Here’s a puzzle for normal people. Nothing but algebra required. Also it’s super practical! We’re going to figure out how much brown sugar you can add to Greek yogurt before you’ve basically turned it into junk food.
Suppose we have some healthy Greek yogurt. According to the nutrition label it has:
\omega grams per serving
\gamma calories per serving
\sigma grams of sugar per serving.
Yes, Greek letters for those, since it’s Greek, and to distinguish from some Go-gurt, marketed to kids, with the following on the nutrition label:
w grams per serving
c calories per serving
s grams of sugar per serving.
There are a couple other things we need to know. First, there are k=3.87 calories per gram of normal sugar and \kappa=3.80 calories per gram of brown sugar. (Normally we round both of those to 4, which would indeed be close enough.) Second, we’ll define “healthiness” in this context as the fraction of calories that are from sugar. So for the Greek yogurt that’s \dfrac{k\sigma}{\gamma} and for the Go-gurt it’s \dfrac{ks}{c}.
Now for the problem. I’ve weighed out on my kitchen scale y grams of Greek yogurt. How many grams, x, of brown sugar do I need to add to make its healthiness match that of Go-gurt?
Partial spoiler but this inspired me to (have AI) make a general tool which I’m calling Sheeq, for “spreadsheet-style UI from an equation”.
You type in any equation at the top and it makes a calculator for you, to compute any of the variables based on the others. So here it is solving this math puzzle (well, numerically solving it):
So, we need the fraction of calories coming from sugar in the resulting mix to match the fraction known for the Go-gurt. The calories from sugar in the mix will be coming from two sources: the part of the original Greek yoghurt calories associated with sugars and those in the added sugar.
First, let’s formalize the number of sugar calories from the Greek yoghurt:
If it’s \sigma grams of sugar per serving, then the portion of y grams gives us \sigma*\frac{y}{\omega} grams of sugar, which gives \sigma*\frac{y}{\omega}*k calories from that part. That’s \frac{\sigma k y}{\omega}
We are adding x g added sugar, which gives us k x calories from added sugar. (Yes, I am pretending the added sugar has the same calories/gram as the sugar in the yoghurt. We don’t know what exactly they counted for sugars in the yoghurt anyways.)
Total calories in the mix are \gamma \frac{y}{\omega} (from the yoghurt) plus the same k x calories from added sugar.
\frac{\frac{\sigma k y}{\omega} + k x}{\frac{\gamma y}{\omega}+ k x} = \frac{k
s}{c}
So, I equated the percentage of the calories from sugar in the mixture to the originally known percentage from sugars in the Go-gurt. I won’t copy all the stages because I am literally using LaTex for the first time and it’s not easy to transcribe it all here, but then I just express x in terms of y from this and arrive at
x = y * \frac{\gamma s - \sigma c}{\omega (c - k s)}
Or did you mean that the solution should include the actual number of grams, based on your weighing the yoghurt-sugar mix together with the spoon?
I literally haven’t done anything related to math for years, probably since the year when I prepared myself and tutored a couple of friends for the GREs about ten years ago. And it’s just fantastic to discover how much my brain became lazier since then. I just looked at Danny’s description of the task and felt the “I don’t want to work” response from the brain. (Which is why I made it personal and decided that I have to conquer it) Funny, isn’t it, how we age (and specialize, probably? I want to think it’s the latter as well, as thinking about complex concepts in literary studies and surrounding humanities doesn’t do that. But I was once proud to be the person to enroll to study literature thanks to the best grade in the math entrance exam half my age ago…).
Nice work! Especially on getting up to speed on LaTeX so fast! The only nitpick (and I think this was too nitpicky for me to have included) is that normal sugar and brown sugar (the kind we’re adding in this example) have an ever-so-slight difference in calories per gram – 3.87 vs 3.8 – what I called \kappa vs k. Also confusingly similar looking variables, I’m now noticing.
Anyway, so my version was this:
\frac{k \sigma y / \omega +\kappa x}{\gamma y / \omega + \kappa x} = \frac{k s}{c}.
The same as yours but those +kx's are +\kappa x's. Then you can turn the algebra crank (cross-multiply and distribute and collect x terms and factor out the x and divide) and end up with this:
x = \frac{y}{\omega}\cdot\frac{k}{\kappa}\cdot\frac{\gamma s - c \sigma}{c - k s}.
Plugging in my numbers for my y = 233.5 grams of Greek yogurt (nutrition label: \omega= 170 grams per serving, \gamma= 120 calories per serving, \sigma= 5 grams of sugar per serving) and my benchmark junk food (nutrition label: c= 150 calories per serving, s= 23 grams of sugar per serving; turns out we don’t need the w= 170 grams per serving) yields x= 46.1 grams of brown sugar. Or leaving it in terms of y (grams of Greek yogurt):
x = 0.197431\cdot y
In other words, weigh out some healthy stuff, increase the weight by 19.7% by adding pure sugar, et voila, junk food. QED. (Questionable Eating Deconstructed?)
This gives us the grams of brown sugar needed. Note that if γks = kσc, then the Greek yogurt already has the same sugar-calorie fraction as Go-gurt, and if γks < kσc, then the Greek yogurt is already “worse” than Go-gurt in terms of our health metric.
Would you like me to create a small calculator to help evaluate this for specific values? We could make it interactive and allow testing different scenarios.
@dreev
Yay, thanks for the confirmation. As for the \kappa / k difference, I am aware that I made a generalization here. But then we don’t really know what they counted for “sugar” in the original yoghurt. ChatGPT suggests that a variety of things can go into this category on the nutritional label, all of them somewhere in the vicinity of ~4 calories/g. Lactose, most likely, in an allegedly “plain” greek yoghurt.
@zzq
Oh yes, they can! *add random elements from the discourse about humans ceasing to think and delegating everything to AI bots* I don’t know what exactly my point is, but it was really weird to catch myself not wanting to use the brain for such a simple math conceptualization, which is evidently within my “GRE math” scope.
By the way, I notice that what I copy-pasted into ChatGPT and Claude lost some formatting, and actually misstated the fraction of calories from sugar. (Both figured out what was meant and weren’t thrown by it.)
Yeah—but my feeling is that it has become like asking “what is 95834977 times 428978191?” Kind of an impressive party trick if you can do it in your head, but not really anything that requires actually solving anything.