A chemist is making a solution that is to be 55% chlorine. He has one solution that is 40% chlorine and another that is 65% chlorine. He wants 100 liters of the final solution. How much of each should he mix?

You know that the volume of the final solution is 100 L. If `x` represents the volume of the 40% solution and `y` represents the volume of the 65% solution, then you can say that

`x + y = 100`

Now you need to focus on the chlorine content. Since the first solution is 40% chlorine, then for `x` liters you would get

`(x)""cancel("L solution") * "40 g chlorine"/(100cancel("L solution")) = (0.4 * x)" g chlorine"`

The same can be said for the 65% solution

`(y)""cancel("L solution") * "65 g chlorine"/(100cancel("L solution")) = (0.65 * y)" g chlorine"`

The final solution has a percent concentration of 55% chlorine, which means that it must contain

`100""cancel("L solution") * "55 g chlorine"/(100cancel("L solution")) = "55 g chlorine"`

The second equation you can write will thus relate the amount of chlorine each solution contributes to the final solution

`0.4x + 0.65y = 55`

This is your system of equations

`{(x+y = 100), (0.4x + 0.65y = 55) :}`

Use the first one to write `x` as a function of `y`

`x = 100 - y`

Now use this expression in the second equation to get the value of `y`

`0.4 * (100 - y) + 0.65y = 55`

`40 - 0.4y + 0.65y = 55`

`0.25y = 15 implies y = 15/0.25 = color(green)(60)`

This means that `x` is equal to

`x = 100 - 60 = color(green)(40)`

So, if you mix 40 L of a 40% chlorine solution with 60 L of a 65% chlorine solution you will get 100 L of a 55% chlorine solution.