1. RATIOS AND PROPORTIONS Created by Leecy Wise and Caitlyn Reese, © Unlimited Learning, Inc. 2015

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3. A ratio is a relationship between two numbers in a defined setting. Huh?! There are 2 boys for every 3 girls in the class There are 2 cups of flour for every 1 cup of sugar in the recipe There are 2 nurses for every 20 patients in the nursing home There are 6 milligrams of drug in every 50 mL of the medicine Well, by “defined setting,” we simply mean that each number represents something, like people or objects, in a certain situation.

4. In fact, there are actually three different ways to write a ratio! Now there has to be a better way to write out a ratio than to say “there are this many of these for every that many of those,” right? 1) We can write this relationship as a fraction 2) We can use the “ratio symbol” (which looks like a colon) Let’s use the example of “two boys for every three girls” from the last slide… 3) Or we can simply use the word “to” between the numbers

5. WOAH there! But how do we know which number goes on top or is written first? Ratios should be written in the same order as the words that express them For example… If someone said that there are “5 physician assistants for every 2 doctors” at a particular clinic, we would write this ratio as:

6. Write the following numbers in the given situations as a proportion in all three of the ways you just saw. Fill in your answers in the table given to you on your Lesson Answer Sheet, then click to check your work. 1)There are 5 trucks for every 7 cars in the parking lot 2)There is only one parent for every 3 kids at the water park 3)There are 8 parks for every 5,000 people in the city 4)There are 68 nurses for every 22 doctors at the hospital 5) There are 96 patients for every 3 providers at the clinic

7. Now check your work! Cross out any incorrect answers and write your corrections on your answer sheet.

8. The two numbers in any ratio need to be in reduced form. This works much like reducing a fraction! For example: Let’s say that 5 out of 10 students in a class have an A in the class. Let’s start out by writing this as a ratio in the fraction form we talked about before: Now look at this like a fraction… IS this in the most reduced form? No! It’s not! What number goes into both 5 AND 10? 5 goes into itself once And 5 goes into 10 two times THIS is now in “reduced form”

9. Write the following ratios as a fraction, and then put the fraction into reduced form. Fill in your answers in the table given to you on your Lesson Answer Sheet, then click to check your work. 1)There are 4 boys for every 6 girls on the team 2)There are 15 players for every 3 referees on the field 3)There are two tables for every 16 people at the party 4)There are 5 doctors for every 100 patients at the clinic 5) There are 750 milligrams of drug in every 5 milliliters of medicine

10. Now check your work! Cross out any incorrect answers and write your corrections on your answer sheet.

11. Actually, ratios (and the next topic, proportions) may be the single most important math concept you will need in your medical career! Check this out…

12. Maybe the best example is medications! Of course you’re going to come across these all the time no matter what medical field you’re in. And guess what? The dosages of ALL medications, whether in pill, powder, or liquid form, uses ratios!!! Do you see the ratio on this label? 250 milligrams (mg) per 5 milliliters (mL) is definitely a ratio! Now if we put this in fraction form we have… Remember to reduce… (5 goes into both numbers) So what does that mean? This means that in every 1 mL of the liquid, there is 50 mg of medicine

13. Another great example of ratios in medicine is Intravenous (in-truh-VEEN-us) fluids, usually called IV fluids for short. IV fluids are not simply water! The typical fluid that is given through an IV in the hospital is called normal saline, which is salt and water, mixed in a specific ratio. IV fluids always have some kind of substance dissolved in them, and the amount of that substance is usually written as a percent. But guess what? Percent is a ratio too!!! For example… Let’s say we have a bag of 2% normal saline. 2% saline just means that there are 2 parts salt to 100 parts water, which is a ratio… 2 parts 100 parts Again, remember to reduce… (2 goes into both numbers) 1 50 So in every bag of normal saline, there is 1 part salt for every 50 parts of water

14. Answer the following questions about medication dosages (Hint: Turn them into a ratio and then convert to most reduced form.) Write your answers on your Lesson Answer Sheet. 1) A full dose of Ibuprofen is 800 mg in four pills. How many mg are there in each pill? 2) There are 5 mg of medicine in 50 mL of cough syrup. How many mL contain just 1 mg of medicine? 3) A bag of IV fluid has 5% potassium. How many parts potassium are there versus parts of water? 4) A bottle of nasal spray has 500 mg of medicine and contains 100 spray doses. How many mg of medicine are in each spray? Make sense? How about…

15. Now check your work! Cross out any incorrect answers and write your corrections on your answer sheet.

16. Let’s say that a physician orders you to give 300,000 units of penicillin IM (intramuscular) to a patient. You find a supply of penicillin in the storage area. However, it states 400,000 units per mL…

17. Now, you know that the amount prescribed is 300,000 units. But how will you figure out how many mL to give the patient when the bottle tells you that there are 400,000 units per 1 mL?

18. Ratios show the relationship between two numbers, as you have learned. Well a proportion is basically a set of two ratios. The numbers in each ratio are different, but… The relationship between the numbers in each set is the same.

19. Basically what this means is that size doesn’t matter in proportions, only relationships matter. This man is a certain height, let’s say 6 feet He also has a certain width, let’s say 2 feet It only means that the relationship between his height and width is the same as the first man. It does NOT mean that they are the same height or width, as you can see…

20. Let’s look at this further… 6 feet 2 feet As we said before, the first man is 6 feet tall and 2 feet wide: Okay, to find the relationship between his measurements, let’s write a ratio: Now reduce, just like before… (2 goes into both numbers)

21. Okay, now what about our smaller man? Remember, he is proportionate to the larger man, meaning the relationship between each man’s height and width is the same Knowing that, what if we now told you that the smaller man is exactly 3 feet tall? 3 feet What does this tell us about his width? 1 foot

22. On the other hand, these two men are NOT proportionate! They have the same height… But VERY different widths… In other words, the relationships between their heights and widths are far from the same

23. As you can see, a proportion is a set of two ratios where the numbers in each one may be different, but when you reduce each ratio, they are actually equal! We call this “ equivalent”

24. Let’s look at a couple more examples of proportions, so we can see how in each one, the ratios are equivalent … Can you tell that these two ratios (or fractions) are equal? 2 goes into both numbers Now it’s exactly equal to the other side!!! If we reduce the right side (3 goes into both top and bottom), we end up with the same thing again!

25. Look at the following sets of ratios, and decide if they are proportions or not. You’ll find each of the following sets of ratios on your answer sheet. Use the free space to reduce your fractions, and then write “yes” if they are proportions (they are equivalent), or “no” if they are not proportions (not equivalent). When you’re finished, click to check your work. 1) 2) 3) 4)

26. 1) 2) 3) 4) Now check your work! Cross out any incorrect answers and write your corrections on your answer sheet. Reduce by 6 Reduce by 5 Yes! Can’t be reduced Reduce by 3 NO! Can’t be reduced Reduce by 3 Yes! Can’t be reduced Reduce by 9 NO!

27. A good understanding of proportions will help A LOT in your health career. So far we’ve learned what they are and how they work…

28. What if we know two things are proportional, but we don’t know one of the numbers in one of the two ratios? The man on the left is 6 feet tall and weighs 142 lbs. The man on the right is 4 feet tall… but how much does he weigh? These two men are proportional.

29. In every proportion problem, the key is to remember that … Why??? Because the fact that the two ratios are equivalent is what makes it a proportion in the first place!!! =

30. To solve a proportion with an unknown number in it, use what is called the “Cross-Multiplication Property.” It works every time! We’ll call it the X rule.

31. In any true proportion (made up of two equivalent ratios) you can cross-multiply, which means multiply the numbers diagonally across the equal sign, and the products you get on either side will be equal… every time! And this is a super easy way to check if any two ratios are equivalent! Multiply across… 2 x 10=20 Now the other way… 5 x 4 = 20 So YES, according to the X rule, this is definitely a true proportion

32. Determine whether or not the following ratios are true proportions using the X rule. Do your work on your Lesson Answer Sheet. If they are true proportions, write “yes,” and if not write “no.” When you are done, click to check your work. 1) 2) 3) 4)

33. 7 X 16 = 112 and 8 x 12 = 112. Yes, this is a true proportion. 3 X 10 = 30 and 5 x 9 = 45. No, this is a NOT a true proportion. 12 X 51 = 612 and 17 x 36 = 612. Yes, this is a true proportion. 2 X 15 = 30 and 3 x 10 = 30. Yes, this is a true proportion. Now check your work! Cross out any incorrect answers and write your corrections on your answer sheet.

34. Now that you know it works, you can solve problems with unknown numbers in them by applying the X rule. Step 1: Set up the KNOWN ratio as a fraction Step 2: Set up the UNKNOWN ratio, using an X to represent the unknown amount How many grams would be in 125 mL of this cough syrup? A cough syrup contains 3 grams (g) of medication in every 15 milliliters (mL). We know that there are 3 grams (top) in every 15 mL (bottom) The question asks how many grams are in 125 mL. So the number of the grams (top) is our unknown number…

35. Now what do you think Step 4 will be, based on what you just learned? Step 3: Now just throw an equal sign in the middle to create a proportion! (We know they are going to be equal, right? It’s the same medicine! All that is different is the amount of liquid…)

36. If you said, “Cross multiply,” you would be absolutely correct! PSST! When we are doing problems with X as the unknown number, a better way to write out multiplication is to just put both numbers in parentheses to avoid confusion! ( ) is the same as ×

37. (3) (125)=(15)(X)(X) Okay, now do the multiplication… (3)(125) = 375 and (15)( X ) = 15 X Now remember, according to the X rule, the two products are the same, right? So put an equal sign between them, and we have… 37515X =

38. Now we just need to solve by getting X by itself… 37515X = Remember, this is 15 times X To get X alone, we need to reverse what is being done to it. Division! Okay, so let’s divide by 15… The two sides of the problem are equal, right? So to make sure it stays equal, anything we do needs to be done to both sides… 375 ÷ 15 is 25 and15X ÷ 15 is 1X (or just X) = And remember, they’re equal!

39. 25 X = So what does that mean for our original problem? (Yes, that was a while ago ) Here is the original problem: How many grams would be in 125 mL of this cough syrup? A cough syrup contains 3 grams (g) of medication in every 15 milliliters (mL). We were able to solve and figure out that X = 25 There are 25 grams of medicine in 125 mL of the cough syrup

40. Now that you understand the steps, let’s go through one more example… Step 1: Set up the KNOWN ratio Step 2: Set up the UNKNOWN ratio, using an X to represent the unknown amount

41. (2)(13.5) Step 3: Set them equal to create a proportion Step 4: Cross multiply = (3)( X ) Which is… 27 = 4.5 X Step 5: Divide on both sides to get X by itself So there are 6 grams of salt in 13.5 liters of the IV solution! 6 = X

42. Now it’s your turn. Find the missing (X) value in the following proportions. Do your work on your Lesson Answer Sheet. When you are done, click to check your answers. Note: It doesn’t matter where the X is placed in the proportion. The process is the same. (Round to two decimal places)

43. Now check your work! If you missed any, figure out the mistake was and correct it on your Answer Sheet. Click again to see the answers to 4-6

44. Now check your work! If you missed any, figure out the mistake was and correct it on your Answer Sheet.

45. A physician ordered you to give 300,000 units of penicillin IM (intramuscular) to a patient. You find a supply of penicillin in the storage area. However, it states 400,000 units per mL. How many units would you give the patient using the supply you found? Do your work on your Lesson Answer Sheet, and then click to check your answer!

46. 400,000 First, set up your known ratio. The problem told us that there are 400,000 units in 1 mL of the medicine. 1 mL Then, set up your unknown ratio. We need to know how many milliliters (X) we will need to get 300,000 units. X mL 300,000 = (Set them equal to make a proportion) (400,000)(X) = (1)(300,000) 400,000X = 300,000 400,000 Now divide on both sides to get X by itself 400,000 X = 0.75 mL

47. Review and Practice Turn in your Lesson Answer Sheet to your coach. Then complete the following worksheets (you should have a print out of them):  Writing and Reducing Ratios  Using Proportions Once you complete each worksheet, ask your coach for the answer key and correct your work. Don’t worry, you’ll only be graded for completion on this part. Finally, click here to review some flash cards with all of the terms you learned in this lesson (you can also play games with the terms!): https://quizlet.com/128869519

48. CONGRATULATIONS! You now have a great introduction to ratios and proportions! Ready to take the quiz?