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Question 1197891: For a certain bathtub, the cold water faucet can fill the tub in 8 minutes. The hot water faucet can fill the tub in 12 minutes. If both faucets are used together, how long will it take to fill the tub?
Do not do any rounding.
Found 3 solutions by Theo, math_tutor2020, greenestamps:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! rate * time = quantity
quantity = 1 full tub.
for the cold water faucet, the equation becomes:
rate * 8 = 1
solve for rate to get rate = 1/8.
for the hot water faucet, the equation becomes:
rate * 12 = 1
solve for rate to get rate = 1/12.
when they are both working together, their rates are additive.
you get (1/8 + 1/12) * time = 1
1/8 * 3/3 = 3/24
1/12 * 2/2 = 2/24
formula becomes:
(3/24 + 2/24) * time = 1 which becomes:
5/24 * time = 1
solve for time to get:
time = 1*24/5 = 24/5 minutes.
that's how long to fill the tub when both faucets are open.
the cold water faucet will fill 1/8 * 24/5 = 3/5 of the tub in 24/5 minutes.
the hot water faucet will fill 1/12 * 24/5 = 2/5 of the tub in 24/5 minutes.
3/5 + 2/5 = 5/5 = 1 = the whole tub in the same 24/5 minutes.
your solution is that it will take 24/5 minutes to fill the tub if both faucets are open.
24/5 = 4.8 minutes in decimal format.
Answer by math_tutor2020(3817)  (Show Source):
You can put this solution on YOUR website!
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 = 4.8 minutes
which is the length of time needed if both faucets are working together.
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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 = 4.8 minutes
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.
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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 = 4.8 minutes
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.
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Answer: 4.8 minutes
Answer by greenestamps(13200)  (Show Source):
You can put this solution on YOUR website!
All the responses you have received to now use the same basic method using rate of work -- e.g., the cold water faucet fills the tub in 8 minutes, so it fills 1/8 of the tub in 1 minute. The method then involves adding fractions with different denominators.
That's a perfectly good method, which you should understand and be able to use. But there is another method that many students find easier.
Consider the LCM of the two given times. The LCM of 8 minutes and 12 minutes is 24 minutes. What could each of the two faucets do in 24 minutes?
The cold water faucet in 24 minutes could fill the tub 24/8 = 3 times; the hot water faucet in 24 minutes could fill the tub 24/12 =2 times.
So in 24 minutes the two faucets together could fill the tub 3+2 = 5 times; that means the number of minutes needed for the two faucets together to fill the one tub is 24/5.
ANSWER: 24/5 minutes, or 4 4/5 minutes, or 4 minutes 48 seconds
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