Question

A tub can be filled in 15 minutes by the hot water faucet and in 12 minutes by the cold water faucet. The drain can empty the tub in 20 minutes. How long will it take the tub to fill if both of the faucets and the drain are open?

Answer 1

rate of hot = 1/15
rate of cold = 1/12
combined rate = 1/15 + 1/12 = 3/20
rate of drain = -1/20

3/20 - 1/20 = 1/x
2/20 = 1/x
x = 10

it will take 10 minutes.

check:
in 10 minutes the amount of hot = 10(1/15) = 2/3 units of the tub
in 10 minutes the amount of cold = 10(1/12) = 5/6 units of the tub
in 10 minutes the amount drained = 10(1/20) = 1/2 units of the tub

combined effect = 2/3 + 5/6 - 1/2
= 1 unit of the tub, i.e. , the tub is full

Answer 2

drain = -1 tub/20 min
hot = 1 tub/15 min
cold = 1 tub/12

1 tub = (1 tub/12min + 1 tub/15 min - 1 tub/20 min)( x min)
1 = (5/60 + 4/60 - 3/60) x
1 = 6 x/60 = x/10
x = 10 min

Answer 3

Now
How much hot water did you waste?

Answer 4

What is 10min*1/20 min? answer, 1/2 tub. That is how much went down the drain.
how much of that was hot water? 1/2 tub* 15/(12+15)=....

Answer 5

or how much hot water did you run?
10 min * 1 tub/15 min = 2/3 tub
How much did you need:
hot = 12 hots for every 15 colds or 12/27 is hot (note more cold than hot)
2/3 - 12/27 = 18/27 - 12/27 = 6/27 = 3/9 of a tub

Answer 6

or about 1/3 tub hot wasted :)