Overview:
Working backwards is another way to solve some math problems. If the result is given at the end of a series, follow each step until the problem is complete. Then check each step to make sure each one is correct.
What Facts Are Known?
As in all math problems, read the problem very carefully to determine what is known. For example, suppose the problem is, “Robert spent one-fifth of his money buying a paperback book, then half of what was left for a haircut. Then he bought lunch for $7. When he got home, he had $13 left. How much did he have originally?” What is known is the final amount, every operation, and each step along the way.
What Are the Steps?
The first step in working backwards is to take the final number and then do the inverse operation. Therefore, is a step involves subtraction, the inverse will be addition, and if it involves division the inverse will be multiplication. Read the problem once again to determine each step. In this problem, the first backwards step will be to add an amount, and the second will be to multiply. The last step will to be to multiply again, then divide.
What Is the Solution?
In this example, if a-7 = 13, then add 7 to each side, so that a -7 +7 = 13+7 , or a = 20. Robert had 20 dollars before he bought lunch. If he spent half of what was left in the next step, the inverse operation is used again, so that (1/2b)2 = 20(2), or 40 dollars before the haircut. Read very carefully for the last step, as one fifth of the money is one fifth of the total amount, rather than one-fifth of what is left. That means that 40 dollars is only 4/5 of the total. Therefore, if 4/5 of c = 40 , 4c= 200. Then c =50. The original amount is 50 dollars.
Checking the Solution
To check the solution, follow the steps forward. If Robert had 50 dollars originally, then a 10-dollar paperback would be one-fifth of the total, because 10 is 1/5 of 50. There was 40 dollars before the haircut, and half of what was left is $20. Then 20-7 is 13 dollars so the steps check. Sometimes, working backwards is a good strategy to solve problems.
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