Bill & Bob are building a new house for their dog, Lyca. Bill alone can finish building in 3 hours while Bob is more delicate and can complete the same task alone in 6 hours.
How many hours does it take for both of them to build Lyca's house?
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Recall:
Rate x Time = Distance
We can use this formula, but substitute Job for Distance:
Rate x Time = Job
Set Job = 1
Therefore the equation simplifies to:
Rate x Time = 1
Bill can do the job in 3 hours so the equation for Bill is:
R x 3 = 1
Solve for R
Bill's rate is: R = 3 1
Similarly, we can find Bob's rate like so:
R x 6 = 1
Solve for R
Bob's rate is: R = 6 1
Combine the both rates:
3 1 + 6 1 = 2 1
Their combined rate is: 2 1
Use Formula again:
R x T = 1
2 1 x T = 1
T = 2
Working at the same time, in 6 hours they would build 3 houses (Bill 2 and Bob 1) Therefore it will take 6/3 or 2 hours to build one.
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In 1 hour, Bill completes 1/3 of the whole work, and Bob completes 1/6 in an hour.
Thus, if they work together, they will complete 1/3+1/6 = 1/2 of the whole work in an hour.
So it will take 1/(1/2) = 2 hours to build Lyca's house.