If 75% of a class answered the first question on a certain test correctly, 55%…
2024
If 75% of a class answered the first question on a certain test correctly, 55% answered the second question on the test correctly, and 20 percent answered neither of the questions correctly, what percentage answered both correctly?
Answer: C. 50 — Concept For two sets A and B, inclusion–exclusion gives P(A ∪ B) = P(A) + P(B) − P(A ∩ B). The complement rule gives P(A ∪ B) = 1 − P(neither). These…
- A.
30
- B.
40
- C.
50
- D.
60
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Show answer & explanation
Correct answer: C
Concept
For two sets A and B, inclusion–exclusion gives P(A ∪ B) = P(A) + P(B) − P(A ∩ B). The complement rule gives P(A ∪ B) = 1 − P(neither). These identities convert the three given percentages into the overlap.
Application
Let A represent students who answered the first question correctly and B represent those who answered the second correctly.
Since 20% answered neither question correctly, the percentage who answered at least one correctly is 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, 25% are in A only and 5% are in B only. Then 25% + 50% + 5% + 20% = 100%, while A totals 75% and B totals 55%, so every given percentage is satisfied.
Result
Therefore, 50% of the class answered both questions correctly.