Nine of anything: nine chairs at R47, nine litres at R23.90, nine months. What is 47 × 9?
Times nine
Nine of anything: nine chairs at R47, nine litres at R23.90, nine months. What is 47 × 9?
423, by the round move: nine is ten less one, so 47 × 9 is 470 less 47. Taking 47 from 470 is easier than it looks if you take 50 and give back 3: 420, 423. For 68 × 9: 680 less 68, which is 680 less 70 plus 2, 612. For 123 × 9: 1,230 less 123, 1,107.
The knowhow
Two things make nine special. First, the digits of any multiple of nine add to nine, or to a multiple of nine: 423 is 4 + 2 + 3 = 9, and 1,107 is 9. That is a free check on every answer, and the reason it works is the digits week's business. Second, the one-digit table has a finger trick: hold up ten fingers, fold the one you are multiplying by, and the fingers to the left are the tens and the ones to the right the units: fold the fourth finger and you see 3 and 6, 36.
And so
The same round move handles 99, 999 and 19: times 100 less once, times 1,000 less once, times 20 less once.
Keep this
Times nine is times ten less once, and the digits of the answer add to nine.
Now try it with your own numbers. Times 9. Times ten, then take the number away once. Check with the digit sum.
a × 9 = a × 10 − a; divisibility by 9 and the digit sum (casting out nines); the finger method for 9× (elementary arithmetic folklore, documented in Benjamin & Shermer 2006). https://en.wikipedia.org/wiki/Casting_out_nines
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