Answer:
let's define:
a = rate at which A works
b = rate at which B works
c = rate at which C works.
We know that A and B can do a piece of work in 10 days, then we have the equation:
(a + b)*10 days = 1 piece of work
B and C can do it in 15 days, then we get:
(b + c)*15 days = 1 piece of work
A and C can do it in 20 days, then we get:
(a + c)*20 days = 1 piece of work
Then we could write it (in a simplified way) as:
(a + b)*10 = 1
(b + c)*15 = 1
(a + c)*20 = 1
remember that the units of each rate will be (piece of work/day)
Now, to solve the system, we need to start by isolating one of the variables in one of the equations, i will isolate a in the first equation.
(a + b)*10 = 1
(a + b) = 1/10
a = (1/10 - b)
Now we can replace this in the third equation, and now our system is:
(b + c)*15 = 1
(1/10 - b + c)*20 = 1
Now we need to isolate another variable, i will isolate b in the first equation.
(b + c)*15 = 1
(b + c) = 1/15
b = 1/15 - c
Now we can replace this in the other equation to get:
(1/10 - ( 1/15 - c) + c)*20 = 1
(1/10 - 1/15 + 2*c)*20 = 1
Let's solve this for c.
(1/10 - 1/15 + 2*c) = 1/20
2*c = 1/20 - 1/10 + 1/15 = 3/60 - 6/60 + 4/60 = 1/60
c = (1/60)/2 = 1/120
With this, we can find the value of b.
b = 1/15 - c = 1/15 - 1/120 = 8/120 - 1/120 = 7/120
And with this we can find the value of a.
a = 1/10 - 7/120 = 12/120 - 7/120 = 5/120
Then we have:
c = 1/120 pieces of work per day.
b = 7/120 pieces of work per day
a = 5/120 pieces of work per day.
Now, to find how much time each one of them needs to complete a piece of work we need to solve:
C) (1/120 pieces of work per day)*T = 1 piece of work
T = (1 piece of work)/(1/120 pieces of work per day) = 120 days
C needs 120 days.
B) (7/120 pieces of work per day)*T = 1 piece of work
T = (1 piece of work)/(7/120 pieces of work per day) = 17.1 days
B needs 17.1 days
A) (5/120 pieces of work per day)*T = 1 piece of work
T = (1 piece of work)/(5/120 pieces of work per day) = 24 days
A needs 24 days
And if the 3 work together, we have:
(a + b + c)*T = 1 piece of work
ignoring the units only for the calculation, we get:
(1/120 + 7/120 + 5/120)*T = 1
(13/120)*T = 1
T = 1/(13/120) = 9.23
The 3 working together need 9.23 days.