A can complete in 2 days as much of a task as B can complete in 3 days and B can complete in 4 days as much of the task as C can complete in 5 days. In what time will A, B and C together complete the task if A alone can complete it in 22 days?

Correct option is C

Given:

A can complete in 2 days as much work as B can in 3 days

B can complete in 4 days as much work as C can in 5 days

A alone can complete the task in 22 days

We are to find the time taken by A, B, and C together to complete the whole task.

Solution:

Let, A's 1 day work =$$\frac{1}{22}$$​​

From the first condition:

A (2 days) = B (3 days)
2A = 3B
$$\frac{A}{B} = \frac{3}{2}$$​​

B =$$\frac{2}{3}A$$​​
B's 1 day work = $$\frac{2}{3} \times \frac{1}{22} = \frac{1}{33}$$​​

From the second condition:

B (4 days) = C (5 days)
4B = 5C
$$\frac{B}{C} = \frac{5}{4}$$​​

C =$$\frac{4}{5}B$$​​
C's 1 day work $$= \frac{4}{5} \times \frac{1}{33} = \frac{4}{165}$$​​

Total work per day =$$\frac{1}{22} + \frac{1}{33} + \frac{4}{165}$$​

Total work per day:

​$$\frac{30 + 20 + 16}{660} = \frac{66}{660} = \frac{1}{10}$$​​

Time taken by A, B, and C together =$$\frac{1}{\frac{1}{10}}$$​ = 10 days

Alternate Method:

2A = 3B
$$\frac{A}{B} = \frac{3\times 5}{2\times 5} =\frac {15}{10}$$ 

4B = 5C
​$$\frac{B}{C} = \frac{5 \times 2}{4\times 2}=\frac {10}8$$

Total Work = 15$$\times 22$$ = 330

Time taken by A, B, and C together = $$\frac {330}{33}$$ = 10 days​