Additional Math Pages & Resources

Monday, October 26, 2009

Why do ink cartridges cost so much?

People often complain It costs so much to print! A new cartridge costs $$$ and they only last a few hundred pages!

Does it really cost a lot to print? Do cartridges really run out quickly? Let's use some math.

Inkjet printers are the least expensive color printers you can buy. They spray dots of ink on the pages to create text and graphics. Black and white dots are generally larger than color dots. They blend together on the paper to give us a color image. Although cheap to buy, these printers can be expensive to use.

Ink is mostly water, but that doesn't mean it's cheap or simple. I went to the HP website and took a look at their Material Safety Data Sheets. Those gave me the rough proportions of ingredients in an inkjet cartridge. This chart shows cyan's ingredients but generally water's about 2/3 of the total volume of ink.

Printer manufacturers use a very small fraction of a liter (known as a picoliter) to measure the size of ink droplets. How large is a picoliter?
  1. milliliter = 1/1,000 of a liter
  2. microliter 1/1,000,000 of a liter
  3. nanoliter 1/1,000,000,000 of a liter
  4. picoliter 1/1,000,000,000,000 of a liter

I pulled a color cartridge off our shelf. It cost us about $40 and contains 69 ml of ink. How many picoliters is that? Drop three zeros and multiply 1 x 69 to get:

69,000,000,000  or 69 billion picoliters. The HP site says each color drop contains 4 picoliters of ink, so that means:

69,000,000,000 ÷ 4 = 17.25 billion droplets.

Distribution of the ink on the page can vary dramatically. HP assumes 5% of the surface area of the paper gets ink. I found some users who estimate 15% coverage for B/W printing, and 50% for color (consumers tend to print lots of family photos in color).

So 17.25 billion divided by ( N [number of droplets possible on a page] x C [% of coverage] x M [mix of color inks to get the color you want]) = P (number of pages)

Hold that thought.

Besides what comes out on the page, there are many other ways your ink can disappear. And I don't mean just the pages where you have made an obvious error, crumpled up the paper and thrown it out.
  • Cleaning - the printer pumps a little bit of ink through the print nozzles to keep them clean. It  does this before and after each print job.
  • Shelf Life - ink is mostly water. Some just disappears, evaporates, or the ink dries up.
  • Calibration / color adjustment - you can run calibration pages yourself, but even if you don't the printer is doing calibration and self-adjustment on a regular basis.
  • Unbalanced color use - not all the ink in each of the multiple colors will be used evenly. If you never print some colors, that ink could dry up or (in a multi-color cartridge) be thrown out.
  • Residual Loss - as HP says, you can't get all the toothpaste out of a tube and you can't get all the ink out of a cartridge.
  • Blended colors -  printers use color inks together with black to get the best solid black tones. You have to select gray-scale options in your software to prevent this from happening.
  • High Quality - there are many levels of quality or print resolution. The higher ones take more ink.
  • Overlap Design - the best printers are designed to give good coverage on the paper, and take into account the seepage or wicking of the color along the paper fibers. Here are HP patent drawings showing various ways they arrange the droplets. (These are greatly enlarged.)

I think this is more complicated than ROCKET SCIENCE. With rockets, we only have to hit the moon or Mars, and we have decades and trillions of dollars.

A $100 inkjet printer has to print pictures of your cute child, adorable aunt or precocious pet in seconds, for pennies.

                                                               Typical family photos

Try to calculate the proportions of each ink color to print these three pictures!

Ignoring the benchmarks and calculations, users have tracked many thousands of pages to find ink alone costs at least $.25-35 per page (color) and $ .08 - .10 (B/W).

Happily a tip from my friend Chris led me to a Dutch group who created a font that takes less ink, and saves you money!

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