General Problem-Solving Steps



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General Problem-Solving Steps

Questions in the Quantitative Reasoning measure ask you to model and solve problems using quantitative, or mathematical, methods. Generally, there are three basic steps in solving a mathematics problem:

Step 1: Understand the problem

Step 2: Carry out a strategy for solving the problem

Step 3: Check your answer

Here is a description of the three steps, followed by a list of useful strategies for solving mathematics problems.

Step 1: Understand the Problem


The first step is to read the statement of the problem carefully to make sure you understand the information given and the problem you are being asked to solve.

Some information may describe certain quantities. Quantitative information may be given in words or mathematical expressions, or a combination of both. Also, in some problems you may need to read and understand quantitative information in data presentations, geometric figures, or coordinate systems. Other information may take the form of formulas, definitions, or conditions that must be satisfied by the quantities. For example, the conditions may be equations or inequalities, or may be words that can be translated into equations or inequalities.

In addition to understanding the information you are given, it is important to understand what you need to accomplish in order to solve the problem. For example, what unknown quantities must be found? In what form must they be expressed?

Step 2: Carry Out a Strategy for Solving the Problem

Solving a mathematics problem requires more than understanding a description of the problem, that is, more than understanding the quantities, the data, the conditions, the unknowns, and all other mathematical facts related to the problem. It requires determining what mathematical facts to use and when and how to use those facts to develop a solution to the problem. It requires a strategy.

Mathematics problems are solved by using a wide variety of strategies. Also, there may be different ways to solve a given problem. Therefore, you should develop a repertoire of problem-solving strategies, as well as a sense of which strategies are likely to work best in solving particular problems. Attempting to solve a problem without a strategy may lead to a lot of work without producing a correct solution.

After you determine a strategy, you must carry it out. If you get stuck, check your work to see if you made an error in your solution. It is important to have a flexible, open mind-set. If you check your solution and cannot find an error or if your solution strategy is simply not working, look for a different strategy.

Step 3: Check Your Answer


When you arrive at an answer, you should check that it is reasonable and computationally correct.

  • Have you answered the question that was asked?
  • Is your answer reasonable in the context of the question? Checking that an answer is reasonable can be as simple as recalling a basic mathematical fact and checking whether your answer is consistent with that fact. For example, the probability of an event must be between 0 and 1, inclusive, and the area of a geometric figure must be positive. In other cases, you can use estimation to check that your answer is reasonable. For example, if your solution involves adding three numbers, each of which is between 100 and 200, estimating the sum tells you that the sum must be between 300 and 600.


  • Did you make a computational mistake in arriving at your answer? A key-entry error using the calculator? You can check for errors in each step in your solution. Or you may be able to check directly that your solution is correct. For example, if you solved the equation 7 times, open parenthesis, 3 x minus 2, close parenthesis, + 4, = 95 for x and got the answer x = 5, you can check your answer by substituting x = 5 into the equation to see that 7 times, open parenthesis, 3 times 5, minus 2, close parenthesis, + 4, = 95.

Strategies


There are no set rules—applicable to all mathematics problems—to determine the best strategy. The ability to determine a strategy that will work grows as you solve more and more problems. What follows are brief descriptions of useful strategies. Along with each strategy one or two sample questions that you can answer with the help of the strategy are given. These strategies do not form a complete list, and, aside from grouping the first four strategies together, they are not presented in any particular order.

The first four strategies are translation strategies, where one representation of a mathematics problem is translated into another.

Strategy 1: Translate from Words to an Arithmetic or Algebraic Representation

Word problems are often solved by translating textual information into an arithmetic or algebraic representation. For example, an “odd integer” can be represented by the expression 2n +1, where n is an integer; and the statement “the cost of a taxi trip is $3.00, plus $1.25 for each mile” can be represented by the expression c = 3 + 1.25m. More generally, translation occurs when you understand a word problem in mathematical terms in order to model the problem mathematically.

Sample Question 1 for Strategy 1: 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.

Sample Question 2 for Strategy 1: Numeric Entry Question.


Working alone at its constant rate, machine A
produces k liters of a chemical in 10 minutes. Working alone at its constant rate, machine B produces k liters of the chemical in 15 minutes. How many minutes does it take machines A and B, working simultaneously at their respective constant rates, to produce k liters of the chemical?

  The answer space for this question is followed by the word minutes.
Explanation

Machine A produces k over 10 liters per minute, and machine B produces k over 15 liters per minute. So when the machines work simultaneously, the rate at which the chemical is produced is the sum of these two rates, which is the fraction k over 10, +, the fraction k over 15, which is equal to k times, open parenthesis, one tenth + one fifteenth, close parenthesis, which is equal to k times, open parenthesis, 25 over 150, close parenthesis, which is equal to k over 6 liters per minute. To compute the time required to produce k liters at this rate, divide the amount k by the rate k over 6 to get the fraction with numerator k and with denominator k sixths = 6. Therefore, the correct answer is 6 minutes (or equivalent).

One way to check that the answer of 6 minutes is reasonable is to observe that if the slower rate of machine B were the same as machine A’s faster rate of k liters in 10 minutes, then the two machines, working simultaneously, would take half the time, or 5 minutes, to produce the k liters. So the answer has to be greater than 5 minutes. Similarly, if the faster rate of machine A were the same as machine B’s slower rate of k liters in 15 minutes, then the two machines would take half the time, or 7.5 minutes, to produce the k liters. So the answer has to be less than 7.5 minutes. Thus, the answer of 6 minutes is reasonable compared to the lower estimate of 5 minutes and the upper estimate of 7.5 minutes.




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