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

Question

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.

BrentGMATPrepNow · Accepted Answer

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

chetan2u · Answer

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.

fskilnik · Answer

?\,\,\,:\,\,\,\left( {{	ext{final}}\,\,S,{	ext{final}}\,\,T} ight)\,\,\,\,\,\underline {{	ext{possible}}}

\left\{ \matrix{
 S\,\,:\,\,\,\,52k\,\,\,\,\, 	o \,\,\,\,\,52k - 38 \hfill \cr 
 T\,\,:\,\,\,\,9k\,\,\,\,\, 	o \,\,\,\,\,9k - 11 \hfill \cr} ight.\,\,\,\,\,\left( {k \ge 1\,\,{\mathop{m int}} } ight)\,\,\,\,\, \Rightarrow \,\,\,\,\,{m{final}}\,\,{m{sum}}\,\,{m{ = }}\,\,61k - 49\,\,\,\,\, \Rightarrow \,\,\,\,\,\left( {{m{final}}\,\,{m{sum}} + 49} ight)\,\,{m{divisible}}\,\,{m{by}}\,\,61\,\,\,\,\left( * ight)

\eqalign{
 & \left( A ight)\,\,532 + 88 + 49 = 669 = 610 + 59\,\,\,\, \Rightarrow \,\,\,\,{m{no}}! \cr 
 & \left( B ight)\,\,794 + 162 + 49 = 1005 = 1220 - 15\,\,\,\, \Rightarrow \,\,\,\,{m{no}}! \cr 
 & \left( C ight)\,\,1106 + 225 + 49 = 1380 = 1220 + 160\,\,\,\, \Rightarrow \,\,\,\,{m{no}}! \cr 
 & \left( D ight)\,\,1418 + 241 + 49 = 1708 = 1220 + 488 = 1220 + 61 \cdot 8\,\,\,\, \Rightarrow \,\,\,\,{m{survivor}}! \cr 
 & \left( E ight)\,\,1728 + 295 + 49 = 2072 = 2440 - 368 = 2440 - 366 - 2\,\,\,\, \Rightarrow \,\,\,\,{m{no}}! \cr}

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

The correct answer is (D).

Regards, 
Fabio.

Gmat20201 · Answer

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

Quick Soln :-

Student : Teacher Ratio is 52:9 
So if students are 52x ,Teachers will be 9x ( x= multiplier)

We are told that 38 students and 11 teachers leave. So we know that 9x>11

Lets take lowest value of x=2 ,( we cannot take 1 since we are given 11 teachers left so number of teachers cannot be 9)

So now if x=2 , students =52*2 = 104 and Teachers = 9*2=18

38 students leave so 104-38=66 students left
11 teachers leave so 18-11=7 teachers left.
 
So number of teachers remaining at school must be multiple of 7

Checking each ans choice, Only D is multiple of 7 .

Ans is option D

Once you understand this it will take less than 1 min. Hope it helps

Abhishek009 · Answer

Must be (D) as -

1418 + 38 = 1456 must be divisible by 52 ; 1456/52 = 28
241 + 11 = 252 must be divisible by 9 ; 252/9 = 28

Hence, Answer must be (D)

BrentGMATPrepNow  is this the method u were hinting ??

BrentGMATPrepNow · Answer

Nice job!
That also works, but testing for divisibility by 52 is time-consuming. 
We can make things easier on ourselves by just testing for divisibility by 4 (if a number is divisible by 52, then it's also divisible by 4.

Cheers,
Brent

Kinshook · Answer

BrentGMATPrepNow 
Given: At a certain school, the student to teacher ratio is 52 to 9.
Asked: If 38 students and 11 teachers leave, which of the following COULD represent the number of students and teachers  remaining  at the school?
Let the students be 52k & teachers be 9k, where k is a positive integer.
After 38 students and 11 teachers leave
Number of students = 52k - 38
Number of teachers = 9k - 11

A) 532 students and 88 teachers
9k - 11 = 88; 9k = 99; k = 11; 52k - 38 = 52*11 - 38 = 534; Incorrect

B) 794 students and 162 teachers
9k - 11 = 162; 9k = 173; k = 173/9 = 19 2/3: Incorrect since k is a positive integer

C) 1106 students and 225 teachers
9k - 11 = 225; 9k = 236; k = 26 2/9; Incorrect since k is a positive integer

D) 1418 students and 241 teachers
9k - 11 = 241; 9k = 252; k = 28; 52k - 38 = 52*28 - 38 = 1418; Correct

E) 1728 students and 295 teachers
9k - 11 = 295; 9k = 306; k = 34; 52k - 38 = 52*34 - 38 = 1730; Incorrect

IMO D

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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.

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