🇺🇸 AMC 8 ⇄ switch contest
1992 AJHSME

Problem 25

Problem 25 · 1992 AJHSME Stretch
Fractions, Decimals & Percents telescoping-product

One half of the water is poured out of a full container. Then one third of the remainder is poured out. Continue the process: one fourth of the remainder for the third pouring, one fifth of the remainder for the fourth pouring, and so on. After how many pourings does exactly one tenth of the original water remain?

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Answer: D — 9.
Show hints
Hint 1 of 3
Don't track how much you POUR OUT — track what STAYS. Pouring out 1/2 leaves 1/2; pouring out 1/3 of what's left leaves 2/3 of it; pouring out 1/4 leaves 3/4. What do you multiply to chain these?
Still stuck? Show hint 2 →
Hint 2 of 3
The amount remaining is a product of survival fractions: 1/2 × 2/3 × 3/4 × …. Before multiplying it all out, look for a pattern in how the tops and bottoms line up.
Still stuck? Show hint 3 →
Hint 3 of 3
Notice each numerator matches the previous denominator — so almost everything cancels in a chain (this ‘telescoping’ collapse leaves just the first top and last bottom).
Show solution
Approach: track what survives; the fractions telescope to a tiny result
  1. Each pouring removes a slice and leaves the rest: keep 1/2, then 2/3 of that, then 3/4, then 4/5, and so on. After k pourings the fraction left is 1/2 × 2/3 × 3/4 × … × k/(k+1).
  2. Watch the cancellation: the 2 on top of 2/3 cancels the 2 on the bottom of 1/2, the 3 on top of 3/4 cancels the 3 below it, and so on down the chain. Everything cancels except the very first top (1) and the very last bottom (k+1), leaving 1/(k+1).
  3. We want exactly 1/10 left, so 1/(k+1) = 1/10 means k+1 = 10, giving k = 9 pourings.
  4. Why this transfers: when a product's numerators reuse the previous denominators, it ‘telescopes’ — the middle all cancels and only the outermost top and bottom remain. Spotting this saves you from multiplying nine fractions by hand.
  5. Sanity check: after 1 pouring you have 1/2; the formula gives 1/(1+1) = 1/2. Good — and 1/10 is reached when the denominator hits 10, i.e. the 9th pouring.
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