# 8.6: Hanging Scales

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

**Hanging Scales – Solve for Unknowns**

**Teacher Notes**

Each problem shows four scales, their contents, and their weights. Students use the data provided in the display as clues to write equations and determine the weight of each block. All scales contain more than one type of block, so weights of blocks cannot be found directly. In some cases, one scale has two sets of the same two blocks and another scale contains that set of blocks and a different block. Students figure out the value of one set of blocks, replace the set of blocks on the other scale with its value, and then determine the weight of the remaining block(s) on the scale.

In other cases, all blocks on one scale are a subset of blocks on a second scale. Students replace the set of blocks with its value in order to find the weight of the other block or blocks 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. \ A: w + x + y + z = 14; \ B: w + w + x = 10; \ C: w + x + y = 9; \ D: w + y + w + y = 10:\!\\
{\;} \quad w = 3 \ pounds, \ x = 4 \ pounds, \ y = 2 \ pounds, \ z = 5 \ pounds\!\\
2. \ A: w + x + y = 16; \ B: w + x + y + z = 24; \ C: w + x + w + x = 24; \ D: x + z + z = 21:\!\\
{\;} \quad w = 7 \ pounds, \ x = 5 \ pounds, \ y = 4 \ pounds, \ z = 8 \ pounds\!\\
3. \ A: y + z + y + z = 28; \ B: x + y + z = 21; \ C: w + w + y = 18; \ D: w + x + y + z = 26:\!\\
{\;} \quad w = 5 \ pounds, \ x = 7 \ pounds, \ y = 8 \ pounds, \ z = 6 \ pounds\!\\
4. \ A: x + x + z = 26; \ B: w + x + z = 23; \ C: x + z + x + z = 34; \ D: w + y + y = 20:\!\\
{\;} \quad w = 6 \ pounds, \ x = 9 \ pounds, \ y = 7 \ pounds, \ z = 8 \ pounds\!\\
5. \ A: y + x + y + x = 26; \ B: x + y + w = 17; \ C: w + x + x = 20; \ D: w + x + y + z = 26:\!\\
{\;} \quad w = 4 \ pounds, \ x = 8 \ pounds, \ y = 5 \ pounds, \ z = 9 \ pounds\!\\
6. \ A: w + y + z = 22; \ B: x + y + z + z = 25; \ C: x + y + z = 20; \ D: w + z + w + z = 26:\!\\
{\;} \quad w = 8 \ pounds, \ x = 6 \ pounds, \ y = 9 \ pounds, \ z = 5 \ pounds\end{align*}

\begin{align*}& \mathbf{Describe}&& \text{There are four scales with blocks}.\\ &&& A: \ y, x, y, x \ \text{blocks weigh} \ 28 \ \text{pounds}.\\ &&& B: \ x, y, z \ \text{blocks weigh} \ 23 \ \text{pounds}.\\ &&& C: \ w, x, y, z \ \text{blocks weigh} \ 28 \ \text{pounds}.\\ &&& D: \ x, x, z \ \text{blocks weigh} \ 25 \ \text{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: y+ x+ y+ x= 28; \ B: x+ y+ z= 23;\\ &&& C: w+ x+ y + z = 28; \ D: x+x+z=25.\\ &&& \text{Look for equations that are related}.\\ \\ &\mathbf{Solve}&& A: y+x+y+x=28. \ \text{There are two of each block, so} \ y+x=14\\ &&& B: x+ y+ z = 23. \ \text{Replace} \ x+ y \ \text{with}\ 14.\\ &&& \qquad 14 +z=23, \ \text{and}\\ &&& \qquad z =23 - 14, \ \text{or} \ 9 \ pounds.\\ &&& D: x+ x+ z = 25. \ \text{Replace} \ z \ \text{with} \ 9.\\ &&& \qquad x+x+9=25, \ 2x=25-9, \ \text{or} \ 16, \ \text{and}\\ &&& \qquad x =16 \div 2, \ \text{or}\ 8 \ pounds.\\ &&& A: x+ y= 14. \ \text{Replace} \ x \ \text{with}\ 8. \ \text{Then} \ 8+y=14.\\ &&& \qquad y=14-8, \ \text{or} \ 6 \ pounds.\\ &&& C: w+x+ y+ z = 28. \ \text{From} \ B, \ x+y+z=23. \ \text{Replace} \ x+ y+z \ \text{with} \ 23;\\ &&& \qquad w +23=28, \ \text{and} \ w=28-23, \ \text{or} \ 5 \ pounds.\\ \\ &\mathbf{Check} && \text{Replace each block with its weight. Check that the totals equal the number}\\ &&&\text{of pounds shown on the scales.}\\ &&& A: 6 + 8 + 6 + 8 =28; \ B: 8 + 6 + 9 =23; \ C: 5 + 8 + 6+9 =28;\\ &&& D: 8+8+9=25.\end{align*}

Write equations. Figure out the weights of the blocks. Same blocks weigh the same.

1.

\begin{align*}w = ? \ pounds, x = \underline{?} \ pounds, y = \underline{?} \ pounds, z = \underline{?} \ pounds\end{align*}

2.

\begin{align*}w = ? \ pounds, x = \underline{?} \ pounds, y = \underline{?} \ pounds, z = \underline{?} \ pounds\end{align*}

3.

\begin{align*}w = ? \ pounds, x = \underline{?} \ pounds, y = \underline{?} \ pounds, z = \underline{?} \ pounds\end{align*}

Write equations. Figure out the weights of the blocks.

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