To complete a task, A and B take 24 days, B and C take 12 days and A, B and C take 6 days. A and C will take:

Correct option is A

Given:

A and B together take 24 days → Work per day = $$\frac{1}{24}$$​​

B and C together take 12 days  Work per day =$$\frac{1}{12}$$​​

A, B and C together take 6 days → Work per day $$= \frac{1}{6}$$​

​Solution:

​$$\frac{1}{A} + \frac{1}{B} = \frac{1}{24}$$​​

​$$\frac{1}{B} + \frac{1}{C} = \frac{1}{12}$$​​

$$\frac{1}{A} + \frac{1}{B} + \frac{1}{C} = \frac{1}{6}$$​

​$$\left( \frac{1}{A} + \frac{1}{B} + \frac{1}{C} \right) - \left( \frac{1}{B} + \frac{1}{C} \right) = \frac{1}{6} - \frac{1}{12} = \frac{1}{12}$$​​

​$$\frac{1}{A} = \frac{1}{12}$$​

Now subtract the first from the third:

​$$\left( \frac{1}{A} + \frac{1}{B} + \frac{1}{C} \right) - \left( \frac{1}{A} + \frac{1}{B} \right) = \frac{1}{6} - \frac{1}{24} = \frac{1}{8}$$

$$\frac{1}{C} = \frac{1}{8}$$​​​

So,

​$$\frac{1}{A} + \frac{1}{C} = \frac{1}{12} + \frac{1}{8} = \frac{2 + 3}{24} = \frac{5}{24}$$​​

Therefore,

A and C together take =$$\frac{24}{5}\text{ days}$$

Alternate Method:

Efficiency of ( A + B + C ) = 4

Efficiency of A = 4 - 2 =2

Efficiency of B = 1 - 2 = -1

Efficiency of C = 4 - 1 = 3

A and C together take =$$\frac{24}{3 + 2}\text{ days} = \frac{24}{5}\text{ days}$$ 

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