What is the Correct Derivation for Refrigeration Efficiency?

[irule]

this is a question from an old exam that I am trying to understand

I don't think you need the passage but I included it anyways along with the explanation of the answer

I specifically don't understand the 2nd step of the algebra where they have they Qc/Qc in the numerator.

http://mysowar.wordpress.com/files/2009/06/refrig-1.jpg
http://mysowar.wordpress.com/files/2009/06/refrig-2.jpg

thanks in advance

 

[CompuChip]

Do you mean where they go from
Qc/(Qh-Qc) to (Qc/Qc)/(1-Qh/Qc)?

They simply divided both numerator and denominator by Qc. The goal here is to rewrite everything not in terms of Qh and Qc separately, but only in their ratio Qc/Qh (or Qh/Qc).

 

[Moo Of Doom]

But they're clearly making a sign error when doing this. Luckily for them, they make the same sign error a second time, so the effects cancel. A correct derivation would be:

$$\frac{Q_C}{Q_H - Q_C} = \frac{1}{Q_H/Q_C - 1} = \frac{1}{T_H/T_C - 1} = \frac{T_C}{T_H - T_C}$$