If the ratio of the number of teachers to the number of

If the ratio of the number of teachers to the number of students is the same in School District M and School District P, what is the ratio of the number of students in School District M to the number of students in School District P ?

Given that \(\frac{T_m}{S_m}=\frac{T_p}{S_p}\), where \(T_m\) and \(S_m\) are the numbers of teachers and students, respectively, in District M, and \(T_p\) and \(S_p\) are the numbers of teachers and students, respectively, in District P.

We need to find the value of \(\frac{S_m}{S_p}\) --> \(\frac{S_m}{S_p}=\frac{T_m}{T_p}\)

(1) There are 10,000 more students in School District M than there are in School District P --> \(S_m=S_p+10,000\). Not sufficient.

(2) The ratio of the number of teachers to the number of students in School District M is 1 to 20 --> \(\frac{T_m}{S_m}=\frac{1}{20}\). Not sufficient.

(1)+(2) Still not sufficient, consider \(S_p=1,000\) and \(S_m=11,000\) (answer 11) AND \(S_p=10,000\) and \(S_m=20,000\) (answer 2).

Answer: E.
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Responding to a pm:



Teacher:Student ratio in M = 1:20. No of teachers in M = m, No of students in M = 20m
Teacher:Student ratio in P = 1:20. No of teachers in M = p, No of students in M = 20p

Mind you, we dont know the value of m and p. All we know is that the teacher student ratio is 1:20 in both.

Ratio of number of students in M: Number of students in P = 20m : 20p = m:p.
We don't know m:p.

You are assuming that both are 20x. How can you say that the multiplier is the same in both the schools? M could have 20 students and 1 teacher while P could have 40 students are 2 teachers. In that case, ratio of number of students = 1:2
M could have 20 students and 1 teacher while P could have 80 students are 4 teachers. In that case, ratio of number of students = 1:4
and so on...

Hi Bunuel,

I'm having a hard time following your reasoning for the scenario where 1&2 are combined. From your answer explanation, it looks like you just accounted for the first statement?

Does the examples used for (1)+(2) violate the second statement in any way? No. The second statement gives the ratio of \(T_m\) to \(S_m\), which is of little use. If \(S_p=1,000\) and \(S_m=11,000\) (\(\frac{S_m}{S_p}=11\)), then according to the second statement \(T_m=550\) AND if \(S_p=10,000\) and \(S_m=20,000\) (\(\frac{S_m}{S_p}=2\)), then according to the second statement \(T_m=1000\).

Does this make sense?
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So the first step is to figure out what the original statement is saying and what it is looking for.
It says that the ratio of teachers so students is the same for school one and school two, so: T(M)/S(M)=T(P)/S(P)
and it wants to know the exact value of S(M)/S(P)

(1) essentially tells us S(M)=S(P)+10,000 but that doesn't tell us anything about the ratios, because S(M) could be 1 or 20,000
(2) tells us that T(M)/S(M)=1/20, thus T(P)/S(P)=1/20, however this tells us nothing about S(M)/S(P), because there could be one teacher in school m and 20 students and 100 teachers in school p and 2,000 students or vice versa.

Both together seems promising, because we know both ratios and that S(M)=S(P)+10,000 however there are still a variety of scenarios that fit that, so the answer is E

Solved this one by testing different values.

Statement (1)
\(\frac{Tm}{Sm}=\frac{3000}{12000} \frac{Tp}{Sp}=\frac{500}{2000}\) --> \(\frac{Sm}{Sp}=\frac{12000}{2000}\)=6

\(\frac{Tm}{Sm}=\frac{10000}{20000} \frac{Tp}{Sp}=\frac{5000}{10000}\) --> \(\frac{Sm}{Sp}=\frac{20000}{10000}\)=2

So we get 2 different ratios here - Not Sufficient

Statement (2) don't give us enough information - see example above

(1) + (2) calculation example above give us different outcomes for the ratios Answer (E)

I am new to preparation so I am confused a bit.

Ratio of M = [T1] / [S1]
Ratio of P = [T2] / [S2]

As said in question [T1] / [S1] = [T2] / [S2]
Considering statement 2
Ratio of M = [1] / [20]
Which mean ratio of P = [1] / [20]

As [T1]/ [S1] = [T2] / [S2]
So [T1] / [T2] = [S1] / [S2] = [20] / [20] = [1] / [1]
So ratio of Students of M to Students of P is 1:1.

Isn't question has just asked the ratio which can be found by statement 2 i.e 1:1? Where I am wrong? Thanks in advance for guidance.

I have answered this question (why stmnt 2 alone is not sufficient) here:
if-the-ratio-of-the-number-of-teachers-to-the-number-of-143680.html#p1239274

Hello karishma
Is it necessary to plug in values in DS questions ? In which cases we should do it because in this question I did not check my answer by plugging in values.

Here is my method:

We know T/S=T'/S'
(T= teachers in school M; S= students in school M) (T'= Teachers in school P; S'=students in school P)

To find: S/S'=?

First statement: (S'+10000)/S'
(S=S'+10000)

So T/(S'+10000) = T'/S'
We have more than one unknown variables. Therefore not sufficient.
BCE

Statement 2: T/S=1/20
Even if we plug this ratio in the above equation, we would have more than one variable.

Thus, insufficient.

T/(S'+10000)= 1/20

(S'+10000)/(S'+10000)= T'/S'

T'/S'= 1/20

Couldn't go beyond this. More than one variable to solve and thus I chose E.

Please let me know if this makes sense.

hi,

and the ratios could still be 1/20 since we have no restrictions for the number of teachers... right ?

Thank you very much.

If the ratio of the number of teachers to the number of students is the same in School District M and School District P, what is the ratio of the number of students in School District M to the number of students in School District P ?

(1) There are 10,000 more students in School District M than there are in School District P.
(2) The ratio of the number of teachers to the number of students in School District M is 1 to 20.

My 2 cents.

First, identify what is given and what is our goal.
Given : Mt / Ms = Pt / Ps
Goal : Find Ms / Ps

1) Ms = 10,000 + Ps --> insufficient, as we don't have any value for Ms or Ps
2) Mt / Ms = 1/20 --> insufficient, as we know nothing about Ps

For checking whether C is the answer,
we need to be careful of premise 2.
Premise 2 is giving us the "ratio", not the value...so we only have the relationship, but no value.
Therefore, as we have no value for Ms, C won't work.
Thus, E.

from both conditions, we have
m number of teacher in M
20m is number of student in M
p is number of teacher in P
20p is number of student in P

if 20m=20p+1000
we can have some value of m, p which satisfy this equation and the ratio of 20m/20p+1000 is variable
p=1
m=1020
p=10
m=1200

it is clear that the ratio is different.

Bunuel, niks18, amanvermagmat




Can I also say we have three different variables : \(S_p\), \(T_m\), \(S_m\)and only two linear equations hence both statements combined are insufficient?
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Hi All ,

I am totally messed up with the second statement ,
According to my understanding of the ratio of number of teachers to the number of student for M is 1:20 then Tm/Sm =Tp/Sp=1/20 which denotes that Sm/Sp = 1/20 ,

Answer should be B

Please help ,

Posted from my mobile device

LeenaSai

From statement 2, Tm/Sm =Tp/Sp=1/20
But we cannot derive Sm/Sp = 1/20 from given info

Multiple all by Sm/Tp
(Tm/Sm) * (Sm/Tp) = (Tp/Sp) * (Sm/Tp) = 1/20 * Sm/Tp
Tm/Tp = Sm/Sp = 1/20 * Sm/Tp

Sm/Sp = 1/20 * Sm/Tp

We don't know the value of Sm/Tp. So B is insuff.

We are looking for the ratio of the number of students in School District M to the number of students in School District P.

Let's denote the number of students in School District M as SM and the number of students in School District P as SP.

We know that the ratio of the number of teachers to the number of students is the same in both districts.

Let's consider the given statements:

(1) There are 10,000 more students in School District M than there are in School District P.

From this statement, we can express the relationship between SM and SP as:

SM = SP + 10,000

However, this statement does not provide any information about the ratio of teachers to students. Therefore, statement (1) alone is not sufficient to determine the ratio of the number of students in School District M to the number of students in School District P.

(2) The ratio of the number of teachers to the number of students in School District M is 1 to 20.

This statement gives us the ratio of teachers to students in School District M, but it does not provide any information about the number of students in either district or the ratio between the two districts. Therefore, statement (2) alone is not sufficient to determine the ratio of the number of students in School District M to the number of students in School District P.

Now, let's consider both statements together:

From statement (1), we have SM = SP + 10,000.

From statement (2), we know the ratio of teachers to students in School District M is 1 to 20, but we still do not have any information about the number of students in School District P or the direct ratio between the two districts.

Therefore, even when considering both statements together, we do not have enough information to determine the specific ratio of the number of students in School District M to the number of students in School District P.

Hence, the answer is that the statements together are not sufficient to determine the ratio of the number of students in School District M to the number of students in School District P.

if statement 1 said "there are 3 times as many students" rather than the arithmetic 10000 more students, would that make both together sufficient?