Arithmetic that survives a clock

The mental-maths test is a gate: below the bar the process ends regardless of everything else on your CV. It rewards a small set of techniques applied without hesitation, not cleverness. The single biggest gain is replacing long multiplication with rounding and correcting.

Why this is tested at all

A trader quoting a market has seconds to work out what a position is worth, what a hedge costs, and whether a price is off. The test is a crude proxy for that, and firms are explicit that it is a filter rather than a ranking: Optiver runs eighty questions in eight minutes, Jane Street around sixty, and the bar sits near seventy per cent correct.

Proposition 1.1

Round to a base, then correct

To multiply, round one factor to the nearest multiple of ten, multiply by that, then add or subtract the correction. Two easy products beat one hard one, and the intermediate numbers stay small enough to hold.

Holds when

Round the factor that lands closest to a multiple of ten.
Keep the sign of the correction in mind before you start, not after.

Example 1.2

Compute $37 \times 64$ without writing anything down.

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Worked solution

Formula $a \times b = a \times b_0 + a \times (b - b_0)$
Substitute $37 \times 64 = 37 \times 60 + 37 \times 4$
Solve $37 \times 60 = 2220$ Three times 37 is 111; add a zero.
$37 \times 4 = 148$
$2220 + 148 = 2368$
Answer $37 \times 64 = 2368$

Sanity check. Roughly $40 \times 64 = 2560$ , and we removed three lots of 64, so a little under 2400 is right.

Equation 1.3

Squares by difference

Pick $d$ so that one factor becomes a round number. This turns a square into a product you can do in one step.

n^2 = (n + d)(n - d) + d^2

$d$: The distance to the nearest convenient round number.

Example 1.4

Compute $47^2$ .

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Worked solution

Formula $n^2 = (n+d)(n-d) + d^2$
Substitute $47^2 = (50)(44) + 9$
Solve $= 2200 + 9$
Answer $47^2 = 2209$

Fraction	Decimal	Percent
$1/2$	$0.5$	$50\%$
$1/3$	$0.3333$	$33.3\%$
$1/4$	$0.25$	$25\%$
$1/6$	$0.1667$	$16.7\%$
$1/7$	$0.142857$	$14.29\%$
$1/8$	$0.125$	$12.5\%$
$1/9$	$0.1111$	$11.1\%$
$1/11$	$0.0909$	$9.09\%$
$1/12$	$0.0833$	$8.33\%$
$1/16$	$0.0625$	$6.25\%$

Table 1.5 · The table worth memorising. Sevenths cycle through the digits

142857

:

2/7 = 0.285714

,

3/7 = 0.428571

, and so on around the same loop.

Proposition 1.6

Build percentages from 10% and 1%

Ten per cent is a decimal shift and one per cent is two. Every other percentage is a short sum of those: $17\%$ is $10\% + 5\% + 2\%$ , and $37.5\%$ is three eighths.

Holds when

For a percentage of an awkward number, reverse it: $17\%$ of $250$ equals $250\%$ of $17$ .

The reversal trick. Because $a\%$ of $b$ equals $b\%$ of $a$ , you can always swap to whichever direction is easier. $4\%$ of $75$ looks awkward; $75\%$ of $4$ is three. This costs nothing to check and saves a surprising amount of time.

Common trap. Treating a percentage fall and the percentage rise that undoes it as the same number. A fall from $50$ to $40$ is $-20\%$ , but getting back from $40$ to $50$ is $+25\%$ . Instead. Always divide by where you started. Under time pressure, say the base out loud before dividing.

Equation 1.7

Doubling time

The rule of 72 approximates $\ln 2 / \ln(1+r)$ . Seventy-two is used rather than the more accurate $69.3$ because it divides cleanly by 2, 3, 4, 6, 8, 9 and 12.

t_{\text{double}} \approx \frac{72}{r}

Example 1.8

The base method near a round number

Compute $97 \times 94$ without writing anything down.

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Worked solution

Formula $(100-a)(100-b) = 100(100 - a - b) + ab$
Substitute $a = 3, \ b = 6$
Solve $100 - 3 - 6 = 91$ Cross-subtract either way; both give 91.
$3 \times 6 = 18$
Answer $9118$

Sanity check. Two digits for the tail, so $18$ not $180$ . Cross-subtracting the *other* way gives $97-6 = 91$ too, which is the built-in check.

Example 1.9

A percentage through a fraction

What is $37.5\%$ of $96$ , and what is $62.5\%$ of the same?

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Worked solution

Formula $37.5\% = \tfrac{3}{8}, \quad 62.5\% = \tfrac{5}{8}$
Substitute $96 \div 8 = 12$
Solve $3 \times 12 = 36$
$5 \times 12 = 60$
Answer $36 \text{ and } 60$

Sanity check. They sum to $96$ , which they must — recognising eighths turned a multiplication into a division by eight and two doublings.

Equation 1.10

Products around a midpoint

When two factors sit the same distance either side of a round number, the product is one square you know minus one small square. $63 \times 57$ is $60^2 - 3^2$ ; $98 \times 102$ is $100^2 - 2^2$ . It only works when the two numbers have the same parity, so that the midpoint is a whole number — otherwise shift one factor by one and correct.

(m - d)(m + d) = m^2 - d^2

$m$: The midpoint, ideally a multiple of ten.
$d$: Half the gap between the two factors.

Example 1.11

A product around a midpoint

Compute $63 \times 57$ .

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Worked solution

Formula $(m - d)(m + d) = m^2 - d^2$
Substitute $m = 60,\ d = 3$
Solve $3600 - 9$
$= 3591$
Answer $3591$

Sanity check. The last digit must be the last digit of $3 \times 7 = 21$ , which is $1$ — and it is. Last-digit checks catch most slips for free.

Proposition 1.12

Dividing by awkward decimals

Division by a decimal is multiplication by its reciprocal, and the common ones are whole numbers or simple fractions. Dividing by $0.25$ is multiplying by $4$ ; by $0.125$ , by $8$ ; by $1.25$ , by $0.8$ ; by $0.8$ , by $1.25$ . Recognise the reciprocal first and the division disappears.

Holds when

$x / 0.25 = 4x$ and $x / 0.125 = 8x$ .
$x / 1.25 = 0.8x$ and $x / 0.8 = 1.25x$ .
$x / 1.5 = \tfrac{2}{3} x$ and $x / 0.75 = \tfrac{4}{3} x$ .

Example 1.13

Division by recognising the reciprocal

Compute $840 / 1.25$ and $66 / 0.75$ .

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Worked solution

Formula $\frac{x}{1.25} = 0.8x, \quad \frac{x}{0.75} = \tfrac{4}{3} x$
Substitute $0.8 \times 840,\quad \tfrac{4}{3} \times 66$
Solve $0.8 \times 840 = 672$
$\tfrac{4}{3} \times 66 = 88$
Answer $672 \text{ and } 88$

Sanity check. Both answers move the right way: dividing by more than one shrinks the number, dividing by less than one grows it.

Number	Square	Number	Square
11	121	21	441
12	144	22	484
13	169	23	529
14	196	24	576
15	225	25	625
16	256	26	676
17	289	27	729
18	324	28	784
19	361	29	841
20	400	30	900

Table 1.14 · Squares from 11 to 30. Squares to thirty should be recall, not arithmetic. Beyond that,

(10a + b)^2 = 100a^2 + 20ab + b^2

builds any two-digit square in three short steps.

Example 1.15

Chaining percentage changes

A price rises $20\%$ and then falls $25\%$ . What is the overall change?

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Worked solution

Formula $(1 + r_1)(1 + r_2) - 1$
Substitute $(1.20)(0.75) - 1$
Solve $0.90 - 1$
$= -0.10$
Answer $-10\%$

Sanity check. Adding the two changes would say $-5\%$ . Percentages compound, and the second change is measured from the higher base.

Common trap — losing a power of ten. Stripping zeros and decimals to make a product easy, and then restoring the wrong number of them. $0.03 \times 4000$ becomes $3 \times 4 = 12$ with the scale mislaid, and answers come out a factor of ten or a hundred off. Instead. Count the powers of ten explicitly as you strip them: $0.03 = 3 \times 10^{-2}$ and $4000 = 4 \times 10^{3}$ , so the product is $12 \times 10^{1} = 120$ . Then sanity-check the magnitude: three per cent of four thousand must be about a hundred.

Check the magnitude and the last digit. Two free checks catch most mental-arithmetic errors. The magnitude check asks whether the answer is roughly the size it must be — $63 \times 57$ is near $60 \times 60 = 3600$ . The last-digit check multiplies only the units digits — $3 \times 7$ ends in $1$ , so the product must too. Running both takes a second and turns a confident wrong answer into a corrected right one.

Pacing under a clock

On an eighty-in-eight test the average item gets six seconds. Any item that will clearly take longer than ten is a candidate to skip and return to: an easy item left unanswered costs as much as a hard one got wrong, and hard items are where time disappears.

Optiver
IMC
Flow Traders

What you need to know

Round to a base and correct; never long-multiply under a clock.
Squares to 40 and the fraction table to sixteenths should be recall, not calculation.
Percentages are built from $10\%$ and $1\%$ , and can be reversed when that is easier.
A percentage change is always measured against where you started.
Estimate first, calculate second: an order-of-magnitude check catches most slips.

Exercise 1.16

What is $4\%$ of $175$ , and what is the fastest route to it?

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Seven. Reverse it: $4\%$ of $175$ is $175\%$ of $4$ , which is $4 + 3 = 7$ . Going the other way you would need $1\%$ of $175 = 1.75$ , then multiply by four — correct, but slower.

Exercise 1.17

Rounding to a base

Compute $86 \times 14$ in your head.

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Round $86$ up to $100$ : $100 \times 14 = 1400$ , then subtract $14 \times 14 = 196$ , giving $1204$ . Equivalently $86 \times 14 = 86 \times 7 \times 2 = 602 \times 2$ . Either way, the last digit $6 \times 4 = 24 \to 4$ confirms the answer ends in $4$ .

Exercise 1.18

Eighths

What are $12.5\%$ and $87.5\%$ of $72$ ?

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$12.5\%$ is one eighth, so $72/8 = 9$ . $87.5\%$ is seven eighths, so $72 - 9 = 63$ . Seeing a percentage as a fraction from the table turns both into a single division.

In the interview

Practise with a clock from day one. The single most common cause of failing this stage is treating it as a maths problem rather than a speed problem: candidates who can do every question given a minute still fail at six seconds each.

Optiver
IMC
Jane Street

After this lesson you should be able to

Why this is tested at all

Pacing under a clock

What you need to know

In the interview