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Strategy 4: Translate from a Figure to an Arithmetic or Algebraic Representation

When a figure is given in a problem, it may be effective to express relationships among the various parts of the figure using arithmetic or algebra.

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

This is a Quantitative Comparison question.


  1. The figure shows triangle PQR, where P is the leftmost vertex of the horizontal side PR and vertex Q is above PR. Point S lies on horizontal side PR. Point S appears to be the midpoint of PR. Line segment QS is drawn from vertex Q to point S. The lengths of PS and SR appear to be equal.
    Quantity A Quantity B
    PS SR

    (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

    From the figure, you know that PQR is a triangle and that point S is between points P and R, so PS is less than PR. and SR is less than PR. You are also given that PQ is equal to PR.However, this information is not sufficient to compare PS and SR. Furthermore, because the figure is not necessarily drawn to scale, you cannot determine the relative sizes of PS and SR visually from the figure, though they may appear to be equal. The position of S can vary along side PR anywhere between P and R. Following are two possible variations of the figure, each of which is drawn to be consistent with the information PQ is equal to PR.

    Variation 1
    Variation 2
    The figure shows triangle PQR, where P is the leftmost vertex of the horizontal side PR and vertex Q is above PR. Point S lies on horizontal side PR. S appears to be closer to R than to P and the length of PS appears to be greater than the length of SR. The figure shows triangle PQR, where P is the leftmost vertex of the horizontal side PR and vertex Q is above PR. Point S lies on horizontal side PR. S appears to be closer to P than to R and the length of PS appears to be less than the length of SR

    Note that Quantity A is greater in Variation 1 and Quantity B is greater in Variation 2. Thus, the correct answer is Choice D, the relationship cannot be determined from the information given.

This is a Numeric Entry question.


  1. RESULTS OF A USED-CAR AUCTION

      Small Cars Large Cars
    Number of cars offered 32 23
    Number of cars sold 16 20
    Projected sales total for cars offered (in thousands) $70 $150
    Actual sales total (in thousands) $41 $120

    For the large cars sold at an auction that is summarized in the table above, what was the average sale price per car?

    $The answer space consists of a dollar sign, followed by a box for the answer.

     

    Explanation

    From the table above, you see that the number of large cars sold was 20 and the sales total for large cars was $120,000 (not $120). Thus the average sale price per car was $120,000 over 20 = $6,000 The correct answer is $6,000 (or equivalent).

    (Note that the comma in 6,000 will appear automatically in the answer box in the computer-delivered test.)

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