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# Of the N students in a certain class, X took a course in biology and Y

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Math Expert
Joined: 02 Sep 2009
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Of the N students in a certain class, X took a course in biology and Y  [#permalink]

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06 Dec 2019, 05:08
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Difficulty:

55% (hard)

Question Stats:

64% (02:22) correct 36% (02:31) wrong based on 41 sessions

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Of the N students in a certain class, X took a course in biology and Y took a course in history, and Z took neither of the courses. If there were students who took both two courses, in terms of N, X, Y and Z, what fraction of the students took both two courses?

(A) $$1 - \frac{X+Y+Z}{N}$$

(B) $$\frac{X+Y+Z}{N} - 1$$

(C) $$\frac{X+Y-Z-N}{N}$$

(D) $$\frac{N+X+Y-Z}{N}$$

(E) $$\frac{N-X-Y+Z}{N}$$

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Re: Of the N students in a certain class, X took a course in biology and Y  [#permalink]

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06 Dec 2019, 05:25
N-Z=X+Y-Both
Both= X+Y-N+Z

Both/N= (X+Y-N+Z)/N

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Re: Of the N students in a certain class, X took a course in biology and Y  [#permalink]

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07 Dec 2019, 02:59
make 2x2 matrix

---B-----NB---total
H--(y-(n-x)-z)---(n-x)-z---y
NH--(n-y)-z---z----n-y
total--x----n-x---n
fraction
(y-(n-x)-z)/n
or say
x+y+z-n/n ;
$$\frac{X+Y+Z}{N} - 1$$
IMO B

Bunuel wrote:
Of the N students in a certain class, X took a course in biology and Y took a course in history, and Z took neither of the courses. If there were students who took both two courses, in terms of N, X, Y and Z, what fraction of the students took both two courses?

(A) $$1 - \frac{X+Y+Z}{N}$$

(B) $$\frac{X+Y+Z}{N} - 1$$

(C) $$\frac{X+Y-Z-N}{N}$$

(D) $$\frac{N+X+Y-Z}{N}$$

(E) $$\frac{N-X-Y+Z}{N}$$

Are You Up For the Challenge: 700 Level Questions
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Joined: 17 Sep 2014
Posts: 332
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GMAT 1: 780 Q51 V45
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Re: Of the N students in a certain class, X took a course in biology and Y  [#permalink]

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07 Dec 2019, 10:24
Bunuel wrote:
Of the N students in a certain class, X took a course in biology and Y took a course in history, and Z took neither of the courses. If there were students who took both two courses, in terms of N, X, Y and Z, what fraction of the students took both two courses?

(A) $$1 - \frac{X+Y+Z}{N}$$

(B) $$\frac{X+Y+Z}{N} - 1$$

(C) $$\frac{X+Y-Z-N}{N}$$

(D) $$\frac{N+X+Y-Z}{N}$$

(E) $$\frac{N-X-Y+Z}{N}$$

Are You Up For the Challenge: 700 Level Questions

If there are two categories A and B, we have the formula Total = A + B - Both + Neither. To go into detail, the A category includes "A only" and "Both A and B". The B category includes "B only" and "Both A and B". So we double-counted "Both" if we just add everything up, hence we take it away once from the total. Here we want to find both,

Both = A + B + Neither - Total = X + Y + Z - N.
Finally, the fraction out of the total of N would be $$\frac{{X + Y + Z - N} }{ N} = \frac{{X + Y + Z} }{ N} - 1$$. The answer would be B.
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Re: Of the N students in a certain class, X took a course in biology and Y  [#permalink]

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07 Dec 2019, 10:45
Bunuel wrote:
Of the N students in a certain class, X took a course in biology and Y took a course in history, and Z took neither of the courses. If there were students who took both two courses, in terms of N, X, Y and Z, what fraction of the students took both two courses?

(A) $$1 - \frac{X+Y+Z}{N}$$

(B) $$\frac{X+Y+Z}{N} - 1$$

(C) $$\frac{X+Y-Z-N}{N}$$

(D) $$\frac{N+X+Y-Z}{N}$$

(E) $$\frac{N-X-Y+Z}{N}$$

Are You Up For the Challenge: 700 Level Questions

N = X + Y - Both +Z
Both = X + Y + Z - N

Fraction of students who took both the courses = (X + Y + Z - N)/N = (X+Y+Z)/N - 1

IMO B
Re: Of the N students in a certain class, X took a course in biology and Y   [#permalink] 07 Dec 2019, 10:45
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