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6.3: Boxes and Boxes Extras

Difficulty Level: At Grade Created by: CK-12
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Extras for Experts - Boxes and Boxes – Interpret pan balances to determine values of variables

Solutions

\begin{align*}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}.\end{align*}1.t=2 pounds; u=4 pounds From D,3u=12, so u=4 pounds. From C,2t=4, so t=2 pounds.

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

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

\begin{align*}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}.\end{align*}4.r=1,2 or 3 pounds;s=10 pounds From B,2s=20, so s=10 pounds. From A,3r<10, or 1,2,3,...,9 pounds. So r=1,2, or 3 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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CK.MAT.ENG.SE.1.Algebra-Explorations-K-7.6.3
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