# The Four-Percent Solution: Explain this math problem to me, because I don't get it.

You have 10 liters of a 10% acid and water solution. How much water do you need to add to get a 4% concentration of acid in water?

So there’s 1 liter of acid, 9 liters of water. So far, so good.

You want to end up with 25 liters of water to 1 liter of acid, right? 1 L/25 L = 0.04, the correct concentration.

So my thinking was that the answer is 16: 25 liters of water required minus 9 liters of water existing in the solution. This gives you a total of 1 liter of acid and 25 liters of water.

Apparently this is incorrect and the answer is actually 15–you’re supposed to subtract 10. To me, this seems like there’s now too much acid in proportion to water–wouldn’t it now be 1 part acid to 24 parts water?

It’s been many years since I last took basic algebra, so could someone please bear with me and explain the correct solution (ba dum bum) so that it makes intuitive sense to me?

Writing out the amounts of water and acid in the 10% solution:

1 = .1*(9 + 1)

writing out the same thing for the 4% solution.

1 = .04*(x + 9 + 1)

In which case x = 25 - 9 - 1 = 15

No, you want to end up with 24 liters of water and 1 liter of acid. Your solution would be 1/26th acid which is less than 4%.

Say you had 100 liters of solution. To be 4% acid, you need 96 liters of water and 4 liters would need to be acid. 25 liters is 1/4 of 100 liters. 1/4 of 96 is 24, 1/4 of 4 is 1. 24 water to 1 acid.

In the unlikely event that further clarification is necessary, the OP is making the common mistake of confusing ratios with percentages. A ratio of water to acid of 24:1 means that there are 25 parts in total, so 1 in 25 parts are acid, giving a percentage of 4 (i.e. 4%).

Notice that the 1 liter of acid represents 10% of the solution, not 10% of the water. (1 is 10% of 10, not 10% of 9.) To be consistent, you want to end up with a solution in which your 1 liter of acid is 25% of the solution, not 25% of the water present.

Oops, I mean 4%. Which is 1 out of 25.

:smack:

thanks!