Question 1197406: SETTING UP AND SOLVING DIRECT AND INVERSE PROPORTIONS
Two sump pumps working at the same rate to drain a flooded basement in 5 1/2 hours. How long would it have taken 3 pumps working at the same rate to drain the basement? I think it is an inverse proportion and I put 2 over 3 = 5 1/2 over x multiplied and got 2x=16 1/2 divided and got 8 1/4. But it is supposed to take less time not more.
Found 6 solutions by Boreal, josgarithmetic, ikleyn, math_tutor2020, greenestamps, MathTherapy:
Answer by Boreal(15235)   (Show Source):
Answer by josgarithmetic(39621)  (Show Source):
Answer by ikleyn(52824)  (Show Source):
Answer by math_tutor2020(3817) (Show Source):
You can put this solution on YOUR website!
5 & 1/2 hours = 5.5 hours
Let's say the task is to pump out 550 gallons of water in total.
I chose this number since it's a multiple of 5.5; but feel free to pick something else and the final answer will still be the same.
Two pumps work at the same rate, so we split the job into two equal parts.
Pump A drains 550/2 = 275 gallons and pump B gets the other 275 gallons.
If pump A needs 5.5 hours to drain 275 gallons, then the unit rate is
275/5.5 = 50 gallons per hour
After each hour, pump A is able to drain 50 gallons of water.
Pump B has the same unit rate as pump A.
If we have 3 pumps working at this same rate, then the combined unit rate is 3*50 = 150.
After each hour, the three pumps collectively drain 150 gallons of water.
This assumes neither pump hinders one another.
Then we can say this:
time = (amount done)/(unit rate)
time = (550 gallons)/(150 gallons per hour)
time = 11/3 hours
It will take the three pumps 11/3 hours to get the job done if they work together, and neither pump hinders one another.
Then let's convert to a mixed number
11/3 = (9+2)/3
11/3 = 9/3+2/3
11/3 = 3+2/3
This means
11/3 hours = 3 hours + 2/3 of an additional hour
1 hour = 60 min
(2/3)*(1 hour) = (2/3)*(60 min)
2/3 hour = 40 min
So 3 hours + 2/3 hour = 3 hours + 40 minutes is the final answer
In terms of purely minutes only, we can say
3 hours + 40 min = 3*60 min + 40 min
3 hours + 40 min = 180 min + 40 min
3 hours + 40 min = 220 min
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Here's an algebraic approach:
x = total number of gallons to drain
If two pumps work together at the same rate, then each pump drains x/2 = 0.5x gallons of water
If the two pumps take 5.5 hours each, then,
unit rate = (amount done)/(time)
unit rate = (0.5x gallons)/(5.5 hours)
unit rate = 0.5x/5.5 gallons per hour
unit rate = 5x/55 gallons per hour
unit rate = x/11 gallons per hour
This is the unit rate for each identical pump.
Triple this unit rate to now include the third pump helping out
3*(x/11) = 3x/11
This represents the combined unit rate.
Then,
time = (amount done)/(unit rate)
time = (x gallons)/( 3x/11 gallons per hour )
time = x*(11/(3x)) hours
time = 11/3 hours
That then converts to 3 hours + 40 minutes as shown above.
The algebraic approach is useful to see that there wasn't anything special with 550 I picked earlier.
But for some students, the numeric approach in the first section is easier to grasp.
Answer by greenestamps(13203)  (Show Source):
You can put this solution on YOUR website!
Since the beginning of your post says "SETTING UP AND SOLVING DIRECT AND INVERSE PROPORTIONS", perhaps the response you were looking for should do that. None of the other responses did that.
This is clearly inverse proportion: more pumps means less time.
The ratio of the number of pumps is 3:2, so the ratio of times should be 2:3. So multiply the given time of 5 1/2 hours by 2/3.
(5 1/2)*(2/3) = (11/2)(2/3) = 11/3
ANSWER: 11/3 hours
Answer by MathTherapy(10555)  (Show Source):
You can put this solution on YOUR website! SETTING UP AND SOLVING DIRECT AND INVERSE PROPORTIONS
Two sump pumps working at the same rate to drain a flooded basement in 5 1/2 hours. How long would it have taken 3 pumps working at the same rate to drain the basement? I think it is an inverse proportion and I put 2 over 3 = 5 1/2 over x multiplied and got 2x=16 1/2 divided and got 8 1/4. But it is supposed to take less time not more.
You're correct in that it's INVERSE VARIATION, because the more pumps are used, the less time it'll take it/them to do the job,
and vice-versa. However, your setup is WRONG!
With P being the number of pumps, k being the CONSTANT of PROPORTIONALITY, and T, the amount of time,
we get: 
 ---- Substituting 2 for P, and T for amount of time taken by both pumps
 ---- Cross-multiplying
With 3 pumps (P) working at the same rate, k, or CONSTANT of PROPORTIONALITY being 11, and T being
amount of time the 3 pumps will take to drain it out, we get: 
3T = 11 ------ Cross-multiplying
Time it'll take the 3 pumps to drain it out, or 
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