SOLUTION: It takes Larry 8 hours to paint his house. Larry started to paint the first 3 hours. His friend Patrick joined him and they both finished it in 2 hours. How long will it take Pa

Question 293528: It takes Larry 8 hours to paint his house. Larry started to paint the first 3 hours. His friend Patrick joined him and they both finished it in 2 hours. How long will it take Patrick to paint the house if he was to do it alone?
I have tried using fractions and the two equation method. However, what gets me is the 3 hours Larry started to paint by himself. Help!
Found 2 solutions by Theo, richwmiller:
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
It takes Larry 8 hours to paint his house.
Larry paints for 3 hours by himself.
His friend Patrick joins him and they finish in 2 hours.
How long would it take Patrick to paint the house himself?

Larry would take 8 hours to paint the house by himself.

His rate of painting is 1/8 of the house per hour.

This is based on the formula R * T = U

The number of units is equal to R is equal to 1.

The time is equal to 8 hours.

Plug those numbers into the equation to get R * 8 = 1

Divide both sides of that equation by 8 to get R = 1/8.

Plug 1/8 into that formula to get 1/8 * 8 = 1 so this rate is good.

in 3 hours Larry has painted 3/8 of the house.

That leaves 5/8 of the house left to do.

Larry and Patrick work together to finish the rest of the house in 2 hours.

Since they are working together their rates are additive.

Larry's rate is 1/8 of the house per hour.

Patrick's rate of painting the house per hour is represented by x.

The formula is Number of Units = Rate per Unit * Time

The number of units is equal to 1 if they had to do the whole house.

Since they only have to do 5/8 of the house, then the number of units is 5/8.

Since they are working together, their rates are combined.

The combines rate is (1/8 + x)

the amount of time they took is 2 hours.

The formula becomes:

5/8 = (1/8 + x) * 2

Simplify by removing parentheses to get:

5/8 = 2/8 + 2*x

Subtract 2/8 from both sides of the equation to get:

5/8 - 2/8 = 2*x

Combine like terms to get:

3/8 = 2*x

Divide both sides of the equation by 2 to get:

3/16 = x

Since x is Patrick's rate, this means that Patrick can paint 3/16 of the house in 1 hour.

This means that the formula for Patrick to do the whole house himself would be:

R * T = U where R = 3/16 and U = 1.

This formula becomes (3/16) * T = 1

Divide both sides of this equation by 3/16 to get T = 1/(3/16) which is the same as T = 16/3 which simplifies to T = 5 and 1/3 hours.

It would take Patrick 5 and 1/3 hours to paint the house by himself.

The key to this type of problem is the number of units that are left to be done.

Answer by richwmiller(17219) (Show Source): You can put this solution on YOUR website!
Larry worked for 3 hours at the rate of 1/8 per hour.
No need for two equations.
Larry already did 3/8 of the job
Pat and Larry will work together for 2 hours, Larry at 1/8 for each hour and Pat at x per hour for 2 hours
3/8+2/8+2/x=1
5/8=2/x=1
2/x=3/8
16=3x
16/3=x
5 1/3=x
So Pat would do the whole house alone in 5 1/3 hours.
Larry had already done 3/8 with pat he did 2/8. So Larry did 5/8 and Pat did 3/8
of the total job.
Larry worked 5 hours total.
5/8+3/8=1

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