◀ ▲ ▶Branches / Fun / Tricks / Example: Multiplying small numbers by 9

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: \[(n-1)\times 10 + (10-n)=10n - 10 + 10 - n=n\times 9. \]

Thank you to the contributors under CC BY-SA 4.0!

Github: