Question : A and B can do a piece of work in 36 days. B and C can do the same work in 60 days. A and C can do the same work in 45 days. In how many days can B alone complete the same work?
Option 1: 45
Option 2: 60
Option 3: 90
Option 4: 120
Correct Answer: 90
Solution :
Let the rates at which A, B, and C can complete the work as $a, b$, and $c$.
These rates are the reciprocals of the time it takes each person to complete the work.
From the problem,
$⇒a + b = \frac{1}{36}$ (A and B can do the work in 36 days) ___(1)
$⇒b + c = \frac{1}{60}$ (B and C can do the work in 60 days) ___(2)
$⇒a + c = \frac{1}{45}$ (A and C can do the work in 45 days) ___(3)
Adding these three equations,
$⇒2a + 2b + 2c = \frac{1}{36} + \frac{1}{60} + \frac{1}{45}$
$⇒a + b + c = \frac{1}{30}$ ___(4)
From equation (3) and (4),
$⇒b = \frac{1}{30} - \frac{1}{45} = \frac{1}{90}$
The time it takes for B to complete the work alone is the reciprocal of B's rate = $\frac{1}{b}$ = 90 days.
Hence, the correct answer is 90.
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