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June 30th, 2016, 07:15 PM   #1
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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?
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June 30th, 2016, 10:19 PM   #2
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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$Let\ w = amount\ of\ mixture\ withdrawn\ and\ replaced\ with\ pure\ water.$

$Syrup\ withdrawn = \dfrac{5w}{8}.$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.
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July 1st, 2016, 07:02 PM   #3
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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.
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July 2nd, 2016, 04:58 PM   #4
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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...
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July 2nd, 2016, 06:07 PM   #5
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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).
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July 4th, 2016, 08:52 AM   #6
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Quote:
Originally Posted by JeffM1 View Post
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.
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.
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July 4th, 2016, 12:38 PM   #7
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Quote:
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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.
You are correct. I was wrong.
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July 7th, 2016, 05:45 PM   #8
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Is the alligation done by an alligator?
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July 7th, 2016, 11:41 PM   #9
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It has been outsourced to crocodiles.
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