Break Even Point for Printing Software Cost

Jul 2010
I'm drawing a blank about how to solve for the break even point here...

Lets say toner for a typical laser printer costs $213 and you can print 7,500 sheets of paper with a mean amount of per sheet toner (amount of toner per sheet that the typical sheet has). A piece of software can save money by reducing the amount of toner used per sheet printed by 20%. The software costs $1.69 per user for 10,000 users for 3 years.

If We have 10,000 users that print with this software and pay the license, for every sheet of paper printed how many sheets will it take before the break even point is reached? If anyone can help me here I would also be interested in how you came to your calculated break even point.
May 2010
Without the machine, the cost of producing x sheets of paper is:

\(\displaystyle \frac{213x}{7500}\)

With the machine, the cost of producing x sheets of paper is
\(\displaystyle \frac{213*0.8*x}{7500} + 16900\)

"break even" is the production level at which the costs are the same (so there has been a gain of zero from switching)

\(\displaystyle \frac{213*0.8*x}{7500} + 16900 = \frac{213x}{7500} \)

Solve for x. Your answer is only valid if the company goes through that much paper in less than 3 years (otherwise, you must add the cost of another license).
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Jul 2010
So the break even is 2,975,352.1 pages printed total, but would be 297.5 pages per user? Thanks a ton for the formula and your help!