Pump A can fill a swimming pool in 6 hours and pump B can fill the same pool in 4 hours. The two pumps began working together to fill an empty pool, but after 40 minutes pump B broke down, and pump A had to complete the job alone. The pool was scheduled to open 4 hours after the pumping started. Was it full by then, or did the opening have to be delayed?

rewording to indicate the rate of completion,

a = 1/6 pool/hr
b = 1/4 pool/hr

they work together for 2/3 hour, so they filled

(1/6 + 1/4)*2/3 = 5/18 pool

So, how long does it take A to fill the other 13/18 pool?

13/18 / 1/6 = 13/3 hour

So, the whole pool took 2/3 + 13/3 = 15/3 = 5 hours to fill, finishing an hour late.

To solve this problem, we'll first calculate the rate at which each pump fills the pool. Then, we'll determine how much of the pool was filled before pump B broke down.

Let's start by finding the rate at which pump A fills the pool. We know that pump A can fill the pool in 6 hours, so its rate of filling is 1 pool per 6 hours, or 1/6 pools per hour.

Similarly, pump B can fill the pool in 4 hours, so its rate of filling is 1 pool per 4 hours, or 1/4 pools per hour.

When working together, the combined rate of pump A and pump B is the sum of their individual rates. Therefore, their combined rate is 1/6 + 1/4 = 2/12 + 3/12 = 5/12 pools per hour.

Since the pumps worked together for 40 minutes, which is 40/60 = 2/3 hour, we can calculate the amount of the pool filled in that time:

Amount filled by both pumps = Rate × Time = (5/12) × (2/3) = 10/36 = 5/18 pools.

Now, pump A needs to complete the remaining portion of the pool alone. Since the pool was scheduled to open 4 hours after pumping started, and the pumps worked for 40 minutes (2/3 hour), this means pump A had 4 - (2/3) = 10/3 hours to complete the job.

The rate of pump A is 1/6 pools per hour, so the amount of the pool filled by pump A during that time is:

Amount filled by pump A = Rate × Time = (1/6) × (10/3) = 10/18 = 5/9 pools.

Adding up the amount filled by both pumps and pump A alone, we have:

Total amount filled = Amount filled by both pumps + Amount filled by pump A
= 5/18 + 5/9 = 5/18 + 10/18 = 15/18 = 5/6 pools.

Since the total amount filled is less than the full volume of the pool (1 pool), the opening would have to be delayed because the pool was not full by the scheduled time of 4 hours after pumping started.

To find out if the pool was full by the time it was scheduled to open, we need to determine the total time it would take for both pumps to fill the pool and compare it to the time available.

First, let's calculate the rate of each pump. Pump A can fill the pool in 6 hours, which means its rate is 1/6 of the pool per hour. Pump B can fill the pool in 4 hours, so its rate is 1/4 of the pool per hour.

When both pumps are working together, their combined rate is the sum of their individual rates. So, the combined rate of Pump A and Pump B working together is 1/6 + 1/4 = 5/12 of the pool per hour.

The pumps worked together for 40 minutes before Pump B broke down. Since there are 60 minutes in an hour, this is equivalent to 40/60 = 2/3 of an hour.

During this time, the pumps filled a portion of the pool at their combined rate. The amount filled is given by the formula: rate × time = portion of the pool filled.

So, during the 40 minutes, the pumps filled a portion of the pool equal to (5/12) × (2/3) = 5/18 of the pool.

Now, let's calculate the remaining time needed for Pump A to fill the pool alone. Since Pump A has a rate of 1/6 of the pool per hour, and we already filled 5/18 of the pool, we need to find the time it takes for Pump A to fill (1 - 5/18) = 13/18 of the pool.

Using the formula: rate × time = portion of the pool filled, we have (1/6) × time = 13/18 of the pool.

Simplifying the equation, we find: time = (13/18) / (1/6) = (13/18) × (6/1) = 13/3 = 4 and 1/3 hours.

Therefore, Pump A needs an additional 4 and 1/3 hours to fill the remaining 13/18 of the pool.

Now, let's add up the time it takes to fill the portion of the pool when both pumps were working and the additional time needed for Pump A to complete the job alone.

Total time = 40 minutes + 4 and 1/3 hours

Converting 40 minutes to hours, we have: 40/60 = 2/3 of an hour.

So, the total time needed to fill the pool is: (2/3) + (4 and 1/3) = 2/3 + 13/3 = 15/3 = 5 hours.

Since the opening of the pool was scheduled 4 hours after the pumping started, and the total time needed to fill the pool is 5 hours, it means the pool was not full by the time it was scheduled to open. Therefore, the opening had to be delayed.

T = t1*t2/(t1+t2) = 6*4/(6+4) = 2.4h = 144 Min. to fill the tank.

40min./144min. * 1 = 5/18 of a tank in 40 min.

1-5/18 = 18/18 - 5/18 = 13/18 to be filled by pump A.

13/18 * 6h = 4 1/3h = Time required for pump A to finish the job. But this is greater than the 4-hour limit to do the job. So the opening had to be delayed.

4 1/3 + 40/60 = 13/3 + 2/3 = 5h = Tot.
to do the job.