Data Sufficiency

When working together at their constant rates, machines A and B can complete a certain task in 4 hours. How many hours would it take machine B to complete the task, when working alone at its constant rate?

1) When working alone at its constant rate, machine A can finish the task in 6 hours.

2) When working alone at their constant rates, it takes machine B 6 hours longer than machine A to finish the task

OA: D

kyuhunl wrote:When working together at their constant rates, machines A and B can complete a certain task in 4 hours. How many hours would it take machine B to complete the task, when working alone at its constant rate?

1) When working alone at its constant rate, machine A can finish the task in 6 hours.

2) When working alone at their constant rates, it takes machine B 6 hours longer than machine A to finish the task

$${T_{A \cup B}} = 4\,{\text{h}}$$
$$? = {T_B}\,\,\,\left[ {\text{h}} \right]$$
$$\left( * \right)\,\,\,\frac{1}{{{T_{A \cup B}}}} = \frac{1}{{{T_A}}} + \frac{1}{{{T_B}}}$$

$$\left( 1 \right)\,\,{T_A} = 6\,{\text{h}}\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,\,\frac{1}{4} = \frac{1}{6} + \frac{1}{{{T_B}}}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{T_B}\,\,{\text{unique}}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{\text{SUFF}}.$$

$$\left( 2 \right)\,\,{T_B} = {T_A} + 6\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,\,\frac{1}{4} = \frac{1}{{{T_A}}} + \frac{1}{{{T_A} + 6}}\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( {**} \right)} \,\,\,\,\,\,{T_A} > 0\,\,{\text{unique}}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{T_B}\,\,{\text{unique}}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{\text{SUFF}}.$$
$$\left( {**} \right)\,\,\,\,\frac{1}{4} = \frac{{2{T_A} + 6}}{{{T_A}\left( {{T_A} + 6} \right)}}\,\,\,\,\, \Rightarrow \,\,\,\, \ldots \,\,\,\, \Rightarrow \,\,\,\,{T_A}^2 - 2{T_A} - 24 = 0\,\,\,\,\,\mathop \Rightarrow \limits^{{\text{roots}}\,\,{\text{product}}} \,\,\,\,\left( {\frac{c}{a} = } \right)\frac{{ - 24}}{1} < 0\,\,\,\, \Rightarrow \,\,\,\,{T_A} > 0\,\,{\text{unique}}$$


We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.

kyuhunl wrote:When working together at their constant rates, machines A and B can complete a certain task in 4 hours. How many hours would it take machine B to complete the task, when working alone at its constant rate?

1) When working alone at its constant rate, machine A can finish the task in 6 hours.

2) When working alone at their constant rates, it takes machine B 6 hours longer than machine A to finish the task

Statement 1:
Case 1: Job = 12 units
Since A and B together take 4 hours to complete the 12-unit job, the combined rate for A+B = 12/4 = 3 units per hour.
Since A alone takes 6 hours to complete the 12-unit job, A's rate alone = 12/6 = 2 units per hour.
B's rate = (combined rate for A+B) - (A's rate alone) = 3-2 = 1 unit per hour.
Since B's rate = 1 unit per hour, B's time to complete the 12-unit job = 12/1 = 12 hours.

Case 1: Job = 24 units
Since A and B together take 4 hours to complete the 24-unit job, the combined rate for A+B = 24/4 = 6 units per hour.
Since A alone takes 6 hours to complete the 24-unit job, A's rate alone = 24/6 = 4 units per hour.
B's rate = (combined rate for A+B) - (A's rate alone) = 6-4 = 2 units per hour.
Since B's rate = 2 units per hour, B's time to complete the 24-unit job = 24/2 = 12 hours.

In each case, B's time alone is THE SAME : 12 hours.
Thus, Statement 1 is SUFFICIENT.

Statement 2:
In Cases 1 and 2, B's time alone (12 hours) is 6 hours longer than A's time alone (6 hours).
Implication:
Statement 2 is conveying the same information as Statement 1.
For B's time alone to be 6 hours longer than A's time alone, A's time alone must be 6 hours.
Since Statement 1 is sufficient -- and Statement 2 conveys the same information as Statement 1 -- Statement 2 must also be SUFFICIENT.

The correct answer is D.