Making Abstract Manipulations

Example:  

The area of a rectangle is 80 square inches. What is the area, in square inches, of a rectangle with twice the length and half the width?

A) 40

B) 80

C) 100

D) 120

Strategy:

When a problem asks you to make changes to unknown values, and to interpret the effect of those changes, assign values.  This works when you have 2 or more variables that are undefined.  Use numbers that are easy to add/multiply and won’t leave you with fractions or decimals after you make your manipulations.

The question is asking you to make changes to unknown values, the initial length and width of the rectangle.  Therefore, assign values for the length and width.  Make sure that your values multiply to be 80, because our initial area must be 80 square inches.  Here, we’ll use 8 and 10 but you could as easily use 20 and 4.

$$l=8\;\;\;w=10\;\;\;l\times w=80$$

Now change the length and width to interpret the rectangle asked about in the question.

$$new\;length=16\;\;\;\;\;\;\;\;\;new\;width=5\;\;\;\;\;\;\;\;\;\;new\;l\times w=80$$

The answer would be B) 80.

Traps

Sometimes, your variable will not be x.  It could be ab, or some other letter.  The strategy works the same no matter what they call the variable.

You might see a constraint on the x’s everywhere, like “x is a positive integer” or “where > 3”.  In that case, instead of using 2, we’d just use 4.  It works the same for any number that isn’t 0 or 1.

Sometimes, you will have two undefined variables.  You can still assign values – just make them both 2 or other appropriate numbers.