Question

An inlet pipe on a swimming pool can be used to fill the pool in 26 hours. The drain pipe can be used to empty the pool in 27 hours. If the pool is 2/3 filled and then the inlet pipe and drain pipe are opened, how many more hours would it take to fill the pool? Round your answer to two decimal places, if needed.

Answer 1

`\boxed{\sf{234 \;hours}}`

Explanation:

Inflow

• Time taken to fill pool by just the inlet pipe = 26 hours
  So in 1 hour, 1/26 of the pool will be filled.
  

• This is the rate at which the pool is filled by inlet pipe alone
  = 1/26 of the pool per hour

Outflow

• Time taken to drain the pool by drain pipe alone = 26 hours
  So in 1 hour, 1/27 of the pool will be filled.

• Rate of draining pool by outlet pipe alone = 1/26 of pool per hour

• With both pipes, open the effective inflow rate is
  inflow rate - outflow rate
  =
  `(1)/(26) - (1)/(27)`

• The LCM of 26 and 27 = 26 x 27 = 702

• Use this as the common denominator and adjust the numerators accordingly
  
  
  `\Rightarrow (1 * 27)/(702) - (1 * 26)/(702)\\\\\Rightarrow (27-26)/(702)\\\\\Rightarrow (1)/(702)`

• So, the two pipes, working in tandem will fill the pool at the rate of
  `\bold{(1)/(702)}` of the pool per hour

• The time taken to fill the entire pool is the reciprocal of this
  (time = 1/rate)
  
• Therefore it will take 702 hours to fill the entire pool with both pipes open
  
• Since the pool is already
  `(2)/(3)` filled, only
  `(1)/(3)` of the pool needs to be filled
  

• Therefore the time taken will be reduced by 1/3 rd
  

• Time taken to fill 1/3 rd of the pool
  
  `= (1)/(3) * 702 = \boxed{234\;hours}`