Mental Math for Quant Trading: Drills and Answers

Mental math for quant trading involves more than quick multiplication. You may need to convert a probability into a price, compare expected payoffs, or update cash and inventory while explaining your reasoning. A fast answer is useful only if you understand its units, sign and assumptions.

This guide provides original arithmetic exercises and a routine for reviewing mistakes. Start without a timer. Once a method is reliable, introduce modest time pressure and track accuracy alongside speed. The timings below are practice choices, not employer assessment rules or hiring benchmarks.

Jane Street's trading interview overview describes comfort with numbers and logic alongside methodical reasoning and clear communication. That supports practicing arithmetic in context; it does not establish a universal mental-math cutoff. Follow the instructions for the assessment or interview you actually receive.

Establish a useful baseline

Work through the eight drill sections below on paper without a calculator. Write each answer and the method you intended to use. Record elapsed time for the whole set, but do not race. Check the solutions only after completing the set.

Separate three outcomes: correct and confident, correct but uncertain, and incorrect. A lucky correct guess needs review. A slower answer reached through a repeatable method gives you something to improve.

If a question makes you stall, note where the stall occurred. You may know multiplication but struggle to retain an intermediate result, or know fractions but reverse the cash-flow sign in a trading example. Those require different practice.

1. Multiply around a convenient anchor

Drill: calculate 47 × 53, 98 × 102 and 39 × 21.

Show answer and explanation

47 × 53 = (50 − 3)(50 + 3) = 2,500 − 9 = 2,491.

98 × 102 = (100 − 2)(100 + 2) = 10,000 − 4 = 9,996.

39 × 21 = 39 × 20 + 39 = 780 + 39 = 819.

The first two use the difference-of-squares identity: (a − b)(a + b) = a² − b². Use it when the numbers sit equally far from a convenient center. The third uses ordinary distribution; forcing it into a symmetric pattern adds work.

Sanity-check the scale. The first product should be slightly below 2,500. The second should be slightly below 10,000. The third should be a little above 780. If your answer violates that direction, inspect it before continuing.

Follow-up: 46 × 54 = 2,500 − 16 = 2,484. Describe why moving farther from the same center lowers the product.

2. Break a percentage into easy parts

Drill: find 15% of 240, 12.5% of 360 and 7.5% of 640.

Show answer and explanation

15% of 240 = 10% + 5% = 24 + 12 = 36.

12.5% = 1/8, so 12.5% of 360 = 360 / 8 = 45.

7.5% of 640 = 5% + 2.5% = 32 + 16 = 48.

Choose the decomposition that fits the number. You can also calculate 7.5% as three quarters of 10%. Here, 10% is 64 and three quarters is 48. A method that is concise for 640 may be awkward for another base.

Always identify the base. “The price rose by 15%” means a percentage of the starting price, not a fixed 15-point increase. Percentages are multiplicative changes; percentage points are differences between percentage levels.

3. Reverse a percentage change

Drill: a quantity falls from 80 to 68. What percentage did it fall? What percentage increase would return 68 to 80?

Show answer and explanation

The decrease is 12 / 80 = 15%.

The required increase is 12 / 68 = 3 / 17, approximately 17.65%. A 15% increase from 68 reaches 78.20, not 80.

The denominator changes when you reverse the move. Work from the actual starting point for each leg. If you remember only one check, use this: a loss followed by an equal percentage gain leaves the quantity below its original level.

For a second drill, increase 100 by 20%, then decrease the result by 20%. You get 120 followed by 96. The total change is −4%, because 1.2 × 0.8 = 0.96.

4. Convert fractions with an honest approximation

Drill: express 3/8, 7/16 and 5/12 as percentages. Which answers terminate exactly?

Show answer and explanation

3/8 = 37.5% and 7/16 = 43.75%. Both terminate.

5/12 = 41.666…%, or approximately 41.67% to two decimal places.

Useful anchors include 1/2 = 50%, 1/4 = 25%, 1/5 = 20%, 1/8 = 12.5% and 1/16 = 6.25%. Build nearby fractions from these instead of memorizing every possible ratio.

Distinguish a rounded answer from an exact one when the precision matters. For 5/12, 40% is a rough estimate and 41.67% is a closer approximation. Neither is the exact fraction. In an interview, say what approximation you are making and why it is sufficient for the decision.

5. Compute a weighted average before net profit

Drill: a game pays 12 points with probability 1/4 and four points otherwise. Entry costs seven points. Find expected payout and expected net profit.

Show answer and explanation

Expected payout = (1/4 × 12) + (3/4 × 4) = 3 + 3 = six points.

Expected net profit = 6 − 7 = −one point.

A quick alternative is to start from the guaranteed minimum payout of four. The high outcome adds eight with probability 1/4, contributing two more points in expectation. Total expected payout is six.

Keep entry costs separate from payouts. The cost occurs every time you enter; it should not be multiplied only by the chance of losing. Also check the range: an expected payout must lie between four and 12 because those are the only two payouts.

Review expected value for the probability setup. Arithmetic practice is more useful once you can explain why those weights belong in the calculation.

6. Read the side of a trade correctly

Drill: starting with zero cash and zero inventory, you quote 19 / 21 per contract. A customer buys three from you, then sells you two. What are your cash and inventory? If each remaining contract settles at 20, what is profit?

Show answer and explanation

The customer buys at your ask, so you sell three at 21: cash +63, inventory −3.

The customer sells at your bid, so you buy two at 19: cash −38, inventory +2.

Total cash is +25 and inventory is −one. Settlement profit is 25 − 20 = five.

The multiplication is easy; the perspective can be harder. Say “my sale” or “my purchase” before updating the ledger. Then use cash + inventory × settlement value. This keeps a short position's settlement payment from turning into a receipt by accident.

Try the same sequence with settlement at 24. Profit becomes one. If settlement is 28, profit becomes −three. Earning a spread on some trades does not remove the risk of the remaining position. Practice a longer ledger in market-making interview questions.

7. Compare alternatives with a bound

Drill: which is larger, 17/41 or 5/12? Try answering without a long decimal expansion.

Show answer and explanation

Both denominators are positive, so compare cross-products: 17 × 12 = 204, while 5 × 41 = 205. Therefore 5/12 is larger.

The exact difference is 5/12 − 17/41 = (205 − 204) / 492 = 1/492, approximately 0.00203, or 0.203 percentage points.

Rounding both fractions to 0.42 would conceal the difference. Ask what precision the question requires before spending time calculating digits. To identify the larger ratio, cross-multiplication answers the question directly.

Do not apply this comparison blindly with unknown denominator signs. The inequality rule depends on those signs. Stating the assumption is part of the solution.

8. Use units to catch a factor-of-100 error

Drill: a hypothetical execution cost is 25 basis points on a $2,000 notional. What is the dollar cost?

Show answer and explanation

One basis point is 0.01 percentage points. Thus 25 basis points = 0.25% = 0.0025 as a decimal multiplier. The cost is 2,000 × 0.0025 = $5.

The key step is the unit conversion. A mental estimate also catches mistakes: 1% of $2,000 is $20, so one quarter of that is $5. An answer of $500 is incompatible with a charge below 1%.

For follow-up practice, 40 basis points on $7,500 is $30. Convert to 0.4%, then multiply. State the notional you are using; multiplying by a per-share price instead of total trade value would answer a different question.

Move from untimed practice to timed practice

Repeat the drills with changed numbers only after you understand the methods. Use three passes:

1. Untimed: identify the operation, calculate, and check sign, scale and units. Write enough working to diagnose an error.
2. Short timed set: choose a five-minute window and attempt a small mixed set. Leave unsolved questions blank when time expires. This is a training constraint you chose, not a passing standard.
3. Spoken set: explain selected calculations to a partner. Include one follow-up that changes the base, probability or settlement value.

Record attempted, correct and total questions separately. If you answer six of eight and all six are correct, accuracy among attempts is 100%, while completion is 75%. Reporting only accuracy hides the missing questions. Reporting only speed hides mistakes.

Compare repeated sets of similar difficulty. A faster multiplication set and a slower percentage set do not establish improvement or regression. Keep methods and question mix reasonably consistent when measuring progress.

Turn mistakes into tomorrow's drill

Use a simple error log: question, your answer, correct answer, error cause and next exercise. Classify the cause precisely: an arithmetic slip, a forgotten intermediate value, an incorrect probability model, a sign error, a unit error or a misread question.

If you treated a customer purchase as your purchase, more multiplication will not fix it. Practice translating trade direction into cash and inventory. If you reversed a percentage using the wrong denominator, practice identifying the base before calculating. If you lost an intermediate value, use fewer decomposition steps or speak the subtotal aloud.

Spend the next session repairing one recurring error, then return to a mixed set to check that the repair survives context changes. The quant interview study plan helps you balance arithmetic with probability, coding and spoken problem solving. Build a preparation routine that produces answers you can verify and explain under follow-up questions.