Bunuel wrote:
If \(a > 0\), \(b > 0\) , and \(c > 0\) , \(a + \frac{1}{b + \frac{1}{c}}\) =
A. \(\frac{a + b}{c}\)
B. \(\frac{ac + bc + 1}{c}\)
C. \(\frac{abc + b + c}{bc}\)
D. \(\frac{a + b + c}{abc + 1}\)
E. \(\frac{abc + a + c}{bc + 1}\)
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 = 1Given expression: a + 1/(b + 1/c) =
1 + 1/(
1 + 1/
1) =
= 1 + 1/2
=
3/2So, when
a = b = c = 1, the given expression evaluates to be
3/2So, the correct answer choice must also evaluate to be
3/2 when we plug in
a = b = c = 1Check 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
=
5/3So, when
a = b = 1 and c = 2 , the given expression evaluates to be
5/3So, 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