ps 117 page 168
I haven't really understood your question. The method used here is the multiplication of numerator and denominator by its conjugate. So if the denominator is √(n+1)-√n then you multiply the numerator and denominator by √(n+1)+√n. And for the denominator use the formula (a-b)(a+b)=a²-b². Hope that clarifies your doubt if not maybe you could rephrase ur question.


PS, 148, page 233
For this problem, I solved the equation to get (10x+20y)/x+y=k and substituted each of the answer choices for k. Since x and y are both positive numbers and x<y you will get a fractional value of x/y in which both will be positive and y will be greater than x. In case of D k=18 you get x/y=1/4 where x<y and both are positive.Hence D is the answer. In case of option A where k=10 you get y=0 and no value for x which is not possible considering we are already told that x and y are +ve.

However, I would still recommend getting familiar with the method used in the OG for this problem as it is an important step in understanding the inequalities chapter. You will need to master that technique to solve inequality problems accurately. Cheers.

You may even want to refer to the Manhattan GMAT OG companion which gives slightly more time saving and easier to comprehend solutions.

Thank you for your answer.

when you have √5 / √5 you multiply both by √5 not for -√5

so why in this case you change the sign and instead of √(n+1)-√n (the denominator) you multiply by √(n+1)+√n? is this a rule?