If x and y are positive real numbers, the operation is defined by

Question

If x and y are positive real numbers, the operation △ is defined by  x△y = \frac{xy}{x + y} . Let a, b, and c, be positive real numbers. Which of the following is equivalent to  \frac{1}{a}△(\frac{1}{b}△\frac{1}{c})  ?

A.  \frac{a + b + c}{abc}

B.  \frac{ab + bc + ac}{abc}

C.  \frac{1}{a + b + c}

D.  \frac{ab + bc + ac}{a + b + c}

E.  \frac{abc}{a + b + c}

gmatophobia · Accepted Answer

x△y = \frac{xy}{x + y} We can find the value of \frac{1}{x} △ \frac{1}{y } Screenshot 2023-08-03 113943.jpg \frac{1}{b}△\frac{1}{c} = \frac{1}{b+c} \frac{1}{b+c}△\frac{1}{a} = \frac{1}{a+b+c} Option C

mmdfl · Answer

Let's call △ = f, just for simplicity (x △ y = f(x,y))
f(x,y)=xy/(x+y) = 1/( 1/x + 1/y). Therefore, f(1/x,1/y)=1/(x+y)

So (1/a) △ ( (1/b) △ (1/c) ) = f( 1/a , f(1/b,1/c))=f( 1/a, 1/(b+c))=1/(a+b+c) =>  (C) is the correct answer.

Paras96 · Answer

1/a△(1/b△1/c)
=1/a△ 
=1/a△ 
=(1/a(b+c))/(1/a+1/(b+c))
=1/(a+b+c)

Hence C

BrushMyQuant · Answer

x△y = \frac{xy}{x + y}  and we need to find the value of  \frac{1}{a}△(\frac{1}{b}△\frac{1}{c})  ?

\frac{1}{b}△\frac{1}{c}  =  \frac{\frac{1}{b} * \frac{1}{c}}{ \frac{1}{b} + \frac{1}{c}}  =  \frac{\frac{1}{bc}}{\frac{b+c}{bc}}  =  \frac{1}{b+c}

Similarly,  \frac{1}{a}  △  (\frac{1}{b}△\frac{1}{c})  =  \frac{1 }{ a + b + c}

So,  Answer will be C 
Hope it helps!

Watch the following video to learn the Basics of Functions and Custom Characters

https://www.youtube.com/watch?v=b7JoMX0gk8M

Khwarizmi · Answer

I wonder if this belongs in the sub-500 difficulty level.

Bunuel · Answer

We do not assign the difficulty level manually. The difficulty level of a question on the site is determined automatically based on various parameters collected from users' attempts, such as the percentage of correct answers and the time taken to answer the question. You can find the difficulty level of a question and its related statistics in the first post. So, the difficulty level of this question is Sub 500 Level based on the timer attempts from the users.

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If x and y are positive real numbers, the operation is defined by

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