Question 1197891
<font color=black size=3>
Method 1


The LCM of 8 and 12 is 24.
I'll stick a 0 at the end to consider a 240 gallon bathtub.
This can be changed to any positive value you want, as the bathtub's capacity has no effect on the final answer. 
I'm only using this to help paint a numeric example scenario.


If the cold water faucet works alone, and the task is to fill the entire 240 gallons, then its unit rate is 240/8 = 30 gallons per minute.
rate = (amount done)/(time)


If the hot water faucet works alone, then its unit rate is 240/12 = 20 gallons per minute.


Combine those unit rates
30+20 = 50


The two faucets working together do so at a combined unit rate of 50 gallons per minute.
After every minute, 50 more gallons of water is in the tub. 


Then we can say:
(unit rate)*(time) = amount done
time = (amount done)/(unit rate)
time = (240 gallons)/(50 gallons per minute)
time = <font color=red>4.8 minutes</font>
which is the length of time needed if both faucets are working together.


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


Method 2


This other approach is possibly quite standard in many algebra classrooms.


The cold water faucet does the job alone in 8 minutes, giving it a unit rate of 1/8 of a job per minute. 
The hot water faucet's unit rate is 1/12 of a job per minute.


Combine those unit rates
1/8 + 1/12
3/24 + 2/24
5/24
The two faucets combine to a unit rate of 5/24 of a job per minute.


Then,
(unit rate)*(time) = amount done
(5/24 of a job per minute)*(x minutes) = 1 full job
(5/24)x = 1
x = 24/5
x = <font color=red>4.8 minutes</font>


Equivalently we are solving this equation
1/8 + 1/12 = 1/x
which is a rewritten version of this
(1/8+1/12)*x = 1
it of course depends on your viewpoint of which you find more intuitive.


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


Method 3


If the cold water faucet can do the job alone and take m minutes, and the hot water faucet works alone for n minutes, then 1/m and 1/n represent their unit rates respectively. 

Add those unit rates up
1/m + 1/n
n/(mn) + m/(mn)
(n+m)/(mn)


So,
(unit rate)*(time) = amount done
(unit rate)*(time) = 1 job
(1/m+1/n)*x = 1
( (n+m)/(mn) )*x = 1
x = (m*n)/(m+n)


Which is a quick convenient formula when dealing with two input sources to see how long it takes if they work together (assuming neither input hinders the other).


In our case, m = 8 and n = 12:
x = (m*n)/(m+n)
x = (8*12)/(8+12)
x = 96/20
x = <font color=red>4.8 minutes</font>


This method is perhaps the fastest assuming you don't worry about the derivation beforehand. The downside is that it's perhaps not as intuitive or obvious at a quick glance. This method is really only recommended if you are in a time crunch such as getting ready for an exam, and you don't mind memorizing yet another formula. Keep in mind that this works for 2 input sources only. If you added a third faucet, then you'll have to derive another formula. 


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


Answer: <font color=red>4.8 minutes</font>
</font>