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# 3.1: Balance the Pans

Difficulty Level: At Grade Created by: CK-12

You have lots of these boxes. If you wanted to make the pans balance, which boxes will you use? Can you solve this problem in more than one way?

### Balance the Pans

In order to balance pans like the one above, use the problem solving steps to help you.

• First, describe what you see and what information you are given.
• Next, identify what your job is and what you are trying to solve. In all of these problems, your job will be to figure out which boxes to put in the pan. You want to make sure that the weight of each pan is the same.
• Third, make a plan for how you will solve.
• Fourth, solve the problem.
• Last, check your answer by making sure that the combinations of boxes you found add up to the amount that you needed.

For the problem above, since the right side weighs 9 pounds, you want to make sure the left side will also weigh 9 pounds.

1. You have lots of these boxes.

Put boxes in the empty pan.

Make the pans balance.

Which boxes will you use?

Can you do it another way?

We will use the problem solving steps to solve this problem.

\begin{align*}& \mathbf{Describe:} && \text{There are 2 pans.}\\ &&& \text{One pan holds a box. The box weighs 8 pounds.}\\ &&& \text{The other pan is empty.}\\ &&& \text{Box}\ A \ \text{weighs 4 pounds.}\\ &&& \text{Box}\ B \ \text{weighs 2 pounds.}\\ &&& \text{There are lots of boxes.}\\ & \mathbf{My \ Job:} && \text{Figure out which boxes to put in the pan.}\\ &&& \text{The boxes have to weigh 8 pounds in all.}\\ & \mathbf{Plan:} && \text{Try to make 8 pounds using}\ \text{A} \ \text{boxes only.}\\ &&& \text{Try using}\ B \ \text{boxes only.}\\ &&& \text{Try using both}\ A \ \text{and}\ B \ \text{boxes.}\\ &\mathbf{Solve:} && \text{These will make the pans balance:}\\ &&& 2 \ A \ \text{boxes weigh 8 pounds.}\\ &&& 4 \ B \ \text{boxes weigh 8 pounds.}\\ &&& \text{One}\ A \ \text{box and}\ 2 \ B \ \text{boxes weigh 8 pounds.}\\ & \mathbf{Check:} && 4 + 4 = 8\\ &&& 2 + 2 + 2 + 2 = 8\\ &&& 4 + 2 + 2 = 8\end{align*}

2. You have lots of these boxes.

Put boxes in the empty pan.

Make the pans balance.

Which boxes will you use?

Can you do it another way?

We will use the problem solving steps to solve this problem.

\begin{align*}& \mathbf{Describe:} && \text{There are 2 pans.}\\ &&& \text{One pan holds a box. The box weighs 6 pounds.}\\ &&& \text{The other pan is empty.}\\ &&& \text{Box}\ C \ \text{weighs 3 pounds.}\\ &&& \text{Box}\ D \ \text{weighs 1 pound.}\\ &&& \text{There are lots of boxes.}\\ & \mathbf{My \ Job:} && \text{Figure out which boxes to put in the pan.}\\ &&& \text{The boxes have to weigh 6 pounds in all.}\\ & \mathbf{Plan:} && \text{Try to make 6 pounds using}\ \text{C} \ \text{boxes only.}\\ &&& \text{Try using}\ D \ \text{boxes only.}\\ &&& \text{Try using both}\ C \ \text{and}\ D \ \text{boxes.}\\ &\mathbf{Solve:} && \text{These will make the pans balance:}\\ &&& 2 \ C \ \text{boxes weigh 6 pounds.}\\ &&& 6 \ D \ \text{boxes weigh 6 pounds.}\\ &&& \text{One}\ C \ \text{box and}\ 3 \ D \ \text{boxes weigh 6 pounds.}\\ & \mathbf{Check:} && 3 + 3 = 6\\ &&& 1 + 1 + 1 + 1 + 1 + 1 = 6\\ &&& 3 + 1 + 1 + 1 = 6\end{align*}

#### Earlier Problem Revisited

We can use the problem solving steps to help us solve this problem.

\begin{align*}& \mathbf{Describe:} && \text{There are 2 pans.}\\ &&& \text{One pan holds a box. The box weighs 9 pounds.}\\ &&& \text{The other pan is empty.}\\ &&& \text{Box}\ E \ \text{weighs 2 pounds.}\\ &&& \text{Box}\ F \ \text{weighs 3 pounds.}\\ &&& \text{There are lots of boxes.}\\ & \mathbf{My \ Job:} && \text{Figure out which boxes to put in the pan.}\\ &&& \text{The boxes have to weigh 9 pounds in all.}\\ & \mathbf{Plan:} && \text{Try to make 9 pounds using}\ \text{E} \ \text{boxes only.}\\ &&& \text{Try using}\ F \ \text{boxes only.}\\ &&& \text{Try using both}\ E \ \text{and}\ F \ \text{boxes.}\\ &\mathbf{Solve:} && \text{These will make the pans balance:}\\ &&& \text{No number of E boxes weigh 9 pounds.}\\ &&& 3 \ F \ \text{boxes weigh 9 pounds.}\\ &&& \text{One}\ F \ \text{box and}\ 3 \ E \ \text{boxes weigh 9 pounds.}\\ & \mathbf{Check:} && 3 + 3 +3 = 9\\ &&& 3 + 2 + 2 + 2 = 9\end{align*}

### Vocabulary

To be equal means to be the same. In this concept, we are trying to make weights equal. This means we are trying to make each side of the pan balance have the same weight.

### Examples

For each problem below, use the problem solving steps. Your job is to put boxes in the empty pan to make the pans balance. Can you solve the problem in more than one way?

#### Example 1

\begin{align*}& \mathbf{Describe:} && \text{There are 2 pans.}\\ &&& \text{One pan holds a box. The box weighs 5 pounds.}\\ &&& \text{The other pan is empty.}\\ &&& \text{Box}\ G \ \text{weighs 1 pounds.}\\ &&& \text{Box}\ H \ \text{weighs 2 pounds.}\\ &&& \text{There are lots of boxes.}\\ & \mathbf{My \ Job:} && \text{Figure out which boxes to put in the pan.}\\ &&& \text{The boxes have to weigh 5 pounds in all.}\\ & \mathbf{Plan:} && \text{Try to make 5 pounds using}\ \text{G} \ \text{boxes only.}\\ &&& \text{Try using}\ H \ \text{boxes only.}\\ &&& \text{Try using both}\ G \ \text{and}\ H \ \text{boxes.}\\ &\mathbf{Solve:} && \text{These will make the pans balance:}\\ &&& 5 \ G \ \text{boxes weigh 5 pounds.}\\ &&& \text{no number of H boxes weigh 5 pounds.}\\ &&& \text{3}\ G \ \text{boxes and}\ 1 \ H \ \text{box weighs 5 pounds.}\\ &&& \text{1}\ G \ \text{box and}\ 2 \ H \ \text{boxes weighs 5 pounds.}\\ & \mathbf{Check:} && 1 + 1 + 1 + 1 + 1 = 5\\ &&& 1 + 2 + 2 = 5\\ &&& 1 + 1 + 1 + 2 = 5\end{align*}

#### Example 2

\begin{align*}& \mathbf{Describe:} && \text{There are 2 pans.}\\ &&& \text{One pan holds a box. The box weighs 10 pounds.}\\ &&& \text{The other pan is empty.}\\ &&& \text{Box}\ I \ \text{weighs 4 pounds.}\\ &&& \text{Box}\ J \ \text{weighs 2 pounds.}\\ &&& \text{There are lots of boxes.}\\ & \mathbf{My \ Job:} && \text{Figure out which boxes to put in the pan.}\\ &&& \text{The boxes have to weigh 10 pounds in all.}\\ & \mathbf{Plan:} && \text{Try to make 10 pounds using}\ \text{I} \ \text{boxes only.}\\ &&& \text{Try using}\ J \ \text{boxes only.}\\ &&& \text{Try using both}\ I \ \text{and}\ J \ \text{boxes.}\\ &\mathbf{Solve:} && \text{These will make the pans balance:}\\ &&& \text{no number of I boxes boxes weigh 10 pounds.}\\ &&& 5 \ J \ \text{boxes weigh 10 pounds.}\\ &&& \text{One}\ I \ \text{box and}\ 3 \ J \ \text{boxes weigh 10 pounds.}\\ &&& \text{Two}\ I \ \text{boxes and}\ 1 \ J \ \text{box weigh 10 pounds.}\\ & \mathbf{Check:} && 2 + 2 + 2 + 2 + 2 = 10\\ &&& 4 + 2 + 2 + 2 = 10\\ &&& 4 + 4 + 2 = 10\end{align*}

#### Example 2

\begin{align*}& \mathbf{Describe:} && \text{There are 2 pans.}\\ &&& \text{One pan holds a box. The box weighs 15 pounds.}\\ &&& \text{The other pan is empty.}\\ &&& \text{Box}\ P \ \text{weighs 3 pounds.}\\ &&& \text{Box}\ Q \ \text{weighs 6 pounds.}\\ &&& \text{There are lots of boxes.}\\ & \mathbf{My \ Job:} && \text{Figure out which boxes to put in the pan.}\\ &&& \text{The boxes have to weigh 15 pounds in all.}\\ & \mathbf{Plan:} && \text{Try to make 15 pounds using}\ \text{P} \ \text{boxes only.}\\ &&& \text{Try using}\ Q \ \text{boxes only.}\\ &&& \text{Try using both}\ P \ \text{and}\ Q \ \text{boxes.}\\ &\mathbf{Solve:} && \text{These will make the pans balance:}\\ &&& \text{no number of Q boxes boxes weigh 15 pounds.}\\ &&& 5 \ P \ \text{boxes weigh 15 pounds.}\\ &&& \text{1}\ P \ \text{box and}\ 2 \ Q \ \text{boxes weigh 15 pounds.}\\ &&& \text{3}\ P \ \text{boxes and}\ 1 \ Q \ \text{box weigh 15 pounds.}\\ & \mathbf{Check:} && 3 + 3 + 3 + 3 + 3= 15\\ &&& 3 + 6 + 6 = 15\\ &&& 3 + 3 + 3 + 6 = 15\end{align*}

### Review

For each problem below, use the problem solving steps. Your job is to put boxes in the empty pan to make the pans balance. Can you solve the problem in more than one way?

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Date Created:
Jan 18, 2013