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# How much cleaning solution should Barbara replace with ammonia?

This is the complete question:
Barbara has a bucket that holds 18 fl. oz. She mixes 14.4 fl. oz. of water and 3.6 fl. oz. of ammonia to make a cleaning solution. She then decides to replace a portion of the cleaning solution to create a cleaning solution with 25% ammonia. How much cleaning solution should be replaced by ammonia?

If I understand this correctly, she takes out a certain amount of  the existing solution and replaces it with ammonia. When she takes out from the original solution she takes out some ammonia. So she is taking out solution containing some ammonia and replacing it with straight ammonia.
She has 3.6 oz of ammonia in an 18 oz solution. 3.6/18=36/180=1/5=20%
She has a 20% solution and wants to add ammonia to get a 25% solution, but again as she takes out solution she is taking out ammonia also; the solution is mixed together. However many ounces of the original solution she takes out(to replace it with straight ammonia), 20% of that is ammonia, which is no longer in the original solution. She wants a 25% ammonia solution so therefore let x=# of oz of ammonia she needs to make a 25% solution and we have
x/18=25/100, and x/18=1/4, and x=4.5 oz. She needs 4.5 oz of ammonia for a 25% ammonia solution.
For every ounce of solution she takes out, 20% of that ounce is ammonia. Therefore for every ounce of solution she takes out she takes out 0.2 oz of ammonia. She has 3.6 oz of ammonia and needs 4.5 oz of ammonia therefore she needs 0.9 oz more of straight ammonia.
We need to determine how many ounces of ammonia she has to put in to actually put in 0.9 oz of ammonia. Don't forget, she loses 0.2 oz of ammonia every time she takes out 1 oz of solution.
therefore our equation is 0.8x=0.9. When she puts in 1 oz of ammonia, she has already lost
0.2 oz of ammonia from the original solution.
0.8x=0.9, 8x=9, and x=9/8 oz of ammonia that must be put in. Let's check our work.
First she takes out 9/8 oz of solution so she lose (9/8)*0.2 oz of ammonia which is
(9/8)(1/5)=9/40 of ammonia lost. She has 3.6 oz of ammonia to begin with and takes out
9/40 oz of ammonia. (3 3/5)-(9/40)=(3 24/40)-(9/40)=3 15/40 or 3 3/8 oz of ammonia left in the solution. She now puts in 9/8 oz of straight ammonia. She has 3 3/8 oz of ammonia and puts in
9/8 oz of ammonia for a total of (3 3/8)+(9/8)=3 12/8=3+1 1/2=4 1/2 oz of straight ammonia.
4.5/18=45/180=1/4=25% solution of ammonia
Final Answer:She must take out 9/8 oz of the original solution and put in 9/8 oz of straight ammonia

100%*x + 20%*(18-x) = 25%*18 ⇒ x= 9/8 oz
Andre, very nice compression. Thank you. I realize my solution is rather lengthy, however, I was thinking of how to explain what was actually happening and to explain what I was doing as I solved the problem. I'm sure you would have explained how you arrived at that equation had you done the problem. When we solve problems, we don't always go down the same path in our solutions. Still, I'm amazed that you put everything I said into one equation. Again, thanks for the comment and the insight into the problem from your perspective.