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.
Sample Question for Strategy 10, Version 1: MultipleChoice – Select One or More Answer Choice Question.
Which two of the following numbers have a product that is between negative 1 and 0?
Indicate both of the numbers.

_{ }negative 20

negative 10

2 to the negative 4 power

3 to the negative 2 power
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 the product negative 20, times 2 to the negative 4 power, the product negative 20, times three to the negative 2 power, the product negative 10, times 2 to the negative 4 power, and, the product negative 10, times 3 to the negative 2 power. The product must also be greater than –1. negative 1. The first product is the fraction negative 20, over 2 to the fourth power, which is equal to the negative of the fraction 20 over 16, which is less than negative 1. the second product is the fraction negative 20, over 3 to the second power, which is equal to the negative of the fraction 20 over 9, which is less than negative 1. and the third product is, the fraction negative 10, over 2 to the fourth power, which is equal to the negative of the fraction 10 over 16, which is greater than negative 1, so you can stop there. The correct answer consists of Choices B (–10) negative 10 and C (2^{–4}). 2 to the negative 4 power.
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.
Sample Question 1 for Strategy 10, Version 2: Quantitative Comparison Question.
Lionel is younger than Maria.
Quantity A: Twice Lionel’s age
Quantity B: Maria’s age

Quantity A is greater.

Quantity B is greater.

The two quantities are equal.

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.
Sample Question 2 for Strategy 10, Version 2: Quantitative Comparison Question.
y = 2 times the square of x, plus 7x, minus 3
Quantity A: x
Quantity B: y

Quantity A is greater.

Quantity B is greater.

The two quantities are equal.

The relationship cannot be determined from the information given.
Explanation
If x = 0, 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 = 1, 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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