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The Art of Scratchwork and Picking Variables

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June 16, 2021

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Josh Jones

“What is the best way to choose variables?” is a question students rarely ask but should. One key to solving many math problems is representing the information in the simplest, accurate, and most natural form. We want the variables or pictures to convey all the relevant information without being cumbersome to write repeatedly or to understand.

The optimal choice of variables:

  1. Is simple, easy to read, and has no unnecessary baggage,
  2. Conveys the relevant information,
  3. Matches the letters to the words (and is not just x and y for words that ideas that have nothing to do with the letters x and y).

For example, suppose we have two entrepreneurs, Alexis and Brandon, and we are to compare their revenues at two different times. How would you represent these quantities?

We could label the individuals A and B, and the time periods 1 and 2. We could call the revenues R_A and R_B, but since we are not interested in any other quantities, we can drop the R and just call the revenues A and B, accomplishing 1).
Also we have to deal with the two time periods and R_A1 R_A2 is cumbersome.
Now since we are dealing with two different time periods, we can call Alexis’s revenue in the first time period A1, Alexis’s revenue in the second time period A2, and so on, so that we have A1, A2, B1, and B2. Now we have variable representation that covers all relevant information, accomplishing 2), but that is not so complex that we need to refer to a key to remember what each variable represented, accomplishing 3).

-Rather than using “x” for every single variable, use the first letter of the word represented. Use L for length, C for cars etc. In the more difficult GMAT problems you’ll have multiple variables and you don’t want to waste time translating between arbitrary letters xyz and concrete words. This also introduces more chance for error.
-Use subscripts like s1 s2 for sales in the 1st time period and sales in the 2nd time period, except if the question is about one variable, such as “What is s2?”. Then just call s2=s and write everything in terms of s.

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