What is the average of a, b and c?

The average of a, b and c is (a + b + c)/3. So, basically to answer the question we need to know the value of a + b + c.

(1) The average of a and b is c --> (a + b)/2 = c --> a + b = 2c. Not sufficient.

(2) The average of b and c is 4 --> (b + c)/2 = 4 --> c = 8 - b. Not sufficient.

(1)+(2) We have that a + b = 2c and c = 8 - b, thus a + b + c = 2c + c = 3c = 3(8 - b). Without knowing the value of c or b we cannot get this value. For example, consider a = 1, b = 5, c = 3 or a = 4, b = 4 and c = 4. Not sufficient.

Answer: E.
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Target question: What is the average of a, b, and c?
NOTE: since the average = (a+b+c)/3, our goal here is to find the SUM a + b + c

So, we can REPHRASE the target question....
REPHRASED target question: What is the value of a + b + c?

Statement 1: The average of a and b is c
This means that (a + b)/2 = c
Multiply both sides by 2 to get: a + b = 2c
There are several values of a, b, and c that satisfy statement 1. Here are two:
Case a: a = 1, b = 1 and c = 2, in which case a + b + c = 1 + 2 + 2 = 5
Case b: a = 1, b = 2 and c = 3, in which case a + b + c = 1 + 2 + 3 = 6
Since we cannot answer the REPHRASED target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: The average of b and c is 4
Since there's no information about a, there's no way to find the value of a + b + c
Since we cannot answer the REPHRASED target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that a + b = 2c
Statement 2 tells us that (b + c)/2 = 4, which means b + c = 8

So, we have the following system:
b + c = 8
a + b = 2c

IMPORTANT: Many students will look at this system and incorrectly conclude that, since we have 2 equations and 3 variables, there's no way to answer the target question. This is not necessarily the case, because the target question does NOT ask us to find the individual values of a, b and c. The target question asks us to find the SUM of a, b, and c. So, for example, if the two equations happened to be a + b + 2c = 5 and 2a + 2b + c = 19, then we could ADD the equations to get 3a + 3b + 3c = 24, and upon dividing both sides by 3, we'd see that a + b + c = 8, in which case we COULD answer the target question despite only having 2 equations with 3 variables.
So, we can't just automatically assume that we don't have sufficient information.

So, let's TEST some values of a, b and c that satisfy BOTH equations.

NOTE: since the equation b + c = 8 doesn't have the variable a, let's use it to first find some values of c and b, then we'll use the other equation, a + b = 2c, to find the corresponding a value.

Case a: b = 1 and c = 7 satisfies the equation b + c = 8 . Now plug these values into the other equation (a + b = 2c) to get: a + 1 = 2(7). Solve to get: a = 13.
So, the values are a = 13, b = 1 and c = 7, which means a + b + c = 13 + 1 + 7 = 21
Case b: b = 0 and c = 8 satisfies the equation b + c = 8 . Now plug these values into the other equation (a + b = 2c) to get: a + 0 = 2(8). Solve to get: a = 16.
So, the values are a = 16, b = 0 and c = 8, which means a + b + c = 16 + 0 + 8 = 24
Since we cannot answer the REPHRASED target question with certainty, the combined statements are NOT SUFFICIENT

Answer =

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(1) a+c=2c
thus avg of a,b,c = a+b+c/3
putting a+b= 2c
avg =c
insuff

(2) no info on a
insuff

combining again, we also not get any unique value

Ans E

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