A roofer requires 10 h to shingle a roof. After the roofer and an apprentice work on a roof for 5 h, the roofer moves on to another job.

Question

A roofer requires 10 h to shingle a roof. After the roofer and an apprentice work on a roof for 5 h, the roofer moves on to another job. The apprentice requires 9 more hours to finish the job. How long would it take the apprentice, working alone, to do the job?

Answer 1

You know the roofer's rate -- 1 roof in 10 hours, of .1 roofs per hour.

 

What you need to figure out is the apprentice's rate.  Call that x.

 

So in the first 5 hours, they together worked at a rate of .1 + x, so they completed 5(.1 + x) of the roof, or .5 + 5x.

 

So the amount left to do was 1 minus that amount:  1 - (.5 + 5x) = .5 - 5x.

 

The apprentice finished that in 9 hours, so his rate was (.5 - 5x) / 9.

 

But we also know that his rate is x, so that gives us an equation: (.5 - 5x) / 9 = x

 

Let's solve that...

 

(.5 - 5x) / 9 = x

.5 - 5x = 9x

.5 = 14x

.5/14 = x

1/28 = x

 

So the apprentice does 1/28 of a roof in one hour.

 

That means that he can do a whole roof in 28 hours on his own.

 

Answer 2

w = rt.  The work in this case is one, because we are roofing 1 house.

 

The roofer can do the job alone in 10 hrs therefore his/her rate is 1/10.

 

The apprentice can do the job alone in x hours therefore his/her rate is 1/x.

 

The work is broken up into parts:

 

1 = 1/10(5) + 1/x(5+9)

1 = 1/2 +14/x

1/2=14/x

x=28 hours