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At a certain school, the student to teacher ratio is 52 to 9

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At a certain school, the student to teacher ratio is 52 to 9

by Brent@GMATPrepNow » Tue Mar 26, 2019 8:14 am

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At a certain school, the student to teacher ratio is 52 to 9. If 38 students and 11 teachers leave, which of the following COULD represent the number of students and teachers remaining at the school?

A) 532 students and 88 teachers
B) 794 students and 162 teachers
C) 1106 students and 225 teachers
D) 1418 students and 241 teachers
E) 1728 students and 295 teachers

Hint: One approach will allow you to solve the question in under 20 seconds.
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At a certain school, the student to teacher ratio is 52 to 9

by fskilnik@GMATH » Wed Mar 27, 2019 5:19 am
Brent@GMATPrepNow wrote:At a certain school, the student to teacher ratio is 52 to 9. If 38 students and 11 teachers leave, which of the following COULD represent the number of students and teachers remaining at the school?

A) 532 students and 88 teachers
B) 794 students and 162 teachers
C) 1106 students and 225 teachers
D) 1418 students and 241 teachers
E) 1728 students and 295 teachers
$$?\,\,\,:\,\,\,\left( {{\text{final}}\,\,S,{\text{final}}\,\,T} \right)\,\,\,\,\,\underline {{\text{possible}}} $$
$$\left\{ \matrix{
S\,\,:\,\,\,\,52k\,\,\,\,\, \to \,\,\,\,\,52k - 38 \hfill \cr
T\,\,:\,\,\,\,9k\,\,\,\,\, \to \,\,\,\,\,9k - 11 \hfill \cr} \right.\,\,\,\,\,\left( {k \ge 1\,\,{\mathop{\rm int}} } \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{final}}\,\,{\rm{sum}}\,\,{\rm{ = }}\,\,61k - 49\,\,\,\,\, \Rightarrow \,\,\,\,\,\left( {{\rm{final}}\,\,{\rm{sum}} + 49} \right)\,\,{\rm{divisible}}\,\,{\rm{by}}\,\,61\,\,\,\,\left( * \right)$$
$$\eqalign{
& \left( A \right)\,\,532 + 88 + 49 = 669 = 610 + 59\,\,\,\, \Rightarrow \,\,\,\,{\rm{no}}! \cr
& \left( B \right)\,\,794 + 162 + 49 = 1005 = 1220 - 15\,\,\,\, \Rightarrow \,\,\,\,{\rm{no}}! \cr
& \left( C \right)\,\,1106 + 225 + 49 = 1380 = 1220 + 160\,\,\,\, \Rightarrow \,\,\,\,{\rm{no}}! \cr
& \left( D \right)\,\,1418 + 241 + 49 = 1708 = 1220 + 488 = 1220 + 61 \cdot 8\,\,\,\, \Rightarrow \,\,\,\,{\rm{survivor}}! \cr
& \left( E \right)\,\,1728 + 295 + 49 = 2072 = 2440 - 368 = 2440 - 366 - 2\,\,\,\, \Rightarrow \,\,\,\,{\rm{no}}! \cr} $$

There is only one survivor, meaning the property (*) is good enough for our purposes!

The correct answer is (D).

Regards,
Fabio.
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by Brent@GMATPrepNow » Wed Mar 27, 2019 5:42 am
Brent@GMATPrepNow wrote:At a certain school, the student to teacher ratio is 52 to 9. If 38 students and 11 teachers leave, which of the following COULD represent the number of students and teachers remaining at the school?

A) 532 students and 88 teachers
B) 794 students and 162 teachers
C) 1106 students and 225 teachers
D) 1418 students and 241 teachers
E) 1728 students and 295 teachers

Hint: One approach will allow you to solve the question in under 20 seconds.
There's a nice integer properties rule that says:
If j is a multiple of d, and k is NOT a multiple of d, then j-k (and j+k) is NOT a multiple of d
For example, since 56 is a multiple of 7, and 11 is NOT a multiple of 7, then 56-11 is NOT a multiple of 7

At a certain school, the student to teacher ratio is 52 to 9.
So, some possible scenarios are as follows:
- there are 52 students and 9 teachers
- there are 104 students and 18 teachers
- there are 156 students and 27 teachers
etc

Notice that the number of students is always a multiple of 52.
Since 52 is a multiple of 4, we can also see that the number of students is always a multiple of 4.
Also notice that the number of teachers is always a multiple of 9.


38 students and 11 teachers leave
Notice that 38 is NOT a multiple of 4
And 11 is NOT a multiple of 9

By the above property, the REMAINING number of students is NOT a multiple of 4
By the above property, the REMAINING number of teachers is NOT a multiple of 9

Which of the following COULD represent the number of students and teachers remaining at the school?
Since the REMAINING number of students is NOT a multiple of 4, we can ELIMINATE answer choices A and E, since 532 and 1728 ARE multiples of 4
Since the REMAINING number of teachers is NOT a multiple of 9, we can ELIMINATE answer choices B and C, since 162 and 225 ARE multiples of 9

By the process of elimination, the correct answer is D

Cheers,
Brent
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by deloitte247 » Sun Mar 31, 2019 4:29 am
Student : Teacher => 52 : 9
Number of student = 52x (multiple of 4)
Number of teacher = 9y (multiple of 9)
38 students and 11 teachers leave the school
38 is not a multiple of 4
11 is not a multiple of 9
Therefore, the remaining number of students are not a multiple of 4
Remaining number of teachers are not a multiple of 9
With this, option A and E will be eliminated because 532 and 1728 are multiples of 4 (number of students remaining)
Option B and C will also be eliminated because 162 and 225 are multiples of 9 (number of teachers remaining)

Therefore, correct answer = option D
1418 students = not a multiple of 4
241 students = not a multiple of 9
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