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

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]