SOLUTION: Problem Page It takes the cold water faucet 4 times as long to fill a bathtub as it does the hot water faucet. Left on together, the cold and hot water faucets take 3 minutes to

Algebra ->  Rate-of-work-word-problems -> SOLUTION: Problem Page It takes the cold water faucet 4 times as long to fill a bathtub as it does the hot water faucet. Left on together, the cold and hot water faucets take 3 minutes to      Log On


   

Question 1135888: Problem Page
It takes the cold water faucet 4 times as long to fill a bathtub as it does the hot water faucet. Left on together, the cold and hot water faucets take
3 minutes to fill the tub. How long will it take the hot water faucet to fill the tub by itself?
Found 3 solutions by josgarithmetic, ikleyn, greenestamps:
Answer by josgarithmetic(39620) About Me  (Show Source):
Answer by ikleyn(52803) About Me  (Show Source):
You can put this solution on YOUR website!
.

            Let me show you three very simple ways to solve this problem.


Solution 1 

Let x be the time to fill the tub by the hot water faucet working alone.

Then the time to fill the tub by the cold water faucet working alone is 4x, according to the condition.


It means that when the hot water faucet works alone, it fills 1%2Fx of the tub volume per minute.


It also means that when the cold water faucet works alone, it fills 1%2F%284x%29 of the tub volume per minute.


We also are given, that when two faucets work simultaneously, they fill 1%2F3 of the tube volume per minute.


In other words,  1%2Fx + 1%2F4x = 1%2F3,   or


    4%2F%284x%29 + 1%2F%284x%29 = 1%2F3,

    5%2F%284x%29 = 1%2F3,   hence,  5*3= 4x,   or   x = 15%2F4 minutes = 33%2F4 minutes = 3 minutes and 45 seconds.    


ANSWER.  It takes 3 minutes and 45 seconds for hot faucet to fill the bathtub working alone.


Solution 2 

Let a be the rate of work of the hot faucet (= which part of the job the hot faucet makes in 1 minute).


Then the rate of work of the cold faucet is a%2F4, according to the condition.


Their combined rate of work is the sum of the individual rates, i.e.  a + a%2F4 = %285a%29%2F4.


It is exactly 1%2F3 of the bathtub volume:  %285a%29%2F4%29 = 1%2F3.


It implies  a = 4%2F%283%2A5%29 = 4%2F15, so the hot faucet fills  4%2F15  of the tub volume per minute.


It means that it requires  15%2F4 minutes = 33%2F4 minutes = 3 minutes and 45 seconds to fill the tub.


The same answer.


Solution 3 

Both faucets combined work together as (1+4) = 5 cold faucets simultaneously, and they fill the tube in 3 minutes.


It means that 1 cold faucet will fill the tube in 3*5 = 15 minutes.


One hot faucet will do it in 4 times faster, i.e. in  15%2F4 minutes = 33%2F4 minutes = 3 minutes and 45 seconds.


The same answer.


A good student should understand and equally possess (have in his hands and in his mind) all three methods.

---------------

It is a standard and typical joint work problem.

There is a wide variety of similar solved joint-work problems with detailed explanations in this site.  See the lessons
    - Using Fractions to solve word problems on joint work
    - Solving more complicated word problems on joint work
    - Selected joint-work word problems from the archive


Read them and get be trained in solving joint-work problems.

Also,  you have this free of charge online textbook in ALGEBRA-I in this site
    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this textbook under the topic
"Rate of work and joint work problems"  of the section  "Word problems".


Save the link to this online textbook together with its description

Free of charge online textbook in ALGEBRA-I
https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson

to your archive and use it when it is needed.


Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


Here is yet another simple way to solve the problem.

As tutor @ikleyn says after showing you three simple methods, you will be a better student if you know several ways of solving a problem. The more tools you have in your tool bag, the more you will be able to accomplish.

The hot water faucet fills the bathtub 4 times as fast as the cold water faucet.
That means in filling the tub together, the hot water faucet does 4/5 of the work.
Then, since the hot water faucet fills 4/5 of the tub in 3 minutes, the amount of time it would require to fill the tub alone is 5/4 of 3 minutes:

%285%2F4%29%2A3+=+15%2F4 or 3 3/4 minutes, or 3 minutes 45 seconds