### Brain Teasers

# Counting Wrappers

Your favourite chocolate bar is having a special promotion, buy 10 get 1 free! To get a free bar, simply collect 10 wrappers and turn them in for 1 free bar. What fraction of a bar is each wrapper worth?

### Hint

Remember that the free bar has a wrapper.### Answer

Each wrapper is worth 1/9 of a bar.10 wrappers are worth 1 free bar, so it appears each wrapper is worth 1/10 of a bar, except that the free bar comes with a wrapper.

10 wrappers are worth 1 free bar (with 1 free wrapper), so each wrapper is worth 1/10 of a bar plus 1/10 of a wrapper. The 1/10 of a wrapper is worth 1/100 of a bar plus 1/100 of a wrapper, and so on. 1/10 + 1/100 + 1/1000 + ... = 0.111... = 1/9.

The answer can be shown with the following algebra, where w is the value of a wrapper, and b is the value of a bar.

10w = b + w (10 wrappers are worth 1 bar with 1 wrapper)

9w = b (subtract w from each side)

w = b/9 (divide both sides by 9)

Each wrapper (w) is worth 1/9 of a bar (b).

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