Simultaneous equations and a little bit of Chemistry

How much of 4 g/l Calcium Chloride solution should you mix with how much of of 9.33 g/l Calcium Chloride solution to obtain 20 liters of 8 g/l Calcium Chloride solution?

This was the question I had to answer yesterday. And believe it or not - my life depended on it!

Rewind to a few days back.

I had this aha moment with regard to my bone pain when I discovered that the vendor was supplying extremely low Calcium acid cans (with 4 g/l of Calcium Chloride as opposed to a regular strength of 8 g/l) for my dialysis. This was causing Calcium to be pulled out from my blood which was, in turn leaching a whole lot of Calcium from my bones causing the bone pain.

I switched to high Calcium acid cans (with 9.33 g/l of Calcium Chloride) to try and correct the deficiency. But this caused my blood Calcium levels to sky rocket which is also not good. In fact it is quite risky to have your blood Calcium above normal. So, I had to switch to a regular Calcium acid can. I ordered these and started using them.

But I also had a whole lot of low Calcium acid cans (4 g/l) and a whole lot of high Calcium acid cans (9.33 g/l). I had to make the regular Calcium acid cans (8 g/l) from these. Which brought me to:

How much of 4 g/l Calcium Chloride solution should you mix with how much of of 9.33 g/l Calcium Chloride solution to obtain 20 liters of 8 g/l Calcium Chloride solution?

Hmmmm. Should be pretty straightforward, I thought.

Let x be the amount of low Calcium acid and y be the amount of high Calcium acid. One can of acid has 20 liters. So:

x + y = 20

Now, to do a Calcium balance:

4x + 9.33y = 8 X 20

4x + 9.33y = 160

When you solve these simultaneous equations you get:

x = 5 and y = 15

Basically I had to mix 5 liters of low Calcium solution with 15 liters of high Calcium solution to get what I wanted!

(P.S. I initially made a mistake and in the second equation, put only 8 on the RHS. I was about to start the mixing with the results of this equation when I thought I should get this checked by someone. This was serious, after all. I called Aditya (a genius and my friend from more than 28 years) and explained the problem to him. After a little bit of a protest - I was making him remember things from eons back - he gave me the correct answers and I also realized the mistake I had made.)