## Weighty Logic

#### Problem

Daddy Bear is 60 kg heavier than Mummy Bear, and Baby Bear is 20 kg heavier than Goldilocks. Given that the lightest and the heaviest weigh 20 kg more than the combined weights of the other two, and the combined weight of all four is 300 kg, how heavy is Goldilocks?

#### Solution

Let the weights of Daddy Bear, Mummy Bear, Baby Bear, and Goldilocks be D, M, B, and G respectively.

We know that D = M + 60 and B = G + 20.

As D + M + B + G = (M + 60) + M + (G + 20) + G = 2M + 80 + 2G = 300 M + G = 110 (equation 1).

However, we must be careful to make any false assumptions about the order of their weights. There are a number of possibilities for which the difference between the heaviest/lightest pair and the other two need to be considered separately.

If Baby Bear is the same weight or heavier than Daddy Bear, we can see that Mummy Bear must be the lightest. Note that is doesn't matter which of Daddy Bear or Goldilocks is the heaviest; they're both the "middle" weights:

In which case, (B + M) (D + G) = (G + 20 + M) (M + 60 + G) = -40.

This is not consistent with the claim that the difference is 20 kg. So we know that Daddy Bear is the heaviest, which leaves two possibilities:

1. Goldilocks is the lightest, and note, once more, that it doesn't matter which of Mummy Bear or Baby Bear are heavier.

(D + G) (B + M) = (M + 60 + G) (G + 20 + M) = 40

2. Mummy Bear is the lightest.

(D + M) (B + G) = (M + 60 + M) (G + 20 + G) = 2M + 40 2G

Only the latter is consistent with the claim that the difference is 20 kg; that is, 2M + 40 2G = 20 M G = -10 (equation 2).

(equation 1) (equation 2): 2G = 120, which means that Goldilocks weighs 60 kg; for completeness it can be shown that M = 50 kg, D = 110 kg, and B = 80 kg.

Problem ID: 218 (30 Mar 2005)     Difficulty: 2 Star

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