Question: If 75 percent of a class answered the first question on a certain test correctly, 55 percent answered the second question on the test correctly, and 20 percent answered neither of the questions correctly, what percent answered both correctly?
(A) 10%
(B) 20%
(C) 30%
(D) 50%
(E) 65%
“If 75 percent of a class answered the first question on a certain test correctly”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Official Guide Quantitative Review". To solve GMAT Problem Solving questions a student must have knowledge about a good number of qualitative skills. GMAT Quant section consists of 31 questions in total. The GMAT quant topics in the problem-solving part require calculative mathematical problems that should be solved with proper mathematical knowledge.
Solution and Explanation:
Approach Solution 1:
Here in the above question, let the number of students be 100. No. of students who answered the first question correctly is equal to 75. Those who didn't equal 25. No. of students who answered the second question correctly equals 55. Those who didn't equals 45. Now the no. of students who didn't answer either of the questions correctly is 20. This indicates that out of the students who weren't able to solve the first question correctly. Also, 20 weren't able to solve the second one either. Then the number of students who weren't able to solve the first question but solved the second question becomes 25-20 = 5. Similarly, among the students who weren't able to solve the second question, 20 did not solve the first one correctly. So the number of students who solved the second question incorrectly but solved the first one correctly becomes 45-20 = 25.
Hence, no. of students who solved both of them = (total students - students who solved the first one - students who solved the second one - students who solved neither of them) = (100 - 25 - 5 - 20) = 50
Therefore, the percentage of students who solved both = students who solved both/total students * 100
This makes 50/100 × 100 = 50%
Hence the required number of students who solved both correctly is 50%.
Correct Answer: D
Approach Solution 2:
This problem can be alternatively solved using sets formula. Let A be the group of students who answered the 1st question correctly while B be the group of students who answered the 2nd question correctly. Let us consider that there are 100 students in the class. (“∩” illustrates intersection)
Given, n(A) = 75 n(B) = 55 , n(A' ∩ B') = 20
n(A'∩B') = 100 - n(A U B) ⇒ n(A U B) = 100- 20 = 80
n(A U B) = n(A) + n(B) - n(A ∩ B) ⇒ 80= 75 + 55 - n(A ∩ B)
n(A ∩ B) = 50
Hence the required number of students who solved both correctly is 50%.
Correct Answer: D
Approach Solution 3:
Let the student count in the going to keep be 100. There were 75 students who successfully answered the first question. those that fell short of 25. There were 55 students who properly answered the second question. There were 45 who didn't. There are now 20 students who misread either of the questions. This shows that some of the students were unable to correctly answer the first question. 20 more people also failed to answer the second problem. Thus, 25-20 = 5 students were able to solve the second question but were unable to do so with the first. In a similar manner, 20 of the students who were unable to answer the second question properly also failed to answer the previous one. Therefore, 45-20 = 25 students correctly answered the first question but incorrectly answered the second.
Therefore, the number of students who correctly answered both of them is equal to (total students - students who correctly answered the first one - students who correctly answered the second one - students who correctly answered neither of them) = (100 - 25 - 5 - 20) = 50.
Thus, the ratio of students who completed both problems to the total number of students is equal to 100.
Thus, 50% is equal to 50/100 x 100.
Therefore, 50% of students must have answered both questions correctly.
Correct Answer: D
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