Given:
A and B can do a work in 8 days
B and C can do a work in 6 days
C and A can do a work in 10 days

To find:
If they all work together, the work will be completed in how many days?

Solution:
Let's assume that the work is to build a wall. And we need to find out how many days it will take for A, B, and C to build the wall together.

Let's calculate the work efficiency of each pair of workers.

Efficiency of A and B:
A and B together can do the work in 8 days
So, in 1 day, they can do 1/8th of the work
Let's assume that A can do x part of the work in a day, and B can do y part of the work in a day.
So, x + y = 1/8 ---(i)

Efficiency of B and C:
B and C together can do the work in 6 days
So, in 1 day, they can do 1/6th of the work
Let's assume that B can do y part of the work in a day, and C can do z part of the work in a day.
So, y + z = 1/6 ---(ii)

Efficiency of C and A:
C and A together can do the work in 10 days
So, in 1 day, they can do 1/10th of the work
Let's assume that C can do z part of the work in a day, and A can do x part of the work in a day.
So, z + x = 1/10 ---(iii)

Now, we have three equations (i), (ii), and (iii) with three unknowns (x, y, and z).
We can solve these equations to find the values of x, y, and z.

Adding equations (i), (ii), and (iii), we get:
2x + 2y + 2z = 1/8 + 1/6 + 1/10
2(x+y+z) = (15+20+24)/240
2(x+y+z) = 59/240
x+y+z = 59/480 ---(iv)

Subtracting equation (i) from (iv), we get:
z = (59/480 - 1/8) = 1/30

Substituting z = 1/30 in equation (iii), we get:
x = (1/10 - 1/30) = 1/15

Substituting z = 1/30 in equation (ii), we get:
y = (1/6 - 1/30) = 1/5

So, the efficiency of A, B, and C is:
A can do 1/15th of the work in a day
B can do 1/5th of the work in a day
C can do 1/30th of the work in a day

Now, let's find out how many days it will take for A, B, and C to build the wall together.

Let's assume that the total work is 1 unit.

In 1 day, A, B, and C together can do:
1/15 + 1/5 +