2$, sliding to the nearest round number: $62^2 = 64 \times 60 + 2^2 = 3840 + 4 = 3844$. It works below the base the same way: $58^2 = 60 \times 56 + 2^2 = 3360 + 4 = 3364$.

3. Multiplying by 5, 25, and 50 through powers of ten

Why it works: $5 = 10/2$, $25 = 100/4$, and $50 = 100/2$, and halving or quartering a number is far cheaper than multiplying it.

$348 \times 5$: append a zero, then halve: $3480 \div 2 = 1740$. $76 \times 25$: $7600 \div 4 = 1900$. $138 \times 50$: $13800 \div 2 = 6900$. Division flips the same lever: $415 \div 5 = 415 \times 2 \div 10 = 830 \div 10 = 83$.

4. The 9 and 11 shortcuts

Why they work: $9 = 10 - 1$ and $11 = 10 + 1$, so multiply by ten and correct with one copy of the number.

$47 \times 9 = 470 - 47 = 423$. $53 \times 11 = 530 + 53 = 583$; the schoolbook digit trick (keep the outer digits, insert their sum: 5, then $5 + 3 = 8$, then 3) is the same algebra in disguise. Watch the carry: $78 \times 11 = 780 + 78 = 858$, because the middle sum $7 + 8 = 15$ pushes a 1 leftward. The idea extends naturally: $23 \times 99 = 2300 - 23 = 2277$.

5. Fraction-decimal anchors

Why they work: the screens recycle a small set of fractions endlessly, so memorized conversions turn division under pressure into pure recall.

Eighths: $1/8 = 0.125$, $3/8 = 0.375$, $5/8 = 0.625$, $7/8 = 0.875$. Sixths: $1/6 \approx 0.1667$ and $5/6 \approx 0.8333$. Sevenths all live on one rotating cycle: $1/7 = 0.142857\ldots$ repeating, and every $k/7$ is a rotation of those same six digits. The entry points run 14, 28, 42, 57, 71, 85 for $1/7$ through $6/7$, which is the multiples of 14 with the last three bumped up by one.

Worked: $5/8$ of 240 is $240 \div 8 = 30$, then $30 \times 5 = 150$. And $3/7$ as a decimal is instant from the cycle: it enters at 42, so $0.428571\ldots$, call it 0.4286.

6. The percentage flip

Why it works: $a\%$ of $b$ equals $ab/100$, which is symmetric in $a$ and $b$, so it also equals $b\%$ of $a$.

16% of 25 feels like work; 25% of 16 is 4 on sight. The non-obvious use is taming ugly rates. A 6.25% fee on $48 flips into 48% of 6.25, and since 6.25 is one sixteenth of 100 you can read it straight off the flip: $48 \div 16 = 3$, so the fee is $3.

7. Successive percentage changes multiply

Why it works: a percentage change rescales the level, so consecutive changes compose as a product, never as a sum.

A stock at $150 rallies 8% and then drops 5%. The combined factor is $1.08 \times 0.95 = 1.026$, so the net move is up 2.6% and the price is $150 \times 1.026 = 153.90$. Stepwise check: $150 \times 1.08 = 162$, then $162 \times 0.95 = 162 - 8.10 = 153.90$. The cross-term form $x + y + xy/100$ gets you there directly: $8 - 5 - 0.40 = 2.60$. The classic trap is up 20% then down 20%: $1.20 \times 0.80 = 0.96$, a 4% loss, not flat.

8. The rule of 72

Why it works: exact doubling time at $r$ percent solves $(1 + r/100)^t = 2$, giving roughly $69.3/r$ years; 72 trades a sliver of accuracy for dividing cleanly by 2, 3, 4, 6, 8, 9, and 12.

At 8% compounded annually, money doubles in about $72 \div 8 = 9$ years. A typical compounding question: how long for $25,000 to become $100,000 at 8%? That is a factor of four, which is two doublings, so about $9 + 9 = 18$ years. The exact factor after 18 years is $1.08^{18} \approx 3.996$, so the estimate is tighter than it has any right to be.

9. Left-to-right addition

Why it works: leading digits carry most of the value, so adding from the left gives a running answer that refines toward exact, instead of a result that only exists once the last carry resolves.

$687 + 246$: hundreds give 800; tens give $80 + 40 = 120$, running total 920; units give $7 + 6 = 13$, final 933. Again with $462 + 379$: hundreds give 700, tens give $60 + 70 = 130$ for 830, units give $2 + 9 = 11$ for 841. On multi-leg questions this is how you carry a running total without ever re-adding a column.

10. Casting out nines

Why it works: every number leaves the same remainder mod 9 as its digit sum, so remainders must stay consistent through any addition or multiplication, and a mismatch proves an error.

Check $47 \times 53 = 2491$ from technique one. Reduce the factors: $4 + 7 = 11 \to 2$ and $5 + 3 = 8$. The product must reduce like $2 \times 8 = 16 \to 7$. The candidate answer reduces as $2 + 4 + 9 + 1 = 16 \to 7$. Consistent, so the answer stands. One honest caveat: the check is blind to transpositions, since 2941 also reduces to 7. It proves errors, never correctness. On a penalty-scored screen it earns its keep on multi-step questions where a copying slip is the likeliest failure mode.

Negative marking changes what you should practice

Score a screen at $+1$ correct, $-2$ wrong, and 0 for a skip, and every answer becomes a bet. If your confidence in an answer is $p$, the expected value of submitting it is $p \times 1 + (1 - p) \times (-2) = 3p - 2$, which is positive only when $p > 2/3$. Below two-thirds confidence, the skip is the trade. This is the same expected-value discipline the rest of the interview tests, applied to your own answer sheet.

The practice consequence is strict ordering: accuracy before speed. A miss is not a lost point, it is a three-point swing against a correct answer and a two-point loss against a skip. Concretely, a candidate who attempts 60 of 80 questions at 95% accuracy scores $57 - 6 = 51$; a faster candidate who attempts all 80 at 85% scores $68 - 24 = 44$. The slower candidate wins by seven. So train at a pace where your error rate sits near zero, then compress the clock. Speed bought with errors is negative expected value, and the test is built to charge you for it.

The 15-minute daily drill

Fifteen focused minutes daily beats a two-hour binge on Sunday, because recall speed is built by frequency, not volume.

Minutes	Block	What you do
3	Anchor recall	Flash through the fraction-decimal table, squares up to $25^2 = 625$, and powers of 2 up to 1024. Recall only, no deriving.
5	One technique	Pick one of the ten above and run 20 to 30 reps, untimed, target zero errors. Rotate through the list across the week.
5	Live sprint	One 2-minute scored sprint, then review every miss while it stings.
2	Error log	Tag each miss: wrong technique, digit slip, or answered below the two-thirds confidence bar.

The error log is the part people skip and the part that moves scores. After two weeks it tells you exactly which technique block deserves the five-minute slot.

How the sprint maps to the real screens

The mental math sprint is built as a rehearsal for these tests, not a casual game. Scoring is native $+1$ correct, $-2$ wrong, so the skip discipline above carries over untouched. It runs three modes: a 2-minute calibration sprint of up to 30 questions, where first runs typically land in the 8 to 14 band; the full 8-minute, 80-question gauntlet in real test proportions, where folklore puts the bar at the big shops around 45; and a 10-minute endurance set of 100 for superday stamina. The question pool regenerates on every run, so there is nothing to memorize, and every sprint is scored against percentile tracking so you watch your position in the distribution rather than a raw number.

Use the 2-minute mode inside the daily drill, the 8-minute mode once or twice a week as a full rehearsal, and the endurance mode in the final week before a real screen. When one question type keeps missing, the worked solutions in the mental math topic cover each technique with graded questions, and the timed screens replicate full company-style formats end to end.

FAQ

How fast do I actually need to be?

The headline pace is six seconds per question, but negative marking means attempts matter less than accuracy. A score of 45 on an 80-in-8 format is strong, and you can reach it by attempting 60 with five misses: $55 - 10 = 45$. Target 95% accuracy at eight seconds per question first; the last two seconds come from drilling, not from rushing.

Are calculators or scratch paper allowed?

Calculators, no. That is the point of the screen. Some firms allow rough paper and some do not, so train fully in your head and treat paper as a bonus. The skill being graded is whether numbers feel like friction or like air.

How long until my scores move?

Most candidates see the biggest jump in the first 15 to 20 sessions. Two to three weeks of the daily drill typically moves a 2-minute calibration score from the low teens into the twenties. When you plateau, the fix is almost always in the error log: one technique is quietly donating points, and it gets the next week of five-minute blocks.

Put it on a clock

Reading about arithmetic is comfortable and useless. The techniques above only pay when they fire in under a second, and that happens through reps under time pressure, not through understanding. Run a 2-minute calibration on the mental math sprint today to get your baseline, drill the gaps through the mental math question bank, then graduate to the company-style timed screens and the full question bank when your accuracy holds at speed. Fifteen minutes a day, three weeks, measurable percentile movement. That is the whole plan.