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

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Updated on Oct 3, 2011

### Practice 2

Using a Venn diagram or using logic with a table can be very helpful when you are solving these types of math word problems. Work through the questions in the practice set that follows. Apply these strategies to help find a solution, and be sure to check your answers.

#### Problems

1. At a middle school, there are 75 students in drama club and 80 students in ski club. If 35 students participate in both drama club and ski club, how many students participate in only ski club?
2. A survey was given to Mrs. Miller's class about the pets they have. Of the 25 students in the class, 10 have a dog, 12 have a cat, and 5 do not have either pet. How many students in the class have both a dog and a cat?
3. Peter, Brenda, and Mike are three siblings in the same family and each likes a different color: red, blue, and green. Brenda's brother likes red. Blue was Mike's favorite color, but now he likes green. Which color does each person like?

#### Solutions

1. Read and understand the question. The question is asking for the number of students that participate in only ski club. The total number of students in drama club, the total number of students in ski club, and the number of students in both groups is given.
2. Devise a plan. Use a Venn diagram to answer this question. The Venn diagram should have a circle for the number of students in drama club that overlaps with a circle for the number of students in ski club. This overlapping section represents the number of students who are involved in both activities.

Carry out the plan. Start the diagram by drawing a rectangle that represents all of the students in the school, and then place the two overlapping circles within the rectangle. The diagram could appear like the one that follows.

Because the number of students who participate in both clubs is 35, place the 35 in the overlapping section between the circles. The 75 students in drama club represent all of the students in drama club, including the 35 who are in both clubs. Therefore, the number in the part of the circle for drama club that does not overlap is 75 – 35 = 40. In the same way, the 80 students in ski club also include the students in both groups. So, the value in the part of the circle for ski club that does not overlap should be 80 – 35 = 45. The Venn diagram should now look like the figure that follows.

Thus, the value in the part of the circle for ski club that does not overlap is the number of students in the ski club only. There are 45 students in only the ski club, and not in both clubs.

Check your work. Add the two numbers that make up each circle to check for accuracy. There are 40 students in drama club only, and 35 students in both clubs. This is a total of 40 + 35 = 75 students in the drama club. There are 45 students in ski club only and 35 students in both clubs for a total of 45 + 35 = 80 students in the ski club. This solution is checking.

3. Read and understand the question. The question is asking for the number of students who have both a dog and a cat. The number of students in the class, along with the number of students without a cat or a dog, the number who have a dog, and the number who have a cat are 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 have a dog that overlaps with a circle for the number of students who have a cat. This overlapping section represents the number of students who have both pets.

Carry out the plan. Start the diagram by drawing a rectangle that represents all of the students in the class, and then place the two overlapping circles within the rectangle. The diagram could appear like the one that follows.

Because the number of students who have neither pet is 5, place a 5 in the diagram inside the rectangle but outside of both circles. Then, subtract 25 – 5 = 20. This is the number of students who have one pet or both pets. Now, add the number of students who have either or both pets: 10 + 12 = 22. Because this is two more than 20, two students were counted in both categories. This means that 2 students have both a cat and a dog. So, the value in the overlapping section should be 2. Thus, there are 10 – 2 = 8 students who have a dog only and 12 – 2 = 10 students who have a cat only. The completed Venn Diagram should now look like the figure that follows.

Check your work. Add the number for each category to make sure that the total number of students in the class is 25. The number of students who do not have either pet is 5, the number who have a dog only is 8, the number with a cat only is 10, and the number with both a cat and a dog is 2: 5 + 8 + 10 + 2 = 25. Because there are 25 students in the class, this solution is checking.

4. Read and understand the question. The question is asking for the favorite color of three different people. Each person likes a different color of red, blue, or green.
5. Make a plan. Make a table that includes the three people in the question along the side. Then, make one column for red, one for blue, and one column for green. 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 favorite color for each person.

Carry out the plan. Start by making the table. A possible table follows.

Use the clue that Brenda's brother likes red. With this information, you can eliminate the fact that Brenda likes red, so place an "X" in that box. The second clue states that Mike used to like blue, but now green is his favorite, so place an "X" in both blue and red for Mike. Therefore, Peter's favorite color must be red and Brenda's favorite is blue. The completed table could look like the one that follows.

Check your work. Check your solution that Mike likes green, Brenda likes blue, and Peter likes red. This satisfies the clue that Mike used to like blue, but his favorite now is green. It also is consistent with the fact that Brenda's brother likes red, so Peter likes red. Then Brenda's favorite color is blue. This solution is checking.

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