Question: Two machines, A and B, each working at a constant rate, can complete a certain task working together in 6 days. In how many days, working alone, can machine A complete the task?

1. The average (arithmetic mean) of the respective times A and B would each take to complete the task working alone is 12.5 days
2. It would take machine A 5 more days to complete the task alone than it would take machine B to complete the task

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are NOT sufficient.

Answer:
Solution with Explanation:
Approach Solution (1):

Given that \(\frac{1}{A}+\frac{1}{B}=\frac{1}{6}\), where A is the time needed for machine A to complete the task working alone and B is the time for the machine B to complete the task working alone

(1) The average (arithmetic mean) of the respective times A and B would each take to complete the task working alone in 12.5 days. This statement implies that A + B = 2 * 12.5 = 25.
Now, since we don’t know which machine works faster than even if we substitute B with \(25-A(\frac{1}{A}+\frac{1}{25-A}=\frac{1}{6})\), we must get two different answers for A and B: A < B and A > B.
Not sufficient

(2) it would take machine A 5 more days to complete the task alone than it would take machine B to complete the task. A = B + 5, so we have that \(\frac{1}{A}+\frac{1}{A-5}=\frac{1}{6}\). From this we can find that A = 2 (not a valid solution since in this case B will be negative) or A = 15.
Sufficient

Correct Option: B

Approach Solution (2):

First define some variables
A = rate of A
B = rate of B
x = time that it takes machine A working alone
y = time that it takes machine B working alone
\(6(A+B)=1\)
\(A+B=\frac{1}{6}\)

Given in S2: It would take machine A 5 more days to complete the task alone that it would take machine B to complete the task
This means x > y
x = y + 5
equation 1:
\(A+B=\frac{1}{6}\)
\(Rate=\frac{Work}{Time}\)
\(A=\frac{1}{5}\)
\(x=y+5\)
\(A=\frac{1}{y+5}\)
\(B=\frac{1}{y}\)
\(A+B=\frac{1}{6}\)
\(\frac{1}{y+5}+\frac{1}{y}=\frac{1}{6}\)
\(\frac{y+y+5}{y(y+5)}=\frac{1}{6}\)
\(\frac{2y+5}{y^2+5y}=\frac{1}{6}\)
\(12y+30=y^2+5y\)
\(y^2-7y-30 = 0\)
\((y-10)(y+3) = 0\)
\(y=10\)
\(or\)
\(y = -3\)

Reject y = -3
x = y + 5
x = 10 + 5
x = 15
Thus statement 2 is sufficient

Correct Option: B

Approach Solution (3):

A + B = 25 and 1/A + 1/B = 1/6 are the results for (1). We obtain 1/A + 1/(25 - A) = 1/6 by putting B = 25 - A into 1/A + 1/B = 1/6. This results in A2 -25A + 150 = 0, which yields A = 10 or A = 15. However, as the solution explains, you actually don't need to do all that if you understand that for (1) we have no method of telling machines A and B apart from one another, thus we must come up with two solutions for A and B: A < B and A > B.

Thus statement 2 is sufficient
Correct Option: B

“Two machines, A and B, each working at a constant rate, can complete a certain task working together in 6 days. In how many days, working alone, can machine A complete the task?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

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