General Problem-Solving Steps

Strategy 8: Search for a Mathematical Relationship

More general than patterns, mathematical relationships exist throughout mathematics. Problems may involve quantities that are related algebraically, sets that are related logically, or figures that are related geometrically. Also, there may be relationships between information given textually, algebraically, graphically, etc. To express relationships between quantities, it is often helpful to introduce one or more variables to represent the quantities. Once a relationship is understood and expressed, it is often the key to solving a problem.

Sample Question 1 for Strategy 8: Quantitative Comparison Question.

Quantity A: x squared, + 1

Quantity B: 2x, minus 1

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

Set up the initial comparison of Quantity A and Quantity B:

x squared + 1, followed by a made up question mark symbol, followed by 2x, minus 1.

Then simplify by noting that the quadratic polynomial x squared, minus 2x, + 1 can be factored:

Step 1: Subtract 2x from both sides to get x squared, minus 2x, + 1, followed by the question mark symbol, followed by negative 1.

Step 2: Factor the left hand side to get open parenthesis, x minus 1, close parenthesis, squared, followed by the question mark symbol, followed by negative 1.

The left hand side of the comparison is the square of a number. Since the square of a number is always greater than or equal to 0, and 0 is greater than negative 1, the simplified comparison is the inequality open parenthesis, x minus 1, close parenthesis, squared, is greater than negative 1 and the resulting relationship is greater than (>). In reverse order, each simplification step implies the inequality greater than (>) in the preceding comparison. Therefore, Quantity A is greater than Quantity B. The correct answer is choice A, Quantity A is greater.

Sample Question 2 for Strategy 8: 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, +, 1 times 35,000, and denominator 2 + 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 12.

Strategy 9: Estimate

Sometimes it is not necessary to perform extensive calculations to solve a problem—it is sufficient to estimate the answer. The degree of accuracy needed depends on the particular question being asked. Care should be taken to determine how far off your estimate could possibly be from the actual answer to the question. Estimation can also be used to check whether the answer to a question is reasonable.

Sample Question 1 for Strategy 9: Quantitative Comparison Question.

Quantity A: 54% of 360

Quantity B: 150

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

This question asks you to compare Quantity A: 54% of 360, and Quantity B: 150.

Without doing the exact computation, you can see that 54 percent of 360 is greater than one half of 360, which is 180, and 180 is greater than Quantity B, 150. Thus, the correct answer is Choice A, Quantity A is greater.

Sample Question 2 for Strategy 9: Multiple-Choice – Select One Answer Choice Question.

A car got 33 miles per gallon using gasoline that cost $2.95 per gallon. Approximately what was the cost, in dollars, of the gasoline used in driving the car 350 miles?

1. 
   $10
   
2. 
   $20
   
3. 
   $30
   
4. 
   $40
   
5. 
   $50
   

Explanation

Scanning the answer choices indicates that you can do at least some estimation and still answer confidently. The car used 350 over 33, gallons of gasoline, so the cost was  open parenthesis, 350 over 33, close parenthesis, times 2.95 dollars. You can estimate the product  open parenthesis, 350 over 33, close parenthesis, times 2.95 by estimating 350 over 33 a little low, 10, and estimating 2.95 a little high, 3, to get approximately 10 times 3 = 30 dollars.

You can also use the calculator to compute a more exact answer and then round the answer to the nearest 10 dollars, as suggested by the answer choices. The calculator yields the decimal 31.287…, which rounds to 30 dollars. Thus, the correct answer is Choice C, $30.