A, B and C can reap a field in 15 ¾ days; B, C and D in 14 days; C, D and A in 18 days; D, A and B in 21 days. In what time can A, B, C and D together reap it?

The correct answer is: 12 3/5 days

Let’s say the total work done to reap the field is represented as W and let’s say the work done by A in 1 day is W_A, work done by B in 1 day is W_B,work done by C in 1 day is W_C and work done by D in 1 day is W_D
Step 1 of 3:
It is given that A, B and C can reap a field in 15 ¾ days or 15.75 days. So we can write
W_A  15.75 + W_B  15.75 + W_C  15.75 = W
W_A  + W_B + W_C =  …….(1)
It is given that  B, C and D can reap a field in 14 days. So we can write
W_B  14 + W_C  14 + W_D  14 = W
W_B + W_C + W_D = …….(2)
It is given that  C, D and A can reap a field in 18 days. So we can write
W_C  18 + W_D  18 + W_A  18 = W
W_C + W_D + W_A = …….(3)
It is given that  D, A and B can reap a field in 21 days. So we can write
W_C  21 + W_D  21 + W_A  21 = W
W_D + W_A + W_B = …….(4)
Step 2 of 3:
Now, adding equations (1), (2), (3) and (4)
3(W_A  + W_B + W_C + W_D) = 
W_A  + W_B + W_C + W_D = ……(4)
Step 3 of 3:
Let’s say the time taken by all of them to complete the work is t. So we can write
W_A  t + W_B  t + W_C  t + W_D  t = W
t(W_A  + W_B + W_C + W_D) = W
Using equation (4)
t  = W
t =  days = 12 days.
Final Answer:
Hence, the time taken by A, B, C and D together to reap the field is 12  days.