Introduction to The Strategies of Working Backward and Using Logical Reasoning in Word Problems
You can only find truth with logic if you have already found truth without it.
—G.K. CHESTERTON (1874–1936)
Mathematics and logic go hand in hand; they are related in many ways. Logical reasoning is used to solve many mathematical problems, and it is also used to explain many mathematical properties. You really cannot have one without the other. This idea will be highlighted in this lesson, which is based on the strategies of showing you how to work backward and use logical reasoning.
Sometimes, the best approach to solving a problem is to begin with the information at the end of a question and work back through the problem. This strategy of working backward will be explained, and you will have the opportunity to practice questions using this strategy. Using logical reasoning is another way to solve word problems, and two specific problem types using reasoning tables and Venn diagrams will be discussed in this section. The practice problems that follow will allow you to apply your skills using each of these strategies.
Working Backward
When you are reading some types of math word problems, the information given in the question can seem like an endless list of facts. In questions such as this, it is sometimes helpful to begin with the last detail given. To apply this technique, start with the last piece of information, and work backward through the problem. Then, check your result by starting at the beginning of the question and working through the operations in the correct order.
Note that you will sometimes need to do the opposite, or inverse, operations when using this strategy. For example, if in the question, 2 was added to a number to get 5 as a result, you will need to subtract 5 – 2 to get the beginning value of 3.
Read through the example that follows, which uses the strategy of working backward. Pay close attention to the inverse operations that are used when you are solving the problem.
Example
Ella sold cookies at a bake sale. During the first hour, she sold onethird of the number of cookies she brought. During the next hour, she sold 10 more. At the end of the sale, she sold half of the remaining cookies. If 10 cookies did not sell, how many cookies did she bring to the sale?
Read and understand the question. The question is asking for the total number of cookies that she began with at the start of the sale. Details about the number of cookies sold during each hour are given throughout the question.
Make a plan. Use the strategy of working backward. Begin with the fact that she had 10 cookies left over at the end of the sale, and work backward from there.
Carry out the plan. Start with the 10 cookies that were left over. At the end of the sale, she sold half of the remaining cookies. Thus, the other half is equal to the 10 cookies left over. Multiply 2 by 10 to double this amount and get 20. Before this, she sold 10 more, so add 20 + 10 = 30. During the first hour, she sold onethird of the number of cookies. This means that she had twothirds remaining, which is equal to 30. If twothirds is 30, then onethird is 15. Add 30 + 15 = 45. She began the sale with 45 cookies.
Check your work. Begin with the solution of 45 cookies and work forward through the question. During the first hour, she sold onethird of the cookies. Onethird of 45 is 15, and 45 – 15 = 30. During the next hour, she sold 10 more cookies: 30 – 10 = 20. At the end of the sale, she sold half of the remaining cookies. Half of 20 is equal to 10, so 20 – 10 = 10. She had 10 cookies remaining at the end of the sale. The solution is checking.
Tip:Be sure to use the inverse (opposite) operations if necessary when using the strategy of working backward. 

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