If 75% of a class answered the first question on a test correctly, 55%…
2026
If 75% of a class answered the first question on a test correctly, 55% answered the second question correctly, and 20% answered neither question correctly, what percentage answered both questions correctly?
Answer: B. 50% — CONCEPTFor two groups A and B, inclusion–exclusion gives P(A ∪ B) = P(A) + P(B) − P(A ∩ B). The subtraction removes the overlap that was counted once in each…
- A.
40%
- B.
50%
- C.
60%
- D.
45%
Attempted by 2678 students.
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Correct answer: B
CONCEPT
For two groups A and B, inclusion–exclusion gives P(A ∪ B) = P(A) + P(B) − P(A ∩ B). The subtraction removes the overlap that was counted once in each group.
The complement rule gives P(A ∪ B) = 100% − P(neither).
APPLICATION
Let A be the students who answered the first question correctly and B the students who answered the second correctly. Then P(A) = 75%, P(B) = 55%, and P(neither) = 20%.
Find the percentage who answered at least one question correctly: P(A ∪ B) = 100% − 20% = 80%.
Apply inclusion–exclusion: 80% = 75% + 55% − P(A ∩ B).
Rearrange: P(A ∩ B) = 75% + 55% − 80% = 50%.
CROSS-CHECK
With 50% in both groups, the first-only group is 75% − 50% = 25% and the second-only group is 55% − 50% = 5%. The four disjoint groups total 25% + 50% + 5% + 20% = 100%, while the two original totals remain 75% and 55%.
Therefore, 50% of the class answered both questions correctly.