Solving mathematical word problems is often an obstacle for an operator in the quest to achieve higher certification. Using a structured approach, long or complex problems can be broken down into simpler, solvable pieces.
For most operators, attaining higher certification is one of their career goals.
In all certification programs the operator must pass a test — and as any operator will tell you, there are math problems on the test. The higher the certification level is, the more complex the math problems are.
For some lucky people, math is easy. For others, math can be intimidating. Listen to people coming out of a test and you may hear, “I did really well on all the questions, but I choked on the math.”
Most people find word problems the most difficult. These types of questions usually can’t be solved by plugging numbers into a single formula. There is no single “super formula” that can be used for word problems. Solving such problems takes a structured approach.
A Word Problem
A word problem is more that just a simple question. It presents several pieces of information about a situation, then asks a question related to that situation. In the shaded box is an example from a previous WEFTEC Operations Challenge process control test.
|This problem gives us a brief description of a plant, some information about its sludge handling process, and cost information related to polymer used in the centrifuge. The question gives us a little bit more information about the centrifuges and then asks about run time.
The Wrong Approach
If you haven’t been trained in solving word problems, your instinct might be to start crunching on the data given in the problem. For example, if you find that the first two pieces of information in the first sentence are a flow and a concentration, you might
| The Problem: The Hogeye WWTP has three primary clarifiers, two oxidation ditches, and four secondary clarifiers. Waste sludge is thickened in a gravity thickener and then pumped to a holding tank where it mixes with unthickened primary sludge. Mixed sludge from the holding tank is pumped to two solid bowl centrifuges. Dry polymer (50 lb bags, $1.25 per lb) is made into a 0.6% solution in a day tank. The polymer solution is pumped into the mixed sludge feed line just prior to the centrifuge inlet. Dewatered solids collect in 20 yard hoppers and are disposed of at the county landfill at a cost of $20 per wet ton. Currently, the plant is sending an average of 60,000 gallons per day of mixed sludge to the centrifuges at a solids concentration of 2.5%.
The Question: Each of the two centrifuges is rated at 600 lb feed mixed sludge per hour, and both are run at the same time. If one breaks down, how many more hours per week will the other one have to be operated to sustain the dewatering process?
use the “pounds formula” to come up with a result. Then, go to the next sentence and calculate what you can from it, and so on, until you have gone through all the data.
This is unstructured problem solving. You are using the data and producing results, but they might not be relevant to the question asked. In the example problem, the first bit of information given is about polymer and how it is dosed. But the question at the end is about centrifuge run time.
A better way to attack a word problem is work backwards. First, read through the problem to get a sense of what it’s all about. If parts seem confusing, read it a couple of times, but don’t worry about how to solve it just yet.
Now focus on the question. What is it asking specifically? Try to compare the question to a task you have done in real life. What information did you need to solve your real life example? What will you need to solve it now?
The key is to start with the question. In our example, the question is asking for hours of run time. Now, reread the problem looking for bits of information that relate to time. In the example, there are two items related to time. One is the feed rate, which is in pounds per hour, and the other is the feed flow, which is in gallons per day.
Let’s focus on the pounds per hour information. The feed rate is the pounds of mixed sludge fed to the press per hour. Intuition suggests that if we can calculate the pounds of sludge, we ought to be able to calculate hours based on the feed rate. We should instinctively think of the pounds formula at this point. For that, we use concentration and flow. This leads us to read the problem again but looking for sludge concentration and flow information. Sure enough, the information is there.
Now we can see how we can use the sludge concentration and flow information, the centrifuge loading rate, and somehow find the answer to the question.
Using Units as Clues
You may have noticed that we focused on units while working through the example. For solving the problem, numbers are not important until the end. We use units to figure out what is needed and how the pieces of information relate. Time units are easily related since we can convert minutes to hours or seconds as needed. If we’re given gallons per minute, that can be converted to gallons per hour without trouble.
Another clue that can be found in the units given in a problem is the word per, as in gallons per hour. The word per almost always means “divided by.” If a unit is gallons per hour, the number of gallons is divided by the number of hours.
A previous Logical Math article, “Making Algebra Less Scary”, takes a closer look at working with formulas.
Even with clues to lead you, word problems can be made very difficult. Including unnecessary information is one way to complicate a problem.
For example, information about pH and dissolved oxygen might be put into a word problem that asks about aeration basin detention time. Often this is done to test whether the test-taker knows which measurements are proper for the calculations. One aspect of solving word problems is realizing that sometimes some information can safely be ignored. Be careful about this, however. Some information might not be used in formulas but may still be needed to determine what formulas are applicable.
Putting it to Paper
Once we’ve figured out how to solve the problem, it’s time to write down the answer. However, on most tests that have word problems, you must show your work. Simply putting “8.4 hours” as the answer won’t be sufficient. Besides, showing your work will make it easier for you to check for mistakes.
The problem should be written out from start to finish, the reverse of how we solved it above. The bits of information should be generated by calculations step by step, with the results from one being used in the next as needed. So the first line on paper for our example would be to convert gallons per day to million gallons per day (mgd):
The next would be to convert percent solids to total suspended solids:
2.5% solids × 10,000 mg/L = 25,000 mg/L
Now we can take the two results and put them in the pounds formula:
25,000 mg/L × 8.34 × 0.060 mgd = 12,510 lb of solids per day
So if a centrifuge can process 600 lb of solids per hour, the needed run time is:
12,510 lb per day ÷ 600 lb per hour = 20.85 hours per day
Since the problem says we have two centrifuges:
20.85 hours per day/2 centrifuges = 10.42 hours per centrifuge per day.
Finally, the question asks how many hours per week, so we multiply by 7 days a week:
10.4 hours per day × 7 days per week = 72.8 hours per week
Once you’ve written out all the steps and entered them into a calculator to get the final result, don’t assume you are finished. It is good practice to compare your answer to the question. Does it make sense?
If the question is about detention time in a clarifier, did you come up with an answer that would be considered reasonable, such as 1 to 6 hours? Be aware that sometimes results are outside of the normal operating ranges. If your answer is 2 minutes or 5 days, then chances are something is wrong.
Looking at our example, we had a result of about 10 hours per day. That is a reasonable run time for a dewatering operation. If we had calculated something greater than 24 hours per day, we would suspect that we did something wrong somewhere. Or if the result had been a fraction of an hour, that should make us want to recheck our work as well.
Another way to check your work is to track the units through a problem. If they properly cancel each other, your answer is probably correct.
Word problems that appear in certification exams can be formidable and appear to be unsolvable. But by working logically backwards from the question, you can find a path from the data to the answer. It will take some time to get comfortable with word problems. Fortunately, your skills will improve with practice.