If you have a container that is 100 units deep and 100 units wide, then you have a total area of 10,000 units[sup]2[/sup]. 200 units[sup]2[/sup] of this is ingredient X, the other 9,800 is water.

Now imagine that the ingredient isn’t mixed in, but is rather lying on the bottom of our square, with the water above it:

```
o o o o o o o o o o
o o o o o o o o o o
o o o o o o o o o o
o o o o o o o o o o
o o o o o o o o o o
o o o o o o o o o o
o o o o o o o o o o
o o o o o o o o o o
o o o o o o o o o o
o o o o o o o o o o
```

If we were to slice it in half vertically and replace the one half with water, like so:

```
o o o o o o o o o o
o o o o o o o o o o
o o o o o o o o o o
o o o o o o o o o o
o o o o o o o o o o
o o o o o o o o o o
o o o o o o o o o o
o o o o o o o o o o
o o o o o o o o o o
o o o o o o o o o o
```

Obviously, we’ll have 100 units[sup]2[/sup] of ingredient to 9,900 units[sup]2[/sup] water, giving us a 1% solution.

So this tells us that a vertical cut of 50% (1/2) shrinks the amount of ingredient by 50%. If we wanted a 0.5% solution, we’d replace 75% (3/4) of the solution with water, and if we wanted a 0.25% solution, we’d replace 87.5% (7/8) with water. We don’t want to shrink it by 50% or 25% nor 12.5% though, we want 0.05/2.0 = 0.025 or 2.5% (1/40). If we cut vertically and replace 97.5% (39/40) of the flask with water, we’ll have a 0.05% ingredient solution.

Since the solution is mixed though, we don’t need to cut vertically or horizontally, or any other particular direction. Any 2.5% units[sup]2[/sup] of the solution added to 97.5% units[sup]2[/sup] water, will create a 0.05% solution.

So pour out 2.5% of your liter of water (reducing it to 97.5%), and pour in the solution until you’re back to a liter, and you’re done.