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# The Strategies of Working Backward and Using Logical Reasoning in Word Problems Study Guide (page 2)

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Updated on Oct 3, 2011

## 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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