Need Clarification on a DS question

[ben wade]

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 the 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.

I chose answer choice as B.
My reasoning was like this -
Given - Tm/Sm = 1/20
We know that - Tm/Sm = Tp/Sp = 1/20.
Altering Tm/Sm = Tp/Sp => Sm/Sp = Tm/Tp = 1/20.

Can anyone point out the error in making this alteration in the ratios?

[Stuart@KaplanGMAT]

Altering Tm/Sm = Tp/Sp => Sm/Sp = Tm/Tp = 1/20.

Can anyone point out the error in making this alteration in the ratios?

You can only substitute things that are identical. You've substituted things that are different.

Using numbers to illustrate:

1/2 = 3/6 = 5/10

You rearraged the first two parts (correctly) to get:

1/2 = 3/6

2/6 = 1/3

then you just added the third part to get the equation:

2/6 = 1/3 = 5/10

which is clearly not true.

[Umar82]

What's the OA . . . C?

[vikram_k51]

I think it is E.There is no clear co-relation b/w the ratios.