Pipes P and Q can completely fill a water tank in 10 hours and 15 hours, respectively. A pipe R can empty a tank filled completely with water in 12 hours. Initially, the tank is empty and only pipes P and Q are opened at 6 a.m. and pipe R is also opened at 9 a.m. By what time will the tank be completely filled?

Question

Accepted Answer

Let the capacity of the tank be 60 units (LCM of 10, 15, 12 for easier calculation).

Pipe P fills 1/10 of the tank per hour = 6 units/hour.
Pipe Q fills 1/15 of the tank per hour = 4 units/hour.
Pipe R empties 1/12 of the tank per hour = 5 units/hour (but negative, as it empties).

From 6 a.m. to 9 a.m. (3 hours), only P and Q are open:
Combined rate = 6 + 4 = 10 units/hour.
In 3 hours: 10 × 3 = 30 units filled.

At 9 a.m., pipe R is also opened. Now, net rate = 6 + 4 - 5 = 5 units/hour.
Tank still to be filled = 60 - 30 = 30 units.
Time required = 30 units / 5 units per hour = 6 hours.

So, from 9 a.m., it takes 6 more hours to fill the tank, i.e., at 3 p.m. But we must add the initial 3 hours (6 a.m. to 9 a.m.), so total time from 6 a.m. is 9 hours.

But let's check the options: The calculation above says 6 a.m. + 9 hours = 3 p.m., but let's double-check:
- 6 a.m. to 9 a.m.: 30 units filled
- 9 a.m. onwards: 5 units/hour
- To fill remaining 30 units: 30/5 = 6 hours
- 9 a.m. + 6 hours = 3 p.m.

However, the correct answer should be 2 p.m. according to the answer key. Let's check if there's a calculation error:

Let's recalculate:
- P: 1/10 per hour
- Q: 1/15 per hour
- R: -1/12 per hour

Combined rate of P and Q: 1/10 + 1/15 = (3/30 + 2/30) = 5/30 = 1/6 per hour
So, in 3 hours: 3 × 1/6 = 1/2 of the tank filled.

Remaining: 1/2 of the tank.

Now, all three pipes are open:
Net rate: 1/10 + 1/15 - 1/12
Find LCM of 10, 15, 12 = 60
- 1/10 = 6/60
- 1/15 = 4/60
- 1/12 = 5/60
Net rate = (6 + 4 - 5)/60 = 5/60 = 1/12 per hour

Time to fill remaining 1/2 tank at 1/12 per hour:
Let t = time required
(1/12) × t = 1/2
=> t = 6 hours

So, from 9 a.m., 6 hours = 3 p.m.

Therefore, the correct answer is 3 p.m.

So, the correct option is 4. 3 p.m.

Answer

Answer

Answer