2.3.3: Solving Problems about Proportional Relationships

Lesson

Let's solve problems about proportional relationships.

Exercise \(\PageIndex{1}\): What Do You Want to Know?

Consider the problem: A person is running a distance race at a constant rate. What time will they finish the race?

What information would you need to be able to solve the problem?

Exercise \(\PageIndex{2}\): Info Gap: Biking and Rain

Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.

If your teacher gives you the problem card:

1. Silently read your card and think about what information you need to be able to answer the question.
2. Ask your partner for the specific information that you need.
3. Explain how you are using the information to solve the problem.
   Continue to ask questions until you have enough information to solve the problem.
4. Share the problem card and solve the problem independently.
5. Read the data card and discuss your reasoning.

If your teacher gives you the data card:

1. Silently read your card.
2. Ask your partner “What specific information do you need?” and wait for them to ask for information.
   If your partner asks for information that is not on the card, do not do the calculations for them. Tell them you don’t have that information.
3. Before sharing the information, ask “Why do you need that information?” Listen to your partner’s reasoning and ask clarifying questions.
4. Read the problem card and solve the problem independently.
5. Share the data card and discuss your reasoning.

Pause here so your teacher can review your work. Ask your teacher for a new set of cards and repeat the activity, trading roles with your partner.

Exercise \(\PageIndex{3}\): Moderating Comments

A company is hiring people to read through all the comments posted on their website to make sure they are appropriate. Four people applied for the job and were given one day to show how quickly they could check comments.

• Person 1 worked for 210 minutes and checked a total of 50,000 comments.
• Person 2 worked for 200 minutes and checked 1,325 comments every 5 minutes.
• Person 3 worked for 120 minutes, at a rate represented by \(c=331t\),
  where is the number of comments checked and is the time in minutes.
• Person 4 worked for 150 minutes, at a rate represented by \(t=\left(\frac{3}{800}\right)c\).

1. Order the people from greatest to least in terms of total number of comments checked.
2. Order the people from greatest to least in terms of how fast they checked the comments.

Are you ready for more?

1. Write equations for each job applicant that allow you to easily decide who is working the fastest.
2. Make a table that allows you to easily compare how many comments the four job applicants can check.

Summary

Whenever we have a situation involving constant rates, we are likely to have a proportional relationship between quantities of interest.

• When a bird is flying at a constant speed, then there is a proportional relationship between the flying time and distance flown.
• If water is filling a tub at a constant rate, then there is a proportional relationship between the amount of water in the tub and the time the tub has been filling up.
• If an aardvark is eating termites at a constant rate, then there is proportional relationship between the number of termites the aardvark has eaten and the time since it started eating.

Sometimes we are presented with a situation, and it is not so clear whether a proportional relationship is a good model. How can we decide if a proportional relationship is a good representation of a particular situation?

• If you aren’t sure where to start, look at the quotients of corresponding values. If they are not always the same, then the relationship is definitely not a proportional relationship.
• If you can see that there is a single value that we always multiply one quantity by to get the other quantity, it is definitely a proportional relationship.

After establishing that it is a proportional relationship, setting up an equation is often the most efficient way to solve problems related to the situation.

Practice

Exercise \(\PageIndex{4}\)

For each situation, explain whether you think the relationship is proportional or not. Explain your reasoning.

1. The weight of a stack of standard 8.5x11 copier paper vs. number of sheets of paper.
2. The weight of a stack of different-sized books vs. the number of books in the stack.

Figure \(\PageIndex{1}\): Paper ream , by Bluesnap. Public Domain. Pixabay. Source.

Figure \(\PageIndex{2}\): Stack of books, by Hermann. Public Domain. Pixabay. Source.

Exercise \(\PageIndex{5}\)

Every package of a certain toy also includes 2 batteries.

1. Are the number of toys and number of batteries in a proportional relationship? If so, what are the two constants of proportionality? If not, explain your reasoning.
2. Use \(t\) for the number of toys and \(b\) for the number of batteries to write two equations relating the two variables.

\(b=\qquad\qquad t=\)

Exercise \(\PageIndex{6}\)

Lin and her brother were born on the same date in different years. Lin was 5 years old when her brother was 2.

1. Find their ages in different years by filling in the table.

   Table \(\PageIndex{1}\)

   Lin's age	Her brother's age
   \(5\)	\(2\)
   \(6\)
   \(15\)
   	\(25\)

2. Is there a proportional relationship between Lin's age and her brother's age? Explain your reasoning.

Exercise \(\PageIndex{7}\)

A student argues that \(y=\frac{x}{9}\) does not represent a proportional relationship between \(x\) and \(y\) because we need to multiply one variable by the same constant to get the other one and not divide it by a constant. Do you agree or disagree with this student?

(From Unit 2.3.2)

Exercise \(\PageIndex{8}\)

Quadrilateral A has side lengths 3, 4, 5, and 6. Quadrilateral B is a scaled copy of Quadrilateral A with a scale factor of 2. Select all of the following that are side lengths of Quadrilateral B.

1. \(5\)
2. \(6\)
3. \(7\)
4. \(8\)
5. \(9\)

(From Unit 1.1.3)