A professor gave his 40 students a test with three questions. Every student answered at least one question. Ten didn't answer the first question. 14 didn't answer the second question. 12 didn't answer the third question. If 18 students answered all three questions, how many answered exactly one question?
Thanks for your help
t = two answers , x = one answer
(3 * 40) - (3 * 18) - 10 - 14 - 12 = 2 t + x
40 - 18 = t + x
solve the system for x
Another way to solve.
Let
A=# of students who answered the first question
B=# of students who answered the second question
C=# of students who answered the third question
Then
we are given
B+C-|B∩C| = 10
A+C-|A∩C| = 14
A+B-|A∩B| = 12
Add three equations
2A+2B+2C-(|A∩B|+|B∩C|+|C∩A|)=36....(1)
But the total number who answered one OR two question is 40-18=22, or
A+B+C-|A∩B|- |B∩C|-|C∩A|=22.....(2)
(1)-(2)
A+B+C=36-22=14
Well, it seems like these students really know how to dodge questions! Let's do some math to solve this mystery.
We know that there are 40 students in total. Out of these students, 10 didn't answer the first question, 14 didn't answer the second question, and 12 didn't answer the third question.
Now, let's find the number of students who answered only one question. We can start by adding up the number of students who didn't answer each question: 10 + 14 + 12 = 36.
Since every student answered at least one question, we can subtract the total students who didn't answer any question from the total number of students: 40 - 36 = 4.
However, we also know that 18 students answered all three questions. So, if we subtract these students from the total number of students who answered only one question, we get 4 - 18 = -14.
Oops! It seems like there's a problem here. According to my calculations, we have a negative number of students who answered exactly one question.
Either there's a mistake in the information given or I need to recalibrate my humor circuits. Let's try again, shall we?
To find the number of students who answered exactly one question, we can use the principle of inclusion-exclusion.
Let's break it down step by step:
1. Total number of students = 40
2. Number of students who didn't answer the first question = 10
3. Number of students who didn't answer the second question = 14
4. Number of students who didn't answer the third question = 12
5. Number of students who answered all three questions = 18
To find the number of students who answered exactly one question:
Step 1: Calculate the number of students who answered at least one question:
a. Number of students who answered the first question = Total number of students - Number of students who didn't answer the first question = 40 - 10 = 30
b. Number of students who answered the second question = Total number of students - Number of students who didn't answer the second question = 40 - 14 = 26
c. Number of students who answered the third question = Total number of students - Number of students who didn't answer the third question = 40 - 12 = 28
Step 2: Calculate the number of students who answered exactly two questions:
a. Number of students who answered both the first and second questions = Number of students who answered the first question + Number of students who answered the second question - Number of students who answered all three questions = 30 + 26 - 18 = 38
b. Number of students who answered both the first and third questions = Number of students who answered the first question + Number of students who answered the third question - Number of students who answered all three questions = 30 + 28 - 18 = 40
c. Number of students who answered both the second and third questions = Number of students who answered the second question + Number of students who answered the third question - Number of students who answered all three questions = 26 + 28 - 18 = 36
Step 3: Now, calculate the number of students who answered exactly one question:
Number of students who answered exactly one question = Number of students who answered the first question - Number of students who answered both the first and second questions - Number of students who answered both the first and third questions = 30 - 38 + 40 = 32
Therefore, there are 32 students who answered exactly one question.
To find the number of students who answered exactly one question, you need to find the total number of students who answered each question individually and then add them up.
Let's break down the information given:
- 10 students didn't answer the first question.
- 14 students didn't answer the second question.
- 12 students didn't answer the third question.
- 18 students answered all three questions.
To find the number of students who answered the first question, subtract the number of students who didn't answer it from the total number of students:
Total students - Students who didn't answer first question = Students who answered first question
40 - 10 = 30 students answered the first question.
Similarly, 26 students answered the second question (40 - 14) and 28 students answered the third question (40 - 12).
To find the number of students who answered exactly one question, you need to add up the number of students who answered each question individually:
Students who answered only the first question + Students who answered only the second question + Students who answered only the third question
To calculate this, you can use the principle of inclusion-exclusion. Start by adding the students who answered each question individually:
Students who answered only the first question = Students who answered the first question - Students who answered all three questions = 30 - 18 = 12 students.
Similarly:
Students who answered only the second question = 26 - 18 = 8 students.
Students who answered only the third question = 28 - 18 = 10 students.
Finally, add up the three values to get the total number of students who answered exactly one question:
Students who answered exactly one question = Students who answered only the first question + Students who answered only the second question + Students who answered only the third question = 12 + 8 + 10 = 30 students.
Therefore, 30 students answered exactly one question.