A bathtub has two taps and a plug hole. The cold tap on its own fills the bathtub in 18 minutes. The hot one on its own fills the bathtub in 15 minutes. The plug hole can drain the bathtub in 10 minutes with the taps off.

How long will the bathtub take to fill if both taps are turned on and the plug left out?

45 minutes.

Method 1

Let us talk about 24 hours as it will simplify the problem. In 24 hours:
The cold tap will be able to fill the bathtub 80 times.
The hot tap will be able to fill the bathtub 96 times.
The drain will be able to drain the bathtub 144 times.

Thus in 24 hours, the bathtub will be filled 80 + 96 – 144 = 32 times (if both taps are turned on with the plug left out)

Thus it will take 45 minutes to fill the tub once.

Method 2

The bathtub filling rate is:
The cold tap will fill the bathtub at 1/18 tub/minute
The hot tap will fill the bathtub at 1/15 tub/minute
The drain will drain the bathtub at 1/10 tub/minute

The nett bathtub fill rate is: 1/18 + 1/15 – 1/10 = 1/45 tub/minute

The water will fill 1/45 of the tub in 1 minute, so the tub will be full in 45 minutes.

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