# Example: Multiplying small numbers by 9

(related to Part: Tricks for Mental Maths)

If you want to multiply $$n\times 9$$, $$n\in\{1,2,\ldots,10\}$$, you do not have to memorize all 10 results. The trick is to look at your hands, find the $$n$$th-finger, and mentally split your 10 fingers into two groups:

• first group being on the left side of your $$n$$th-finger (can be none, if $$n=1$$),
• second group being on the right side of your $$n$$th-finger (can be none, if $$n=10$$).

The following figures demonstrate this principle for $$n=1,2,3$$ and $$4$$, respectively:

As you can see, the blue-marked fingers are the ones of the result, the orange-marked fingers are the tenner part of the result. The results above are:

• $$1\times 9=9\Rightarrow$$ zero tenner and nine ones in the left-upper figure,
• $$2\times 9=18\Rightarrow$$ one tenner and eight ones in the right-upper figure,
• $$3\times 9=27\Rightarrow$$ two tenner and seven ones in the left-bottom figure,
• $$4\times 9=36\Rightarrow$$ three tenner and six ones in the right-bottom figure.
• ...
• $$n\times 9\Rightarrow$$ $$n-1$$ tenner and $$10-n$$ ones

#### Why does it work?

This works, since $$n-1$$ tenner and $$10-n$$ ones means: $(n-1)\times 10 + (10-n)=10n - 10 + 10 - n=n\times 9.$

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