Working Backwards to Solve Word Problems

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  • 0:03 Working Backwards
  • 1:02 Solving Our Candy Example
  • 2:14 When to Use Working Backwards
  • 4:00 Lesson Summary
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Lesson Transcript
Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

There are many ways to go about solving word problems in mathematics. This lesson will illustrate a particular solving process called working backwards. We'll see how and when to use this process to solve word problems.

Working Backwards

Suppose Dan is handing out candy. He give Susan 8 pieces, but she says she doesn't need that many, so she gives him 3 pieces back. He gives Mandy 7 pieces, and lastly, he gives James 5 pieces. After doing this, Dan has 4 pieces left for himself. Based on all the information, can you tell me how many pieces of candy Dan started out with?

There are a number of ways to go about this problem, but we're going to talk about one solving process in particular, and that is working backwards. To solve a problem by working backwards, we basically want to undo the problem step-by-step. We start at the end of the problem and work through to the beginning. In other words, we do as the name of this solving process suggests - we work backwards!

I don't know about you, but solving processes like this always make more sense to me when I see them in action. Let's explore our opening example to illustrate how to use working backwards to solve a problem.

Solving Our Candy Example

We want to know how many pieces of candy I started out with. Like we said, to work backwards to solve, we start at the end of the problem and undo it one step at a time. At the end of the problem, Dan had 4 pieces of candy left for himself, so this is where we'll start.

Right before Dan had 4 pieces left, he gave James 5 pieces. To undo this, we add 5 pieces to the 4 Dan has left, and 4 + 5 = 9, so now Dan has 9 pieces of candy. The next thing to undo is giving Mandy 7 pieces. To do this, we add those 7 pieces to my Dan's 9 pieces, and 7 + 9 = 16, so Dan has 16 pieces of candy. Before this, Dan gave Susan 8 pieces of candy, but she gave him 3 pieces back, making this one is a little trickier to undo. Since 8 - 3 = 5, this means Dan gave Susan 5 pieces of candy in all.

Thus, to undo this step, we add 5 pieces of candy to Dan's running total, giving him 16 + 5 = 21. This tells us that Dan had 21 pieces of candy to start with. That's not so hard, is it?

When to Use Working Backwards

As we said, there are multiple ways to go about solving word problems, so it is useful to know when the process of working backwards is the best solving option. There are certain characteristics of word problems that indicate working backwards should be used to solve the problem.

The first of these characteristics is that the end of the problem will have a result that is very clear, whereas the beginning of the problem will be less obvious. Consider our candy example. At the beginning of the problem, it jumps right into Dan handing out candy. It doesn't specify how much candy Dan has to hand out. However, at the end of the problem, it clearly states that Dan had 4 pieces left over for himself. This is a good indication that working backwards is a good process to use.

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