Searching for SolutionsWorking Backward
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Strategy 7 - Working Backward
Working backward is a tricky strategy. It is used when you aren't given the information that would fit in the beginning of a problem.
For example: You went to a water park with a friend. You spent half of your money on admission to the park. You forgot to bring a towel, so you spent half the money that was left on a new one. Then you had $4 left, which you spent on lunch. How much money did you take to the water park? (Passport to Mathematics, Book 1, McDougal Littell, 1999, p. 34-35.)
Solution: In this problem, you know how much money was spent at the end --> $4 for lunch; however, you spent money on a towel and money on admission, and at the end of the day, you had no money left. To solve this sort of problem, you have to work from the end of the problem back to the beginning.
 | At the end of the day, you had $0 left. |
| At lunch time, you spent the $4 that you had left from the other purchases. |
| Before lunch, you bought a towel. To purchase the towel, you spent half the money you had. If you were left with $4 after purchasing the towel, and you used 1/2 to buy the towel, you must have had $8 before you bought the towel. ($4 x 2 = $8) |
| The cost of admission was half of the amount that you brought for the day. According to the story, you spent half on admission and from the prior step (the towel information), we know you had $8. That means that admission was double this amount ($8 x 2 = $16). Therefore, you must have taken $16 to the water park. |
| To check this problem, insert the answer we just got ($16) into the original problem and see if the outcome is the same. |
Here's one for you to try: Melba makes bookmarks to sell. She gave a dozen to Kim to sell for her. Then she divided equally what she had left among herself and two charity groups. She now has 18 bookmarks. How many bookmarks did she originally have? (Passport to Mathematics, Book 1, McDougal Littell, 1999, p. 35.)
Now give this one a try: You leave the house in the morning for school with some money in your pocket. On the bus, your friend asks you for the $2 you owe her, and you pay her back. While at school, you spend $3 for your lunch. After school, you and a friend split the bill for something to eat. The bill is $6. On the way home, your neighbor pays you the $5 that he owes you for mowing his lawn last weekend. Finally, that night you empty your pockets and you have $8. How much money did you have when you left for school? (Passport to Mathematics, Book 1, McDougal Littell, 1999, Practice Workbook 1.7)
Activity 1: For the next set of problems, the missing information is not at the beginning, so read carefully!
| Mrs. Fox, a school cook, has $49.90 left from her paycheck. She paid a 4-week newspaper bill, spent $46.28 for food, and deposited $99.50 in her savings account. If her paycheck is $213.40, what is the weekly cost of the newspaper? (Mathematics Plus 6, Harcourt Brace & Company, 1989, p. 126.) |
| Mrs. Chu has 18 shelves on which to display bottles of her blueberry syrup. A year after she went into business, 2 shelves were added. Three years later, the number of shelves tripled. How many shelves were originally available for the bottles of her blueberry syrup? (Mathematics Plus 7, Harcourt Brace & Company, 1989, p. 169.) |
Activity 2: Solving problems like these can get tricky. You are probably thinking that this is one of those strategies that can only be used in a textbook situation. Well, here is an example of a real-life problem that involves working backward.
Getting Places on Time
| When traveling by airplane, you should generally arrive at the airport 45 minutes before your scheduled flight. |
| Begin by identifying how long it will take you to get to the airport from your home. Go to the the Directions area of MapQuest.org. Fill in the information about your home address in the Starting Location area. The Ending Location is the Baltimore Washington International Thurgood Marshall Airport. In the Place section, type BWI Airport. Click Get Directions to view the directions and travel time needed to arrive at BWI. |
| Go to the Baltimore Washington International Thurgood Marshall Airport web site to learn about flight information. |
| Click on the link entitled "Airlines and Schedules" to see the schedules for flight arrivals and departures. |
| Click on "Access the Flight Schedules" and enter a city or state name and travel date. Click on Search. |
| Select a flight. |
| Work backward to find out what time you should leave your house to arrive at the airport 45 minutes before your plane leaves. (Hint: If there is bad weather or the road conditions are poor, you will have to add additional time. This may add an extra step in your equation.) |
| Fill out the following chart with your information. Then choose another starting location and flight information and work backward to solve this problem. You may continue trying out travel plans as time permits. |
| Home Location | Distance from BWI (miles and time) | Departure Location | Flight Name - Flight Number - Departure Time | What time do you need to leave from home in order to arrive 45 minutes before your flight takes off? (Show all work!) |
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© Searching for Solutions - A Web-Based Problem-Solving Unit - Developed by Judy Campf, e-Learning Specialist.
