Eleven of something at R47 each. The till is slow. What is 47 × 11, before it rings up?
Times eleven
Eleven of something at R47 each. The till is slow. What is 47 × 11, before it rings up?
517, and the trick is a doorway. Multiplying by 11 is multiplying by 10 and adding the number once more: 470 + 47. But there is a faster shape. Write the two digits of 47 with a gap: 4 _ 7. Put their sum in the gap: 4 + 7 is 11, which is more than 9, so write the 1 and carry the other 1 into the 4: 5 1 7. When the digits add to 9 or less there is no carry at all: 35 × 11 is 3, then 3 + 5, then 5: 385.
The knowhow
Why it works is the split move. 47 × 11 is 47 × 10 plus 47 × 1, and adding 470 and 47 lines the 4 and the 7 up in the middle column, where they add, while the outer digits stay where they are. The carry is the ordinary carry of that column.
And so
Three-digit numbers work the same way with two gaps: 234 × 11 is 2, then 2 + 3, then 3 + 4, then 4: 2574. It is the first trick most people learn, and it is the one that shows every trick is a picture of an ordinary sum.
Keep this
Times eleven: the digits stay, their sum goes in the middle, and a carry when the sum passes nine.
Now try it with your own numbers. Two-digit numbers times 11. Split the digits, put their sum in the middle, carry if it passes 9.
Benjamin & Shermer (2006), Secrets of Mental Math, ch. 1, the 11 rule; distributive law a × 11 = a × 10 + a. https://en.wikipedia.org/wiki/Multiplication_algorithm
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