A does half as much work as B in three-fourth of the time. If together they take 66 days to complete the work, how many days shall B alone take to do it?

Correct option is C

Given:
A does half as much work as B in three-fourth of the time
Together, A and B take 66 days to complete the work
Solution:
Let time taken by B alone be x days
Work done by B per day = $$\frac 1x$$​​

Work done by A per day = ($$\frac 12$$​) work in ($$\frac 34$$​)x time
​$$\text{Work done by A per day} = \frac{\frac{1}{2}}{\frac{3}{4}x} = \frac{2}{3x}$$​​
Together, work done per day = work by A + work by B = $$\frac{2}{3x} + \frac{1}{x} = \frac{5}{3x}$$​​
Work done together per day = $$\frac{1}{66}$$​
​$$\frac{5}{3x} = \frac{1}{66} \implies 5 \times 66 = 3x \implies 330 = 3x \implies x = 110$$​​
B alone will take 110 days to complete the work. 

Alternative:  

A does half as much work as B in three-fourth of the time 

It means if B does  4B work in 4 (unit time ) , A does 3A Time in which is half of 4B  

4B = $$2 \times 3A$$​​ 

A  : B 

2  :  3 

Efficiency Ration = 2  :  3 

Total work =5$$\times 66$$  = 330 ​

B alone complete the work =$$\frac{330}{3} = 110$$ days​