SOLUTION: maria can paint the walls of an apartment in 8 hours. After she has worked 3 hours, Pat joins her and they finish the job in 2 hours more. How long would it take Pat to do the who

Question 1134856: maria can paint the walls of an apartment in 8 hours. After she has worked 3 hours, Pat joins her and they finish the job in 2 hours more. How long would it take Pat to do the whole job without the aid of Maria?

Found 2 solutions by Theo, MathTherapy:
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
let m = the rate that maria paints the apartment.
let p = the rate that pat paints the apartment.

rate * time = quantity of work completed.

maria can paint the apartment in 8 hours.

that leads to m * 8 = 1, where...

m is the rate that maria works.
1 is the quantity of work completed which is 1 painted apartment.

from m * 8 = 1, solve for m to get m = 1/8.

this means maria can paint 1/8 of the apartment in 1 hour.

after she had worked 3 hours, maria has painted 3/8 of the apartment.

that means that 5/8 of the apartment still needs to be painted.

when they work together, the rate that maria works and pat works is additive.

together they finish the apartment in 2 hours.

that leads to (m + p) * 2 = 5/8 which becomes (1/8 + p) * 2 = 5/8 because maria'a rate is equal to 1/8.

simplify this equation to get 1/8 * 2 + 2 * p = 5/8.

simplify further to get 2/8 + 2 * p = 5/8.

subtract 2/8 from both sides of this equaiton to get 2 * p = 3/8

solve for p to get p = (3/8) / 2 = 3/16

that's the rate that pat can paint the apartment at.

confirm by replacing p with 3/16 in the equation of (1/8 + p) * 2 = 5/8

equation becomes (1/8 + 3/16) * 2 = 5/8

simplify to get 2/8 + 6/16 = 5/8

since 6/16 is the same as 3/8, you get 2/8 + 3/8 = 5/8 which is true, confirming the determination that pat's rate of painting the apartment is correct.

to find how long it would take pat to do the work itself, go back to rate * time = quantity formula and replace p with 3/6 and q with 1 to get 3/16 * time = 1
solve for time to get time = 1 * 16/3 = 16/3 = 5 and 1/3 hours.

it would take pat 5 and 1/3 hours to completely paint the apartment by herself.

that's your solution.

Answer by MathTherapy(10555) (Show Source): You can put this solution on YOUR website!

maria can paint the walls of an apartment in 8 hours. After she has worked 3 hours, Pat joins her and they finish the job in 2 hours more. How long would it take Pat to do the whole job without the aid of Maria?

Maria does the job in 8 hours, or does of job in 1 hr
Pat does the job in P hours, or does of job in 1 hr
Maria works 3 + 2, or 5 hours, while Pat works for 2 hours
We then get:

5P + 16 = 8P ----- Multiplying bu LCD, 8P
16 = 8P - 5P
16 = 3P
P, or time Pat takes, alone =
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