SOLUTION: What percentage of a 35% solution of alcohol in water should be replaced by pure alcohol to give a solution containing 75% alcohol? Hint. Let the original solution 100 units. ​

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Question 1144672: What percentage of a 35% solution of alcohol in water should be replaced by pure alcohol to give a solution containing 75% alcohol? Hint. Let the original solution 100 units. ​
Found 2 solutions by josgarithmetic, greenestamps:
Answer by josgarithmetic(39620) About Me  (Show Source):
You can put this solution on YOUR website!
Similar to DRAIN & REPLACE lessons
https://www.algebra.com/my/drain-and-replace-antifreeze.lesson?content_action=show_dev

Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


The numbers don't work out "nicely" in this problem; that makes it a good example to use to show that there is a very simple alternative to the traditional method for solving this kind of "mixture" problems.

First a traditional algebraic solution....

You are mixing x units of pure (100%) alcohol with (100-x) units of 35% alcohol to get 100 units of 75% alcohol. The equation showing the amounts of alcohol in the two ingredients and in the mixture is then

1.00%28x%29%2B0.35%28100-x%29+=+0.75%28100%29

Solve for x to find the number of units of the original solution that need to be replaced with pure alcohol.

x+%2B+35+-+0.35x+=+75
0.65x+=+40
x+=+40%2F0.65+=+4000%2F65+=+800%2F13

Since we used 100 units, that number of units is also the percentage of the original solution that needs to be replaced with pure alcohol -- which is what the problem asked for.

ANSWER: (800/13) percent of the original 35% solution needs to be replaced with pure alcohol to get a 75% alcohol mixture.

Now for the alternative method which gets you very quickly to the same answer, with far less effort....

(1) The target percentage, 75%, is 40/65 = 8/13 of the way from the 35% of the original solution to the 100% of what is being added. (Look at the three percentages -- 35, 75, and 100, on a number line; 75 is 8/13 of the way from 35 to 100.)

(2) That means 8/13 of the final mixture has to be the 100% pure alcohol that is being added.

The fraction 8/13 as a percentage is (800/13) percent.

Same answer -- with far less work....