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

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CEO
Joined: 12 Sep 2015
Posts: 3990
At a certain school, the student to teacher ratio is 52 to 9.  [#permalink]

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25 Mar 2019, 10:59
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Top Contributor
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Difficulty:

95% (hard)

Question Stats:

47% (02:58) correct 53% (02:37) wrong based on 60 sessions

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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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Re: At a certain school, the student to teacher ratio is 52 to 9.  [#permalink]

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25 Mar 2019, 21:22
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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.

Although the equation will be $$38+x=52a......11+y=9a$$
Now we have to substitute the options in x and y..
A short cut would be to play around with number properties and eliminate wrong choices..

(I) 11+y = 9a
This tells us that when we add 11 to teachers strength, it should be a multiple of 9.
Add the digits to check.. 88 means 8+8=16 and 16+11 =27 ok... 162 and 225 add up to 9, so adding 11 will give you an answer that is not divisible by 9.
241 adds up to 7 and 295 adds up to 16, so they are also ok..
Property of divisibility by 9 is that the sum of digits is divisible by 9
so we are left with A, D and E.

(II) Now 38+x=52a..
52a is a multiple of 4, but 38 is not, so x has to be even but not a multiple of 4.
let us look at the choices
A. 532.. 32 is divisible by 4... so out
D. 1418...18 is not divisible by 4.. so OK
E. 1728 .. 28 is divisible by 4... so out
Property of divisibility by 4 is that the last two digits are divisible by 4

Only D is left.
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Joined: 12 Sep 2015
Posts: 3990
Re: At a certain school, the student to teacher ratio is 52 to 9.  [#permalink]

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26 Mar 2019, 09:20
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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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Re: At a certain school, the student to teacher ratio is 52 to 9.  [#permalink]

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27 Mar 2019, 06:26
1
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!

Regards,
Fabio.
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Re: At a certain school, the student to teacher ratio is 52 to 9.   [#permalink] 27 Mar 2019, 06:26
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