Strategy 10: Trial and Error

Version 1: Make a Reasonable Guess and then Refine It

For some problems, the fastest way to a solution is to make a reasonable guess at the answer, check it and then improve on your guess. This is especially useful if the number of possible answers is limited. In other problems, this approach may help you at least to understand better what is going on in the problem.

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

This is a Multiple-Choice – Select One or More Answer Choices Question.

  1. Which two of the following numbers have a product that is between –1 and 0?

    Indicate both of the numbers.

    (A) −20
    (B) −10
    (C) two raised to the power negative 4
    (D) three raised to the power negative 2

     

    Explanation

    For this question, you must select a pair of answer choices. The product of the pair must be negative, so the possible products are negative twenty, times, two raised to power negative four, negative twenty, times, three raised to power negative two, negative ten, times, two raised to power negative fourand negative ten, times, three raised to power negative two. The product must also be greater than −1. The first product is The fraction with numerator negative 20 and denominator 2 to the power 4, equals the negative fraction with numerator 20 and denominator 16, which is less than negative 1., the second product is The fraction with numerator negative 20 and denominator 2 to the power 3, equals the negative fraction with numerator 20 and denominator 9, which is less than negative 1., and the third product is, The fraction with numerator negative 10 and denominator 2 to the power 4, equals the negative fraction with numerator 10 and denominator 16, which is greater than negative 1., so you can stop there. The correct answer consists of Choices B (−10) and C two raised to the power negative 4.

Version 2: Try More Than One Value of a Variable

To explore problems containing variables, it is useful to substitute values for the variables. It often helps to substitute more than one value for each variable. How many values to choose and what values are good choices depends on the problem. Also dependent on the problem is whether this approach, by itself, will yield a solution or whether the approach will simply help you generate a hypothesis that requires further exploration using another strategy.

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

This is a Quantitative Comparison question.


  1. Lionel is younger than Maria.
    Quantity A Quantity B
    Twice Lionel's age Maria's age

    (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

    If Lionel's age is 6 years and Maria's age is 10 years, then Quantity A is greater, but if Lionel's age is 4 years and Maria's age is 10 years, then Quantity B is greater. Thus, the relationship cannot be determined. The correct answer is Choice D, the relationship cannot be determined from the information given.

This is a Quantitative Comparison question.


  1. y = 2 times the square of x + 7x, minus 3
    Quantity A Quantity B
    x 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

    If x is equal to zero then y = 2 times, open parenthesis, 0 squared close parenthesis, +,7 times open parenthesis 0, close parenthesis, minus 3, which is equal to negative 3 so in this case, x is greater than y but if x is equal to one then y = 2 times, open parenthesis, 1 squared, close parenthesis, + 7 times, open parenthesis, 1 close parenthesis, minus 3, which is equal to 6 so in that case, y is greater than x Thus, the correct answer is Choice D, the relationship cannot be determined from the information given.

    Note that plugging numbers into expressions may not be conclusive. However, it is conclusive if you get different results after plugging in different numbers: the conclusion is that the relationship cannot be determined from the information given. It is also conclusive if there are only a small number of possible numbers to plug in and all of them yield the same result, say, that Quantity B is greater.

    Now suppose there are an infinite number of possible numbers to plug in. If you plug many of them in and each time the result is, for example, that Quantity A is greater, you still cannot conclude that Quantity A is greater for every possible number that could be plugged in. Further analysis would be necessary and should focus on whether Quantity A is greater for all possible numbers or whether there are numbers for which Quantity A is not greater.

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