Rate problem

[beater]

If Samson is filling a bathtub with COLD water, it will take him 6 minutes and 40 seconds, and if he fills it with HOT water, it will take him 8 minutes. If draining the tub takes 13 minutes 20 seconds, how many minutes will it take to fill up the bath tub with both HOT and COLD water running while the plug is out, so the water is constantly draining?

a. 16
b. 12
c. 8.6
d. 5
e. 4.75

[Ian Stewart]

If Samson is filling a bathtub with COLD water, it will take him 6 minutes and 40 seconds, and if he fills it with HOT water, it will take him 8 minutes. If draining the tub takes 13 minutes 20 seconds, how many minutes will it take to fill up the bath tub with both HOT and COLD water running while the plug is out, so the water is constantly draining?

a. 16
b. 12
c. 8.6
d. 5
e. 4.75

I find it easiest to do rates problems by converting every worker to the same amount of time:

Cold fills one tub in 6 2/3 mins
Hot fills one tub in 8 mins
Drain empties one tub in 13 1/3 mins

So, using 40 mins for each worker:

Cold fills 6 tubs in 40 mins
Hot fills 5 tubs in 40 mins
Drain empties 3 tubs in 40 mins

So in 40 mins, all working together, they'd fill 6+5-3 = 8 tubs. To fill one tub, it will take 1/8th as long, so 5 minutes is the answer.

[jessicamuniz]

Hi Ian,
I have a question, why did you used 40 miuntes for each worker?
Thanks in advance

[Ian Stewart]

Hi Ian,
I have a question, why did you used 40 miuntes for each worker?
Thanks in advance

It is, of course, easiest to work with whole numbers (to find a small whole number to use, I multiplied 13 1/3 by 3 to get 40). You could use any other number you like, of course, but 40 is quite convenient here, since then you don't have any fractions.

[Mclaughlin]

how did you get 40 when multiply 13 1/3 by 3. I got 39.9 and if i had proceeded with that, what a mess it would've been!

[Ian Stewart]

how did you get 40 when multiply 13 1/3 by 3. I got 39.9 and if i had proceeded with that, what a mess it would've been!

Yes, that would lead to a mess. If you got 39.9, then you were multiplying 13.3 by 3, and not 13 1/3 by 3. 1/3 is not equal to 0.3. 13 1/3 = 40/3, so 3*(13 1/3) = 40.