Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I may be over-complicating things, but something doesn't seem right (and I swear this isn't homework, I'm friggin 30 years old :P).

I want to see what it will cost me to make a TWO liter of Coca-Cola using coke syrup and CO2 (assume the water is free). Assume CO2 costs 0.21usd per liter of water (25usd for 120 liters).

The coke syrup needs to be diluted at a "5.4:1" ratio with the carbonated water. So how much will it cost me to make my own 2-liter?

Here's what I've done so far:

syrup cost per liter = C / ((G * D) * 3.785)

where

  • C is the cost of G gallons of coke syrup
  • D is the dilution (5.4), and
  • 3.785 converts gallons to liters

For a 65.00usd box of 5 gallons of coke syrup, I get a cost of 0.64usd per liter. Along with CO2 cost, that makes it 1.69usd for a 2-liter.

That seems very, very costly, given that 2-liters in the store usually cost much less. If I'm wrong, where am I going wrong? If I'm right, why don't restaurants use 2-liters instead of fountain machines to increase their profits/lower their costs? I know they charge a lot for a glass of coke (and refills), not a 2-liter, but it still seems like it would make them more money.

I put that equation into an Excel sheet, and it seems like the syrup would have to be 30usd in order to get the cost down to 1.00usd/2-liter, and that doesn't even seem like much of a gain over buying 2-liter bottles from a grocery (especially with the equipment overhead costs). I spoke to a restaurant manager, and his cost is 56usd for a 5-gallon box, which puts the per 2-liter cost at $1.51 by my calculations, so that's when I suspected I was doing something wrong.

share|improve this question
1  
I seem to get something slightly different. 5 gallons of syrup is 18.925 liters; at USD65, that's 3.4346 per liter of syrup. 10 liters of syrup and 54 liters of carbonated water make 64 liters of coke. This is 34.35 for the syrup, 11.34 for the water, giving 45.69 for 64 liters. Dividing by 32, I get about 1.43 for two liters, not your 1.69. –  Arturo Magidin Dec 31 '10 at 19:36
1  
Arturo has taken into account that a liter of syrup is mixed with 5.4 litres of water to produce 6.4 litres of coke. From a chemistry standpoint, I'm not sure if that answer is true. You can dissolve a huge amount of salt in water without increasing it's volume. I'm sure this applies to sugar to some extent. If this is true, then 5.4 litres of water plus 1 litre of syrup may produce a mixture with volume LESS THAN 6.4 litres. Thus the answer would be somewhere between 1.43 and 1.69. Sorry, it’s my lack of science knowledge that has stumped me on this question. –  user3180 Dec 31 '10 at 22:48
1  
@Joe: I suspect that the Coke syrup does not substantially dissolve into the water and the quantities are additive, or at least quite close to it. –  Isaac Dec 31 '10 at 23:16
1  
I've personally analyzed the pricing of packaged Diet Coke versus buying the syrup and carbonating the water. Even neglecting the cost of the carbonated water, the cost per liter for the syrup is around the same or slightly higher than the cost per liter of the packaged soda when it's on sale (unless you can get a better price on the syrup than Sam's Club or similar). The only reasons I can think of to go forward with a home fountain system are the environmental savings of not transporting the soda to/from the store and not packaging the soda. –  Isaac Dec 31 '10 at 23:18
2  
Welcome! Glad to see some practical applications of this site :-) –  Aryabhata Dec 31 '10 at 23:31

1 Answer 1

up vote 5 down vote accepted

With a $5.4:1$ ratio of carbonated water to syrup, assuming that the quantities are additive (mixing 5.4 units of carbonated water with 1 unit of syrup yields 6.4 units of the mixture), I get: $$5\text{ gallons of syrup}\times\frac{6.4\text{ gallons of mixture}}{1\text{ gallon of syrup}}\times\frac{3.785\text{ liters}}{1\text{ gallon}}\approx 121\text{ liters of mixture}.$$ At $\$65$ per box, $$\frac{\$65}{121\text{ liters}}\approx\$0.54\text{ per liter}.$$ This is all ignoring the cost of carbonating the water, because around me, a 2-liter bottle of Coke goes on sale for 99 cents with enough regularity that it already doesn't make sense to mix at home.

share|improve this answer
1  
You forgot to add the cost of the 5.4 gallons of carbonated water to the $\$65$ per box of syrup. The assumed cost of the carbonated water is $\$0.21$ per liter. –  Arturo Magidin Dec 31 '10 at 23:30
1  
@Arturo: I added a note that I was skipping that cost and why (and probably did so at the same time you were commenting). –  Isaac Dec 31 '10 at 23:34

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.