One-third of a work is finished in four days by P and one-fourth of the same work is finished in six days by Q. P and Q together work for six days and the rest of the work is finished by R in two days. R alone can finish the whole work in _____ days.

Correct option is D

Given:
P finishes $$\frac 13$$​ of work in 4 days

Q finishes $$\frac 14$$​ of work in 6 days
P + Q work together for 6 days
Remaining work finished by R in 2 days
Solution:
Total work = LCM of denominators (3,4) = 12 unit
P does$$\frac 13$$​ of 12 = 4 units in 4 days
P’s 1 day work = $$\frac 44$$​ = 1 unit
Q does$$\frac 14$$​of 12 = 3 units in 6 days
Q’s 1 day work = $$\frac 36$$​= $$\frac 12$$​ unit
(P+Q) 1 day work = 1 + $$\frac 12$$​= $$\frac 32$$​ units
Work done by P+Q in 6 days = 6 ×$$\frac 32$$​ = 9 units
Remaining work = 12 − 9 = 3 units
R finishes 3 units in 2 days => R’s 1 day work =  $$\frac 32$$​ units
Time for R alone to finish 12 units = 12 ÷$$\frac 32$$​ = 12 × $$\frac 23$$​ = 8 days