How much am I saving (if anything) by using a Sodastream?

I think we need to think about the total cost per 12 oz. of soda. And let's use root beer as an example. We'll pit sodastream diet root beer against a can of diet A&W from Meijer.

The Cost of Sodastream

Let:

• $\hat{Y}$ denote total estimated cost
• $x$ be the quantity of sodastream root beer we wish to produce
• $a$ and $b$ be coefficients denoting the cost of syrup and $\mathrm{CO_2}$, respectively

$$\hat{Y}_{sodastream} = x\left(a_{syrup} + b_{\mathrm{CO_2}}\right)$$

The Cost of Syrup

$$ a_{syrup} = \left(\textrm{355mL of soda}\right) \times \left(\frac{\$5.99}{\textrm{440mL syrup}}\right) \times \left( \frac{\textrm{15mL syrup}}{\textrm{240mL soda}} \right) $$

$$ a_{syrup} = \frac{\$0.3021}{\mathrm{355mL of soda}} $$

The cost of $\mathrm{CO_2}$

$$ b_{CO_2} = \left( \textrm{355mL of soda} \right) \left( \left( \frac{\$29.99 - \left( \textrm{\$10 cylinder recycle discount} \right)} {\mathrm{cylinder}} \right) \right) \times \left( \frac{\mathrm{cylinder}}{\textrm{60000mL of soda}} \right) $$

$$ b_{\mathrm{CO_2}} = \frac{\$0.1183}{\textrm{355mL of soda}} $$

Total cost of a "can" of sodastream

$$ \hat{Y}_{sodastream} = \textrm{355mL of soda} \times \left( \frac{\$0.3021 + \$0.1183}{\textrm{355mL of soda}} \right) $$

$$\textrm{sodastream price} = \frac{\$0.420}{\mathrm{can}}$$

References

Total cost of a can of diet A&W root beer

$$ \left( \frac{\$11.00}{\textrm{4 cases}} \right) \times \left( \frac{\textrm{1 case}}{\textrm{12 cans}} \right) $$

$$ \textrm{can price} = \frac{\$0.229}{\textrm{can}} $$

References

Conclusion

At just 23 cents per can, it seems drastically cheaper to buy A&W diet root beer (whose taste I prefer) to continuing to operate my sodastream (which costs 42 cents per can).