[eddyq]

I have a sprayer that has a ratio of 17 :: 1 meaning 17 quarts of water from the hose will empty 1 quart of solution in the sprayer.

I have a chemical that needs to be mixed 200 :: 1 meaning 200 parts of water to 1 part of chemical.

I need to make a solution from the chemical then add that to the sprayer such that the chemical is diluted 200 :: 1 as it comes out of the sprayer.

I figure this: The mixture before adding to the sprayer would be 1 :: x. So my formula would be 1/x * 1/17 = 1 / 200 and x would be 11.76.

Did I do it correctly?

[mikasaur]

Yes, you have calculated it correctly.

(1/x)*(1/17)=(1/200)
(1/17x)=(1/200)
x = 200/17

x = 11.76

You would need to add a mixture with a 1 :: 11.76 ratio to achieve a 1 :: 200 ratio as it comes out of the sprayer.

[eddyq]

I thought about this more last night. I wonder if I need to use 1/18 and 1/201. I think this because a 1 :: 1 ratio would be 1/2 water and 1/2 chemical and a 1 :: 2 ratio would be 1/3 chemical and 2/3 water. So it seems the denominator should be n+1. Is this thinking flawed?

[mikasaur]

No, I don't believe you need to use N+1. You're overthinking it.

As you say, you need to achieve a dilution of 200 :: 1, i.e. for every 200 units of water, 1 unit of chemical comes out. You are actually trying to achieve a situation where if you sprayed 200 units of solution you wouldn't quite get 1 unit of chemical. You're just trying to get the ratios correct.

Someone else might able to explain less hand-wavy and more mathematically. But your initial thinking was correct.

[eddyq]

Thanks for your response. Maybe an algebraic example would help me.

Suppose I want 200 to 1 and my sprayer is 1 to 1. Then my algebra doesn't work:

1 / x * 1 / 1 = 1 / 200 which gives x = 200. That would mean I would make a solution of 1 to 200 and mix it in with a sprayer that cuts it in half but that would give an output of 1 to 400.

 

[mikasaur]

You know what, I think you might be right! 

For a simple example, let's say that your sprayer sprays at 1::1. And you need to achieve an output of 2::1 H₂O::chemical.

That means the sprayer will put out 1 unit of H₂O and 1 unit of solution. If we say that x is the amount of chemical in the solution, then amount of H₂O coming out of the sprayer is:

1+(1-x)

If you think of the sprayer as spraying two streams, the first 1 above is the pure H₂O stream and the second (1-x) term is the amount of water coming from the solution.

The amount of chemical coming out of the sprayer is:

x

So the ratio of water::chemical coming out of your 1::1 sprayer is:

(1+(1-x))::x

If we want to achieve 2::1 we can do:

(1+(1-x))/x = 2/1

Solving for x will tell you how many units of chemical you want per unit of water. You'll see your n+1 terms appear when you solve, I believe.

You can do the same for the actual question.

Of course, I was so sure of the original answer and now I think you were right to question it. It would be nice if someone else could chime in; this is a simple but tricky little math question!