Question 285536
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.