# Spinal Tap

My downstairs sink has two taps and a plug hole. The cold tap on its own fills the sink in 18 minutes, the hot one in 9 minutes. The plug hole can drain the sink in 15 minutes with the taps off. How long will the sink take to fill if I leave both taps on with the plug left out?

### 10 guesses to Spinal Tap

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• Roolstar

10 minutes

Every minute:
The cold tap fills 1/18th of the tub
The hot tab fills 1/9th of the tub
The drain empties 1/15th of the tub

=> In one minute 1/18+1/9-1/15 = 1/10th of the tub is filled
=> In 10 minutes the tub’s gonna be full

• Dude

Correct, Rool!

In 24 hours there are 1,440 minutes. So, in 24 hours the cold tap can fill 1440 ÷ 18 = 80 baths. Also, in 24 hours the hot tap can fill 1440 ÷ 9 = 160 baths. Finally, in 24 hours the plug hole can empty 1440 ÷ 15 = 96 baths. So, in 24 hours we have 80 + 160 – 96 = 144 baths. So each bath would take 1440 ÷ 144 = 10 minutes to fill.

You explained it beautifully and you’re today’s winner.

• John

The sink will take only 10 minutes to fill the if both the taps are left open.My explanation is same as Roolstar.
OK here’s a brainteaser for you too ! –
http://www.iqtestexperts.com/brainteasers/

• max

using the speed distance and time formula we can figure this out. we only know the time but if we make up a volume then the speed will be incorrect but the time witch is what we want will be correct so if we make up that the volume = 100 then put into a calculator (100 \ 9) + (100\18) – (100\15) = a. then we do distance(100) over speed(a) 100\a = 10. NOTE: 9 and18 and 15 were times given in the riddle. so 10 is our awser.

• Dude

You are correct too Max.

• dude

max is so smart

• Andy

Unfortunately this teaser is a little more complex. The water flows out of the taps at a constant rate, yet the water will flow out of the plug hole at an increasing rate – as the height of the water in the sink increases (Torricelli’s law of efflux).

We can start by combining the taps – i.e. they fill the sink in 6 minutes when both on. If we use x to denote the ratio of actual water level to level at the top of the basin we can relate the change in x with time to filling time:

1) dx/dt (filling) = 1/6

2) dx/dt (emptying) = -K.sqrt(x)

The basin empties in 15 minutes and the change in x goes from K (sink full) to 0 (sink empty). The equation for efflux is the same form as velocity and displacement for uniform acceleration (dx/dt = sqrt(2.a.d) so we can conclude that the average rate of decrease in x is K/2. This can also be expressed in terms of time to empty:

3) K/2 = 1/15 hence K = 2/15

Combining emptying and filling and substituting for K we get:

dx/dt = (1/6)-(2/15).sqrt(x)

Integrating to get x(t) and substituting x=1 (ie sink is full) the answer is 30 mins.

For more details – a similar question can be found in “200 Puzzling Physics Problems”, published by Cambridge University Press.

• Dude

Thank you for pushing me this morning to learn about Torricelli’s law of efflux and how it relates to Bernoulli’s Principle.

• lesleyrocksu

If you have a sink with 4 taps on each side and you turn on each on individually the first tap will take 15 minutes to fill the sink, the 2nd 30 minutes, 3rd 45 minutes and the final tap 60 minutes (remember each tap is the only tap on at the time starting with an empty sink). How long would it take to fill the sink if all 4 taps were turned on at the same time?

This has caused arguements and i want to know how to work it out

• Andy

lesleyrocksu – just to be absolutely clear, the sink is just filling up – there is no drain open?

If it is just filling up, the individual flows from each tap are as follows:

F1 = V/15, F2 = V/30, F3 = V/45, F4 = V/60. V = Volume of the sink.

Switching on all the taps the Total Flow = F1 + F2 + F3 + F4 = V x (1/15 + 1/30 + 1/45 + 1/60)

Time to fill up a volume V = V/Flow = V/Total Flow = 1/(1/15 + 1/30 + 1/45 + 1/60)

Time = (180/25) = 7.2 minutes