SOLUTION: It takes Albert and Billy 8 hours to shovel snow off a driveway together. Albert shovels snow slower than Billy. Billy would only take 12 hours to shovel the snow on his own. How m

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Question 817289: It takes Albert and Billy 8 hours to shovel snow off a driveway together. Albert shovels snow slower than Billy. Billy would only take 12 hours to shovel the snow on his own. How many hours would it take Albert to shovel the snow on his own?
Answer by jsmallt9(3759) (Show Source):
You can put this solution on YOUR website!
The key to these rate of work problems is to focus not on how long it takes someone to complete the whole job (which is what you are usually given) but on how much of the job that person/group can do in a single hour.
Let a = the number of hours Albert would take shoveling on his own. Then
1/a = the fraction of the job which Albert can complete in 1 hour.
12 = the number of hours Billy would take shoveling on his own. Then
1/12 = the fraction of the job which Billy can complete in 1 hour.
8 = the number of hours the time is takes for both together to finish. Then
1/8 = the fraction of the job they can complete together in 1 hour.

We can solve this with an equation that says that the sum of the individual rates is the rate of the two of them together:

Now we just solve this equation. It is probably easiest if we eliminate the fractions. This can be done all at once if we multiply each side of the equation be the lowest common denominator (LCD) of all the denominators. The LCD of a, 12 and 8 is 24a. So we multiply each side by 24a:

As we multiply (using the Distributive Property on the left, of course) all the denominators will cancel out:

Subtract 2a:

So Albert would take 24 hours to finish if he worked by himself.