Lesson 3

Lesson 3: Solving Rate Problems

Focus

Have you ever had to exchange your money for a different currency? Maybe you have travelled to another country and needed to buy some of that country’s currency? Perhaps you have relatives in another country and they have sent you some of their money? Maybe you had to exchange Canadian dollars for currency in peso, pound, euro, zloty, franc, real, lev, yuan, colon, birr, gourde, rupee, shilling, or yen.

You have probably heard news anchors on TV mention how the Canadian dollar rose or fell against the U.S. dollar or another nation’s currency on any given day. As you saw in Lesson 1, the value of a nation’s currency is expressed as a rate that compares it to another nation’s currency. For example, the following rates show how much one Canadian dollar was worth on a given day compared to each of the following currencies:

• 1:00 Cdn : 1.004 17 U.S.
• 1.00 Cdn : 0.628 190 GBP
• 1.00 Cdn : 12.1396 MXN
• 1.00 Cdn : 45.5222 INR

So, on this particular day, one Canadian dollar is worth 1.004 17 U.S. dollars, 0.628 190 Great Britain pounds, 12.1396 Mexican pesos, and 45.5222 Indian rupees. Exchange rates are constantly changing as they are determined by the market.

When going on a vacation or moving to a different country, having that nation’s currency is important. People also take the exchange rate into account when they are making purchases in another country. If you need to transfer a large amount of funds from one currency to another, the exchange rate is particularly important. A small change in the exchange rate can have a big influence on the total cost of exchanging a large amount of money.

This lesson will help you answer the following inquiry question:

• How can different strategies be used to analyze and solve problems that involve rates?

Assessment

• Lesson 3 Assignment
• Module 6 Project: Part 1

All assessment items you encounter need to be placed in your course folder.

Save a copy of the Lesson 3 Assignment to your course folder. You will receive more information about how to complete the assignment later in this lesson.

In this lesson you will also be directed to work on Part 1 of the Module 6 Project. You will find a link later in this lesson that will take you to the Module 6 Project.

Materials and Equipment

• graphing calculator

Explore

Currency is not the only unit that might be different in another country. For instance, some countries use different systems of measurement. When making a purchase of an item in a different country, you may have to apply multiple rates to make an accurate comparison.

The strategy that Jeff used to compare the cost of filling up his truck in Canada versus filling his truck up in the United States is similar to the unit rate multiplication strategy you used in previous lessons to solve problems. The difference was that, in Jeff’s solution, multiple rates needed to be used. To solve the problem, Jeff had to convert the volume of the gas tank to the same unit of measurement and the cost of gas to the same currency.

Self-Check 1

A lot of the problems you have solved in this module have required you to use rates. For instance, you compared rates to find the best buy or to do conversions. Sometimes you need to use rates to solve for an unknown.

Functions can also be written to help solve problems involving rates. Linear functions represent situations where a rate of change is constant.

Consider the following situation.

Rajesh’s sister, Akuti, is preparing for the move by finding a new cellphone plan. She intends to keep in contact with her friends through texting. She has two choices for an add-on texting plan: a pay-per-use plan and a texting bundle. Both plans only charge for outgoing text messages. Incoming texts are free. You can view Akuti’s Options to see the two plans.

Akuti could also have used a graph to determine which plan was the better deal. Recall that linear functions can be entered into your graphing calculator or into a spreadsheet program. Begin by entering the two equations that Akuti developed to represent the two plans.

Since Akuti has a limit on what she can spend on the texting plan, you will also enter a third equation to represent that limit: Y3 = 40.

Before pressing the “Graph” button, you need to ensure that your window settings are set up correctly. This ensures that all three lines will show up on the screen.

• The domain of the graphs—represented by t in your equation—has to be greater than zero. (You can’t send a negative number of text messages.)
  
  
• The range of the graphs, which represents the total cost of the plan, has to be greater than zero.

A potential window setting is shown for a maximum of 800 text messages sent and a maximum cost of $100.

Now you can press the “Graph” button to show the graphs of the three lines.

From the previous graph you can determine which plan allows the most text messages to be sent for $40. The values on the x-axis (the horizontal axis) represent the number of text messages sent. The graph that intersects the horizontal line, Y3 (total cost of $40) at a higher value on the x-axis (further to the right), shows the plan that is the better buy. This plan allows the most text messages for $40.

You can see from the calculations Akuti made, and from the graph, that the text-bundle plan is the better deal for Akuti. To determine how many more text messages can be sent, you can use the intersect feature on your graphing calculator.

The text-bundle plan allows for 100 more text messages to be sent for $40 than the pay-per-use plan.

Try This 1

Using your graphing calculator, determine the maximum number of texts that Akuti could send in order for the pay-per-use plan to be the less expensive option. Note that, in this case, Akuti doesn’t have a $40-a-month budget.

You will begin with the same two equations since the cellphone plans are the same, but there is not a limit on what she will spend. The key to this question is to use your graph to find the point at which the pay-per-use plan costs less. Answer

Share 1

Working with a partner, discuss what other factors Akuti would need to consider before deciding on a texting plan.

Connect

Lesson Assignment

Lesson 3 Summary

As you have seen in this lesson, there are a variety of strategies you can use to solve problems involving multiple rates. These same strategies work when using rates to solve for an unknown. You might

• use unit analysis
• estimate using proportional reasoning
• use equivalent ratios
• use unit rates
• use a function
• use a graph

Depending on the situation, choose the strategy that will work the best for you.

You may use these math skills many times in daily life.

• Being able to convert between the imperial and metric systems is a critical skill as Canadian society has not fully changed to the metric system.
  
  
• Since Canada shares close ties with the Unites States, currency conversion may be needed if you purchase something from the U.S. over the Internet or plan to make a trip there.
  
  
• Many people have cellphones, so analyzing phone plans to save money is important.

Whether you are watching television, surfing the Internet, or shopping for groceries, sound math skills help you make wise consumer decisions.