It's better to solve this one part to part:

= \(b + \frac{1}{c}\)

=\(\frac{bc + 1}{c}\)

Then

=\(\frac{1}{(bc + 1) / c}\)

= \(\frac{c}{bc + 1}\)

last part :

= a + \(\frac{c}{bc +1}\)

=\(\frac{c + abc + a}{bc +1}\)

The best answer is E.

assume values for a,b,c and put values of abc in options

We're looking for an expression that is equivalent to the given expression.
So, if we find the value of the given expression for certain values of a, b and c, then the correct answer choice must also equal the same value for the same value of values of a, b and c.

So, let's see what happens when a = b = c = 1

Given expression: a + 1/(b + 1/c) = 1 + 1/(1 + 1/1) =
= 1 + 1/2
= 3/2

So, when a = b = c = 1, the given expression evaluates to be 3/2
So, the correct answer choice must also evaluate to be 3/2 when we plug in a = b = c = 1

Check the answer choices....
A. (a + b)/c = (1 + 1)/1 = 2. No good. We need 3/2. ELIMINATE.

B. (ac + bc + 1)/c = (1 + 1 + 1)/1 = 3. No good. We need 3/2. ELIMINATE.

C. (abc + b + c)/bc = (1 + 1 + 1)/1 = 1. No good. We need 3/2. ELIMINATE.

D. (a + b + c)/(abc + 1) = (1 + 1 + 1)/(1 + 1) = 3/2. Perfect!! KEEP

E. (abc + a + c)/(bc + 1) = (1 + 1 + 1)/(1 + 1) = 3/2. Perfect!! KEEP


Okay, we have two possible answers: D or E.
So, we must test one more set of values.



Let's see what happens when a = b = 1 and c = 2
Here, the given expression: a + 1/(b + 1/c) = 1 + 1/(1 + 1/2) =
= 1 + 1/(3/2)
= 1 + 2/3
= 5/3

So, when a = b = 1 and c = 2 , the given expression evaluates to be 5/3
So, the correct answer choice must also evaluate to be 5/3 when we plug in a = b = 1 and c = 2

Check the REMAINING answer choices....
D. (a + b + c)/(abc + 1) = (1 + 1 + 2)/(2 + 1) = 4/3. No good. We need 5/3. ELIMINATE.

E. (abc + a + c)/(bc + 1) = (2 + 1 + 2)/(2 + 1) = 5/3. Perfect!! KEEP

Answer: E

Cheers,
Brent
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