If Tom, George and Mario work together, they can paint a 600 ft2 room in 4 hours. When only George and Mario work together, it takes 8 hours to paint the room. Tom and George, working together, take 6 hours to paint the room. How long would it take Tom to paint the room alone? Hint: Write the system of equations shown below in augmented form, then solve the system for T (Tom's painting rate). Next, use the equation Painted Area = Painting Rate× Time and solve for Time to determine how long it would take Tom to paint the room alone. 4(T+G+M)=600 8(0+G+M)=600 6T+G+0)=600
Accepted Solution
A:
We call:
T = Tom's paiting rate [tex]( \frac{ ft^{2} }{h} )[/tex] G = George's paiting rate [tex]( \frac{ ft^{2} }{h} )[/tex] M = Mario's paiting rate [tex]( \frac{ ft^{2} }{h} )[/tex]
As the first statement says that they can work together, then:
T + G + M = Painting rate working together, that is the surface area they paint per hour.
Since the problem says that they last four hours painting a [tex] 600ft^{2} [/tex] room, then we need to multiply the amount of hours times the painting rate that is equal the surface area of the room, so:
4(T + G + M) = 600
Applying a similar reasoning, the George and Mario's painting rate is:
0 + G + M Note that 0 implies that Tom didn't work in this case.
As they last eight hours painting the room, then:
8(0 + G + M) = 600
The same reasoning is applied to Tom and George working together. In this case 0 implies that Mario didn't work, so:
6(T + G + 0) = 600
We have a system of linear equations in three variables T, G, M:
4(T+G+M)=600 8(0+G+M)=600 6(T+G+0)=600
Solving:
(1) T + G + M = 150 (2) G + M = 75 (3) T + G = 100
Substitute (3) in (1): 100 + M = 150 ∴ M = 50
Replace M in (2): G + 50 = 75 ∴ G = 25
Substitute G in (3) T + 25 = 100 ∴ T = 75
Then, the painting rates are: Tom = 75 [tex]( \frac{ ft^{2} }{h} )[/tex] George = 25 [tex]( \frac{ ft^{2} }{h} )[/tex] Mario = 50 [tex]( \frac{ ft^{2} }{h} )[/tex]
Solving for Time to determine how long it would take Tom to paint the room alone:
Painted Area = Painting Rate×Time, so:
[tex]Time Tom = \frac{Painted Area}{Painting Rate (T)} [/tex] [tex]Time Tom = \frac{600}{75} = 8 hours [/tex]