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June 30th, 2016, 07:15 PM  #1 
Newbie Joined: Mar 2016 From: India Posts: 6 Thanks: 0  Alligation and Mixture
I have some problem in solving this question. Please explain the process/formula to reach the solution also. Q 1. A vessel is filled with liquid, 3 parts of which are water and 5 parts syrup. How much of the mixture must be drawn off and replaced with water so that the mixture may be half water and half syrup? 
June 30th, 2016, 10:19 PM  #2 
Senior Member Joined: May 2016 From: USA Posts: 1,122 Thanks: 465 
I don't like this problem because it is not clear that what is being asked is that the SAME quantity of liquid is to fill the vessel before and after the mixture is diluted. The problem is not well worded and so is confusing. $Let\ v = volume\ of\ vessel.$ $Syrup\ in\ original\ mixture = \dfrac{5}{5 + 3} * v = \dfrac{5v}{8}.$ $Water\ in\ original\ mixture = \dfrac{3v}{8}.$ $Let\ w = amount\ of\ mixture\ withdrawn\ and\ replaced\ with\ pure\ water.$ $Syrup\ withdrawn = \dfrac{5w}{8}.$ $Syrup\ after\ dilution = \dfrac{v}{2} = water\ after\ dilution.$ $So\ \dfrac{5v}{8}  \dfrac{5w}{8} = \dfrac{v}{2} \implies 5v  5w = 4v \implies w = 0.2v.$ Lets check using water. $\dfrac{3v}{8}  \dfrac{3w}{8} + w = \dfrac{3v}{8}  \dfrac{3 * 0.2v}{8} + \dfrac{8 * 0.2v}{8} = \dfrac{3v  0.6v + 1.6v}{8} = \dfrac{4v}{8} = \dfrac{v}{2}.$ If you had 8 liters of the mixture, you would have 5 liters of syrup mixed in. Remove 1.6 liters of the mixture. That leaves 6.4 liters containing 4 liters of syrup. Now add 1.6 liters of water. You have 8 liters of diluted mixture containing 4 liters of syrup. 
July 1st, 2016, 07:02 PM  #3 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,853 Thanks: 1078 Math Focus: Elementary mathematics and beyond 
We want the remaining fraction of syrup to be 1/2, so we write 5/8 * x = 1/8 where x equals the quantity to be removed (and 5/8  1/8 = 4/8 = 1/2). Hence x = 1/5, so we must remove 1/5 of the mixture and replace that with water. It shouldn't hurt to do a sanity check: 3/8 * 1/5 = 3/40, 3/8  3/40 + 1/5 = 20/40 = 1/2, so the work holds. 
July 2nd, 2016, 04:58 PM  #4 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 13,112 Thanks: 911  Me to the rescue...!
Boil the mixture so the water evaporates. We now have syrup only: 5/8 of vessel. Remove 1/8 of syrup: now 1/2 syrup. Fill with water... 
July 2nd, 2016, 06:07 PM  #5 
Global Moderator Joined: Dec 2006 Posts: 19,511 Thanks: 1743 
How much of the syrup in the mixture must somehow be replaced with an equal amount of water so that the mixture becomes half water and half syrup? As the mixture is 3 parts water and 5 parts syrup, the answer is a fifth of the syrup. Hence the answer to the original question is a fifth (or 20% if you prefer). 
July 4th, 2016, 08:52 AM  #6 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 894  I disagree. The problem says that the amount drawn off is replaced by water so it is saying that we have the same total quantity of liquid.

July 4th, 2016, 12:38 PM  #7 
Senior Member Joined: May 2016 From: USA Posts: 1,122 Thanks: 465  
July 7th, 2016, 05:45 PM  #8 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 894 
Is the alligation done by an alligator?

July 7th, 2016, 11:41 PM  #9 
Senior Member Joined: Apr 2014 From: UK Posts: 883 Thanks: 323 
It has been outsourced to crocodiles.


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