Bunuel wrote:
In a state bar exam with two sections, 35% of candidates failed one section, 42% failed the other section, and 15% failed both sections. If 5000 candidates took the exam, how many passed one section but not both?
A. 650
B. 2150
C. 2350
D. 4150
E. 4500
We note that “the number of candidates who passed one section but not both” is equivalent to “the number of candidates who failed one section but not both.” This is because the candidates who failed both sections do not satisfy the condition of having passed one section, and the candidates who failed neither section will have passed both sections and therefore are not included in the group of people in which we’re interested.
We can create the following equation:
Total # people = # who failed one section + # who failed the other section - # who failed both sections + # who failed neither section
5,000 = 0.35(5,000) + 0.42(5,000) - 0.15(5,000) + N
5,000 = 1,750 + 2,100 - 750 + N
5,000 = 3,100 + N
1,900 = N
Notice that 750 in the equation above is the the number of people who failed both sections and N = 1,900 is the number of people who failed neither section. In other words, 1,900 people passed both sections. Since we want the number of people who pass only one section, we need to exclude both 750 and 1,900 from 5,000. Therefore, there are 5,000 - 750 - 1,900 = 2,350 people who pass one section but fail the other.
Alternate Solution:
We note that “the number of candidates who passed one section but not both” is equivalent to “the number of candidates who failed one section but not both.”
We can use the formula:
#who failed one section but not both = (#who failed one section - #who failed both) + (#who failed the other section - #who failed both)
We note that:
#who failed one section = 0.35(5,000) = 1750
#who failed the other section = 0.42(5,000) = 2100
#who failed both = 0.15(5,000) = 750
Thus,
#who failed one section but not both = (1750 - 750) + (2100 - 750) = 1000 + 1350 = 2350.
Answer: C
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