Question 1204617: A tub can be filled with hot water in 15 minutes, or with cold water in 12 minutes. The tub can also be emptied when the plug is pulled out, in 10 minutes. Farley ran the hot water in his tub for 6 minutes, then both the cold and hot water together for 3 minutes, and then with the two taps still running, he accidentally pulled the plug. How long did it take, in minutes, for the tub to fill after the plug was pulled?
Found 3 solutions by ikleyn, mananth, math_tutor2020:
Answer by ikleyn(52847)   (Show Source):
You can put this solution on YOUR website! .
This problem was solved at the forum several years ago.
See the solution by the tutor @Bundy under this link
https://www.algebra.com/algebra/homework/Rate-of-work-word-problems/Rate-of-work-word-problems.faq.question.1147570.html
Answer by mananth(16946)  (Show Source):
You can put this solution on YOUR website! A tub can be filled with hot water in 15 minutes, or with cold water in 12 minutes. The tub can also be emptied when the plug is pulled out, in 10 minutes. Farley ran the hot water in his tub for 6 minutes, then both the cold and hot water together for 3 minutes, and then with the two taps still running, he accidentally pulled the plug. How long did it take, in minutes, for the tub to fill after the plug was pulled?
A tub can be filled with hot water in 15 minutes,
it fills 1/15 of tub in 1 minute
Farley ran the hot water in his tub for 6 minutes,
6/15 of the tub was filled with hot water
with cold water fills in 12 minutes.
it fills 1/12 of tub in 1 minute cold water
then he ran both the cold and hot water together for 3 minutes,
In 1 minute both taps on 1/12 +1/15 of the tub is filled= 9/60 = 3/20
In 3 minutes they fill 9/20 of the tub
Total water in tub = 6/15 + 9/20 = 51/60
Balance to be filled = 1-51/60 = 9/60 of the tub
In 1 minute both taps on 1/12 +1/15 of the tub is filled= 9/60 = 3/20
emptied when the plug is pulled out, in 10 minutes
drains 1/10 of tub in 1 minute
When both taps and drain are on it will fill
3/20-1/10 = 1/20 of the tub
1/20 of tub is filled in 1 minute
9/60 in how many minutes
(9/60) /(1/20)= 3 minutes
Answer by math_tutor2020(3817)  (Show Source):
You can put this solution on YOUR website!
Let's find the LCM of {15,12,10}
To do so, find the prime factorization of each value first.
10 = 2*5
12 = 2*2*3
15 = 3*5
The unique primes are 2, 3, 5
2 shows up at most twice, making 2^2 part of the LCM.
3 and 5 show up at most once per value, so 3*5 is also part of the LCM
LCM = 2^2*3*5 = 4*15 = 60
List out the multiples of 10, 12, and 15 to verify that the LCM is indeed 60. The LCM is useful in work rate problems because we'll be dividing this value over the stated numbers. It's much better to get whole number results.
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Let's say the tub's full capacity is 60 gallons.
The tub can be completely filled with hot water in 15 minutes.
The rate is 60/15 = 4 gallons per minute.
Formula is: rate = (amount done)/(time)
This is when the cold water tap is turned off.
Or if we go with cold water, then it takes 12 minutes
Rate = 60/12 = 5 gallons per minute
This is when the hot water tap is turned off.
The drain rate is 6 gallons per minute because 60/10 = 6.
Farley ran the hot water for 6 minutes.
At a rate of 4 gallons per minute, there would be 6*4 = 24 gallons of hot water in the tub.
Then he runs hot and cold together for 3 minutes.
Before these 3 minutes are up, I'm assuming Farley did not pull the plug.
The combined rate of hot and cold is 4+5 = 9 gallons per minute, so that's an extra 9*3 = 27 gallons of water.
24+27 = 51 gallons so far.
He needs 60-51 = 9 gallons to finish filling the tub.
Farley pulls the plug to drain the water.
I'm assuming Farley pulls the plug after those 3 minutes are up.
The net fill rate is 4+5-6 = 3 gallons per minute when accounting for both hot and cold, along with the drain as well. Think of it like a tug-of-war in which the hot/cold combo wins out.
Ultimately the bath tub is getting filled despite the drain trying to do the opposite.
If x is the number of extra minutes, after the plug is pulled, then 3x represents the amount of extra water needed (since 3 gallons per min was that net rate calculated above)
Set this equal to the 9 gallons we need and solve for x.
3x = 9
x = 9/3
x = 3
Answer: 3 minutes
More practice with a similar problem
https://www.algebra.com/algebra/homework/coordinate/Linear-systems.faq.question.1204620.html
and
https://www.algebra.com/algebra/homework/Polynomials-and-rational-expressions/Polynomials-and-rational-expressions.faq.question.1197891.html
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