A and B can do a piece of work in 10 days, B and C in 15 days and A and C in 12 days. A,B and C work together to finish the work. If they are paid 15000, find the share of each?
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.
1/a + 1/b = 1/10
1/b + 1/c = 1/15
1/a + 1/c = 1/12
subtract 2 from 1
1/a - 1/c = 1/30
1/a + 1/c = 1/12
add them
2/a = 7/60
1/a = 7/120
similarly, 1/b = 1/24 and 1/c = 1/40
1/a is the share of the work a can do in a day.
Now set up a proportion to allocate the money
1/a : 1/b : 1/c = 7/120 : 1/24 : 1/40
= 7:5:3 = 7000:5000:3000
and those are the shares of the $15000