Lesson 34 Using Reasoning to Solve Problems

In this lesson you will use logical reasoning to solve such problems as finding a missing digit in a sum or using a pattern to find a missing number or equation. In one special type of problem, you might be asked to figure out what different sets have in common. For these problems, you will have to know about Venn diagrams. Venn diagrams consist of circles that stand for sets of elements. These elements can be numbers, letters, people, animals, and many other things. In this lesson you will use logical reasoning to solve such problems as finding a missing digit in a sum or using a pattern to find a missing number or equation. In one special type of problem, you might be asked to figure out what different sets have in common. For these problems, you will have to know about Venn diagrams. Venn diagrams consist of circles that stand for sets of elements. These elements can be numbers, letters, people, animals, and many other things.

These circles are usually placed inside a rectangle that is the boundary of all the elements under consideration. Study this diagram: These circles are usually placed inside a rectangle that is the boundary of all the elements under consideration. Study this diagram:

The inside of the rectangle stands for all 8 th graders. The three circles show that some of the 8 th graders play in the orchestra, some play in the band, and some sing in the chorus. Notice the overlapping parts. For example, the part of the diagram where the circle for band overlaps the circle for chorus means that there are 8 th graders who take part in both band and chorus. Here is a sample question that could be asked using this diagram. The inside of the rectangle stands for all 8 th graders. The three circles show that some of the 8 th graders play in the orchestra, some play in the band, and some sing in the chorus. Notice the overlapping parts. For example, the part of the diagram where the circle for band overlaps the circle for chorus means that there are 8 th graders who take part in both band and chorus. Here is a sample question that could be asked using this diagram.

Example 1 Which of these generalizations can be made from the Venn diagram on the previous slide? Which of these generalizations can be made from the Venn diagram on the previous slide? A All chorus members are in the band. A All chorus members are in the band. B All 8 th graders are in either the band or orchestra. B All 8 th graders are in either the band or orchestra. C No chorus member plays in the orchestra and band. C No chorus member plays in the orchestra and band. D Some chorus members play in the orchestra and band. D Some chorus members play in the orchestra and band.

Strategy: Study the diagram as you examine each statement. Strategy: Study the diagram as you examine each statement. Statement A: Not true. If it were true, then the entire circle for the chorus would be inside the circle for the band. Statement A: Not true. If it were true, then the entire circle for the chorus would be inside the circle for the band. Statement B: Not true. There are 8 th graders in the chorus who are not in the band or orchestra. There are also 8 th graders who are not in any of the three activities. Statement B: Not true. There are 8 th graders in the chorus who are not in the band or orchestra. There are also 8 th graders who are not in any of the three activities.

Statement C: Not true. The circle for chorus overlaps the circles for the band and the orchestra. Statement C: Not true. The circle for chorus overlaps the circles for the band and the orchestra. Statement D: This statement is true. The circle for the chorus overlaps the circle for the orchestra and the circle for the band. Statement D: This statement is true. The circle for the chorus overlaps the circle for the orchestra and the circle for the band.

Solution The answer is D. The answer is D. The next example is another type of logical reasoning: working a problem backward. The next example is another type of logical reasoning: working a problem backward.

Example 2 If you know that the average of 4 numbers is 13, and the sum of the first three numbers is 40, then what is the fourth number? If you know that the average of 4 numbers is 13, and the sum of the first three numbers is 40, then what is the fourth number? Strategy: Remember that the average of 4 numbers is the sum of the numbers divided by 4. Strategy: Remember that the average of 4 numbers is the sum of the numbers divided by 4.

Step 1: Find the sum of the 4 numbers. This is the backwards part. You know the average is 13, so you find the sum by multiplying 13 by 4. The sum is 13 x 4 = 52. Step 1: Find the sum of the 4 numbers. This is the backwards part. You know the average is 13, so you find the sum by multiplying 13 by 4. The sum is 13 x 4 = 52. Step 2: Subtract 40 from the sum. Step 2: Subtract 40 from the sum. 52 – 40 = – 40 = 12

Solution The fourth number is 12. The fourth number is 12.

Example 3 Given these two statements, which conclusion can you make? All pencils come with an eraser. Donna has a yellow pencil. Given these two statements, which conclusion can you make? All pencils come with an eraser. Donna has a yellow pencil. A Donna’s pencil does not have an eraser. A Donna’s pencil does not have an eraser. B Donna’s pencil has an eraser. B Donna’s pencil has an eraser. C Donna does not have a pencil. C Donna does not have a pencil. D No conclusion can be made. D No conclusion can be made.

Strategy: Read the two statements carefully. As you can see, Donna has a pencil. Don’t let the “yellow” part of Statement 2 fool you. Donna’s pencil must have an eraser, since Statement 1 says all pencils have erasers. Strategy: Read the two statements carefully. As you can see, Donna has a pencil. Don’t let the “yellow” part of Statement 2 fool you. Donna’s pencil must have an eraser, since Statement 1 says all pencils have erasers.

Solution The conclusion is B. The conclusion is B.