Sam can do a job in X hours. Denise can do the same job in x+2 hours. How long will it take Sam and Denise to do the job together? (Express your answer in terms of x.)
Rate Sam works : ( 1 job ) / x hrs
Rate Denise works : ( 1 job ) / ( x + 2) hrs
Rate together : 1 / x + 1 / ( x + 2 ) =
To solve this problem, we need to find the amount of work done by Sam in 1 hour and the amount of work done by Denise in 1 hour. Then we can add up their rates to find the combined rate at which they work together, and finally, divide the total amount of work by their combined rate to find the time it takes for them to complete the job together.
Let's start by finding Sam's rate. We know that Sam takes X hours to complete the job. Therefore, Sam's rate is 1/X, which means he completes 1/Xth of the job in one hour.
Similarly, Denise takes x+2 hours to complete the job. Therefore, Denise's rate is 1/(x+2), which means she completes 1/(x+2)th of the job in one hour.
Now, we can add up their rates to find their combined rate when working together. Adding their rates gives us 1/X + 1/(x+2).
To find the time it takes for them to complete the job together, we need to divide the total work (which is 1, the whole job) by their combined rate. So the equation becomes:
1 = (1/X + 1/(x+2)) * T
Where T represents the time it takes for them to complete the job together. To solve for T, we can multiply both sides by (X * (x+2)) to get rid of the fractions:
X * (x+2) = (x+2) + X * T
Now, we can solve for T by isolating it:
X * x + 2X = x + 2 + X * T
X * x + X * T = x + 2 - 2X
X * (x + T) = x + 2 - 2X
T = (x + 2 - 2X) / X
Therefore, it will take Sam and Denise (x + 2 - 2X) / X hours to do the job together.