The Secret To Wastewater Math

Put the calculator aside for a moment. Learn how to factor units into your solutions to math problems on licensing exams.

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In my years as a wastewater operator trainer, I find that math is by far the most anxiety-producing topic for those preparing to take certification exams.
But why? For most wastewater math problems, all you need to do is add, subtract, multiply or divide. Everybody knows how to do that, right? Usually the problem is not with the actual math — it is with setting up the problem and picking out the right formula and the right units.

When us old timers were kids learning to do math, there was no such thing as a calculator — all we had were pencil and paper. So we had to write everything down. This gives us a big edge over whippersnappers who grew up with calculators, because they can’t plug units into the device. If you just punch numbers into a calculator, you’ll come up with an answer, but if you’re not careful, you won’t know if you stopped at the right point or whether you’re in the right units.

A better approach

Here is how I suggest approaching wastewater math. Grab a pencil and a piece of paper and shove that calculator just out of reach. Set up your problem based on the units, and the units will tell you how to do the math — whether to multiply or divide.

For example, say you have a flow of 10 cubic feet per second and you want to know how many gallons per minute that is. First clue: Any time you see that word per, think divide. So you can write the problem out this way:

10 cubic feet/second = X gallons/minute

Now, how do you get from cubic feet per second to gallons per minute? Well, you can do math with units just like you do numbers. In this problem, there are two units to convert — cubic feet to gallons, and seconds to minutes. In each case you will use a conversion factor.

A basic principle of algebra is that if you divide something by itself in an equation, you cancel it out. For every conversion factor, there will be two associated units:

The unit in the original problem that you want to cancel out.

The unit you want to end up with in the answer.

You take the unit you want to cancel out and put in the formula wherever it works for canceling — either above or below the line that signifies division. Then you take the unit you want to end up with and put it on the opposite side of the line from where you put the first unit.

So, in our problem, let’s first deal with cubic feet and gallons: You want to cancel out cubic feet and be left with gallons. If you follow the procedure above, your problem now looks like this:

10 cubic feet/second x gallons/cubic foot = X gallons/minute

Now notice that the unit “cubic feet” appears both above and below the line. You are dividing it by itself and therefore can cancel it in both places, as shown below.

10 cubic feet/second x gallons/cubic foot = X gallons/minute

So you have cancelled out cubic feet and are left with gallons — but there is also a number associated with the conversion factor for going from cubic feet to gallons. You should receive a formula sheet with your exam that includes a table of conversion factors. Look down that table until you see something with gallons on one side of the equal sign and cubic feet on the other. You will see a number associated with one of those units. In this case, it is 7.48.

But do you multiply or divide by 7.48?  Which unit did that 7.48 go with? In this case, there are 7.48 gallons per cubic foot, so you would multiply. And now your problem looks like this:

10 cubic feet/second x 7.48 gallons/cubic foot = X gallons/minute

Don’t reach for that calculator yet. You still need to convert from gallons per second to gallons per minute. You won’t need your table of conversion factors to know to get there are 60 seconds per minute. So, follow the same procedure for units described earlier. Now your equation looks like this:

10 cubic feet/second x 7.48 gallons/cubic foot x 60 seconds/minute = X gallons/minute

Notice that now you have seconds both above and below the line — which means you can cancel it out. And your equation becomes:

10 cubic feet/second x 7.48 gallons/cubic foot x 60 seconds/minute = X gallons/minute

Now, pick up your calculator, do the multiplication, and you have your answer:

10 cubic feet/second x 7.48 gallons/cubic foot x 60 seconds/minute = 4,488 gallons/minute

Those old friends

So, do you see how setting up a formula to cancel out units will automatically tell you whether to punch the multiply or divide button on your calculator? Nowadays on the job you can set up spreadsheets to do almost all your math for you — but until you can bring your computer into the exam with you (not likely anytime soon), I suggest you bring the good old pencil and paper.

About the author

Steve Duerre is a pollution control specialist with the Minnesota Pollution Control Agency. He can be reached at steve.duerre@state.mn.us.   



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