SOLUTION: In a class, 85% of the students passed a test. 40% of the students who failed the test received a grade lower then 30. 20% of the students who passed the test received a grade high

Question 1201991: In a class, 85% of the students passed a test. 40% of the students who failed the test received a grade lower then 30. 20% of the students who passed the test received a grade higher than 90. Then, what percent of students scored between 30 and 90, inclusive?
(A) 68%
(B) 71%
(C) 74%
(D) 77%
(E) 80%
Found 2 solutions by Theo, ikleyn:
Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
85% passed the test.
15% failed.
40% of the students who failed received a grade lower than 30.
that means 60% of the student who failed received a grade greater then or equal to 30.
20% of the students who passed the test received a grade higher than 90.
that means 80% of the students who passed received a grade lower than or equal to 90.

since 85% passed the test, then 80% * 85% received a grade less than or equal to 90.
since 15% failed the test, then 60% * 15% received a grade greater than or equal to 30.
you have:
.8 * .85 = 68% passed the test and received a grade less than or equal to 90.
.6 * .15 = 9% failed the test and received a grade greater than or equal to 30.
total percent that received a grade between 30 and 90 would be 68% + 9% = 77%.

to see if this makes sense, assume that 1000 students took the test.
85% passed, so 850 passed and 150 failed.
40% of the student who failed the test gor a grade lower than 30.
that would be .4 * 150 = 60 who got a greae lower than 30.
that leaves 90 who failed and got a grade greater than or equal to 30.
20% of the students who passed the test got a grade higher than 90.
.2 * 850 = 170 who got a grade higher than 90.
that leaves 680 who passed and got a grade less than or equal to 90.
total between 30 and 90 would be 680 + 90 = 770
770/1000 = .77 = 77%.
this confirms the solution is correct.
looks like selection D is your answer.

Answer by ikleyn(52776)   (Show Source): You can put this solution on YOUR website!
.
In a class, 85% of the students passed a test. 40% of the students who failed the test
received a grade lower then 30. 20% of the students who passed the test
received a grade higher than 90. Then, what percent of students scored between 30 and 90, inclusive?
(A) 68%
(B) 71%
(C) 74%
(D) 77%
(E) 80%
~~~~~~~~~~~~~~~~~~~~~~~

        This problem admits short, clear and strightforward solution, as shown below.

85% of the students passed a test; hence, 15% failed. 40% of these 15% received a grade lower then 30. Hence, 0.4*0.15 = 0.06, or 6%, are out of our consideration. 20% of 85%, or 0.2*0.85 = 0.17 = 17%, received a grade higher than 90. These 17% also are out of our consideration. It is clear that these 6% set and 17% set are disjoint, since they are on opposite ends of the 100% set. Hence and therefore, the ANSWER to the problem's question is 100% - 6% - 17% = 77%.

Solved.

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