Question 1064379
x is the rate of the first pipe.
y is the rate of the second pipe.


rate * time = quantity of work produced.


when they work together their rates are additive


therefore, (x + y) * 6 = 1


simplify this to get 6x + 6y = 1


when they both work for 3 hours, they fill 1/2 the pool.


one of them is then shut shut and the other takes 4 hours to fill the pool.


assuming the rate of the pipe that fills half the pool in 4 hours is x, then you get:


4x = 1/2


this says that the second pipe, working at a rate of x amount of the pool in one hour, takes 4 hours to fill half the pool.


if you multiply both sides of that equation by 2, you will see that the second pipe takes 8 hours to fill the pool by itself.


to find out how long it takes the first pipe to fikll the pool by itself, you have 2 equations that need to be solved simultaneously.


they are:


6x + 6y = 1
4x = 1/2


solve for x to get x = 1/8.


replace x in the first equation with 1/8 to get 6/8 + 6y = 1


simplify to get 6/8 + 6y = 8/8


subtract 6/8 from both sides of the equation to get 6y = 2/8.


solve for y to get y = (2/8)/6 = 2/48 = 1/24


the rate of the pipe that was shut off is 1/24 of the pool in one hour.


you get x = 1/8 and y = 1/24


simplify to get x = 3/24 and y = 1/24


the combined rate is 4/24 = 1/6.


go back to the original equation to get (x + y) * y = 1 which becomes 1/6 * 6 = 1 which becomes 1 = 1.


the solution is good


the rate of the pipe that was shut off after 3 hours is 1/24 of the pool in one hour.
that rate was originally y that we solved to get y = 1/24.


the rate of the pipe that was left open to finish the job in 4 hours is 1/8 of the pool in one hour.
that rate was originally x that we solved to get x = 1/8.