Math Olympiad Strategies Working Backwards

Money Wise 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 each of the three people had $24. How much money did each have to begin with? This problem has 4 stages. The only amount that is known is in the fourth stage, which is the final outcome. Let’s consider the stages in reverse order and make a table: STAGE ACTION Ashley Ben Chris SUM 4 Each ends with $24 $24 $72 3 Chris gave Ashley and Ben half of what each will have in stage 4 $12 $48 2 Ben gave Ashley and Chris half of what each will have in stage 3. $6 $42 1 Ashley gave Ben and Chris half of what each will have in stage 2. $39 $21

Inverse Flow What number belongs in the START circle of this “flow chart”? START ? DIVIDE BY 2 SUBTRACT 5 MULTIPLY BY 8 END 32 To find the starting number use the inverse of each operation in reverse order: 32 divided by 8 plus 5 multiply by 2 = ? The number in the START circle is 18.

Work Backwards Strategy Homework NAME:______________________________ Due:____________________________ I went into a store and spent half of my money and then $20 more. I went into a second store and spent half of my remaining money and then $20 more. Then I had no money left. How much money did I have when I went into the first store? What number belongs in the START box of this “flow chart” ? At the end of one school day a teacher had 17 crayons left. The teacher remembered giving out 14 of all her crayons in the morning, getting 12 crayons back at recess, and giving out 11 crayons after lunch. How many crayons did the teacher have at the start of the day? I have a Magic Money Box that will double any amount of money placed in it and then add $5 to the doubled amount. Today I placed a certain amount of money in the box. The box then gave me a new amount, which I placed back into the box. This time the box gave me $43. How much money did I first place in the Magic Money Box? A cooperative farm has three subdivisions, A, B, and C; the subdivisions lend equipment to each other as needed. In the beginning A lent B and C as many reapers as each then had. Several months later B lent A and C as many reapers as each had then. The following spring C lent A and B as many repers as each then had. Each subdivision then had 16 reapers. How many reapers did each have to begin with? START ? MULTIPLY BY 3 ADD 20 DIVIDE BY 2 END 40