A professor gave his 40 students a test with three questions. Every student answered at least one question. Ten student did not answer the first question, 14 did not answer the second question, and 12 did not answer the third question. If 18 students answered all three questions, how many answered exactly one question?
idk if non tutors can answer questions or not, so sorry if i'm stepping outta line, but 14 is the answer. could u help me out with the question i just posted?
Lots of our students are helped by other students. You are welcome to help answer questions any time you are sure of the answer. But also remember, it helps students to know HOW to find the answers so they'll know how to the same type of problem next time.
oh ok thx i'll remember that next time
To find the number of students who answered exactly one question, we need to analyze the given information step by step.
First, let's sum up the number of students who did not answer each question:
- 10 students did not answer the first question.
- 14 students did not answer the second question.
- 12 students did not answer the third question.
Now, let's find the total number of students who answered at least one question. We can use the principle of inclusion-exclusion for this:
Total = Students answered Q1 + Students answered Q2 + Students answered Q3 - Students answered (Q1 and Q2) - Students answered (Q1 and Q3) - Students answered (Q2 and Q3) + Students answered (Q1, Q2, and Q3).
We know that 18 students answered all three questions, so we can revise the equation as follows:
Total = Students answered Q1 + Students answered Q2 + Students answered Q3 - Students answered (Q1 and Q2) - Students answered (Q1 and Q3) - Students answered (Q2 and Q3) + 18
We also know that there are 40 students in total, so:
Total = 40
Substituting this value into the equation, we get:
40 = Students answered Q1 + Students answered Q2 + Students answered Q3 - Students answered (Q1 and Q2) - Students answered (Q1 and Q3) - Students answered (Q2 and Q3) + 18
We can simplify the equation by combining the intersections:
40 = Students answered Q1 + Students answered Q2 + Students answered Q3 - 2(Students answered (Q1 and Q2)) - 2(Students answered (Q1 and Q3)) - 2(Students answered (Q2 and Q3)) + 18
Now, let's analyze the information we have:
- We want to find the number of students who answered exactly one question.
- We know that there are 18 students who answered all three questions, and each student can only answer a maximum of three questions.
- Therefore, the sum of the answer counts for each question must add up to 40 - 18 = 22.
Let's denote the number of students who answered exactly one question for Q1, Q2, and Q3 as X, Y, and Z, respectively.
From the given information, we can deduce the following equations:
Students answered Q1 = X + 2(Students answered (Q1 and Q2)) + 2(Students answered (Q1 and Q3)) + 18
Students answered Q2 = Y + 2(Students answered (Q1 and Q2)) + 2(Students answered (Q2 and Q3)) + 18
Students answered Q3 = Z + 2(Students answered (Q1 and Q3)) + 2(Students answered (Q2 and Q3)) + 18
Now we can substitute these equations back into the total equation:
40 = X + Y + Z + 2(Students answered (Q1 and Q2)) + 2(Students answered (Q1 and Q3)) + 2(Students answered (Q2 and Q3)) + 18 - 2(Students answered (Q1 and Q2)) - 2(Students answered (Q1 and Q3)) - 2(Students answered (Q2 and Q3)) + 18
Simplifying the equation further:
40 = X + Y + Z + 36
Now, rearrange the equation:
X + Y + Z = 4
Since we are looking for the number of students who answered exactly one question, the answer is 4.