Multiplication · level 3

4-digit multiplication, reference 10 000

Four-digit numbers clustered around 10 000 are the showpiece of the method: two small circles, one cross-add, and a product of eight digits appears.

Worked example: 9985 × 10021

reference number 10,000

9,985−15

10,021+21

cross-add: 9,985 +21 = 10,006

× reference: 100,060,000

circles: −15 × +21 = −315

= 100,059,685

Step by step

1
Choose the reference number: 10,000
9985 is − 15 · 10021 is + 21
Write each number with its difference from the reference in a circle.
2
Cross-subtract (or cross-add) diagonally
9985 + 21 = 10,006
The other diagonal gives the same answer: 10021 − 15 = 10,006.
3
Multiply that result by the reference number
10,006 × 10,000 = 100,060,000
4
Multiply the two circled numbers
(−15) × (+21) = −315
5
Subtract it from the running total
100,060,000 − 315 = 100,059,685

9985 × 10021 = 100,059,685

The theory behind it

Intuition

Multiplication is easiest near a round number. You multiply the friendly number, then repair the difference — the repair term is always (how far off) × (the other factor).

Common pitfalls

×Losing the sign of the correction: rounding up means you subtract, rounding down means you add.
×Correcting against the wrong factor — the repair multiplies the factor you did not round.

In the interview

Every case round is a multiplication: users × price, volume × margin, market × share. Speed here buys you thinking time for the so-what.