General Problem-Solving Steps

Strategy 11: Divide into Cases

Some problems are quite complex. To solve such problems you may need to divide them into smaller, less complex problems, which are restricted cases of the original problem. When you divide a problem into cases, you should consider whether or not to include all possibilities. For example, if you want to prove that a certain statement is true for all integers, it may be best to show that it is true for all positive integers, then show it is true for all negative integers, and then show it is true for zero. In doing that, you will have shown that the statement is true for all integers, because each integer is either positive, negative, or zero.

Sample Question 1 for Strategy 11: Quantitative Comparison Question.

Quantity A: The least prime number greater than 24

Quantity B: The greatest prime number less than 28

1. 
   Quantity A is greater.
   
2. 
   Quantity B is greater.
   
3. 
   The two quantities are equal.
   
4. 
   The relationship cannot be determined from the information given.
   

Explanation

For the integers greater than 24, note that 25, 26, 27, and 28 are not prime numbers, but 29 is a prime number, as are 31 and many other greater integers. Thus, 29 is the least prime number greater than 24, and Quantity A is 29. For the integers less than 28, note that 27, 26, 25, and 24 are not prime numbers, but 23 is a prime number, as are 19 and several other lesser integers. Thus, 23 is the greatest prime number less than 28, and Quantity B is 23. Thus, the correct answer is Choice A, Quantity A is greater.

Sample Question 2 for Strategy 11: Multiple-Choice – Select One or More Answer Choices Question.

Which of the following integers are multiples of both 2 and 3 ?

Indicate all such integers.

1. 
   8
   
2. 
   9
   
3. 
   12
   
4. 
   18
   
5. 
   21
   
6. 
   36
   

Explanation

You can first identify the multiples of 2, which are 8, 12, 18, and 36, and then among the multiples of 2 identify the multiples of 3, which are 12, 18, and 36. Alternatively, if you realize that every number that is a multiple of 2 and 3 is also a multiple of 6, you can identify the choices that are multiples of 6. The correct answer consists of Choices C (12), D (18), and F (36).

Strategy 12: Adapt Solutions to Related Problems

When solving a new problem that seems similar to a problem that you know how to solve, you can try to solve the new problem by adapting the solution—both the strategies and the results—of the problem you know how to solve.

If the differences between the new problem and the problem you know how to solve are only surface features—for example, different numbers, different labels, or different categories—that is, features that are not fundamental to the structure of the problem, then solve the new problem using the same strategy as you used before.

If the differences between the new problem and the problem you know how to solve are more than just surface features, try to modify the solution to the problem you know how to solve to fit the conditions given in the new problem.

Sample Question 1 for Strategy 12: Multiple-Choice – Select One or More Answer Choices Question.

Each employee of a certain company is in either Department X or Department Y, and there are more than twice as many employees in Department X as in Department Y. The average (arithmetic mean) salary is $25,000 for the employees in Department X and $35,000 for the employees in Department Y. Which of the following amounts could be the average salary for all of the employees of the company?

Indicate all such amounts.

1. 
   $26,000
   
2. 
   $28,000
   
3. 
   $29,000
   
4. 
   $30,000
   
5. 
   $31,000
   
6. 
   $32,000
   
7. 
   $34,000
   

Explanation

One strategy for answering this kind of question is to find the least and/or greatest possible value. Clearly the average salary is between $25,000 and $35,000, and all of the answer choices are in this interval. Since you are told that there are more employees with the lower average salary, the average salary of all employees must be less than the average of $25,000 and $35,000, which is $30,000. If there were exactly twice as many employees in Department X as in Department Y, then the average salary for all employees would be, to the nearest dollar, the following weighted mean,

the fraction with numerator 2 times 25,000, plus, 1 times 35,000, and denominator 2 plus 1, which is approximately 28,333 dollars

where the weight for $25,000 is 2 and the weight for $35,000 is 1. Since there are more than twice as many employees in Department X as in Department Y, the actual average salary must be even closer to $25,000 because the weight for $25,000 is greater than 2. This means that $28,333 is the greatest possible average. Among the choices given, the possible values of the average are therefore $26,000 and $28,000. Thus, the correct answer consists of Choices A ($26,000) and B ($28,000).

Intuitively, you might expect that any amount between $25,000 and $28,333 is a possible value of the average salary. To see that $26,000 is possible, in the weighted mean above, use the respective weights 9 and 1 instead of 2 and 1. To see that $28,000 is possible, use the respective weights 7 and 3.

Note: This question also appears as a sample question for Strategy 8.

Sample Question 2 for Strategy 12: Multiple-Choice – Select One or More Answer Choices Question.

Which of the following could be the units digit of 57 to the power n, where n is a positive integer?

Indicate all such digits.

1. 
   0
   
2. 
   1
   
3. 
   2
   
4. 
   3
   
5. 
   4
   
6. 
   5
   
7. 
   6
   
8. 
   7
   
9. 
   8
   
10. 
    9
    

Explanation

The units digit of 57 to the power n is the same as the units digit of 7 to the power n for all positive integers n. To see why this is true for n = 2, compute 57 to the power 2 by hand and observe how its units digit results from the units digit of 7 to the power 2. Because this is true for every positive integer n, you need to consider only powers of 7. Beginning with n = 1 and proceeding consecutively, the units digits of 7, 7 to the power 2, 7 to the power 3, 7 to the power 4, and 7 to the power 5 are 7, 9, 3, 1, and 7, respectively. In this sequence, the first digit, 7, appears again, and the pattern of four digits, 7, 9, 3, 1, repeats without end. Hence, these four digits are the only possible units digits of 7 to the power n and therefore of 57 to the power n. The correct answer consists of the four choices B, D, H, and J, which are 1, 3, 7, and 9, respectively.

Note: This question also appears as a sample question for Strategy 7.

Strategy 13: Determine Whether a Conclusion Follows from the Information Given

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.

Sample Question 1 for Strategy 13: Quantitative Comparison Question.

_{It is given that . w is greater than 1.}

Quantity A: 7w minus 4

Quantity B: 2w + 5

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 of Quantity A and Quantity B:

7w minus 4, followed by a made up question mark symbol, followed by 2w + 5.

Then simplify:

Step 1: Subtract 2w from both sides and add 4 to both sides to get 5w, followed by the question mark symbol, followed by the number 9.

Step 2: Divide both sides by 5 to get w, followed by the question mark symbol, followed by the fraction 9 over 5.

The comparison cannot be simplified any further. Although you are given that  w is greater than 1, you still don’t know how w compares to  the fraction 9 over 5, or 1.8. For example, if w = 1.5, then w is less than 1.8, but if w = 2, then w is greater than 1.8. In other words, the relationship between w and the fraction 9 over 5 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.

Sample Question 2 for Strategy 13: Multiple-Choice – Select One or More Answer Choices Question.

This question is based on the following 3-column table. The table shows the annual percent change in the dollar amount of sales at five retail stores from 2006 to 2008. The headers for the columns are: Column 1, Store; Column 2, Percent Change from 2006 to 2007; Column 3, Percent Change from 2007 to 2008. There are five rows of data in the table.

Annual Percent Change in Dollar Amount of Sales
at Five Retail Stores from 2006 to 2008

Store	Percent Change from 2006 to 2007	Percent Change from 2007 to 2008
P	10	negative 10
Q	negative 20	9
R	5	12
S	negative 7	negative 15
T	17	negative 8

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.

Explanation

For Choice A, since the only data given in the table accompanying the question 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 0.93B, and the dollar amount for 2008 is given by 0.85, times, 0.93, times, B which is 0.7905B. Note that this represents a percent decrease of 100, minus, 79.05 = 20.95 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 1.05C and the dollar amount for 2008 is given by 1.12, times, 1.05, times, C which is 1.176C. 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 also appears as a sample question for Strategy 14.

Strategy 14: Determine What Additional Information Is Sufficient to Solve a Problem

Some problems cannot be solved directly from the information given, and you need to determine what other information will help you answer the question. In that case, it is useful to list all the information given in the problem, along with the information that would be contained in a complete solution, and then evaluate what is missing. Sometimes the missing information can be derived from the information given, and sometimes it cannot.

Sample Question for Strategy 14: Multiple-Choice – Select One or More Answer Choices Question.

This question is based on the following 3-column table. The table shows the annual percent change in the dollar amount of sales at five retail stores from 2006 to 2008. The headers for the columns are: Column 1, Store; Column 2, Percent Change from 2006 to 2007; Column 3, Percent Change from 2007 to 2008. There are five rows of data in the table.

Annual Percent Change in Dollar Amount of Sales
at Five Retail Stores from 2006 to 2008

Store	Percent Change from 2006 to 2007	Percent Change from 2007 to 2008
P	10	negative 10
Q	negative 20	9
R	5	12
S	negative 7	negative 15
T	17	negative 8

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.

Explanation

For Choice A, since the only data given in the table accompanying the question 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 0.93B, and the dollar amount for 2008 is given by 0.85, times, 0.93, times, B which is 0.7905B. Note that this represents a percent decrease of 100, minus, 79.05 = 20.95 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 1.05C and the dollar amount for 2008 is given by 1.12, times, 1.05, times, C which is 1.176C. 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 also appears as a sample question for Strategy 13.

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