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Extras for Experts - Boxes and Boxes – Interpret pan balances to determine values of variables

Solutions

1. \quad t = 2 \ \text{pounds}; \ u = 4 \ \text{pounds}\!\\{\;} \quad \ \text{From} \ D, 3u = 12, \ \text{so} \ u = 4 \ \text{pounds}.\!\\{\;} \quad \ \text{From} \ C, 2t = 4, \ \text{so} \ t = 2 \ \text{pounds}.

2. \quad v = 3 \ \text{pounds}; w = 1 \ \text{pounds}\!\\{\;} \quad \ \text{From} \ H, v < 5 \ \text{or} \ 1, 2, 3 \ \text{or} \ 4 \ \text{pounds}.\!\\{\;} \quad \ \text{From} \ G, v = 3w, \ \text{so} \ v \ \text{must be a multiple of} \ 3.\!\\{\;} \quad \ \text{Then} \ 3 = 3w \ \text{and} \ w \ \text{is} \ 1 \ \text{pound}.

3. \quad y = 4, 8, \ \text{or} \ 12 \ \text{pounds}; z = 1, 2, \ \text{or} \ 3 \ \text{pounds}\!\\{\;} \quad \ \text{From} \ J, y < 16 \ \text{or} \ 1, 2, 3, ..., 15 \ \text{pounds}.\!\\{\;} \quad \ \text{From} \ K, y = 4z, \ \text{so} \ y = \ \text{must be a multiple of} \ 4.\!\\{\;} \quad \ \text{The possible multiples of} \ 4 \ \text{that are less than} \ 16 \ \text{are} \ 4, 8, \ \text{and} \ 12.\!\\{\;} \quad \ \text{If} \ y = 4, 8, \ \text{or} \ 12, \ \text{then} \ z = 1, 2, \ \text{or} \ 3 \ \text{pounds}.

4. \quad r = 1, 2 \ \text{or} \ 3 \ \text{pounds}; s = 10 \ \text{pounds}\!\\{\;} \quad \ \text{From} \ B, 2s = 20, \ \text{so} \ s = 10 \ \text{pounds}.\!\\{\;} \quad \ \text{From} \ A, 3r < 10, \ \text{or} \ 1, 2, 3, ..., 9 \ \text{pounds. So} \ r = 1, 2, \ \text{or} \ 3 \ \text{pounds}.

All weights are whole numbers of pounds.

What could be the weights? Tell how you figured it out.

All weights are whole numbers of pounds.

What could be the weights? Tell how you figured it out.

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