Problem 3 · AMC 8 Stretch
Core
Logic & Word Problems
Arithmetic & Operations
work-backwardlogical-reasoning
You have only a \(5\)-liter bucket and an \(11\)-liter bucket and as much water as you want. How can you end up with exactly \(7\) liters in the big (\(11\)-liter) bucket?
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Answer: 7 liters (make 1 L, then top off the small bucket to pour off exactly 4 L)
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Hint 1 of 4
Work backward from the goal. You want \(7\) liters in the \(11\)-liter bucket. How much EMPTY space is left in that bucket when it holds \(7\)? (\(11-7\).)
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Hint 2 of 4
There are \(4\) liters of empty space. You could create exactly \(4\) empty liters by pouring from a full \(11\)-liter bucket into the small bucket — but only if the small bucket already had \(1\) liter in it (so it can take just \(4\) more).
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Hint 3 of 4
So now you only need to make exactly \(1\) liter. Try filling the big bucket and pouring out \(5\) liters twice: \(11-5-5=1\) liter is left over.
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Approach: Working backward, then doing the steps forward
- Work backward: \(7\) liters in the big bucket leaves \(4\) liters of empty space. To pour off exactly \(4\) liters into the \(5\)-liter bucket, the small bucket must already hold \(1\) liter (so it only has room for \(4\) more). And to get exactly \(1\) liter, notice \(11-5-5=1\).
- Now the forward steps. Fill the \(11\)-liter bucket. Pour into the \(5\)-liter bucket and dump it out; do this twice. After pouring out \(5+5=10\), the big bucket holds \(1\) liter.
- Pour that \(1\) liter into the empty \(5\)-liter bucket. Fill the \(11\)-liter bucket again (now it holds \(11\)).
- Pour from the big bucket to fill the small bucket the rest of the way. The small bucket had \(1\), so it takes \(4\) more. The big bucket now has \(11-4 = 7\) liters. Done!
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