Multiplying Numbers Near a Round Base

Squaring a number near a round base is a special case of a more general trick: multiplying any two numbers that both happen to sit close to the same round number, like $97 \times 103$ or $48 \times 53$. Doing these longhand is slow; doing them via their distances from a shared base turns them into small, easy arithmetic almost every time.

The trick: distances from a shared base

Pick a convenient base $B$ (usually a multiple of 10 or 100) close to both numbers. Write each number as the base plus or minus a small offset: $x = B + a$, $y = B + b$. Then:

$$xy = B(x+y-B) + ab .$$

In plain English: multiply the base by the sum of the two offsets-from-base combined back into the numbers (which is just $x+y-B$, easy since it's near $B$), then add the product of the two small offsets — a small correction that's easy to compute since $a$ and $b$ are small.

A cleaner way to remember it when both offsets are known: $xy = B^2 + B(a+b) + ab$, but in practice the fastest path is usually to compute $B \times (x + y - B)$ directly since that's one multiplication by a round number, then add $ab$.

Worked example 1: 97 × 103

Base $B = 100$. Offsets: $a = -3$ (since $97 = 100 - 3$), $b = 3$ (since $103 = 100+3$). Then $x + y - B = 97 + 103 - 100 = 100$, so:

$$97 \times 103 = 100 \times 100 + (-3)(3) = 10{,}000 - 9 = 9{,}991 .$$

This is the difference-of-squares pattern in disguise — when offsets are equal and opposite, $xy = B^2 - a^2$ directly, since $97 \times 103 = (100-3)(100+3) = 100^2 - 3^2$.

Worked example 2: 48 × 53 (unequal offsets)

Base $B = 50$. Offsets: $a = -2$ (48 = 50 − 2), $b = 3$ (53 = 50 + 3). Compute $x + y - B = 48 + 53 - 50 = 51$, so:

$$48 \times 53 = 50 \times 51 + (-2)(3) = 2{,}550 - 6 = 2{,}544 .$$

$50 \times 51$ is instant (half of 5,100), and the correction is a tiny single-digit multiplication — far faster than multiplying 48 by 53 directly.

What this means in practice

This trick shows up constantly when estimating notional values, position sizes, or quick P&L calculations in an interview — numbers in trading questions are frequently "near round" by design, since interviewers pick them that way to test exactly this skill. Combined with the squaring shortcut, it's the difference between visibly struggling through two-digit multiplication and answering before the interviewer finishes writing the number down.