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?
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).
