Math: Formula?

 Two pipes can be used to fill a swimming pool. 

 When the first pipe is closed, the second pipe can fill the pool in 14 hours. 

 When the second pipe is closed, the first pipe can fill the pool in 6.5 hours. 

 How long (in hours) will it take to fill the pool if both pipes are open?

  1/14 + 1/6.5 = 1/x

  x ≈ 4.43902 hours

  It will take 4.44 hours (The answer is rounded to two decimal places).

1/14 + 1/6.5 = 1/x

41/182 = 1/x

41x = 182

x = 4.44 hours answer//

r₁: rate for the pipe No. 1

r₂: rate for the pipe No. 2

When the first pipe is closed, the second pipe can fill the pool in 14 hours: → r₂ = pool/14

When the second pipe is closed, the first pipe can fill the pool in 6.5 hours: → r₁ = pool/6.5

When the first pipe is open and when the second pipe is open, you add the rates together.

r = r₁ + r₂

r = (pool/6.5) + (pool/14)

r = [(14 * pool) + (6.5 * pool)]/(6.5 * 14)

r = (20.5 * pool)/91

r = (20.5/91) * pool

r = pool/(91/20.5)

The necessary time to fill the pool, with the 2 pipes are open at the same time is: 91/20.5

≈ 4.44 hours → to go further

= 91/20.5

= (82 + 9)/20.5

= (82/20.5) + (9/20.5)

= 4 + (9/20.5) ← this result represents hours

= 4 h + (9/20.5) h → you know that: 1 hour = 60 min

= 4 h + [(9/20.5) * 60 min]

= 4 h + (540/20.5) min

= 4 h + [(533 + 7)/20.5] min

= 4 h + [(533/20.5) + (7/20.5)] min

= 4 h + [26 + (7/20.5)] min

= 4 h + 26 min + (7/20.5) min → you know that: 1 min = 60 s

= 4 h + 26 min + [(7/20.5) * 60 s]

= 4 h + 26 min + (420/20.5) s

= 4 h + 26 min + [(410 + 10)/20.5] s

= 4 h + 26 min + [(410/20.5) + (10/20.5)] s

= 4 h + 26 min + 20 s + (10/20.5) s

= 4 h + 26 min + 20 s + ≈ 0.4878 s

= 4 h + 26 min + 20.4878 s

The rate r₂ at which the second pipe pumps water is 1 pool / 14 hours. Thus, we have

r₂= 1/14

Similarly, the rate r₁ at which the first pipe pumps water is 1 pool / 6.5 hours, and thus, we have

r₁ = 1 / 6.5

We want the time t it will take to fill the pool with both pipes open. The amount of water pumped by both pipes within t hours is given by r₁t and r₂t respectively.

Thus, we have

r₁t + r₂t = 1

Solving for t gives

t = 1 / (r₁ + r₂)