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\n" ); document.write( "I will show several ways to solve the problem. Since having more tools at your disposal makes you a better problem solver, I would recommend looking at and trying to understand all of them.
\n" ); document.write( "(1) 20% of 20 liters is 4 liters; since you are adding water (0% salt) you will still have 4 liters of salt after adding the water. 12.5% as a fraction is 1/8, so the 4 liters of salt you have must be 1/8 of the final salt solution. That means the final solution is 32 liters; and that means you added 12 liters of water.
\n" ); document.write( "ANSWER: 12 liters
\n" ); document.write( "(2) Traditional algebra...
\n" ); document.write( "You are mixing 20 liters of 20% salt to x liters of 0% salt to end up with (20+x) liters of 12.5% salt:
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\n" ); document.write( "ANSWER: 12 liters
\n" ); document.write( "(3) Alligation (yes, with an \"i\") -- this method is apparently taught in pharmacology.
\n" ); document.write( "You asked for a diagram. This method in fact uses a diagram....
\n" ); document.write( "Here is the diagram for your problem:
\r\n" ); document.write( "\r\n" ); document.write( " 20 12.5\r\n" ); document.write( " 12.5\r\n" ); document.write( " 0 7.5
\n" ); document.write( "The \"20\" and \"0\" in the first column are the percentages of salt in the two ingredients.
\n" ); document.write( "The \"12.5\" in the middle of the diagram is the percentage of the mixture.
\n" ); document.write( "The \"12.5\" and \"7.5\" in the third column are the differences -- calculated diagonally -- between the numbers in the first column and the number in the second column: 12.5 is the difference between 0 and 12.5, and 7.5 is the difference between 20 and 12.5.
\n" ); document.write( "The two numbers in the third column give you the RATIO in which the two ingredients must be mixed. In this problem, that ratio is 12.5:7.5 = 5:3. So you need 5 parts of the 20% salt solution to 3 parts of water.
\n" ); document.write( "Given that there are 20 liters of the 20% salt solution, find the number of liters of water using a proportion:
\n" ); document.write( "5:3 = 20:x
\n" ); document.write( "60 = 5x
\n" ); document.write( "x = 12
\n" ); document.write( "ANSWER: 12 liters
\n" ); document.write( "(4) Finally, here is what I think is the easiest and fastest way to solve any 2-part mixture problem like this. It is closely related to the previous method, but to me the calculations are much simpler.
\n" ); document.write( "You are starting with a 20% salt solution and adding 0% salt until you have a mixture that is 12.5% salt. Think of that as \"starting at 20 and walking towards 0, stopping when you get to 12.5\".
\n" ); document.write( "Mentally, or using a number line if it helps, observe/calculate that 12.5 is 3/8 of the way from 20 to 0.
\n" ); document.write( "That means 3/8 of the final mixture is what you are adding. (You started at 20 and walked towards 0, but you stopped when you were only 3/8 of the way there; so 3/8 of the mixture is what you are adding.)
\n" ); document.write( "Since 3/8 of the mixture is the water you are adding, the original 20 liters of 20% salt solution is 5/8 of the mixture.
\n" ); document.write( "Then, similar to the previous method, you have a proportion to solve: knowing that 5/8 of the mixture is 20 liters; you know 3/8 of the mixture which is the water you are adding is 12 liters.
\n" ); document.write( "ANSWER: 12 liters
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