Gina and Maggie working together can do their MI-4 project in 24 hours. If Gina works alone
for 8 hours and Maggie then finishes the job in 30 hours, how many hours would it take each
working alone to do the project? [Assume both work at a constant rate].
suppose that 8 of Maggie's hours coincided with Gina's 8 hours
that would complete 1/3 (8/24) of the project
Maggie then finishes the remaining 2/3 in 22 hours (30 - 8)
so Maggie would take 33 hours alone
... she does 24/33 or 8/11 working 24 hours with Gina
Gina does 3/11 (1 - 8/11) in 24 hours
... so it would take her 88 hours to do it alone (24 / (3/11))
To solve this problem, we can use the concept of work rates. Let's start by assigning variables to the rates at which Gina and Maggie work.
Let G represent Gina's work rate, and M represent Maggie's work rate.
We are given the following information:
- Gina and Maggie together can do the MI-4 project in 24 hours. This means their combined work rate is 1/24 of the project per hour: (G + M) = 1/24.
- Gina works alone for 8 hours. Therefore, the work she completes is 8G.
- After Gina's 8 hours, Maggie finishes the job in 30 hours. This means Maggie's work rate is such that she completes 1/24 of the project per hour. Since Gina already completed 8G of the work, Maggie needs to complete the remaining (1 - 8G) of the work in 30 hours. This can be expressed as M = (1 - 8G)/30.
We can now use these equations to solve for the values of G and M.
Multiplying the first equation by 30 to eliminate fractions, we get:
30(G + M) = 30(1/24)
30G + 30M = 5/2
Substituting the value of M from the second equation into this equation:
30G + 30((1 - 8G)/30) = 5/2
30G + 1 - 8G = 5/2
22G + 1 = 5/2
22G = 5/2 - 1
22G = 5/2 - 2/2
22G = 3/2
G = (3/2) / 22
G = 3/44
Now that we have the value of G, we can substitute it back into the second equation to find M:
M = (1 - 8G)/30
M = (1 - 8(3/44))/30
M = (1 - 24/44)/30
M = (20/44)/30
M = 20/(44 * 30)
Now, we have the work rates for both Gina and Maggie. To find the individual times required to complete the project, we can use the equation Time = Work/Rate.
Time taken by Gina alone = Work/ Gina's rate = 1/ G
Time taken by Maggie alone = Work/ Maggie's rate = 1/ M
Substituting the values we found:
Time taken by Gina alone = 1/(3/44) = 44/3 hours
Time taken by Maggie alone = 1/(20/(44 * 30)) = 66 hours
Therefore, Gina would take 44/3 hours (approximately 14.67 hours) and Maggie would take 66 hours to complete the project individually.