In some problems, you are given information and a statement describing a possible conclusion, which may or may not follow from the information. You need to determine whether or not the conclusion is a logical consequence of the information given.
If you think that the conclusion follows from the information, try to show it. Using the information and any relevant mathematical relationships, try to reason step-by-step from the information to the conclusion. Another way to show that the conclusion follows from the information, is to show that in all cases in which the information is true, the conclusion is also true.
If you think that the conclusion does not follow from the information, try to show that instead. One way to show that a conclusion does not follow from the information is to produce a counterexample. A counterexample is a case where the given information is true but the conclusion is false. If you are unsuccessful in producing a counterexample, it does not necessarily mean that the conclusion does not follow from the information — it may mean that although a counterexample exists, you were not successful in finding it.
• This strategy is used in the following two sample questions.
This is a Quantitative Comparison question.
Quantity A Quantity B
(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.
Set up the initial comparison:
Step 1: Subtract 2w from both sides and add 4 to both sides to get
Step 2: Divide both sides by 5 to get
The comparison cannot be simplified any further. Although you are given that you still don't know how w compares to or 1.8. For example, if then but if then In other words, the relationship between w and cannot be determined. Note that each of these simplification steps is reversible, so in reverse order, each simplification step implies that the relationship cannot be determined in the preceding comparison. Thus, the relationship between Quantities A and B cannot be determined. The correct answer is Choice D, the relationship cannot be determined from the information given.
This is a Multiple Choice — Select One or More Answer Choices question.
Annual Percent Change in Dollar Amount of Sales at Five Retail Stores from 2006 to 2008
from 2006 to 2007
from 2007 to 2008
Based on the information given, which of the following statements must be true?
Indicate all such statements.(A) For 2008 the dollar amount of sales at Store R was greater than that at each of the other four stores.
(B) The dollar amount of sales at Store S for 2008 was 22 percent less than that for 2006.
(C) The dollar amount of sales at Store R for 2008 was more than 17 percent greater than that for 2006.
For Choice A, since the only data given in the figure above are percent changes from year to year, there is no way to compare the actual dollar amount of sales at the stores for 2008 or for any other year. Even though Store R had the greatest percent increase from 2006 to 2008, its actual dollar amount of sales for 2008 may have been much smaller than that for any of the other four stores, and therefore Choice A is not necessarily true.
For Choice B, even though the sum of the two percent decreases would suggest a 22 percent decrease, the bases of the percents are different. If B is the dollar amount of sales at Store S for 2006, then the dollar amount for 2007 is 93 percent of B, or and the dollar amount for 2008 is given by which is Note that this represents a percent decrease of percent, which is not equal to 22 percent, and so Choice B is not true.
For Choice C, if C is the dollar amount of sales at Store R for 2006, then the dollar amount for 2007 is given by and the dollar amount for 2008 is given by which is Note that this represents a 17.6 percent increase, which is greater than 17 percent, so Choice C must be true.
Therefore, the correct answer consists of only Choice C (The dollar amount of sales at Store R for 2008 was more than 17 percent greater than that for 2006).
Note: This question and explanation also appear as an example of Strategy 14.