Fast Division and Reciprocals

Division is the operation most people are slowest at mentally, because unlike multiplication it doesn't decompose cleanly — there's no easy "carry the digit" shortcut for $840 / 24$ the way there is for multiplying two numbers. The fix used by fast mental calculators is to almost never actually divide: convert the division into a multiplication by a memorized (or quickly estimated) reciprocal instead.

Division is multiplication by the reciprocal

$$\frac{a}{b} = a \times \frac{1}{b} .$$

In plain English: if you already know $1/b$ as a decimal, dividing by $b$ becomes a multiplication, which is the operation your brain is faster and more reliable at. This only pays off if you have a stock of reciprocals memorized — the same way the fractions-to-decimals library works, but inverted.

Worth having memorized: $1/3 = 0.333$, $1/4 = 0.25$, $1/6 = 0.1667$, $1/7 \approx 0.1429$, $1/8 = 0.125$, $1/9 = 0.111$, $1/11 \approx 0.0909$, $1/12 \approx 0.0833$, $1/13 \approx 0.0769$.

Worked example 1: 840 / 24

Long division here means tracking remainders through several steps. Instead, notice $24 = 8 \times 3$, so $\frac{1}{24} = \frac{1}{8}\times\frac{1}{3} = 0.125 \times 0.3333 \approx 0.04167$. Then:

$$840 \times 0.04167 \approx 840 \times \frac{1}{24} = 35 .$$

Or more directly: $840 / 24 = 840/8 / 3 = 105/3 = 35$ — breaking the divisor into easy factors and dividing by each in turn is often faster than reaching for a reciprocal at all, but the reciprocal framing is what to fall back on when the divisor doesn't factor nicely. Both routes land on the same answer; which one is faster depends on whether the divisor happens to factor cleanly.

Worked example 2: 5,300 / 13 by reciprocal estimate

13 doesn't factor nicely and doesn't divide 5,300 cleanly, which is exactly when the reciprocal trick earns its keep. Recall $1/13 \approx 0.0769$. Then:

$$5{,}300 \times 0.0769 \approx 407.6 .$$

Sanity check: $13 \times 400 = 5{,}200$, and the remaining $100/13 \approx 7.7$, giving $407.7$ — consistent. In an interview, landing within a percent or two of the exact answer via the reciprocal, then refining with a quick sanity check, beats a slow, error-prone long division every time.

What this means in practice

Quick reciprocal-based division is what lets you compute things like Sharpe ratios, win/loss ratios, or price-to-earnings multiples on the spot rather than visibly grinding through long division at the whiteboard. It also pairs naturally with factoring: before reaching for a memorized reciprocal, always check whether the divisor breaks into small factors you can divide by sequentially — that's usually the fastest path of all.