WORKING BACKWARDS Math Olympiad Lesson 3 Some problems

WORKING BACKWARDS Math Olympiad Lesson #3

Some problems involve a sequence of actions: the final result of the action is known, an you are asked to determine the beginning conditions of the problem. An effective way to solve this type of problem is to consider the actions in reverse order. Let’s take a look at some examples:

Ashley gave Ben and Chris as much money as each had. Then Ben gave Ashley and Chris as much money as each already had. Then Chris gave Ashley and Ben as much money as each had. Then the three people had $24. How much money did each have to begin with? The problem has 4 stages: 1. Ashley gives Ben and Chris as much money as each has. 2. Ben gives Ashley and Chris as much money as each has. 3. Chris gives Ashley and Ben as much money as each has. 4. Each now has $24. The only amount of money that is known is the last stage, so sta there and work backwards. Making a table or chart may help.

As you work backwards, keep two facts in mind. 1. Each time a person gives each of the others “as much money as each had”, each Of the others will have had ½ as much money as in the succeeding stage. 2. NOTE that the total amount of money can be computed from the final stage since They all have $24 each, and this total remains the same throughout each stage. Stag e Action Ashley Ben Chris TOTAL SUM: 4 Each ends with $24 $24 $72 3 Chris gave Ashley and Ben ½ of what each will have in stage 4 $12 $48 $72 2 Ben gave Ashley and Chris half of what each will have in stage 3. $6 $42 $24 $72 1 Ashley gave Ben and Chris ½ of what each will have in stage 2 $39 $21 $12 $72

Inverse Flow Start ? Divide by 2 Subtract 5 Multiply by 8 End 32 What number belongs in the start circle of this flow chart? To find the starting number, begin with the ending number and work backwards. Use the inverse of each operation along the way.

Mrs. Winthrop went to a store, spent half of her money and then $10 more. She went to a second st spent half of her remaining money and then $10 more. But she then had no money left. How much Money did she have to begin with when she went to the first store?