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

\(?\,\,\,:\,\,\,\left( {{\text{final}}\,\,S,{\text{final}}\,\,T} \right)\,\,\,\,\,\underline {{\text{possible}}}\)

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

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


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

The correct answer is (D).

Regards,
Fabio.

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

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 ??

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

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