In a state bar exam with two sections, 35% of candidates failed one

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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SOLUTION
This overlapping sets problem derives much of its difficulty from the fact that the information is given in terms of "failed" while the question asks for "passed." But this also provides you with a way to derive a logical shortcut: if you set up a Venn Diagram with the given information:

You may see that using the inside circles will tell you how many people failed exactly one test: those who failed only X but not Y plus those who failed only Y but not X must therefore have passed exactly one section. Since "X only" can be calculated as "X total" minus "Both" (and Y can be calculated similarly, you can fill in those partial circles: with failed x=35% and failed Y=42% and Both=15%

You may see that using the inside circles will tell you how many people failed exactly one test: those who failed only X but not Y plus those who failed only Y but not X must therefore have passed exactly one section. Since "X only" can be calculated as "X total" minus "Both" (and Y can be calculated similarly, you can fill in those partial circles: failed only X=35-15=20% and failed only Y=42-15=27%

And then know that the 20% who failed X and passed Y plus the 27% who failed Y and passed X sums to 47%. And 47% of 5000 is 2350, so the answer is C.

If you don't see that shortcut you can still get to work in a more classic Venn construct. Here it is best to begin working using the "failed" information, as it is already provided for you in classic Venn Diagram friendly form. If you account for that given information in a diagram it will again look like:
Since the Venn formula is X + Y - Both + Neither = Total

35 + 42 - 15 + Neither = 100

62 + Neither = 100

Neither = 38. So 38% of people failed neither exam.

What does that mean in the context of the question? The people who passed neither exam thus passed both exams. So if 38% passed both and 15% failed both, then the remaining 47% passed exactly one exam. 47% of 5000 is 2350, so answer choice C is correct.
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IMO C
35% of 5000=1750
42% of 5000=2100
30% of 5000=1500
Now we have to subtract twice of 15% from sum of 1750 and 2100 to get candidates who passed in only one subject.

IMO B
65% (5000) = 3250
58% (5000) = 2900
15% failed both sections = 750

Required no. (Students who passed in one section but not both)
=3250 - 750 = 2500 OR
= 2900- 750 = 2150

Hence, B

Please correct me if I am thinking in the wrong direction.
Thank you

Sent from my SM-N9200 using GMAT Club Forum mobile app

khushbumodi

from where u get above highlighted part??

thanks

Hello

Nothing appears to be highlighted.
Please resend the message

Thanks

Sent from my SM-N9200 using GMAT Club Forum mobile app

He calculated the pass percentage rates for each section instead of the failed percentage rates.

Hi khushbumodi

You can Use Double Matrix to Solve this Problem, which is pretty easy, Solution Image attached

With the pass data you have got all the people who have passed one , two or both. You have to minus the passed in both to get the answer. This way it is more time consuming to solve.

Posted from my mobile device

I used a Venn diagram but took a different route (no percents after initial calculation).

I focused on calculating "failed only one."

From percentages given, find the numbers of
candidates who failed first, second, or both sections

With those three sets of numbers, we have "captured" the population
in whom we are interested, because

Passed both are excluded entirely (we want pass one)
Passed neither (passed 0 = failed both) get subtracted (we want pass one)
"Failed only one" is equivalent to "passed only one."

We are down to two sets that mirror one another.
Members of either set failed one section only.
Those members simultaneously also passed one section only.

If you failed one section but not the other, then
you also passed one section but not the other

METHOD
Calculate the actual numbers of candidates who
• failed Section #1
• failed Section #2
• failed BOTH sections

NUMBER of candidates who failed one or two sections
Section #1, 35% failed: (.35*5,000) = 1,750 TOTAL
Section #2, 42% failed: (.42*5,000) = 2,100 TOTAL
Both sections, 15% failed: (.15*5,000) = 750

STEPS
Draw two intersecting circles
Write totals for S #1 and S #2 outside

Next steps:
1) Inside first. Put 750 in the gray overlap (failed both #1 and #2) = 750

2) S #1 ONLY? Pink shading
S #1 ONLY = (Section 1 TOTAL - overlap). S #1 ONLY: (1,750 - 750) = 1,000

3) S #2 ONLY? Green shading. (2,100 - 750) = 1,350
S #2 ONLY = (Section 2 TOTAL - overlap)

4) Add the ONLY parts: (1,000 + 1,350) = 2,350 candidates
failed one section but not the other, and thus passed one section but not the other

Answer C

EDIT: AweG , of the great username, I meant to give you kudos.
By "different route" I mean that once I figured out the number who
failed one only, I did not extrapolate percentages as you did. +1
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Off top of head:

2 circles. Failed sect 1 = 35% - 15% = 20%.
Failed sect 2 = 42% - 15% = 27%.
20% + 27% = 47%.
47% of 5000 is slightly less than 50% of 5000 = 2500. 2350 is closest answer.

Hii,
Till table I am able to get.But how to come to 1350+1000? why not 1900+750? I am little confused..

Bunuel or VeritasKarishma could u provide your 2 cents to this question?

Failed one = 35%
Failed other = 42%
Failed Both = 15%

Failed at least one section = 35 + 42 - 15 = 62%

So 62 - 15 = 47% failed EXACTLY one section. If they failed exactly one section, then they passed exactly one section too.

47% of 5000 is 3% less than half which means 3*50 less than 2500 which gives 2350.

Answer (C)
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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.
Didnt really get this line. for Example : If 60 students passed one section and not the other. Ho can we say these 60 students failed one section and not the other

Reply:

Suppose the first section is verbal and the second section is quantitative. “candidates who passed one section but not both” include 1) candidates who only pass verbal, and 2) candidates who only pass quantitative. For the first group, since they only passed verbal and since there are two sections in total, it must be true that these candidates failed quantitative. Similarly, the candidates in the second group must have failed verbal. Thus, we can rewrite the groups as 1) candidates who only fail quantitative, and 2) candidates who only fail verbal. As you can see, “candidates who passed one section but not both” and “candidates who failed one section but not both” correspond to the exact same group of people.

Regarding your example, if 60 students passed one section and not the other; then these 60 students failed one section and not the other. Notice that “not passing a section” and “failing a section” is the same thing; just like “not failing a section” and “passing a section” is the same thing.
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