Chapter 11Work Backwards With most other strategies, you work forward through the information in a problem. In some situations, such as planning a work schedule based on a given completion date, it is easier to work backwards. Much of algebra is based on working backwards as well. One of the most difficult aspects of working backwards is keeping track of a problem’s information and organizing it in a meaningful way.
Working Backwards Mr. Collins will come to San Diego this weekend for a business conference. By coincidence, some of his relatives (with kids) are also coming for vacation. They planned to visit Sea World on Sunday morning and then drive to Anaheim in the afternoon. If Mr. Collins has to fly back to NYC at 10:30pm, tell him when he should leave Sea World.
Example 1 Mr. Phil T. Rich left half of his estate to his wife, $300,000 to his daughter, half of what was left to his butler, half of what remained for the care of his exotic tropical fishes, and the remaining $80,000 to charity. What was the total value of his estate?
The table below keeps track of the amount of money left after each step, we then work backwards.
The table below keeps track of the amount of money left after each step, we then work backwards. 1,240,000 620,000 320,000 160,000 80,000 0
Example 2 Dad was paid on Wednesday. On Thursday morning, my brother borrowed half of Dad’s money to open a checking account, because he was always short of money. On Friday I needed some money for a date, so I borrowed one-third of what remained. Sis came along next and borrowed half of the remaining money. Dad then went to gas up the car and used three-quarters of the rest of his money and he had only $5 left. He wondered where it all went so fast. How much money did he start with?
The table below keeps track of the amount of money that Daddy has left in his wallet after each step
The table below keeps track of the amount of money that Daddy has left in his wallet after each step 120 60 40 20 5
Example 3 After dinner, three friends at a restaurant paid their bill and noticed a bowl of mints on the front counter. Sean took one-third of the mints but returned 4 because he had a momentary pang of guilt. Faizah then took one-fourth of what was left but returned 3 for similar reasons. Eugene then took half of the remainder but returned 2. (He felt no pangs of guilt, he just didn’t want mints that looked like they had been slobbered on.) The bowl only had 17 mints left when the raid was over. How many mints were originally in the bowl?
The table below keeps track of the number of candies left in the bowl after each step,
The table below keeps track of the number of candies left in the bowl after each step, 48 32 36 27 30 15 17
1. Losing Streak A man competing on a game show ran into a losing streak. First he bet half of his money on one question, and lost it. Then he lost half of his remaining money on another question. Then he lost $300 on another question. Then he lost half of his remaining money on another question. Finally he got a question right and won $200. At this point, the show ended and he had $1200 left. How much did he have before his losing streak began?
2. Generosity Phil Anthropist likes to give away money. One day he was feeling especially generous, so he went to the park with a wad of money. He gave $100 to a man feeding pigeons. He then gave half of his remaining money to a child licking an ice cream cone. He then gave $50 to the balloon seller. He then bought a hot dog, and paid for it with a $20 bill. "Keep the extra for a tip," he said to the hot-dog seller. Then he gave half of his remaining money to someone giving a sermon on a soapbox. At this point, he had $3 left and stuck it under the collar of a stray cat. How much money did Phil have when he started his good deeds?
4. The Mall My sister loves to go shopping. Yesterday she borrowed a wad of money from Mom and went to the mall. She began her excursion by spending $18 on a new DVD. Then she spent half of her remaining money on a new dress. Then she spent $11 taking herself and her friend out to lunch. Then she spent one-third of her remaining money on a book. On her way home she bought gas for $12 and spent one-fourth of her remaining money on a nail polish at the convenience store. Finally, she slipped me $2 when she got home, and gave Mom back $10 in change. Mom was furious and demanded an explanation of where the money went. What did Sis tell her? (List the items and amount spent on each item.)
6. Hockey cards Jack and Jill were admiring each other's hockey cards. Jack had quite a few cards, and Jill had some of her own. Jack was trying to impress her, so Jack gave Jill as many hockey cards as she already had. Jill then gave Jack back as many cards as he had left. Jack, not to be outdone by Jill's generosity, gave her back as many cards as she had left. This left poor Jack with no cards and left Jill with 40 cards altogether. How many hockey cards did each of them have just before these exchanges took place? (You might want to try acting this out in conjunction with working backwards.)
7. Donuts Four customers came into a bakery called Totally Excellent Donuts. The first one said, "Give me half of all the donuts you have left, plus half a donut more." The second customer said, "Give me half of all the donuts you have left, plus half a donut more." The third customer said, "Give me three donuts." The last customer said, "Give me half of all the donuts you have left, plus half a donut more." This last transaction emptied the display case of donuts. How many donuts were there to start with? And how many donuts did each customer really get?
2. Crossing the River again Two young kids Alyse, Jeremy, and their Grandpa and Grandma were going to the train station to catch a train. Before they could get to the station, they must cross a river through a narrow bridge. It was a dark evening and the bridge was very dangerous to use without lights. Fortunately Grandpa had a pocket flashlight but it was just bright enough for at most two people to use at a time. The bridge was also not short enough for them to throw the flashlight back, so some one must carry the flashlight back for the other two people to use. Alyse could cross in 2 minutes, Jeremy could cross in 1 minute, Grandpa needed 10 minutes to cross, and Grandma needed 5 minutes.If two people were walking together, the faster one must slow down for the slower one because no body was strong enough to carry another person. The train was going to leave in 18 minutes and yet they did manage to cross the river using that bridge in 17 minutes. How did they do that?