I am seriously lost...?

Question

One pipe can fill a tank in 5 hours less than another; together they fill the tank in 5 hours. How long would it take each pipe alone to fill the tank?


I had someone try to help me but the answers are not making sense to me.. on my paper it says it should be in decimal form but they got... a whole number.. please someone explain♥

Answer 1

You have to use rates

P1 = 1/x
P2 = 1/(x-5)

P1+P2 = 1/5

1/x + 1/(x-5) = 1/5

multiply through by 5(x)(x-5)

5x-25 + 5x = x^2-5x

x^2-15x+25=0

a=1, b= -15, c=25

x = [15+/- sqrt(225-100)]/2
x = [15+/-sqrt(125)]/2
x = [15+/-5sqrt(5)]/2

x = 13.1 or x = 1.9

Time for pipe 1 = 13.1 and Time for pipe 2 = 8.1

or

Time for pipe 1 = 1.9 and Time for pipe 2 = -3.1 This cannot be true (can't have negative time).

13.1 and 8.1 are the correct answers

Answer 2

It takes the other pipe, Pipe 2, x hours to fill the tank. Pipe 1 takes x - 5 hours to fill the tank. That means that Pipe 1 fills at a rate of (1 tank) / (x - 5 hr) and Pipe 2 fills at a rate of (1 tank) / (x hr). Together, they fill at a rate of (1 tank) / (x - 5 hr) + (1 tank) / (x hr) = (1 tank) / (5 hr). This is addition of fractions. (1 / (x- 5)) + (1/x) = (x + x - 5) / ((x - 5)x) = (2x - 5) / (x^2 - 5x), and we know this is equal to 1/5. So (2x - 5) / (x^2 - 5x) = 0.2 ==> 0.2x^2 - x = 2x - 5 ==> 0.2x^2 - 3x + 5 = 0. You can use the quadratic formula to solve for x. x = (3 +/- sqrt(3^2 - 4*0.2*5)) / 2*0.2 = (3 +/- sqrt(5)) / 0.4 = 13.1 or 1.9. But it can't be 1.9, since this is the time for Pipe 2 to fill the tank, while Pipe 1 fills the tank in 5 hours less, which would be a negative amount of time and therefore impossible if x were 1.9. Therefore, x = 13.1. Pipe 2 fills the tank in 13.1 hours and Pipe 1 fills the tank in 13.1 - 5 = 8.1 hours.

(Note that the answerer below is wrong. How could the two pipes filling the tank together take longer than the faster pipe alone?)

Answer 3

Pipe one fills the tank in x hours.
Pipe two fills the tank in x-5 hours

Pipe one fills 1/x of the tank per hour
Pipe two fills 1/(x-5) of the tank per hour
Working together they fill (1/x + 1/(x-5)) of the tank per hour. Since it takes 5 hours to fill the tank,
(1/x + 1/(x-5)) = 1/5

Get a common denominator
1(x-5)/x(x-5) + x/x(x-5) = 1/5
(2x -5)/(x)(x-5) = 1/5
5(2x-5)/(x)(x-5) = 1
5(2x-5) = x(x-5)
10x - 25 = x^2 - 5x
x^2 -15x + 25 = 0

x = (15 +/- sqrt(225 - 100))/2
= (15 +/- sqrt(125))/2
= 7.5 +/- 2.5*sqrt(5)
= 7.5 +/- 5.59
= 13.09 or 1.9

Obviously it can't be 1.9

So 13.09 hr and 13.09-5 or 8.09 hr

You can check this.

1/13.09 + 1/8.09 = .2 or 1/5

Answer 4

One pipe fills in 7.5 hours, the other in 2.5 hours. Averaged together (7.5+2.5)/2 = 5 hours. 7.5-2.5 = 5 hours.