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.

Using Logical Reasoning
Two common strategies you can use in logical reasoning questions are Venn diagrams and reasoning with a table. Each strategy is demonstrated in the following examples.
Example 1: Using a Venn Diagram
Of 65 ninth graders at a high school, 40 take Russian and 30 take German. If 18 students take both Russian and German, how many ninth graders do not take either language?
Read and understand the question. The question is asking for the number of ninth graders who do not take Russian or German. The total number of ninth grade students in the school, as well as the number who take Russian, German, or both, is given.
Make a plan. Use a Venn diagram to answer this question. The Venn diagram should have a circle for the number of students who take German, which overlaps with a circle for the number of students who take Russian. This overlapping section represents the number of students who take both languages.
Carry out the plan. Start the diagram by drawing a rectangle that represents all of the ninth graders, and then place the two overlapping circles within the rectangle. The diagram could appear like the one shown here.
Because the number of students who take both is 18, place the 18 in the overlapping section between the circles. The 40 students taking Russian represent all of the students taking the language, including the 18 who take both languages. Therefore, the number in the part of the circle for Russian that does not overlap is 40 – 18 = 22. In the same way, the circle for the 30 students taking German also includes the students taking both languages. So, the value in the part of the circle for German that does not overlap should be 30 – 18 = 12. The Venn diagram should now look like the following figure.
Now, add the three values in the diagram to get the total number of students who take one or both of the languages: 22 + 18 + 12 = 52. Subtract this amount from the total number of ninth graders at the school: 65 – 52 = 13. Thirteen students do not take either language. The number 13 is placed inside the rectangle but outside of either circle, as shown in the figure.
Check your work. Add the number for each category to make sure that the total number of ninth graders is 65. The number of students who take Russian only is 22, the number of students who take German only is 12, the number of students who take both languages is 18, and the number of students who do not take either language is 13: 22 + 12 + 18 + 13 = 65. Since each ninth grader at the school is in one of these four categories, this solution is checking.
Tip:
When solving problems using a Venn diagram, be sure to subtract the number of objects that include more than one category from the total number in the category. If this is not done, the objects will be counted more than once in the problem.

Example 2: Reasoning with a Table
Tim, Curt, and Kara are brothers and sister in the same family. Kara is younger than Curt. Tim is not the youngest or the oldest. Assuming none of the children are twins or triplets, what is the order from youngest to oldest?
Read and understand the question. The question is asking for the birth order of three children from the same family. The children are not twins or triplets.
Make a plan. Make a table that includes the three people in the question along the side. Then, make one column for the youngest child, one for the middle child, and one column for the oldest child. Use the clues in the question to place an "X" in any box where the possibility can be eliminated. Use the process of elimination to figure out the order the children were born.
Carry out the plan. Start by making the table. A possible table follows.
Use the clue that Kara is younger than Curt. With this information, you can eliminate the fact that Kara is the oldest, so place an "X" in this box. You can also reason that Curt is not the youngest, so place an "X" in this box. The table at this point could look like the one that follows.
The second clue states that Tim is not the youngest or the oldest, so place an "X" in both of these boxes. Tim is the middle child. Therefore, the youngest child must be Kara and Curt is the oldest child. The completed table could look like the following one.
Check your work. Check your solution that Kara is the youngest, Tim is the middle child, and Curt is the oldest. This satisfies the clue that Kara is younger than Curt. It also is consistent with the fact that Tim is not the oldest or the youngest. This leaves Curt as the oldest child in the family. This solution is checking.
Find practice problems and solutions for these concepts at The Strategies of Working Backward and Using Logical Reasoning in Word Problems Practice Questions.
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