# 8.5: Hanging Scales

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

**Hanging Scales – Write Equations and 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, one of the scales has a double set of two blocks. Students figure out the value of a single set of these two blocks. A second scale contains a single set of the same two blocks and one other block. Students must 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 determined, 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 remaining 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 + x + z = 20; \ B: x + y + z = 15; \ C: x + y + y = 14\!\\ {\;} \ \quad x = 4, \ y = 5, \ z = 6\!\\ 2. \quad D: x + y + z = 19; \ E: y + z + y + z = 24; \ F: x + x + z = 24\!\\ {\;} \ \quad x= 7, \ y = 2, \ z = 10\!\\ 3. \quad G: x + y + z = 20; \ H: x + y + x + y = 26; \ I: z + z + y = 22\!\\ {\;} \ \quad x= 5, \ y = 8, \ z = 7\!\\ 4. \quad J: x + y + z = 22; \ K: x + x + y = 25; \ L: x + z + x+ z= 26\!\\ {\;} \ \quad x= 8, \ y = 9, \ z = 5\!\\ 5. \quad M: y + x + y + x = 32; \ N: z + z + x = 28; \ P: x+ y+ z = 25\!\\ {\;} \ \quad x= 10, \ y = 6, \ z = 9\!\\ 6. \quad Q: x + y + z = 28; \ R: y + z + y+ z= 34; \ S: x + x+ y= 32\!\\ {\;} \ \quad x= 11, \ y = 10, \ z = 7\end{align*}

**Hanging Scales – Write Equations and Solve for Unknowns**

\begin{align*}& \mathbf{Describe:} && \text{There are three scales with blocks.}\\ &&& \text{A: Two} \ x \ \text{and two} \ y \ \text{blocks. They weigh 26 pounds.}\\ &&& \text{B: One} \ x, \ \text{one} \ y, \ \text{and one} \ z \ \text{block. They weigh 22 pounds.}\\ &&& \text{C: One} \ x \ \text{and two} \ z \ \text{blocks. They weigh 24 pounds.}\\ & \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 + x + y = 26; \ B: x + y + z = 22; \ C:x + z + z = 24\\ &&& \text{Solve the equations.}\\ & \mathbf{Solve:} && A: x + y + x + y = 26. \ \text{There are two of each block, so} \ x + y = 13\\ &&& \text{B}: (x + y) + z = 22. \ \text{Replace} \ (x + y) \ \text{with} \ 13.\\ &&& \quad \quad 13 + z = 22, \ \text{and}\\ &&& \quad \quad z = 22 - 13, \ \text{or} \ 9 \ \text{pounds.}\\ &&& C: x + z + z = 24. \ \text{Replace each} \ z \ \text{with} \ 9.\\ &&& \quad \quad x + 18 = 24, \ \text{and}\\ &&& \quad \quad x = 24 - 18, \text{or} \ 6 \ \text{pounds}\\ &&& A: x + y = 13. \ \text{Replace} \ x \ \text{with} \ 6. \ \text{Then}\ 6 + y = 13.\\ &&& \quad \quad y = 13 - 6, \ \text{or} \ 7 \ \text{pounds.}\\ & \mathbf{Check:} && \text{Replace each block with its weight. Check that the total equal the number of}\\ &&& \text{pounds shown on the scales.} \\ &&& A: 6 + 7 + 6 + 7 = 26; \ B: 6 + 7 + 9 = 23; \ C: 6 + 9+ 9 = 24.\end{align*}

Write equations. Figure out the weights of the blocks.

*Extra for Experts:***Hanging Scales – Write Equations and Solve for Unknowns**

Write equations. Figure out the weights of the blocks.

**Solutions:**

\begin{align*}1. \quad A: x + x + y + y = 36; \ B: x + y + z = 27; \ C: x + z + z = 30\!\\ {\;} \ \quad x= 12, \ y = 6, \ z = 9\!\\ 2. \quad D: x + y + z = 24; \ E: z+ z + y + y= 32; \ F: x + x + y = 25\!\\ {\;} \ \quad x= 8, \ y = 9, \ z = 7\!\\ 3. \quad G: x + y + y= 29; \ H: x + y + z = 26; \ I: x+ x+ z + z = 32\!\\ {\;} \ \quad x= 9, \ y = 10, \ z = 7\!\\ 4. \quad A: x + z + z = 27; \ B: x + y + x + y = 36; \ C: x + y + z = 26\!\\ {\;} \ \quad x= 11, \ y = 7, \ z = 8\!\\ 5. \quad D: y + z + y + z = 28; \ E: x + y + z = 26; \ F: x + x + y = 32\!\\ {\;} \ \quad x= 12, \ y = 8, \ z = 6\!\\ 6. \quad G: x + y + z = 28; \ H: y + y + z = 25; \ I: x + x + z + z = 42\!\\ {\;} \ \quad x= 10, \ y = 7, \ z = 11\end{align*}

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