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Strategy 5: Simplify an Arithmetic or Algebraic Representation

Arithmetic and algebraic representations include both expressions and equations. Your facility in simplifying a representation can often lead to a quick solution. Examples include converting from a percent to a decimal, converting from one measurement unit to another, combining like terms in an algebraic expression and simplifying an equation until its solutions are evident.

• This strategy is used in the following two sample questions.

This is a Quantitative Comparison question.


  1. y is greater than 4
    Quantity A Quantity B
    The fraction with numerator 3y +2 and denominator 5 y

    (A) Quantity A is greater.
    (B) Quantity B is greater.
    (C) The two quantities are equal.
    (D) The relationship cannot be determined from the information given.

     

    Explanation

    Set up the initial comparison:

    The fraction with numerator 3y + 2 and denominator 5, followed by a question mark symbol, followed by y.

    Then simplify:

    Step 1: Multiply both sides by 5 to get  3y +2, followed by the question mark symbol, followed by 5y

    Step 2: Subtract 3y from both sides to get  2, followed by the question mark symbol, followed by 2y

    Step 3: Divide both sides by 2 to get  1, followed by the question mark symbol, followed by y

    The comparison is now simplified as much as possible. In order to compare 1 and y, note that you are given the information y is greater than 4 (above Quantities A and B). It follows from y is greater than 4 that y is greater than 1 or 1 is less than y so that in the comparison 1, followed by the question mark symbol. the placeholder The question mark symbol. represents less than (<): 1 is less than y..

    However, the problem asks for a comparison between Quantity A and Quantity B, not a comparison between 1 and y. To go from the comparison between 1 and y to a comparison between Quantities A and B, start with the last comparison, 1 is less than y. and carefully consider each simplification step in reverse order to determine what each comparison implies about the preceding comparison, all the way back to the comparison between Quantities A and B if possible. Since step 3 was "divide both sides by 2," multiplying both sides of the comparison 1 is less than y. by 2 implies the preceding comparison 2 is less than 2y thus reversing step 3. Each simplification step can be reversed as follows:

    • Reverse step 3: multiply both sides by 2.
    • Reverse step 2: add 3y to both sides.
    • Reverse step 1: divide both sides by 5.

     

    When each step is reversed, the relationship remains less than (<), so Quantity A is less than Quantity B. Thus, the correct answer is Choice B, Quantity B is greater.

This is a Numeric Entry question.

  1. A merchant made a profit of $5 on the sale of a sweater that cost the merchant $15. What is the profit expressed as a percent of the merchant's cost?

    Give your answer to the nearest whole percent.

    The answer space consists of a box for the answer followed by the % sign.%

     

    Explanation

    The percent profit is Open parenthesis, 5 over 15, close parenthesis, times 100, which equals 33.333..., which equals 33 point 3 with a bar over it. percent, which is 33%, to the nearest whole percent. Thus, the correct answer is 33% (or equivalent).

    If you use the calculator and the Transfer Display button, the number that will be transferred to the answer box is 33.333333, which is incorrect since it is not given to the nearest whole percent. You will need to adjust the number in the answer box by deleting all of the digits to the right of the decimal point.

    Also, since you are asked to give the answer as a percent, the decimal equivalent of 33 percent, which is 0.33, is incorrect. The percent symbol next to the answer box indicates that the form of the answer must be a percent. Entering 0.33 in the box would give the erroneous answer 0.33%.