We have some of these mini cans of ginger ale in the house this week. I am not sure where they came from; only that my wife bought them. Normally we only have sparkling water around, not pop (nor soda, nor…)

So I’m looking at the can instead of grading like I should be and I notice the “25% fewer calories than regular ginger ales” claim.

And I think what any skeptical consumer ought to think. *Sure fewer calories in the mini can. Duh.*

Then I see this:

They have controlled for the size of the can. Nice. This one has 60 calories per 7.5 fl. oz. Regular ginger ales have 90 calories per 7.5 fl. oz.

I am briefly satisfied. And impressed.

But wait! 60 is 25% less than 90? ARGH!

Two possible explanations:

- 25% means
*at least* 25%, and Seagram’s chose this nice simple number over the more complicated .
- It really is exactly 25%. But we know that calorie counts are rounded to the nearest 10 calories.

This second explanation leads to a sort of lovely task. How can we characterize the set of possible calorie counts for 7.5 fl. oz. of Seagram’s and of regular ginger ale so that, (a) the counts round to 60 and 90, and (b) one number is exactly 25% less than the other?

**Extra credit:** Which standards for mathematical practice are you using as you solve?

**Double extra credit:** Which of my abbreviated list of standards for mathematical practice (see below) are you using as you solve? And which was I using as I gazed at my can of ginger ale?

**Prof. Triangleman’s Abbreviated List of Standards for Mathematical Practice.**

**PTALSMP 1: Ask questions.** Ask why. Ask how. Ask whether your answer is right. Ask whether it makes sense. Ask what assumptions you have made, and whether an alternate set of assumptions might be warranted. Ask what if. Ask what if not.

**PTALSMP 2: Play.** See what happens if you carry out the computation you have in mind, even if you are not sure it’s the right one. See what happens if you do it the other way around. Try to think like someone else would think. Tweak and see what happens.

**PLALSMP 3: Argue. **Say why you think you are right. Say why you might be wrong. Try to understand how someone else sees things, and say why you think their perspective may be valid. Do not accept what others say is so, but listen carefully to it so that you can decide whether it is.

See also my Desmos graph of this relationship.

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Love the PTALSMP! Can I quote them?

Of course you may. The elaborations are first drafts banged out in about 8 minutes of blogging, so suggestions for revisions are accepted.

I think that perhaps you are giving Seagram’s FAR too much credit, but the problem you pose is way more interesting than just poking fun at them. PTALSMP 1 reminds me of the notice and wonder language. I would wonder how to write an expression to account for your claim of being within 10 calories. For the 90 calorie measure this would mean, I think, that the ‘true’ calorie measure of original would lie in the domain interval (85, 95) while the ‘true’ calorie measure of the smaller can would be in the range of (55, 65). So now I want a relationship where my range element (call it y for originality’s sake) is 0.75 x where x is my clever name for the calorie in the original soda. If I wasn’t prepping my kiddos to bed, I’d play with this. So that’s for later…

This is a good application problem for systems of linear inequalities. Here is my attempt at a graph: https://www.desmos.com/calculator/ix9hpbpjlf

Dave

That looks fantastic. Clearer than my garbled explanation above…

I’ll believe they rounded. I would give the Coca-Cola company, owner of Seagram’s, a lot of credit for knowing how to maximize their profits. Finishing mrdardy’s argument, the possible values of 0.75x for x in (85,95) are those x in the interval (63.75,71.25). This barely overlaps with the other region, so the answer to your question would be that the smaller can would need to have between 63.75 and 65 calories while the larger would have between 85 and 86.7 calories.

By rounding the larger count up to 90, then, they are rounding up by about as much as they can. And by rounding down the smaller count down to 60, they are rounding the smaller can down as much as possible. So both kinds of rounding contribute to about as large a discrepancy between the calorie counts as possible.

How did they arrange this? Notice the absolutely weird volume of the can, 7.5 fluid ounces, which they are apparently required to use when describing the calorie density: “per 7.5 fluid ounce (222 mL)”. By comparison, I think standard coke cans are 12 fluid ounces. It’s a good bet 7.5 was selected to ensure the rounding led to as big a discrepancy between calorie counts as possible. So make the problem more interesting — what size of can will maximize the percentage difference between the two calorie counts? If you allowed calorie counts to get very small, I think this might have no answer. But I believe the FDA requires rounding only for calorie counts over 50, and presumably the can size must be in a certain range for consumers to be willing to buy. So fix a unit, like “half fluid ounce”, and determine what can size between, say between 6 ounces and 12 ounces, will maximize the percentage difference between calories.

Wow – what a terrific, detailed analysis. I am thinking that this is worthy of an investigation by my precalc class sometime soon. Between this and the oreo problem we should have systems covered!

I don’t know how to edit my previous comment, so here’s an amendment: it would seem from the OP that regulations require them to use actual calorie density, not rounded ones, when saying something like “25% fewer calories”, so optimizing the percentage difference as I described wouldn’t do any good.

But people, especially with soda, are fixated on the actual calorie numbers. And people especially concerned with them are going to be buying the smaller cans. So I’d believe they’d arrange to have the calorie count of their soda and the competition (“regular ginger ales?”) get rounded to make as large an ABSOLUTE difference in calorie counts as possible. People who really care about calories will see that 60 and 90 written on the side of the can and make a decision based on that.

I’m afraid that I think the most likely reason is that the person/people deciding what to do there didn’t know enough math to really figure it out, but had the number sense to know they were safe with 25%, or just made a mistake.

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