# 6.1: Boxes and Boxes

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

Look at the pictures of the pan balances below. What do you see? If all boxes have whole number weights, can you figure out the weight of box m? Can you say anything about the weight of box r? In this concept, we will learn how to use problem solving steps to help us to describe the relative weights of objects given in pan balances.

### Guidance

In order to determine the relationship between the 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 the relative weights of the two boxes that have letters on them. - Third, make a
**plan**for how you will solve. Figure out what information each pan balance gives you. - Fourth,
**solve**the problem. - Last,
**check**your answer.

#### Example A

What could be the weights? Tell how you figured it out. All weights are whole numbers of pounds.

**Solution:**

We can use problem solving steps to help.

\begin{align*}& \mathbf{Describe} && \text{There are two pan balances,} \ \text{C} \ \text{and} \ \text{D}.\!\\ &&& \text{C}: \text{The pans are not level.}\!\\ &&& \text{D}: \text{The pans are level.}\!\\ &&& \text{There are two different boxes,} \ p \ \text{and} \ n. \ \text{There is an} \ 8 \ \text{pound box on} \ \text{D}.\\ & \mathbf{My \ Job} && \text{Figure out the weights of} \ p \ \text{and} \ n. \ \text{Decide if other weights are possible}.\!\\ & \mathbf{Plan} && \text{C}: \ 2 \ p \ \text{weighs less than} \ n.\!\\ &&& \text{D}: \text{The pan with} \ n \ \text{weighs the same as} \ 8 \ \text{pounds}.\!\\ &&& \qquad \text{Start with this fact}.\!\\ & \mathbf{Solve} && \text{C}: \ 2p < 8 \ \text{pounds so} \ p = 1, 2 \ \text{or} \ 3 \ \text{pounds}\!\\ &&& \text{D}: \ n = 8\\ & \mathbf{Check} && \text{C}: \ 1+1 <8; 2+2<8; 3+3<8\end{align*}

#### Example B

What could be the weights? Tell how you figured it out. All weights are whole numbers of pounds.

**Solution:**

We can use problem solving steps to help.

\begin{align*}& \mathbf{Describe} && \text{There are two pan balances,} \ \text{E} \ \text{and} \ \text{F}.\!\\ &&& \text{E}: \text{The pans are level.}\!\\ &&& \text{F}: \text{The pans are not level.}\!\\ &&& \text{There are two different boxes,} \ r \ \text{and} \ s. \ \text{There is a} \ 12 \ \text{pound box on} \ \text{F}.\\ & \mathbf{My \ Job} && \text{Figure out the weights of} \ r \ \text{and} \ s. \ \text{Decide if other weights are possible}.\!\\ & \mathbf{Plan} && \text{E}: \ 1 \ r \ \text{weighs the same as} \ 3 s.\!\\ &&& \text{F}: \text{The pan with} \ 4 s \ \text{weighs less than} \ 12 \ \text{pounds}.\!\\ &&& \qquad \text{Start with this fact}.\!\\ & \mathbf{Solve} && \text{F}: \ 4s < 12 \ \text{pounds so} \ s = 1 \ \text{or} \ 2 \ \text{pounds}\!\\ &&& \text{E}: \ \text{If} \ s=1, \ \text{then} \ r=3 \ \text{pounds. If} \ s=2, \ \text{then} \ r=6 \ \text{pounds.}\!\\ & \mathbf{Check} && \text{E}: \ 3=1+1+1 \ \text{or} \ 6=2+2+2\!\\ &&& \text{F}: \ 12>1+1+1+1 \ \text{or} \ 12>2+2+2+2\end{align*}

#### Example C

What could be the weights? Tell how you figured it out. All weights are whole numbers of pounds.

**Solution:**

We can use problem solving steps to help.

\begin{align*}& \mathbf{Describe} && \text{There are two pan balances,} \ \text{G} \ \text{and} \ \text{H}.\!\\ &&& \text{G}: \text{The pans are level.}\!\\ &&& \text{H}: \text{The pans are not level.}\!\\ &&& \text{There are two different boxes,} \ q \ \text{and} \ t. \ \text{There is a} \ 6 \ \text{pound box on} \ \text{G}.\\ & \mathbf{My \ Job} && \text{Figure out the weights of} \ q \ \text{and} \ t. \ \text{Decide if other weights are possible}.\!\\ & \mathbf{Plan} && \text{G}: \ 3 \ q \ \text{weighs the same as} \ 6 \ \text{pounds}.\!\\ &&& \text{H}: \text{The pan with} \ 2t \ \text{weighs the same as} \ 4q.\!\\ &&& \qquad \text{Start with the first fact}.\!\\ & \mathbf{Solve} && \text{G}: \ 3q=6 \ \text{pounds so} \ q=2 \ \text{pounds}\!\\ &&& \text{H}: \ 2t=4q \ \text{so} \ 2t=4(2)=8 \ \text{pounds. Therefore,} \ t=4 \ \text{pounds.}\\ & \mathbf{Check} && \text{G}: \ 2+2+2=6\\ &&& \text{H}: \ 4+4=2+2+2+2\end{align*}

#### Concept Problem Revisited

We can use problem solving steps to help.

\begin{align*}& \mathbf{Describe} && \text{There are two pan balances,} \ \text{A} \ \text{and} \ \text{B}.\!\\ &&& \text{A}: \text{The pans are level.}\!\\ &&& \text{B}: \text{The pans are not level.}\!\\ &&& \text{There are two different boxes,} \ m \ \text{and} \ r. \ \text{There is a} \ 6 \ \text{pound box on} \ \text{B}.\\ & \mathbf{My \ Job} && \text{Figure out the weights of} \ m \ \text{and} \ r. \ \text{Decide if other weights are possible}.\!\\ & \mathbf{Plan} && \text{A}: \ 3 \ r \ \text{weighs the same as} \ m.\!\\ &&& \text{B}: \text{The pan with} \ m \ \text{weighs less than} \ 6 \ \text{pounds}.\!\\ &&& \qquad \text{Start with this fact}.\!\\ & \mathbf{Solve} && \text{B}: \ m < 6 \ \text{pounds so} \ m = 1, 2, 3, 4, \ \text{or} \ 5 \ \text{pounds}\!\\ &&& \text{A}: \ \text{Since} \ m = 3r, \ \text{then} \ m \ \text{is a multiple of} \ 3. \ \text{So} \ m = 3 \ \text{and} \ r = 1.\!\\ & \mathbf{Check} && \text{A}: \ 3 = 1 + 1 + 1\!\\ &&& \text{B}: \ 6 > 3\end{align*}

### Vocabulary

To be ** equal** means to be the same. When pans are

**then the weights of the two pans are**

*balanced***. To be**

*equal***greater than**means to be bigger. To be

**means to be smaller. When two pans are not**

*less than***then the weight of one pan is is**

*balanced***the weight of the other pan.**

*greater than*### Guided Practice

For each problem, what could be the weights of the two lettered boxes? Tell how you figured it out. All weights are whole numbers of pounds.

1.

2.

3.

**Answers:**

1. From J, \begin{align*}2v=12\end{align*}, so \begin{align*}v=6\end{align*} pounds. From K, since \begin{align*}v\end{align*} is heavier than \begin{align*}3u,\end{align*} \begin{align*}3u<6,\end{align*} and \begin{align*}u<2.\end{align*} So, \begin{align*}u=1\end{align*} pound.

2. From L, \begin{align*}3x<9\end{align*} and \begin{align*} x<3 \end{align*}. So \begin{align*}x=1\end{align*} or \begin{align*}2\end{align*} pounds. From M, \begin{align*}w=2x.\end{align*} Since \begin{align*}x=1\end{align*} or \begin{align*}2,\end{align*} \begin{align*}w=2\end{align*} or \begin{align*}4\end{align*} pounds.

3. From N, \begin{align*}3y=15,\end{align*} so \begin{align*}y=5\end{align*} pounds. From P, \begin{align*}y>2z,\end{align*} so \begin{align*} 6>2z. \end{align*} Then \begin{align*}z=1\end{align*} or \begin{align*}2\end{align*} pounds.

### Practice

For each problem, what could be the weights of the two lettered boxes? Tell how you figured it out. All weights are whole numbers of pounds.

### Image Attributions

Students determine the relative weights of objects from the relationships displayed in pan balances. Students use problem solving steps to help them.