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# 5.1: Jars and Jars

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Look at the pictures of the pan balances below. What do you see? Can you figure out the weight of jar z? Can you say anything about the weight of jar x? 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 jars that have letters of them.
• Third, make a plan for how you will solve. See if you can figure out the exact weight of one jar first. Then, describe what you know about the weight of the other jar.
• Fourth, solve the problem.

#### Example A

What is the weight of jar p? What could be the weight of jar m? Tell how you figured it out.

Solution:

We can use problem solving steps to help us.

$& \mathbf{Describe} && \text{I see two pan balances. I see two jars} \ m \ \text{and} \ p.\\ &&& \text{Pan Balance} \ D: \ \text{The pans are not balanced}.\\ &&& \text{Pan Balance} \ C: \ \text{The pans are balanced. One pan has a} \ 5 \ \text{pound jar}.\\& \mathbf{My \ Job} && \text{Figure out a weight for} \ p. \ \text{Then figure out possible weights for m.}\\& \mathbf{Plan} && \text{Pan Balance} \ D: \ \text{The pan with} \ m \ \text{is lower so,} \ m \ \text{is heavier than} \ p.\\&&& \text{Pan Balance} \ C: \text{Use the fact that} \ p \ \text{weighs} \ 5 \ \text{pounds}.\\& \mathbf{Solve} && \text{Pan Balance} \ C: \ p=5 \ \text{pounds.}\\&&& \text{Pan Balance} \ D: \ m \ \text{has to weigh more than} \ 5 \ \text{pounds}.\\& \mathbf{Check} && \text{Pan Balance} \ C: \ p=5 \ \text{pounds}.\\& \mathbf{Check} && \text{Pan Balance} \ A: \ m \ \text{is more than 5 pounds.}$

#### Example B

What is the weight of jar k? What could be the weight of jar j? Tell how you figured it out.

Solution: We can use problem solving steps to help us.

$& \mathbf{Describe} && \text{I see two pan balances. I see two jars} \ k \ \text{and} \ j.\\ &&& \text{Pan Balance} \ G: \ \text{The pans are not balanced}.\\ &&& \text{Pan Balance} \ H: \ \text{The pans are balanced. One pan has a} \ 6 \ \text{pound jar}.\\& \mathbf{My \ Job} && \text{Figure out a weight for} \ k. \ \text{Then figure out possible weights for } \ j.\\& \mathbf{Plan} && \text{Pan Balance} \ G: \ \text{The pan with} \ j \ \text{is lower, so} \ j \ \text{is heavier than} \ k.\\&&& \text{Pan Balance} \ H: \text{Use the fact that} \ k+k \ \text{weighs} \ 6 \ \text{pounds}.\\& \mathbf{Solve} && \text{Pan Balance} \ H: \ k+k=6 \ \text{pounds, so} \ k = 3 \ \text{pounds}.\\&&& \text{Pan Balance} \ G: \ j \ \text{has to weigh more than} \ 3 \ \text{pounds}.\\& \mathbf{Check} && \text{Pan Balance} \ H: \ 3+3=6 \ \text{pounds}.\\&&& \qquad \qquad \qquad \qquad \ k=3 \ \text{pounds}.\\& \mathbf{Check} && \text{Pan Balance} \ G: \ j \ \text{is more than 3 pounds.}$

#### Example C

What is the weight of jar y? What could be the weight of jar z? Tell how you figured it out.

Solution:

We can use problem solving steps to help us.

$& \mathbf{Describe} && \text{I see two pan balances. I see two jars} \ y \ \text{and} \ z.\\ &&& \text{Pan Balance} \ K: \ \text{The pans are not balanced}.\\ &&& \text{Pan Balance} \ J: \ \text{The pans are balanced. One pan has 2} \ 2 \ \text{pound jars}.\\& \mathbf{My \ Job} && \text{Figure out a weight for} \ y. \ \text{Then figure out possible weights for z.}\\& \mathbf{Plan} && \text{Pan Balance} \ K: \ \text{The pan with} \ y \ \text{is lower so,} \ y \ \text{is heavier than} \ z.\\&&& \text{Pan Balance} \ J: \text{Use the fact that} \ y \ \text{weigh} \ 2 + 2 =4 \ \text{pounds}.\\& \mathbf{Solve} && \text{Pan Balance} \ J: \ y = 4 \ \text{pounds.}\\&&& \text{Pan Balance} \ K: \ z \ \text{has to weigh less than} \ 4 \ \text{pounds}.\\& \mathbf{Check} && \text{Pan Balance} \ J: \ y=2+2=4 \ \text{pounds}.\\& \mathbf{Check} && \text{Pan Balance} \ K: \ z \ \text{is less than 4 pounds.}$

#### Concept Problem Revisited

We can use problem solving steps to help us.

$& \mathbf{Describe} && \text{I see two pan balances. I see two jars} \ x \ \text{and} \ z.\\ &&& \text{Pan Balance} \ A: \ \text{The pans are not balanced}.\\ &&& \text{Pan Balance} \ B: \ \text{The pans are balanced. One pan has a} \ 12 \ \text{pound jar}.\\& \mathbf{My \ Job} && \text{Figure out a weight for} \ z. \ \text{Then figure out possible weights for x.}\\& \mathbf{Plan} && \text{Pan Balance} \ A: \ \text{The pan with} \ z \ \text{is lower so,} \ z \ \text{is heavier than} \ x.\\&&& \text{Pan Balance} \ B: \text{Use the fact that} \ z+z+z \ \text{weighs} \ 12 \ \text{pounds}.\\& \mathbf{Solve} && \text{Pan Balance} \ B: \ z + z + z = 12 \ \text{pounds, so} \ z = 4 \ \text{pounds}.\\&&& \text{Pan Balance} \ A: \ x \ \text{has to weigh less than} \ 4 \ \text{pounds}.\\& \mathbf{Check} && \text{Pan Balance} \ B: \ 4 + 4 + 4 = 12 \ \text{pounds}.\\&&& \qquad \qquad \qquad \qquad \ z=4 \ \text{pounds}.\\& \mathbf{Check} && \text{Pan Balance} \ A: \ x \ \text{is less than 4 pounds.}$

### Vocabulary

To be equal means to be the same. When pans are balanced then the weights of the two pans are equal . To be greater than means to be bigger. To be less than means to be smaller. When two pans are not balanced then the weight of one pan is is greater than the weight of the other pan.

### Guided Practice

What could be the weight of the jars? Tell how you figured it out.

1.

2.

3.

1. x is 6 pounds; v is more than 3 pounds

2. w is 4 pounds; t is less than 8 pounds

3. a is 7 pounds; b is less than 7 pounds

### Practice

What could be the weight of the jars? Tell how you figured it out.

Jan 18, 2013

May 27, 2014