To find the share of each, we need to determine the individual contributions of A, B, and C towards the work.
Let's first assign variables to represent the individual rates at which they work:
- Let's assume A's work rate is represented by a (in work per day).
- B's work rate will be represented by b (in work per day).
- C's work rate will be represented by c (in work per day).
Using these variables, we can set up the following equations based on the given information:
1. The combined work rate of A and B is 1 work per 10 days:
a + b = 1/10
2. The combined work rate of B and C is 1 work per 15 days:
b + c = 1/15
3. The combined work rate of A and C is 1 work per 12 days:
a + c = 1/12
Now, let's solve these equations to determine the individual work rates:
First, add equations 1 and 3 to eliminate the variable b:
(a + b) + (a + c) = 1/10 + 1/12
2a + b + c = 1/10 + 1/12
Similarly, add equations 2 and 3 to eliminate the variable b:
(b + c) + (a + c) = 1/15 + 1/12
a + 2c + b = 1/15 + 1/12
Now, we have a system of linear equations:
2a + b + c = 1/10 + 1/12
a + 2c + b = 1/15 + 1/12
To solve this system, we can use methods like substitution or elimination. Elimination seems like a convenient approach here.
Subtract equation 1 from equation 2 to eliminate the variable a:
(a + 2c + b) - (2a + b + c) = (1/15 + 1/12) - (1/10 + 1/12)
Simplifying the equation, we get:
a + b + c = 1/15 + 1/12 - 1/10
Now, substitute the value of (a + b) from equation 1 into this equation:
1/10 + c = 1/15 + 1/12 - 1/10
Simplify further:
c = 1/15 + 1/12 - 1/10 - 1/10
Now, we have the value of c. Substitute it back into equations 1 and 3 to find the values of a and b:
From equation 1:
a + b = 1/10 - c
From equation 3:
a + c = 1/12
Solve these equations to find the values of a and b.
Once we have the values of a, b, and c, we can determine their respective shares by calculating the portion of the total work done by each person:
A's share = (a)/(a + b + c) * 15000
B's share = (b)/(a + b + c) * 15000
C's share = (c)/(a + b + c) * 15000
Plug in the values of a, b, c, and calculate the shares.