A and B can do a piece of work in 10 days, B and C in 15 days, A and C in 12 days. A,B and C work together to finish the work. If they are paid $1500, how should the money be divided?
Let their times to do the job alone be A, B, and C days respectively
A and B can do a piece of work in 10 days -->
1/A + 1/B = 1/10
A + B = AB/10
10A + 10B = AB
B and C in 15 days:
1/B + 1/C = 1/15
B+C = BC/15
15B + 15C = BC
A and C in 12 days
12A + 12C = AC
Not the nicest triplet of equations to solve, but how about
A = 120/7
B = 24
C = 40
so the ratio of their times
= (7/120) : 1/24 : 1/40
= 7 : 5 : 3
So the cost should be split in that ratio, or
(7/15)(1500) : (5/15)(1500) : (3/15)(1500)
= $700 : $500 : $300
Thank you so much. :D :D . I am very grateful towards you... :)
You are welcome, Happy New Years
Thanks... can you help me with this question as well?
A and B take 18 days to complete a work. A left the work after 10 days and B takes 12 days more to complete the work. In how many days can A complete the whole work?
To solve this problem, we can use the concept of "work rates" and set up a system of equations. Let's start by assigning variables to the work rates of A, B, and C.
Let's say the work rate of A is x units per day.
The work rate of B will be y units per day.
The work rate of C will be z units per day.
According to the problem, we know that:
A and B can do a piece of work in 10 days, so their combined work rate is (x + y) units per day.
B and C can do the same work in 15 days, so their combined work rate is (y + z) units per day.
A and C can do the same work in 12 days, so their combined work rate is (x + z) units per day.
Now let's set up the equations based on the work rates:
Equation 1: (x + y) * 10 = 1 (work unit)
Equation 2: (y + z) * 15 = 1 (work unit)
Equation 3: (x + z) * 12 = 1 (work unit)
To solve this system of equations, we need to find the values of x, y, and z.
Multiplying Equation 1 by 3, Equation 2 by 2, and Equation 3 by 5, we get:
3(x + y) = 3/10 (work unit)
2(y + z) = 2/15 (work unit)
5(x + z) = 5/12 (work unit)
Now, simplifying these equations, we get:
3x + 3y = 1/10
2y + 2z = 1/15
5x + 5z = 1/12
We can rewrite these equations in terms of y:
3x + 3y = 1/10 ---> Equation 4
2y + 2z = 1/15 ---> Equation 5
5x + 5z = 1/12 ---> Equation 6
Now, let's solve this system of equations using any preferred method (substitution, elimination, or matrix method). Let's use substitution for simplicity:
From Equation 5, we can express z in terms of y:
2y + 2z = 1/15
2z = 1/15 - 2y
z = 1/30 - y/15 ---> Equation 7
Now substitute the value of z in terms of y from Equation 7 into Equation 6:
5x + 5z = 1/12
5x + 5(1/30 - y/15) = 1/12
5x + 1/6 - y/3 = 1/12
5x - y/3 = 1/12 - 1/6
5x - y/3 = 1/12 - 2/12
5x - y/3 = -1/12
Now, rewrite this equation without fractions:
60x - 4y = -1 ---> Equation 8
Now we have two equations for x and y:
3x + 3y = 1/10 ---> Equation 4
60x - 4y = -1 ---> Equation 8
Now solve these two equations simultaneously to find the values of x and y. By solving the equations, we get:
x = 1/600
y = 1/3000
Now that we have the values of x and y, we can substitute them into one of the previous equations to find the value of z. Let's use Equation 7:
z = 1/30 - y/15
z = 1/30 - (1/3000) / 15
z = 1/30 - 1/45000
z = 1499/45000
Now that we have the values of x, y, and z, we can find the work rates for A, B, and C.
The work rate of A = x = 1/600 units per day
The work rate of B = y = 1/3000 units per day
The work rate of C = z = 1499/45000 units per day
The total work done is equal to 1 unit, as given in the problem. Therefore, the combined work rate of A, B, and C is:
Total work rate = (work rate of A) + (work rate of B) + (work rate of C)
= 1/600 + 1/3000 + 1499/45000
= 1/600 + 1/3000 + 1/30
= 5/3000 + 1/3000 + 100/3000
= (5 + 1 + 100) / 3000
= 106/3000
Now, to find the share of each individual, we divide their work rate by the total work rate and multiply it by the total payment.
For A:
Share of A = (work rate of A) / (total work rate) * Total payment
= (1/600) / (106/3000) * 1500
= (1/600) * (3000/106) * 1500
= 5 * 15
= 75
For B:
Share of B = (work rate of B) / (total work rate) * Total payment
= (1/3000) / (106/3000) * 1500
= (1/3000) * (3000/106) * 1500
= 5 * 15
= 75
For C:
Share of C = (work rate of C) / (total work rate) * Total payment
= (1499/45000) / (106/3000) * 1500
= (1499/45000) * (3000/106) * 1500
= 75 * 15
= 1125
Therefore, the money should be divided as follows:
A: $75
B: $75
C: $1125