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A law school admissions office sent letters to each of its 200 applica

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A law school admissions office sent letters to each of its 200 applica  [#permalink]

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New post 24 Nov 2016, 02:42
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A law school admissions office sent letters to each of its 200 applicants, but because of an error in the mail room the acceptance and denial letters did not all go to the proper recipients. 40% of those who should have received denial letters received acceptance letters instead, and 10% of those who were supposed to receive acceptances received denial letters. If 160 applicants received acceptance letters, how many applicants who should have received acceptance letters instead received denial letters?

A. 16
B. 20
C. 24
D. 28
E. 32

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Re: A law school admissions office sent letters to each of its 200 applica  [#permalink]

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New post 24 Nov 2016, 03:41
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Bunuel wrote:
A law school admissions office sent letters to each of its 200 applicants, but because of an error in the mail room the acceptance and denial letters did not all go to the proper recipients. 40% of those who should have received denial letters received acceptance letters instead, and 10% of those who were supposed to receive acceptances received denial letters. If 160 applicants received acceptance letters, how many applicants who should have received acceptance letters instead received denial letters?

A. 16
B. 20
C. 24
D. 28
E. 32


Call \(x\) is the number of applicants who should have received acceptance letters
Call \(y\) is the number of applicants who should have received denial letters
We have \(x+y=200\)

10% of those who were supposed to receive acceptances, or \(0.1x\), received denial letters. This leads to 90% of those who were supposed to receive acceptances, or \(0.9x\), received acceptances letters.
40% of those who should have received denial letters, or \(0.4y\), received acceptance letters instead.

Since there are 160 applicants received acceptance letters, we have \(0.9x+0.4y=160\).

Now we have:
\(\begin{cases}
x+y=200\\
0.9x+0.4y=160\\
\end{cases}\)

Solve this we have \(x=160\) and \(y=40\).

The number of applicants who should have received acceptance letters instead received denial letters is \(0.1x=16\).

The answer is A
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Re: A law school admissions office sent letters to each of its 200 applica  [#permalink]

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New post 24 Nov 2016, 02:57
Bunuel wrote:
A law school admissions office sent letters to each of its 200 applicants, but because of an error in the mail room the acceptance and denial letters did not all go to the proper recipients. 40% of those who should have received denial letters received acceptance letters instead, and 10% of those who were supposed to receive acceptances received denial letters. If 160 applicants received acceptance letters, how many applicants who should have received acceptance letters instead received denial letters?

A. 16
B. 20
C. 24
D. 28
E. 32

I would go with B.160 applicants received the letters which represent the 40 percent of the total number of students.So 100 percent is 400 students.Now,the ratio of letters sent to applicants to total applicants is 200/400.It implies that the applicants who should have received acceptance letters instead received denial letters would be 400*10%*200/400=20

Please correct me if I am incorrect.
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Re: A law school admissions office sent letters to each of its 200 applica  [#permalink]

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New post 25 Jul 2018, 17:34
Bunuel wrote:
A law school admissions office sent letters to each of its 200 applicants, but because of an error in the mail room the acceptance and denial letters did not all go to the proper recipients. 40% of those who should have received denial letters received acceptance letters instead, and 10% of those who were supposed to receive acceptances received denial letters. If 160 applicants received acceptance letters, how many applicants who should have received acceptance letters instead received denial letters?

A. 16
B. 20
C. 24
D. 28
E. 32


We can let r = the number of applicants who should have received acceptance letters; thus, (200 - r) = the number of applicants who should have received denial letters.

Thus, 0.9r = the number of applicants who should have received acceptance letters and received acceptance letters, and 0.1r = the number of applicants who should have received acceptance letters but received denial letters instead. Similarly, 0.6(200 - r) = the number of applicants who should have received denial letters and received denial letters, and 0.4(200 - r) = the number of applicants who should have received denial letters but received acceptance letters instead.

Therefore, we can create the following equation for the total number of applicants who received acceptance letters:

0.9r + 0.4(200 - r) = 160

9r + 4(200 - r) = 1600

9r + 800 - 4r = 1600

5r = 800

r = 160

Since 10% of the 160 applicants who should have received acceptance letters received denial letters, 16 of them received denial letters.

Answer: A
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Re: A law school admissions office sent letters to each of its 200 applica  [#permalink]

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New post 14 Oct 2018, 14:47
Bunuel wrote:
A law school admissions office sent letters to each of its 200 applicants, but because of an error in the mail room the acceptance and denial letters did not all go to the proper recipients. 40% of those who should have received denial letters received acceptance letters instead, and 10% of those who were supposed to receive acceptances received denial letters. If 160 applicants received acceptance letters, how many applicants who should have received acceptance letters instead received denial letters?

A. 16
B. 20
C. 24
D. 28
E. 32

Another very nice problem in which "blending" the k technique and the grid (double-matrix) "shields" the problem fast and clear!

(At least to my students) Study this problem asking yourself why start putting in the grid the 10k "value" was a smart move.

Image

\(? = {1 \over {10}}\left( {200 - 10k} \right) = 20 - k\)

\(160 - {2 \over 5}\left( {10k} \right) = {9 \over {10}}\left( {200 - 10k} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,160 - 4k = 180 - 9k\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,5k = 20\)

\(? = 16\)


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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Re: A law school admissions office sent letters to each of its 200 applica  [#permalink]

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New post 14 Oct 2018, 15:05
Let us assume the actual candidates to be denied as D and those accepted as A

Now as per question:
0.4D + 0.9 A=160
0.4(200-A) +0.9A=160 {As A+D=200}
Hence A=160

Applicants who should have received acceptance letters instead received denial letters =0.1*A =16

Hence Answer is A
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Re: A law school admissions office sent letters to each of its 200 applica   [#permalink] 14 Oct 2018, 15:05
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