### Remember Your Units!

Did you ever have one of those anal teachers in high school math or science who would take off points if you did not include the correct units? So an answer of 7 would only receive partial credit when the answer was 7 inches. Although this practice likely seemed frustrating at the time, I hope to provide some method behind this madness “ or specifically how awareness of units can help you on the GMAT.

My appreciation of units first began during college. I was a chemistry major in college, and as part of my major I had to take physics. The topics in physics never came naturally for me so I was always looking for little tricks that would lead me towards a correct answer. One trick I found that was surprisingly effective was to just combine the numbers in the way such that the answer was in the appropriate units. For example if the question asked for an acceleration (the rate at which speed is changing or the second derivative of distance for the calculus-inclined), I knew that acceleration is always in the form of units of distance / units of time^2 (e.g. meters/ seconds^2). So unless I combined the numbers in a way that resulted in these units as the answer “ for example by dividing a speed in meters per second by a time in seconds “ I knew I had done something wrong.

Since units are not required on the GMAT, I find many students exclude them entirely from their note taking and calculations. But keeping track of units, while it may cost a little time, can help lead you towards right answers and prevent you from doing illogical algebra.

First, remember that any variable you create from a word problem there are some sort of units implied, whether you write them out explicitly or not.

Example 1: Chris is currently 5 years older than Dave. In four years, Chris will be 6 years younger than twice Dave’s age at the time.

If I create a variable C to represent Chris’ age now, that variable has the units of years attached to it. A corollary to this point (not totally related to units) is that a single variable can only represent one thing “ in this case Chris’ age now (not Chris’ age at another point in time). Once you have established the units attached to a variable, it makes sense to start thinking about how you can combine one variable with another variable or other numbers.

Addition and subtraction can only be used to combine likes. It makes sense to subtract $5 from $15. It makes sense to add cats to dogs if you are counting the total number of animals. Try to think of a context where it would make sense to add a dollar value to a number of dogs.

Example 2: Of the 350 people who received a flu vaccine, 70 percent experienced soreness at the injection site, 30 percent experiences swelling at the injection site, and 10 percent experienced a mild fever. 1/5 of the people experienced none of these side effects and 55 people experienced exactly 2 side effects. How many people experienced exactly one side effect?

I am going to walk you through an approach to solving. Try to find my mistake.

Let x = the number of people who experienced 1 side effect.

Let y = the number of people who experienced 3 side effects.

Since 1/5 of the people experienced no side effects, only 4/5*350 = 280 people experienced any side effects.

Therefore, x + 55 (people with 2 side effects) + y = 280. [Equation 1]

I can also calculate the total number of side effects experienced by all people.

.7* 350 (soreness) + .3*350 (swelling) + .1* 350 (fever) = 385 side effects

Therefore, x + 2*55 + y = 385 [Equation 2]

Two equations and two variables should be solvable, but in this case you will see you do not get an answer. Can you identify my mistake?

Let’s think about our variables. Both x and y are have the units people. This unit works in Equation 1 because I am counting the number of people. But what is the unit for the total, 385, in equation 2? It is actually side effects. I cannot simply add x and y, which have the units of people, into an equation that is totaling side effects.

So how do you change the units on a variable? You multiply or divide by another number (sometimes called a conversion factor) to get to the desired units. For example, a rate of speed could be represented in the units miles per hour. If I know John drove 200 miles in 4 hours, I can calculate his speed by dividing the number of miles by the number of hours: 200 miles/ 4 hours = 50 miles/hour. My answer is in the correct units.

Let’s go back to our previous example. In equation 2, the total was in side effects. How would I convert x or y (with units of people) into units of side effects? I could multiply by a conversion factor with the units of side effects per person. The corrected form or equation 2 is below.

Multiplying my variables by the conversion factors has made it possible to add together the number of side effects. Now, most people would simply write the equation as below without being explicit about the units.

x + 55*2 + 3y = 385 (all in terms of side effects)

x + 3y = 275 [equation 2 revised]

I am now down to simple algebra and no longer need to worry about units. To finish off this example, I can now use the revised form or equation 2 and equation 1 to solve for x.

x + y = 225

x + 3y = 275

Therefore, 2y = 50. That means y = 25 and x = 200.

While you may not find it necessary to be explicit about your units in all cases, if you ever find yourself getting confused by your equations, I encourage you to consider units as a way to guide you to logical algebra.

And if you find this method helpful don’t forget to send that thank you card to your 10^{th} grade geometry teach.

Note: In example 1, C (Chris’ current age) = 12 years.

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