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Boxes and Boxes - Interpret pan balances to determine values of variables

Teacher Notes

In each problem, two pan balances are shown with boxes of different types. All boxes of the same type have the same weight and all are whole numbers of pounds. Students figure out the relative weights of objects from the relationships displayed by the pan balances. To begin, be sure that students recognize differences in the types of boxes. Draw students’ attention to the positions of the pans. If the two pans are not at the same level, then one pan, the lower one, is heavier. If the pans are level, then they hold equal weight. One of the difficulties students often experience is recognizing that if one box (call it A) has the same weight as, for example, two others, (B + B), then the one box (A) is heavier. Point out to students that lb stands for pounds, and that all boxes with the same letter weigh the same number of pounds.

Solutions

1. \quad n = 8 \ \text{pounds}; \ p = 1, 2, \ \text{or} \ 3 \ \text{pounds}.\!\\{\;} \quad \ \text{From} \ D \ \text{Box} \ n \ \text{weighs} \ 8 \ \text{pounds}.\!\\ {\;} \quad \ \text{From} \ C, \ \text{since} \ 2p \ \text{is less than} \ 8, \ \text{then} \ p \ \text{is less than} \ 4. \ \text{So} \ p = 1, 2, \ \text{or} \ 3 \ \text{pounds}.

2. \quad s = 1 \ \text{or} \ 2 \ \text{pounds}; \ r = 3 \ \text{or} \ 6 \ \text{pounds}.\!\\{\;} \quad \ \text{From} \ F, \ 4s < 12, \ \text{so} \ s < 3 \ \text{or} \ 1 \ \text{or} \ 2 \ \text{pounds}.\!\\{\;} \quad \ \text{From} \ E, \ r = 3s, \ \text{so} \ r = 3 \ \text{or} \ 6 \ \text{pounds}.

3. \quad q = 2 \ \text{pounds}; \ t = 4 \ \text{pounds}\!\\{\;} \quad \ \text{From} \ G, \ 3q = 6, \ \text{so} \ q = 2 \ \text{pounds}.\!\\{\;} \quad \ \text{From} \ H, \ 4q = 4 \times 2, \ \text{or} \ 8 \ \text{pounds}.\!\\{\;} \quad \ \text{So} \ 2t = 8 \ \text{and} \ t = 4 \ \text{pounds}.

4. \quad v = 6 \ \text{pounds}; \ u = 1 \ \text{pound}\!\\{\;} \quad \ \text{From} \ J, \ 2v = 12, \ \text{so} \ v = 6 \ \text{pounds}.\!\\{\;} \quad \ \text{From} \ K, \ \text{since} \ v \ \text{is heavier than} \ 3u, 3u < 6, \ \text{and} \ u < 2. \ \text{So} \ u = 1 \ \text{pound}.

5. \quad x = 1 \ \text{or} \ 2; w = 2 \ \text{or} \ 4 \ \text{pounds}\!\\{\;} \quad \ \text{From} \ L, 3x < 9 \ \text{and} \ x < 3. \ \text{So} \ x = 1 \ \text{or} \ 2 \ \text{pounds}.\!\\{\;} \quad \ \text{From} \ M, w = 2x. \ \text{Since} \ x = 1 \ \text{or} \ 2, w = 2 \ \text{or} \ 4 \ \text{pounds}.

6. \quad y = 5 \ lb; \ z = 1 \ \text{or} \ 2 \ \text{pounds}\!\\{\;} \quad \ \text{From} \ N, 3y = 15, \ \text{so} \ y = 5 \ \text{pounds}.\!\\{\;} \quad \ \text{From} \ P, y > 2z, \ \text{so} \ 5 > 2z. \ \text{Then} \ z = 1 \ \text{or} \ 2 \ \text{pounds}.

All weights are whole numbers of pounds.

What could be the weights of Boxes m and r?

& \mathbf{Describe} && \text{There are two pan balances,} \ \text{A} \ \text{and} \ \text{B}.\!\\ &&& \text{A}: \text{The pans are level.}\!\\&&& \text{B}: \text{The pans are not level.}\!\\ &&& \text{There are two different boxes,} \ m \ \text{and} \ r. \ \text{There is a} \ 6 \ \text{pound box on} \ \text{B}.\\& \mathbf{My \ Job} && \text{Figure out the weights of} \ m \ \text{and} \ r. \ \text{Decide if other weights are possible}.\!\\& \mathbf{Plan} && \text{A}: \ 3 \ r \ \text{weighs the same as} \ m.\!\\&&& \text{B}: \text{The pan with} \ m \ \text{weighs less than} \ 6 \ \text{pounds}.\!\\&&& \qquad \text{Start with this fact}.\!\\& \mathbf{Solve} && \text{B}: \ m < 6 \ \text{pounds so} \ m = 1, 2, 3, 4, \ \text{or} \ 5 \ \text{pounds}\!\\&&& \text{A}: \ \text{Since} \ m = 3r, \ \text{then} \ m \ \text{is a multiple of} \ 3. \ \text{So} \ m = 3 \ \text{and} \ r = 1.\!\\& \mathbf{Check} && \text{A}: \ 3 = 1 + 1 + 1\!\\&&& \text{B}: \ 6 > 3

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.

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.

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