1. Ratios and Proportions REVIEW CONCEPTS

2. What is a ratio? A ratio is a comparison of two numbers. Ratios can be written in three different ways: a to b a : b ab NOTE: Ratios must be expressed in lowest terms. 3 to 4 3 : 4 34

3. How to simplify ratios. Simplify ratios the same way you simplify fractions: Find a common factor. 16:4 = 4:1 18:24 = 3:4 125:25 = 5:1

4. Example 1: Part to Whole In a 100g sample of copper alloy, there is 55g of copper and 45g of tungsten. What is the ratio of copper alloy to tungsten? The ratio of copper alloy to tungsten is 20:9. Tungsten : Copper Alloy 45 : 100 9 : 20

5. Example 2: Part to Part In a 100g sample of copper alloy, there is 55g of copper and 45g of tungsten. What is the ratio of copper to tungsten? The ratio of copper to tungsten is 11:9. Copper : Tungsten 55:45 11:9

6. Proportions A proportion is an equation that equates two ratios.

7. Cross-Product Property 2x6=12 3x4 = 12 ad = bc Therefore:

8. Example 3: Step 1: Cross Multiply Step 2: Multiply Step 3: Divide

9. Example 4: Direct Proportion In a 100g sample of copper alloy, there is 55g of copper and 45g of tungsten. How much copper is in a 120g sample of copper alloy? Step 1: Set up the proportion Original Copper Original Alloy New Copper New Alloy

10. Example 4: Continued… Step 2: Cross Multiply Step 3: Solve for the unknown Therefore, there are 66g of copper in a 120 sample of copper alloy.

11. Example 5: Direct Proportion The ratio of tungsten to copper in a sample of copper alloy is 9:20. If you have a sample of copper alloy that has is 220g of copper, how many grams of tungsten is in the sample? Step 1: Set up the proportion Ratio of Tungsten Ratio of Copper Amount of Tungsten Amount of Copper

12. Example 5: Continued… Step 2: Cross Multiply Step 3: Solve for the unknown Therefore, there are 99g of tungsten in the sample of copper alloy.

13. Example 6: Direct Proportion

14. Indirect Proportion An indirect proportion is a comparison between two ratios that are INVERSELY proportional This means that an increase in one quantity leads to a decrease in the other quantity When we solve for inverse proportions, we need to invert one of our ratios

15. Example 1: Indirect Proportion If it takes 3 carpenters 30 days to build one house, how many days would it take 5 carpenters to build the same house? Let’s think about this to see if it is a direct or indirect proportion. If we INCREASE the number of carpenters, will the amount of time it takes to build the house decrease?

16. Example 1: Indirect Proportion (Cont) If it takes 3 carpenters 30 days to build one house, how many days would it take 5 carpenters to build the same house? Setting up the proportion if it was direct: But because it is indirect: We switch the two values on the bottom

17. Example 1: Indirect Proportion (Cont) If it takes 3 carpenters 30 days to build one house, how many days would it take 5 carpenters to build the same house? Solving for X yields 18 days

18. Example 2: Indirect Proportion If it takes 8 carpenters 26 days to complete a project, how many days would it take 11 carpenters to complete the same project? ORIGINAL: INVERTING: X = 18.9 Days

19. Other Examples A journeyman takes 2 hours to install a door. An apprentice takes 3.5 hours to do the same job. How long would it take them do install a door if they were working together? Let’s try to understand what this question is asking The first thing we need to do is to figure out how much of the door each of them can complete PER hour The journeyman takes 2 hours for 1 door. So in 1 hour he completes 0.5 of the door (1/2) The apprentice takes 3.5 hours for 1 door. So in 1 hour he completes 0.2857 of the door (1/3.5)

20. Example Cont Combining the two, JourneymanApprentice 0.5 +0.2857 = 0.7857 This means they complete 0.7857 of the door in 1 hour (60 minutes). Setting up the proportion: X = 76.365 minutes

21. Example Cont X = 76.365 minutes We don’t want to leave our answer in minutes so we will use a proportion to convert to hours. X = 1.27275 hours 0.27275 * 60 minutes = 16.365 minutes Therefore, it would take the apprentice and the journeyman 1 hour and 16 minutes to install a door if they are working together

22. Trying another one A journeyperson takes 3 hours to install two doors. An apprentice takes 6.75 hours to install 2 doors. If they are working together to install two doors, how long will it take them? Try it yourself!

23. Trying another one A journeyperson takes 3 hours to install two doors. An apprentice takes 6.75 hours to install 2 doors. If they are working together to install two doors, how long will it take them? Journeyperson: 2/3 = 0.667 doors per hour Apprentice = 2/6.75 = 0.296 doors per hour 0.667 + 0.296 = 0.963 doors per hour 0.963X = 120 X = 124.6 minutes X = 2.077 hours 0.077*60 = 4.62 minutes Therefore, it would take them 2 hours and 5 minutes to install two doors