Pipe A takes 4⁄5 of the time required by pipe B to fill an empty tank individually. When an outlet pipe C is also opened simultaneously with pipes A and B, it takes 4⁄5 more time to fill the empty tank than it takes when only pipe A and pipe B are opened together. If it takes 40 hours to fill the tank when all the three pipes are opened simultaneously, in what time (in hours) will pipe C empty the full tank, operating alone?

Correct — D. The stem is an image: A takes 4⁄5 of B's time, adding outlet C makes the fill take 4⁄5 more time than A and B alone, all three together take 40 hours, and C alone has to be timed.

Let B alone take b hours, so A takes 4b⁄5 hours.
Rate of A and B = 5⁄(4b) + 1⁄b = 9⁄(4b)
Time for A and B = 4b⁄9 hours

4⁄5 more time multiplies that by 9⁄5:
(4b⁄9) × (9⁄5) = 4b⁄5 hours, and this is the given 40

4b⁄5 = 40 → b = 50, so B alone takes 50 h and A alone 40 h.

Subtract the rates:
C's rate = (rate of A and B) − (rate of all three)
= 9⁄200 − 1⁄40 = 9⁄200 − 5⁄200 = 4⁄200 = 1⁄50

C alone empties the full tank in 50 hours → option (d)

Why the others are wrong

• (a)
  45 — If C emptied in 45 hours the three together would fill at 9⁄200 − 1⁄45 = 41⁄1800, taking about 43.9 hours. The stem fixes that time at 40, so a 45-hour outlet drains too little.
• (b)
  75 — 75 hours makes C far too weak: 9⁄200 − 1⁄75 = 19⁄600, a fill time of about 31.6 hours. A slower outlet means a faster fill, so this pushes the answer the wrong way.
• (c)
  65 — At 65 hours the net rate is 9⁄200 − 1⁄65 = 385⁄13000, giving a fill of about 33.8 hours. Reaching exactly 40 hours needs the outlet to remove 4⁄200 of the tank an hour, which is 1⁄50.

Pipe sums are rate sums. A filling pipe adds 1⁄(its time) each hour, an outlet subtracts its own 1⁄(its time), and what fills the tank is the net rate.

This stem hides both times behind ratios. A takes 4⁄5 of B's time, which makes A the faster pipe, not the slower one.

And the outlet stretches the fill by 4⁄5 more time, so the new time is 1 + 4⁄5 = 9⁄5 of the old one — not 4⁄5 of it.

That second phrase is where the sum is won or lost. Read 4⁄5 more as a shortening and every step afterwards runs backwards.

Key facts

• An outlet pipe is a negative rate, so the net filling rate is the sum of the inlet rates minus the outlet rate.
• Taking 4⁄5 more time means the new time is 9⁄5 of the old time, not 4⁄5 of it.
• Here A alone takes 40 hours, B alone 50 hours, and A with B 200⁄9 hours.
• C alone empties in 50 hours, a rate of 1⁄50 that exactly cancels B's 1⁄50.

Common traps

• Reading 4⁄5 more time as a time of 4⁄5, which shortens the fill instead of lengthening it.
• Taking A's time as 5⁄4 of B's because A is named first in the sentence.
• Adding C's rate rather than subtracting it, when C empties the tank.

No directly related past PYQ was found.