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012Speed round: convert 3/32, 7/16 and 11/64 to decimals in your head, and explain the pattern you used.Belvedere TradingChicago · 2021
Try it first
What is 3/32 as a decimal?
Show the worked solution
3/32 = 0.09375, 7/16 = 0.4375 and 11/64 = 0.171875. Every denominator here is a power of two, so the unit fraction is a chain of halvings: 1/2 = 0.5, 1/4 = 0.25, 1/8 = 0.125, 1/16 = 0.0625, 1/32 = 0.03125, 1/64 = 0.015625. Find the rung, then multiply by the numerator. Each decimal ends exactly, because 2 divides a power of 10.
Why do powers of two give clean decimals?
Think of cutting a one-metre ribbon in half again and again: 50 cm, 25 cm, 12.5 cm, 6.25 cm. Each cut adds at most one digit to the length. A fraction terminates in decimal exactly when its denominator has no prime factors other than 2 and 5, so every power-of-two fraction ends, and 1 over 2 to the n has exactly n decimal places. That tells you before you start that 11/64 will have six digits after the point.
Each rung of the halving ladder is half the one above, from 0.5 down to 0.015625 for 1/64, so 3/32 is three of the 0.03125 rung, 0.09375, and 11/64 is eleven of the 0.015625 rung, 0.171875. How do you do 11/64 without losing a digit?
Split the numerator into pieces you already know. 11/64 is 8/64 + 2/64 + 1/64, which is 1/8 + 1/32 + 1/64: 0.125 + 0.03125 + 0.015625 = 0.171875. Or take 11 x 0.015625 as 10 x 0.015625 plus one more, 0.15625 + 0.015625. Either way you add numbers you have memorised instead of dividing. 7/16 works the same way as 1/2 - 1/16, 0.5 - 0.0625 = 0.4375.
The relationship1/8, 1/32, 1/64 rungs of the halving ladder 11 = 8 + 2 + 1 the numerator written in binary What it says in wordsWrite the numerator as a sum of powers of two and add the matching rungs.Why do trading firms test this?
Some bond and futures markets have long quoted prices in 32nds and 64ths of a point, and option deltas and odds come up as fractions all day. A trader who converts 3/32 at the speed of reading reacts to a price while a slower colleague is still dividing. The same round usually mixes in products such as 38 x 42, which is 40 squared minus 2 squared, 1,596: the test is spotting structure that turns long arithmetic into one step.
Where candidates lose it
Candidates try long division under pressure and drop or add a zero: 0.9375 for 3/32 is a common slip, and it is actually 15/16. Knowing the ladder by heart removes the division entirely.
The second loss is rounding. The question asks for the decimal, and 0.094 or 0.17 sounds careless when the exact answer is short and available. Give all the digits, then the rounded figure if asked.
What the interviewer asks next
- What is 13/128 as a decimal?
- Now 38 x 42 in your head, and say the trick you used.
- Convert 0.859375 back to a fraction.
Asked at Belvedere Trading, Trading, Chicago, 2021 (Wall Street Oasis):
3/32 mental math, 38*42, crossing the bridge in the shortest amount of time
037Without paper: work out 56 x 56 and 73 x 74, and say the shortcut you used for each.Akuna CapitalChicago · 2025
Try it first
What is 56 x 56?
Show the worked solution
56 x 56 = 3,136 and 73 x 74 = 5,402. For 56 squared, split it as 50 + 6: 2,500, plus two strips of 300, plus 36. For 73 x 74, anchor both on 70: 4,900, plus 70 x 7 = 490, plus 3 x 4 = 12. A second route checks each: (60 - 4) squared = 3,136, and 73.5 squared minus a quarter = 5,402.
What is the shortcut for squaring a two-digit number?
Tiling a floor that is 56 tiles on each side, you would lay the big 50 by 50 block first, then two thin strips along the edges, then a small corner. Splitting a number into a round base plus a small part turns one hard product into one easy square and a few small ones: (a + b) squared = a squared + 2ab + b squared. For 56: 2,500 + 2 x 300 + 36 = 3,136. You can also go down from the next round number: (60 - 4) squared = 3,600 - 480 + 16, again 3,136.
56 squared splits into a 2,500 block, two 300 strips and a 36 corner, total 3,136; 73 x 74 splits into 4,900, 280, 210 and 12, total 5,402, the same answer as 73.5 squared minus a quarter. What changes when the two numbers differ, as in 73 x 74?
When two numbers share a tens digit, anchor both on it. For (70 + 3)(70 + 4), the product is 70 squared, plus 70 times the sum of the units, plus the product of the units: 4,900 + 490 + 12 = 5,402. The midpoint route gives the same: numbers equally spaced around 73.5 multiply to 73.5 squared minus the square of the half gap, 0.25, and 73.5 squared is 4,900 + 490 + 12.25. Another quick path: 73 x 74 = 73 squared + 73 = 5,329 + 73.
The relationshipa the round base, 70 b, c the small parts, 3 and 4 What it says in wordsMultiply the round parts, add the round part times the sum of the small parts, then add the small product.Timed tests reward a fixed routine more than cleverness. Pick one decomposition, say the partial products in order, and check with a second route only if time allows. The last digit is a free check: 6 x 6 ends in 6 and 3 x 4 ends in 2, so 3,136 and 5,402 pass. A good habit is to sanity check the size too: 56 squared must sit between 50 squared, 2,500, and 60 squared, 3,600.
Where candidates lose it
The usual slip in 56 squared is adding one strip of 300 instead of two, giving 2,836, or dropping the 36. The area picture makes both errors visible: a square has two strips and a corner.
On a timed screen the other loss is switching methods halfway. Commit to the split, say each partial product, then add. Checking the last digit costs a second and catches most slips.
What the interviewer asks next
- Work out 97 x 103 in your head.
- What is 35 squared, and what is the trick for squares ending in 5?
- Estimate 48 x 52 without multiplying directly.
Asked at Akuna Capital, Prop Trading, Chicago, 2025 (Wall Street Oasis):
The mental math problems which were timed, one example was the 56*56
