Four hikers are carrying packs and all four packs weigh different amounts. When the packs are put on a scale, two at a time, the pairs of packs weigh the following amounts: 34 kg, 37 kg, 39 kg, 41 kg, 43 kg, 46 kg. What is the greatest possible weight of the heaviest pack?
+6248 \(\text{let (a,b,c,d) be the packs sorted from lightest to heaviest}\\ \text{the following four equations must hold}\\ a+b=34\\ a+c=37\\ b+d=43\\ c+d=46\)
\(\text{we then have a pair of equations which can be satisfied in two ways}\\ b+c=39 \text{ and } a+d=41\\ \text{or}\\ b+c=41 \text{ and }a+d = 39\)
\(\text{Solving the first set of 6 equations gets }\\ (a,b,c,d) = (16, 18, 21, 25)\\ \text{Solving the second set gets }\\ (a,b,c,d) = (15, 19, 22, 24)\\ \text{and thus 25 is the heaviest possible weight for pack d}\)
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