# 7.8: Hanging Scales

**At Grade**Created by: CK-12

**Hanging Scales - Solve for Unknowns**

**Teacher Notes**

Each problem shows three scales, their contents, and their weights. Students use the data provided in the display as clues to determine the weight of each block. All of the scales contain more than one type of block, so weights of blocks cannot be found directly. In all cases, all of the blocks on one of the scales are on a second scale. The second scale contains at least one other block. Students replace the set of blocks with their value in order to find the weight of the other block on the second scale. Once the value of one block is known, urge students to record its value on all blocks of that type on all scales. This will enable students to figure out the weights of the other blocks. Encourage students to check solutions by replacing each block with its weight and comparing the total weight with the scale indicator.

**Solutions**

\begin{align*}1. \quad A: x + z = 10; \ B: x + x + z = 13; \ C: x + y + y = 15\!\\
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{\;} \quad \ x = 3, \ y = 6, \ z = 7\!\\
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2. \quad D: x + x + y = 9; \ E: x + x + y + y = 14; \ F: x + y + z = 10\!\\
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{\;} \quad \ x = 2, \ y = 5, \ z = 3\!\\
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3. \quad G: y + z = 11; \ H: y + y + z = 14; \ I: x + x + z = 16\!\\
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{\;} \quad \ x = 4, \ y = 3, \ z = 8\!\\
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4. \quad J: x + x + y + z = 19; \ K: x + x + z + z = 22; \ L: x + y + z = 16\!\\
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{\;} \quad \ x = 3, \ y = 5, \ z = 8\!\\
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5. \quad M: y + y + z = 20; \ N: y + z = 14; \ P: x + x + z = 22\!\\
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{\;} \quad \ x = 7, \ y = 6, \ z = 8\!\\
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6. \quad Q: x + z + z = 17; \ R: x + x + y = 25; \ S: x + y + z + z = 24\!\\
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{\;} \quad \ x = 9, \ y = 7, \ z = 4\end{align*}

\begin{align*}& \mathbf{Describe} && \text{There are three scales with blocks.}\\ &&& A: \ \text{one} \ x \ \text{and} \ 2 \ y \ \text{blocks. They weigh} \ 14 \ \text{pounds.}\\ &&& B: \ \text{one} \ x \ \text{and} \ 3 \ y \ \text{blocks. They weigh} \ 18 \ \text{pounds.}\\ &&& C: \ \text{one} \ y \ \text{and} \ 2 \ z \ \text{blocks. They weigh} \ 14 \ \text{pounds.}\\ &&& \text{All blocks on} \ A \ \text{are also on} \ B.\\ & \mathbf{My \ Job} && \text{Use the scales as clues. Figure out the weights of the blocks.}\\ & \mathbf{Plan} && \text{Write equations, one for each scale.}\\ &&& A: x + y + y = 14; \ B: x + y + y + y = 18; \ C: \ y + z + z = 14.\\ &&& \text{Solve the equations.}\\ & \mathbf{Solve} && A: x + y + y = 14\\ &&& B: (x + y + y) + y = 18. \ \text{Replace} \ (x + y + y) \ \text{with} \ 14.\\ &&& 14 + y = 18, \ \text{and}\\ &&& y = 18 - 14, \ \text{or} \ 4 \ \text{pounds.}\\ &&& A: \text{Replace} \ y + y \ \text{with} \ 4 + 4, \ \text{or} \ 8.\\ &&& x + 8 = 14, \ \text{and}\\ &&& x = 14 - 8, \ \text{or} \ 6 \ \text{pounds}\\ &&& C: y + z + z = 14. \ \text{Replace} \ y \ \text{with} \ 4. \ \text{Then} \ 4 + z + z = 14.\\ &&& z + z = 14 - 4, \ \text{or} \ 10.\\ &&& z = 10 \div 2, \ \text{or} \ 5 \ \text{pounds.}\\ & \mathbf{Check} && \text{Replace each block with its weight. Check that the total equals the}\\ &&& \text{number of pounds shown on the scales.}\\ &&& A: 6 + 4 + 4 = 14; \ B: 6 + 4 + 4 + 4 = 18; \ C: 4 + 5 + 5 = 14.\end{align*}

Write equations. Figure out the weights of the blocks.

Write equations. Figure out the weights of the blocks.