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Question 285536: I need help figuring out this problem:
Melanie can paint a certain barn by herself in x days. Her helper melissa can paint the same barn by herself in 2x days. write a rational expression for the fraction of the barn that they complete in one day by working together. Evaluate the expression for x=5.
would it be written as 2x+5x
Found 2 solutions by Theo, richwmiller:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! Melanie paints the barn in x days.
Melissa paints the barn in 2x days.
The formula to use is rate * time = units
The number of units is equal to 1 (the barn)
Melanie can paint the barn in x days, so the formula for her would be:
r * x = 1
For Melanie, r = 1/x which is the amount of the barn she can paint in one day.
Melissa can paint the barn in 2x days, so the formula for her would be:
r * 2x = 1
For Melissa, r = 1/2x which is the amount of the barn she can paint in one day.
When they work together, their rates are combined (added together).
Formula for them working together would be:
(1/x + 1/2x) * T = 1
Their combined rate is (1/x + 1/2x).
Since 1/x is equivalent to 2/2x, then their combined rate becomes:
(2/2x + 1/2x) * T = 1 which becomes:
3/2x * T = 1
Their combined rate is 3/2x which means that they can paint 3/2x of the barn in one day.
If x = 5, then the following happens:
Melanie's rate is 1/5.
Melissa's rate is 1/10.
Their combined rate is 1/5 + 1/10 = 3/10.
It would take Melanie 5 days to paint the barn working alone because 5 * 1/5 = 1.
It would take Melissa 10 days to paint the barn working alone because 10 * 1/5 = 1
It would take both of them 10/3 days to paint the barn working together because 10/3 * 3/10 = 1
In 10/3 days, Melanie has painted 1/5 * 10/3 = 2/3 of the barn.
In 10/3 days, Melissa has painted 1/10 * 10/3 = 1/3 of the barn.
Together they have painted 2/3 + 1/3 of the barn = 1 of the barn = the whole barn.
Your answer is:
Their combined rate is 1/x + 1/2x = 2/2x + 1/2x = 3/2x.
If x is equal to 5, then their combined rate is 1/5 + 1/10 = 2/10 + 1/10 = 3/10.
It would not be 2x + 5x.
In this type of problem, always think of the basic formula and how it fits the problem.
The basic formula is rate * time = units.
If Melanie can paint the barn in x days, then time is x.
The formula becomes rate * x = units.
To get her rate, you have to divide both sides of this equation by x to get rate = units / x which puts the x in the denominator.
Since units is 1, then we get rate = 1/x for Melanie.
For Melissa, we got rate = 1/2x because she takes twice as long as Melanie.
When they asked you what happens when x = 5, you needed to replace x with 5 in the equations as I did above.
1/x + 2/x became 1/5 + 1/10.
Answer by richwmiller(17219)  (Show Source):
You can put this solution on YOUR website! r/x+r/2x=1
2r+r=2x
3r=2x
r=2x/3
in one day they can do 3/(2x)
It takes them r days working together.
3r/2=x
3r=2x
if x=5
r/5+r/10=1
2r+r=10
3r=10
r=10/3 =3 1/3 days
if one can do it in 5 days the other in 10 days together they can do it in 3 1/3 days
in one day one does 1/5 and the other 1/10
2/10+1/10 they can do 3/10 in one day working together.
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