A and B can do a piece of work in 12 days B and C in 15 days C and A in 20 days. In how many days can C alone do it?

Efficiency of A + B = 33.33% = 100/3
Ratio of efficiency of A and B = 3 : 1
? Efficiency of A = (3/4) x 33.33 = 25%
? Number of days taken by A = 100/25 = 4

Efficiency of A + B = 100/(20/3) = 15%
No. of days ( A:B ) = 4 : 5
Efficiency ( A:B ) = 5 : 4

? Efficiency of B = (4/9) x 15 = 20/3 = 6²/₃%

A : B : C
Ratio of efficiency 3 : 1 : 2
Ratio of number of days { 1/3 : 1/1 : 1/2 }
or 2 : 6 : 3
Hence, (a) is correct [ Time = 1/ Efficiency ]

Efficiency of A = 100/24 = 4.16%
Efficiency of B = 100/30 = 3.33%
Efficiency of (A+ B+C) =100/12 = 8.33%

? Efficiency of C = (8.33) - (4.16 + 3.33) = 0.83%
? Number of days required by C to complete the work alone
= 100/0.83 = 100/(5/6) = 120 days
(Note : 5/6 = 0.833)

Efficiency of A = 10%
Efficiency of B = 6.66%

? Required number of days = 100/16.66 = 6 days

Efficiency of (A + B) = 12.5%
Efficiency of (B + C) = 8.33%
Efficiency of (C + A) = 12.5%

? Efficiency of [(A + B) + (B + C) + (C + A)] = 33.33%
? Efficiency of (A + B + C) = 16.66%
? No. of days Required by A, B and C together = 100/16.66 = 6 days

(A + B + C)' S 1 day's work = (1/24) + (1/6) + (1/12) = 7/24
? (A + B + C) can do the entire work in = 24/7
= 3³/₇ days

Women's 1 day's work = 1/6
(Women + Daughter)'s 1 day's work = 1/4
? Daughter's 1 day's work = 1/4 - 1/6 = 1/12
So, the daughter can do the work in 12 days.

(P + Q)'S 1 day's work = 1/6
(Q + R)'s 1 day's work = 1/10
(P + R)'s 1 day's work = 1/5
On adding, 2 (P + Q + R)'s 1 day's work = 1/6 + 1/10 + 1/5 = 7/15
( P + Q + R)'s 1 day's work = 7/30
? (P + Q + R)'s can do the entire work in = 30/7 = 4²/₇ days

As the efficience of Romil is twice that of Sanjay, Romil, will take x h for completeion of the work, when Sanjay is taking 2x h.
Given, 2x - x = 4
? x = 4
Hence, they take = 1/(1/4 + 1/8) = 8/3 h