Module 5.2 Work Problems

1. Module 5.2 Work Problems By Dr. Marcia L. Tharp

2. Introduction • In today’s world of work people are asked to work in groups to complete projects or a task. For this reason it is important to know how long it will take a group of people working together to complete a job or task such as taking an inventory, xeroxing a report, painting walls or cleaning a building. These type of problems in which two or more people or sometimes machines work together to complete a job or task are called work problems. To solve them we need to understand some key ideas.

3. Big Idea #1 • Work problems are represented using fractional equations. The part of each job or task done by each worker is usually totaled to equal one whole job. The number 1 represents one whole job.

4. For example:

5. Check your understanding:

6. Solution:

7. You can check your work by placing your answer in to the fraction equation. • Joe’s part of job done + Mary’s part of job done = 1 whole job. • + = 1 • or = 1 • or 1 = 1 • Since both sides are equal our answer is correct.

8. Big Idea #2 • The rate at which a person or machine • works to complete a task is based on • the time it takes them to complete the task. • In general if a person can complete a task • in t hours, then the rate they work at is

9. For example:

10. Check your understanding: • Julia can complete a task in 4 hours. What is her rate per hour?

11. Solution:

12. Big Idea #3 • To determine the part of a job completed by a • person or machine multiply the rate a person • works by the time they spend doing the work.

13. For example: • What part of the job can Bonita complete • if she can complete the whole job in • 4 hours and she only works on it for • 3 hours?

14. Solution: • Using our formula we know:

15. Check your understanding: • Megan can complete a job in 12 • hours. How much of the job will • she complete if she only works • on it for 5 hours? • It’s your turn now! Think about • this before you answer.

16. Solution: • Using the formula we have:

17. Now lets put these Big Ideas together to solve some work problems. But first here is the pattern we will use to solve these problems.

18. A Pattern For Problem Solving • 1) List the important information. • 2) Determine what the problem asks you to look for. • 3) Assign a variable to what you are looking for. • 4) Guess the answer using a table so that you • become familiar with how the problem works. • 5) Set up an equation. • 6) Solve the equation. • 7) Check the result by replacing your number in the equation you set up.

19. Example 1 • Curtis and Nadine work as a team at a car wash. Working alone it takes Curtis 3 hours to wash and detail a van. If Nadine worked alone it would take her 4 hours to wash and detail a van. Their boss wants to know how long will it take then to wash and detail a van together?

20. 1) List the important information. • Curtis works 3 hours to wash and detail the van. • Nadine works 4 hours to wash and detail the van.

21. 2) Determine what the problem asks you to look for. • We need the time it takes Curtis and Nadine to wash and detail the van together. • (Hint: look at the question “?” sentence.) • 3) Assign a variable to what you • are looking for. • Let t= the time it takes Nadine and Curtis • to wash & detail the van together

22. 4) Guess the answer using a table below to become familiar with the problem. • Click here to check your guess by changing the number in the first column on the spreadsheet.

23. 5) Set up an equation. We first need to have the part of the job Curtis and Nadine each did separately. We can put this in a table to get it.

24. Now complete the equation below by filling in the part of the job completed by each person who worked.

25. 6) Solve the equation. • Multiply each side of the equation by the LCD, 12.

26. Answer:

27. Is that your final answer?(Step 8 Check the result.) • Well to be sure we will check by putting our answer in to the fraction equation.

28. We finish simplifying it. • Since both sides are equal the answer is correct. • We have no doubt that our final answer is correct.

29. Example 2 • Kareem and Jody are in a hurry to fill their Jacuzzi. So they use both the hot and cold-water faucets. With the hot water on the Jacuzzi can be filled in 2 hours. If the cold water is used alone it will take 3 hours to fill the Jacuzzi. How long will it take to fill a Jacuzzi using both faucets together?

30. Step 1 List the important information. • It takes 2 hours to fill the Jacuzzi using the hot water faucet alone. • It takes 3 hours to fill the Jacuzzi using the cold-water faucet alone.

31. Step 2 Determine what the problem asks you to look for. • We need the time it takes to fill the Jacuzzi using • both the hot and cold water faucets. Step 3 Assign a variable to what you are looking for. Let t = the time it takes to fill the Jacuzzi using both the hot and cold water faucets.

32. Step 4 Guess the answer using a table to become familiar with problem. • To make your guess click on the link to the spreadsheet end change the first column.

33. Step 5 Set up the equation • Hint: Make a table to show the part the job each faucet completes in one hour. • See if you can fill in the table below.

34. Here is what to put in the table.

35. Step 5 Set up the equation.

36. Step 6-Solve the equation.

37. Step 7 - Check your answer by replacing it in the equation you set up in step 5.

38. Simplify this equation. • This is true because both sides are equal.

39. Example 3 • Two computers are available to process • a batch of data. The slower computer • can process it in 30 minutes. If both computers are used at the same time • they can process it in 10 minutes. How • long will it take the faster computer to • do the whole job alone?

40. Step 1 List important information. • The slower computer completes the job • in 30 minutes. • The 2 computers working together complete • the job in 10 minutes.

41. Step 2Determine what we are looking for. • How long will it take the faster computer • to complete the job alone? Step 3 Assign a variable to it. Let t = the time it takes the faster computer to complete the job.

42. Step 3 Put what you are looking for in variable form. • Let t = how long it will take the faster computer to complete the job alone.

43. Step 4Guess the answer using a table to become familiar with the problem. • Click on this link to make your guess by using the spreadsheet.

44. Computer Rate Time Part of job completed Slower Faster Step 5 Set up the equation.First fill in a rate time chart.

45. Computer Rate Time Part of job completed Slower Faster Step 5 Set up the equation.Remember rate = 1/time it takes to do the whole job

46. Computer Rate Time Part of job completed Slower 10 min. Faster 10 min. Step 5 Set up the equation.Next fill in the time.

47. Computer Rate Time Part of job completed Slower 10 min. Faster 10 min. Step 5 Set up the equation.Now multiply rate x time to find the part of the job completed.

48. Computer Rate Time Part of job completed Slower 10 min. Faster 10 min. Step 5 Set up the equation. Use the last column to build your equation. Slower Computer + Faster Computer = 1 whole job + = 1

49. Step 6Solve the equation. • Multiply by the LCD, 3t 1) 3t + 3t = 3t 1 t + 30 = 3t 30 = 2t 15 = t It takes 15 minutes for the faster computer to complete the job.

50. Step 7 Check your answer by replacing it in the equation you set up in step 5. • . Both sides are the same. So we have the correct solution.