Determine whether the following statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
In a survey, 140 students were taking mathematics, 50 were taking psychology, and 20 were taking neither. Thus, 210 students were surveyed.
Choose the correct answer below.
A. The statement is true.
B. The statement is false. A true statement would be "In a survey, 140 students were taking mathematics, 50 were taking psychology, and 20 were taking neither. Thus, 170 students were surveyed."
C. The statement is false. A true statement would be "In a survey, 140 students were taking mathematics, 50 were taking psychology, and 20 were taking neither. Thus, at most 210 students were surveyed."
Answer (2)
The total number of students surveyed can be calculated by adding the number of students in each category (mathematics, psychology, and neither) and subtracting the number of students taking both mathematics and psychology.
If no students take both mathematics and psychology, the total is 140 + 50 + 20 = 210 .
If some students take both, the total is less than 210.
Therefore, the statement is false, and a true statement would be "at most 210 students were surveyed." OC
Explanation
Analyze the problem Let's analyze the given information.
We have:
140 students taking mathematics
50 students taking psychology
20 students taking neither
The statement claims that 210 students were surveyed. We need to determine if this is true or false. The key here is to consider whether any students are taking both mathematics and psychology.
Set up the equation Let M be the number of students taking mathematics, P be the number of students taking psychology, and N be the number of students taking neither. Let B be the number of students taking both mathematics and psychology.
We are given: M = 140 P = 50 N = 20
The total number of students surveyed is given by: T o t a l = M + P + N − B T o t a l = 140 + 50 + 20 − B T o t a l = 210 − B
If B = 0 (no students take both subjects), then the total number of students is 210. However, if 0"> B > 0 , then the total number of students is less than 210.
Determine the correct statement Since it's possible that some students take both mathematics and psychology, the statement that exactly 210 students were surveyed is not necessarily true. The maximum number of students surveyed is 210, which occurs when no students take both subjects. Therefore, the statement is false.
The correct statement would be: "In a survey, 140 students were taking mathematics, 50 were taking psychology, and 20 were taking neither. Thus, at most 210 students were surveyed."
Final Answer The statement is false. The correct statement is: "In a survey, 140 students were taking mathematics, 50 were taking psychology, and 20 were taking neither. Thus, at most 210 students were surveyed."
Examples
Imagine you're planning activities for a school club. 140 students are in the math club, 50 are in the psychology club, and 20 aren't in either. To know how many students you need to cater to, you need to consider that some students might be in both clubs. If no one is in both, you'd cater to 210 students. But if some students are in both, you'd cater to fewer. This is why 'at most' is important.
The statement is false because it does not account for students who might be enrolled in both mathematics and psychology. A true statement would be that at most 210 students were surveyed, considering the overlap. Therefore, the correct choice is C.
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