Math HELP????

Question

Here's a math problem i found in the SAT Bookelt, help please?
3) L@ K = L + K/L. What is L @ (L @ K)
A] L + 2k + K/4
B] 2L +K/4 + 1
C] L
D] L+ 1 + K/L squared
E] L/K

If you could show me ur thought process, that would be great.
Thanks

Answer 1

This problem defines the operator @ by the rule that

L @ K = L + K/L

For instance, 2 @ 3 = 2 + 3/2 = 5 1/2.

In order to evaluate L @ (L @ K), we must expand the expression from "inside" to "outside". First,

L @ (L @ K) = L @ (L + K/L)

then

L @ (L + K/L) = L + (L + K/L) / L

This is the correct answer but it can be simplified:

L + (L + K/L) / L = L + (L/L) + ((K/L)/L) =
= L + 1 + K/L^2.

This is answer (D).

Answer 2

The answer is D) L + 1 + K / L squared.

L @ (L @ K) = L @ (L + K / L), when we substitute using the original equation.
Let's rephrase the original equation just to make it easier to see.
A @ B = A + B/A
Let A = L and let B = (L + K / L)
Then L @ (L + K / L) = L + ((L + K / L) / L)
= L + (L/L) + (K/L squared)
= L + 1 + K / L squared