1. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Section 4.1 Rates and Unit Rates

2. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Objectives o Write rates as fractions o Solve proportional equations o Calculate unit rates

3. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Rate A rate is a fraction used to compare two quantities that are not necessarily in the same units.

4. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 1: Writing Rates Suppose you and your friends are deciding where to spend Friday night. Laser tag offers the following deals: $13.00 per person for 2 games or $18.00 per person for 3 games. However, the county fair is offering $20.00 wristbands allowing all ‑ you ‑ can ‑ ride access for the night. Since both options sound appealing, you let your friends decide where to spend the evening. To make the decision easier, you would like to present them with rates for the evening’s choices based on time.

5. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 1: Writing Rates (cont.) Write down a rate for each choice based on time if it takes 30 minutes to play a game of laser tag or if you could stay at the fair from 8 p.m. until 10 p.m. Solution Since we are asked to write rates based on time, we will write each option as a fraction with price in the numerator of the fraction and time in the denominator.

6. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 1: Writing Rates (cont.) Consider the laser tag option first. If it takes 30 minutes to play one game of laser tag, the denominator is calculated by multiplying the number of games played by 30.

7. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 1: Writing Rates (cont.) For the fair, the rate will be based on staying at the fair from 8 p.m. until 10 p.m., which is 2 hours of unlimited rides.

8. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 2: Identifying Rates Foods in Bulk offers 30 pounds of candy for $64.49 on its website. Which of the following fractions represents the rate given by their website?

9. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 2: Identifying Rates (cont.) Solution There are actually two correct ways to write the rate from the website: answers b. and d. You can either write the fraction in pounds of candy per price in dollars like answer b., or price in dollars per pounds of candy like answer d. Both a. and c. are incorrect because neither compares the amount of candy to the price of the candy.

10. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 3: Working with Miles per Gallon As a college graduation present, Katie’s parents helped her buy her first new vehicle. She chose one that claims a fuel efficiency of 29 miles per gallon (mpg) on the highway. Check the actual mpg on Katie’s new car if she drove 309 miles on 11 gallons of gas.

11. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 3: Working with Miles per Gallon (cont.) Solution Begin by setting up a ratio of the actual miles Katie drove on 11 gallons of gas. We’ll put the gallons in the denominator so that we can have a comparison of the actual mpg to the manufacturer’s mpg.

12. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 3: Working with Miles per Gallon (cont.) The question asks how many miles did she drive on 1 gallon? We can simply divide here to find that, as follows. Whether or not Katie drove all 309 miles on the highway, she still managed to get close to the manufacturer’s mpg rate of 29 mpg.

13. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Skill Check #1 If rice costs $7.99 for 5 pounds, determine the cost per pound of the rice. Answer: $1.598 per pound

14. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Proportional Equation A proportional equation consists of two rates set equal to one another.

15. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 4: Using Proportional Equations How much gasoline should Katie have used after driving 243 miles in her new car that boasts 29 mpg? Set up and use a proportional equation to find the answer. Solution Begin by setting up the miles per gallon rate that was claimed by the manufacturer as a fraction. Remember to put miles in the numerator and gallons in the denominator.

16. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 4: Using Proportional Equations (cont.) We need to find an equivalent fraction that has 243 miles in the numerator rather than 29 miles. We can use a variable, such as x, to represent the number of gallons of gasoline used for 243 miles. That gives us the rate of

17. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 4: Using Proportional Equations (cont.) Now we can set the two rates equal to one another to form a proportional equation. Note that both rates are in the same form, with miles in the numerator and gallons in the denominator.

18. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 4: Using Proportional Equations (cont.) To solve a proportional equation like this, recall that we use algebra to isolate the unknown quantity x. In other words, we want to get x on one side of the equation by itself. Begin by multiplying both sides of the equation by each of the denominators in order to remove the fractions. In this case, multiply by 1x, or simply x.

19. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 4: Using Proportional Equations (cont.) So, Katie should have used approximately 8.38, or about gallons of gas after driving 243 miles.

20. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Unit Rate A unit rate is a rate comparing two measured quantities, one of which is a single unit written in the denominator of the fraction.

21. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 5: Finding Unit Rates Find the unit rate for each of the following types of wood mulch at the local garden center. a.Pine bark mulch: $3.77 for 3 cubic feet b.Aromatic cedar mulch: $3.98 for 2 cubic feet c.Evergreen red mulch: $3.33 for 2 cubic feet Solution To find the unit rates for each mulch, we first set up each rate as a fraction; that is,

22. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 5: Finding Unit Rates (cont.) a.Pine bark mulch: $3.77 for 3 cubic feet b.Aromatic cedar mulch: $3.98 for 2 cubic feet c.Evergreen red mulch: $3.33 for 2 cubic feet

23. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 5: Finding Unit Rates (cont.) Now that we have a rate for each type of mulch, we need to convert them to unit rates. To do this, we divide both parts of the fraction by the amount in the denominator. a.Pine bark mulch: or $1.26 per cubic foot

24. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 5: Finding Unit Rates (cont.) b.Aromatic cedar mulch: or $1.99 per cubic foot c.Evergreen red mulch: or $1.67 per cubic foot Once converted to unit rates, we can see that the pine bark mulch is the least expensive mulch in our list.

25. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Skill Check #2 Helium gas (He) was placed in a container fitted with a porous membrane. The helium effused through the membrane at the rate of 1.5 L/24 hr. Find the unit rate of effusion per hour for the helium. Answer: 0.0625 L/1 hr

26. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 6: Using Unit Rates Emma is traveling to Italy for spring break. She is preparing herself for all the shopping she has planned by looking at the exchange rate between dollars and euros. To ensure that she does not overspend while there, she wants to have in mind the equivalent dollar amounts for euro prices. Table 1 shows a section of the exchange rates that Emma found listed on the Internet.

27. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 6: Using Unit Rates (cont.) Table 1: Exchange Rates EURO vs.1 EURin EUR American Dollar1.3510.740192 Argentine Peso5.622510.177856 Brazilian Real2.25420.443617 British Pound0.84191.18779 Canadian Dollar1.33280.7503

28. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 6: Using Unit Rates (cont.) a.What is the unit exchange rate for American dollars to euros? b.What is the unit exchange rate for euros to American dollars? c.If Emma is considering taking $200 for spending money, approximately how many euros will she have to spend?

29. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 6: Using Unit Rates (cont.) d.Once in Italy, Emma falls in love with a leather bag priced at €200. She knows that it is over her budget, but wants to know by how much. Use the exchange rate to find out approximately how much the bag would cost in American dollars.

30. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 6: Using Unit Rates (cont.) Solution a.The format of the table is very similar to many others you will find online. Reading it correctly is half the battle. The column labeled “1 EUR” gives the amount of corresponding currency that would be equivalent to one euro. If we want a rate of American dollars to euros, the unit rate for euros should be in the denominator.

31. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 6: Using Unit Rates (cont.) Reading down the first column, we can see that there are 1.351 dollars for each euro. We have

32. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 6: Using Unit Rates (cont.) b.This time we need the rate for euros to American dollars, so the dollar amount will be in the denominator. The second column labeled “in EUR” provides the number of euros equivalent to one unit of currency listed. Therefore, we have

33. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 6: Using Unit Rates (cont.) c.To answer this question, we will set up a proportional equation to solve. We know that Emma has $200 and wants to know the equivalent amount of euros. Writing a rate with the unknown amount of euros as x gives us We can use the dollars/euros unit rate we found in part a. to set up a proportional equation.

34. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 6: Using Unit Rates (cont.) Solving for x, we have the following. So, Emma will have approximately €148 to spend on her trip to Italy.

35. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 6: Using Unit Rates (cont.) d.In this situation, we want to change from euros to dollars. We can set up a proportional equation as we did in part c. and solve it. This time, however, we will use the unit rate found in part b., euros per dollar, since we are given euros to begin with. The unit rate found in part b. is

36. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 6: Using Unit Rates (cont.) Setting up a proportional equation using €200 and x as the equivalent amount in dollars gives us Solving for x we have the following.

37. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 6: Using Unit Rates (cont.) This means that the leather bag costs $270.20 (in American dollars). Since her original budget was $200, buying this bag would put her approximately $70 over her budget.