Carlo,Clyde,and Marky working together can do a job in 6 days. Carlo and Marky can do can do the job in 8 days. Clyde and Marky can do the job in 9 days. find how long it will take each guy working alone to do the job.
marko's rate --- 1/x
carlo's rate --- 1/y
clyde's rate --- 1/z
combined rate of all three
= (xy + xz + yz)/(xyz)
xyz/(xy + xz + yz) = 6
xyz = 6((xy + xz + yz) **
Carlo and Marko
1/(1/x + 1/y) = 8
xy/(x+y) = 8
xy = 8(x+y) ***
clyde and marko
1/(1/x + 1/z) = 9
xz = 9(x+z) ****
that's a messy set of equations,
ran it through Wolfram and got this
http://www.wolframalpha.com/input/?i=solve+xyz+%3D+6(xy+%2B+xz+%2B+yz)+,+xy+%3D+8(x%2By)+,+xz+%3D+9(x%2Bz)
To solve this problem, we can assign variables to represent the work rate of each person. Let's say:
- Carlo's work rate is "C" (in job per day).
- Clyde's work rate is "L".
- Marky's work rate is "M".
Given that Carlo, Clyde, and Marky can complete the job together in 6 days, we can create the equation:
1/6 = C + L + M (equation 1)
It is also given that Carlo and Marky can complete the job in 8 days, so we have another equation:
1/8 = C + M (equation 2)
Similarly, Clyde and Marky can complete the job in 9 days, so we have:
1/9 = L + M (equation 3)
Now, we can solve this system of equations to find the work rate of each person.
First, let's solve equations 2 and 3 using substitution. Rearrange equation 2 to solve for C:
C = 1/8 - M (equation 4)
Now substitute equation 4 into equation 3:
1/9 = L + M (equation 3)
1/9 = L + (1/8 - M)
1/9 = L + 1/8 - M
To add L and 1/8, find a common denominator of 72:
1/9 = (9L + 8)/72 - M
72/9 = (9L + 8)/72 - M
8 = 9L + 8 - 72M
Simplify:
0 = 9L - 72M
Divide both sides by 9:
0 = L - 8M (equation 5)
Now, let's solve equations 1 and 5 using substitution.
Substitute equation 4 into equation 1:
1/6 = (1/8 - M) + L + M (equation 1)
1/6 = 1/8 + L - M + M
1/6 = 1/8 + L
Simplify:
1/6 - 1/8 = L
4/24 - 3/24 = L
1/24 = L (equation 6)
Now substitute equation 6 into equation 5:
0 = (1/24) - 8M
0 = 1 - 192M
Simplify:
192M = 1
M = 1/192
Therefore, Marky's work rate is 1/192 of the job per day.
Substitute M = 1/192 into equation 4:
C = 1/8 - (1/192)
C = 24/192 - 1/192
C = 23/192
Therefore, Carlo's work rate is 23/192 of the job per day.
Substitute M = 1/192 into equation 6:
L = 1/24
Therefore, Clyde's work rate is 1/24 of the job per day.
To find how long each person will take working alone, we can calculate the reciprocal of their respective work rates:
Carlo: 192/23 ≈ 8.35 days.
Clyde: 24 days.
Marky: 192 days.
Therefore, it will take Carlo approximately 8.35 days, Clyde 24 days, and Marky 192 days to complete the job individually.