1. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Section 4.2 Ratios

2. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Objectives o Write and interpret ratios

3. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Ratio A ratio is a fraction used to compare two measured quantities whose units are the same type.

4. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Writing Ratios In mathematics there are three accepted ways to write ratios. written with words, “1 to 2” written with a colon, “1 : 2” written as a fraction, “ “

5. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 1: Writing Ratios On a university campus, the fall enrollment shows that the new freshman class consists of 321 women and 214 men. Write the ratio of men to women. Solution Since the ratio we are writing is men : women, the number of men must come first. So, the ratio of men to women on the campus is

6. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 1: Writing Ratios (cont.) Often, it is the case that ratios are written in a reduced form, which is usually the simplest form of the fraction. Since both numbers are divisible by 107 here, the ratio of men to women is equivalent to 2 : 3.

7. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Equivalent Ratios Equivalent ratios are ratios that express the same relationship.

8. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Equivalent Ratios In the previous example, the ratio 214 : 321 was shown to be equivalent to the reduced ratio 2 : 3 by dividing both numbers by 107. We could also say the ratio 214 : 321 is equivalent to 428 : 642 by multiplying both numbers by 2. In fact, every ratio has an unlimited number of equivalent ratios that can be found by using either multiplication or division.

9. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Skill Check #1 A local radio station says that out of every 10 listeners, 6 are women and 4 are men. Write the ratio of women to men listeners for the radio station in lowest terms. Answer: 3 : 2

10. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 2: Writing Ratios According to Fortune’s 2011 list of the top “40 Under 40,” the up-and ‑ coming top 40 adults in America includes 15 women. 1 a.Write the ratio of the number of women on the list to the total number of adults on the list in all three notations. b.Write the ratio of women to men on the list. 1 Fortune, http://www.fortune.com

11. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 2: Writing Ratios (cont.) Solution a.We were asked to write the ratio of women : total number of adults, so we write the number of women first. We were also asked to provide the ratio in all three notations. They are the following. with words: 15 to 40 with a colon: 15 : 40 with a fraction: All three are read the same way, “fifteen to forty.”

12. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 2: Writing Ratios (cont.) The ratios tell us that 15 out of every 40 adults on the list are women. We can write an equivalent ratio by dividing each number by 5. The simplified ratio is then 3 to 8, which tells us that 3 out of every 8 adults on the list are women.

13. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 2: Writing Ratios (cont.) b.To calculate the ratio of women to men, we need to know the number of men and the number of women on the list. The study states that there were a total of 40 people on the list, of which 15 were women. This means that 25 of them were men. Remember that, since we were asked to write a ratio of women : men, we write the number of women first, followed by the number of men.

14. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 2: Writing Ratios (cont.) with words: 15 to 25 with a colon: 15 : 25 with a fraction: Notice that the total number of people on the list (40) does not appear in the ratio of women to men at all. In this example, both 15 and 25 are divisible by 5, so we can once again simplify our ratio to an equivalent one by dividing both numbers by 5. The simplified ratio of women to men is 3 : 5.

15. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 3: Writing and Interpreting Ratios Table 1 shows a study by an online health journal claiming that, although fish has long been a food of choice for helping to control heart disease, farm-raised tilapia might actually be dangerous to the heart. Omega fatty acid ratios in farm-raised tilapia actually give undesirable amounts of omega-6 acids. In farm- raised tilapia, the ratio of these potentially detrimental long-chain omega ‑ 6 fatty acids to the beneficial long- chain omega-3 fatty acids averaged about 11 : 1.

16. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 3: Writing and Interpreting Ratios (cont.) Table 1 shows a sample of the number of fatty acids in a tablespoon of different types of fish oil. Write a ratio for each so that they can be compared to the given ratio for farm ‑ raised tilapia.

17. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 3: Writing and Interpreting Ratios (cont.) Table 1: Number of Omega Fatty Acids in a Tablespoon of Fish Oil Fish Oil Long-Chain Omega-6 Fatty Acids Long-Chain Omega-3 Fatty Acids Herring391509 Salmon924657 Sardine2393096 Cod Liver1272557 Menhaden1593624 Source: EFA Education. “Essential Fats in Food Oils.” http://efaeducation.org/essential.html

18. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 3: Writing and Interpreting Ratios (cont.) Solution In the information given, we were told that the ratio 11 : 1 for farm-raised tilapia was the ratio of long-chain omega-6 fatty acids to that of long-chain omega-3 fatty acids. So, for comparison we need to list the omega-6 fatty acids first in each of the new ratios. For each ratio, all we need to do is write the numbers in the order they appear in the table, since the omega-6 values are listed first.

19. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 3: Writing and Interpreting Ratios (cont.) Table 2: Ratio of Omega Fatty Acids in Fish Oil Fish Oil Long-Chain Omega- 6 Fatty Acids Long-Chain Omega- 3 Fatty Acids Ratio Herring39150939:1509 Salmon92465792:4657 Sardine2393096239:3096 Cod Liver1272557127:2557 Menhaden1593624159:3624

20. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 3: Writing and Interpreting Ratios (cont.) Now that we’ve written the ratios, to be able to compare them to the ratio of 11 : 1 for farm ‑ raised tilapia, we need to reduce them all to a common form. Since the farm-raised tilapia has a “1” on the right side of the ratio, we’ll divide each ratio by the right-hand number so that it also will have a 1. You can think of this as being in the same vein as converting to a unit rate.

21. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 3: Writing and Interpreting Ratios (cont.) Table 3: Unit Ratio of Omega Fatty Acids in Fish Oil Fish OilRatioUnit Ratio Herring39:1509(39/1509):(1509/1509)0.0258:1 Salmon92:4657(92/4657):(4657/4657)0.0198:1 Sardine239:3096(239/3096):(3096/3096)0.0772:1 Cod Liver127:2557(127/2557):(2557/2557)0.0497:1 Menhaden159:3624(159/3624):(3624/3624)0.0439:1 Tilapia 11:1

22. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 3: Writing and Interpreting Ratios (cont.) Now that all the ratios are in a common form, we can see that there is at least a hundred-fold difference between the fatty acid ratios of tilapia and the other fish.

23. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 4: Using Ratios for Scaling Using the given map, answer the following questions. a.If the distance on the map between point A and point B measures 3 inches, what is the actual distance between these points? b.If you wanted to take an 11-mile walk one day, what distance would that be on the map, in inches?

24. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 4: Using Ratios for Scaling (cont.) Solution a.Because the distance on the map is 3 inches, and the scale of the map is 1 inch = 5 miles, we can calculate the actual distance x between points A and B by setting up a proportional equation. In the equation, we'll put inches in the numerator and miles in the denominator of each ratio. Note that it makes no difference which unit we choose to place in the numerator.

25. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 4: Using Ratios for Scaling (cont.) We can now solve the equation for x by multiplying each side of the equation as shown. Therefore, the actual distance between the points is 15 miles.

26. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 4: Using Ratios for Scaling (cont.) b.Again, we use the scale of 1 inch = 5 miles to find the distance on the map. The proportional equation will have the variable in the numerator this time since we are looking to find the distance in inches the map would show.

27. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 4: Using Ratios for Scaling (cont.) Thus, the distance of your walk would measure 2.2 inches on the map.

28. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 5: Scaling Ratios Suppose we know that the distance from Chicago, Illinois, to Albany, New York, is 816 miles. On the map, the distance from Chicago to New York is 1.4 inches. Use this ratio to determine how far it is from Chicago to St. Louis, Missouri, if the distance on the map between these two cities is 0.5 inches.

29. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 5: Scaling Ratios (cont.) Solution We can use the distances given in the scenario to help us set up ratios in an equation that we can solve. First of all we are told that the actual distance between Chicago and Albany is 816 miles and that the map distance is 1.4 inches. So the ratio for Chicago to Albany is

30. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 5: Scaling Ratios (cont.) We can then fill in the ratio for Chicago to St. Louis based on what we know.

31. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 5: Scaling Ratios (cont.) Notice we were careful to put the actual distances between cities in the numerator of each ratio while the map distances are in the denominator. We chose this orientation so that the distance we are looking to find is once again in the numerator. The proportional equation is then

32. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. We can solve this proportional equation as follows. Therefore, the distance from Chicago to St. Louis is approximately 291 miles. Example 5: Scaling Ratios (cont.)

33. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 6: Using Multiple Ratios Raw mixed nuts make a healthy and delicious snack. An online store promises raw cashews, raw almonds, and raw Brazil nuts in a mixed bag of nuts. They go even further to say that the ratio of cashews, almonds, and Brazil nuts is 2 : 3 : 5 in every bag. Suppose that there are 5 ounces of cashews in a bag. How many ounces of almonds and Brazil nuts will the bag have?

34. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 6: Using Multiple Ratios (cont.) Solution We are told that the ratio 2 : 3 : 5 is that of cashews : almonds : Brazil nuts, and that there are 5 ounces of cashews. Using the first part of the ratio, cashews : almonds, we can find the amount of almonds that will be in the bag.

35. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 6: Using Multiple Ratios (cont.) Setting up a proportional equation using the variable x for the unknown amount of almonds gives us the following proportional equation to solve.

36. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Using algebra, we get x on one side of the equation by itself as follows. So, there are 7.5 ounces of almonds in the bag. Example 6: Using Multiple Ratios (cont.)

37. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 6: Using Multiple Ratios (cont.) Now, we can use the cashews : Brazil nuts ratio of 2 : 5 to find the amount of Brazil nuts in the bag. Use the same method to find x.

38. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 6: Using Multiple Ratios (cont.) So, there are 12.5 ounces of Brazil nuts in the bag. In fact, we now know that, for this particular bag, the ratio 2 : 3 : 5 of cashews : almonds : Brazil nuts is the same as the ratio 5 : 7.5 : 12.5.

39. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 7: Using Ratios to Find Quantities In a hardware store, bags of miscellaneous nails are sold at a discount price. If a bag containing 300 nails has a 4 : 5 : 6 ratio of nails in decreasing lengths (long, medium, short), how many of each nail should the bag have?

40. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 7: Using Ratios to Find Quantities (cont.) Solution In this situation, the ratio 4 : 5 : 6 indicates how to partition every set of 4 + 5 + 6 = 15 nails. Each set of 15 nails should have 4 long, 5 medium, and 6 short nails, as shown in the following ratios.

41. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 7: Using Ratios to Find Quantities (cont.) We can use these ratios to set up a proportional equation for each length of nail and solve to find the amounts needed. We then have the following 3 proportional equations. Long nails:

42. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 7: Using Ratios to Find Quantities (cont.) Medium nails: Short nails:

43. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 7: Using Ratios to Find Quantities (cont.) Note that as a check, 80 long nails + 100 medium nails + 120 short nails adds up to 300, the total number of nails in each bag.