Let’s call the two concentrated vinegars A and B. Instead of dealing with the 240ml, let’s instead think about what proportion of each vinegar we would want to use. If we used half of each, we’d end up with a vinegar concentration of (5%+15%)/2 which is 10%. That’s too high for our goal.
What about (5%+5%+15%)/3? That’s 8.33% which is too low for our goal.
What if we used a fifth of A, a fifth of B, a fifth of A, a fifth of B, and another fifth of A? We’d get (5%+15%+5%+15%+5%)/5 which is 9%. That’s 240/5*3 (144ml) of A and 240/5*2 (96ml) of B. I sort of stumbled upon this by chance while thinking about examples of distributions, but we should probably have a more strategic method of dealing with the issue.
Let’s write an equation to describe the vinegar concentrations. You may automatically want to say we have two unknown amounts of vinegar we plan to use for A and B:
x millimetres of A times it’s concentration of 5% plus y millilitres of B times it’s concentration 15% should equal 240ml times a 9% concentration
x * 5% + y * 15% = 240*9%
That is, x and y are the volumes of vinegar A and B that we need to use to make 15% vinegar. Of course, we don’t know them yet (and are trying to determine them) so we write them as variables x and y. Unfortunately, we have two unknowns and therefore we cannot solve this equation!
There’s a missing story our equation doesn’t tell. Our chef only wants 240ml of the concoction, and therefore we are choosing between the trade offs of each. If I use 240ml of A, I can’t add any of B because the chef only wants 240ml. If I only use 120ml of A, how much of B will I add? 120! Yes. And if I only add 40ml of A? Well, then I’ll need to add 180ml of B. So in a way, we can’t choose how much of the other vinegar to add - because it’s whatever amount we aren’t using of A. Therefore, y=240-x because the volume of y is the remaining volume we still need to add!
Let’s rewrite our equation subbing in y:
5%*x+ 15%*(240-x)= 240*9%
The percentage signs are bothering me so let’s multiply through by 100.
5x+15(240-x)=240*9
I’m gonna divide by 5 to make my life easier.
x+3(240-x)=48*9
x+720-3x=432
-2x=-288
x=144ml
Therefore, y=240-144=96ml.
To summarise, we had two equations:
5%*x+ 15%*(y)= 240*9% (1)
y=240-x (2)
Which were combined to solve for x, the concentration of A. We used the solution for x to find y using equation (2). We knew equation (2) because y is the compliment of x, and x plus y must equal 240ml.