How to never mess up your unit conversions anymore

Units are a peculiar thing. In engineering, we spend a ton of time with them. Still, most of us never really think about them properly. One of many side effects of that is, that converting units into each other has led to hilarious and catastrophic miscalculations in the past (often both).

So today, let’s have a closer look at unit conversions. To this end, consider the following calculation done in a spreadsheet tool. We try to calculate the energy demand of your dishwasher per activation.

Ok, in principle, this looks about right. The energy demand is the average power demand of your dishwasher multiplied by the duration of the washing program. The result in our example would be 3000 Wh (Watt Hours).

A follow-up question might be the price for this amount of energy. Luckily, there are great sources like Statista where we can find average electricity prices for different countries of the world. As I am currently in Germany, I find that I pay 0.36 USD/kWh.

This is great and all, but before we can find out that I pay 0.9€ per dishwasher activation, we need to do quite some unit conversion. And this is always a tedious task that always gets wrong somehow. Reason enough to give some tips on how to get these things right.

Getting unit conversions right

In our example, we deal to start with Wh but need to convert them into kWh to calculate costs using the numbers we found online. The conversion factor in our case is 1000. But do we need to divide our value (3000) or do we need to multiply it? The result would be rather different and to us, choosing between the two options feels like plugging in a USB device.

The good news is, that we can ease the pain if we follow a few rules. The trick is to treat the unit conversion as a fraction. The specific steps are the following:

1. Write out the conversion as a fraction that equals 1
2. Multiply your value with the fraction
3. Cancel out the result, including units

For our example, this process looks somehow like this. First, we build the fraction:

As we know, we can multiply anything with “1” without changing the result, so we can do the following to convert 3000 Wh into kWh like so:

This, still, looks a bit weird. Luckily, we have the last step to go. Canceling out everything leaves us with the following result:

Note: If we had done this wrong (“upside-down”), the calculation would have looked like this, and we would have not been able to cancel the units out at the end. Not helpful, but at least obviously wrong.

Using this result, we can now calculate that using my dishwasher costs me ~3*0.36=1,08 USD. However, we were interested in Euros. This is a great opportunity to show how the process illustrated here also works for “multi-stage” conversions. Let’s convert 0.36 USD/kWh to EURO/Wh to illustrate the process.

First, we need to find the conversion factors and write them as a fraction (step 1):

Then, we multiply our value with the fractions (step 2):

And lastly, cancel everything out (step 3):

Using this now, we can directly calculate the price of one dishwasher activation and arrive at (as stated earlier) 0.9 € per activation.

Conclusion

The approach presented here is by no means new. However, it illustrates how engineers “bend” mathematics to suit their needs better. Strictly speaking, they created their own language to be able to use their domain knowledge within calculations. In our next post, we will investigate this idea further and potentially see other ways in which inventing new languages can be beneficial.