Strategies to Simplify Word Problems Study Guide (page 2)
Introductino to Strategies to Simplify Word Problems
Leaders are problem solvers by talent and temperament, and by choice.
—HARLAN CLEVELAND (1918–2008)
You have tackled some challenging problems so far, each one made easier by using various strategies. In this lesson, the strategies of solving a simpler problem and guess and check will be two more to add to your list. Each strategy will be explained and examples given to help guide you. Remember, there are many ways to solve most problems. Having a variety of strategies to choose from can only help you on this road to word-problem solving success!
Solving a Simpler Problem
With some word problems, the numbers used within the problem may be difficult to use. They may be too large for one of the previously mentioned strategies. Or, the numbers given in the question may be complicated, and should be simplified to make the question easier to handle. One way to manage this situation is to make the problem easier to solve by using simpler values than the ones given. Take, for example, the question that follows.
In a certain school district, there are 294 students who are bused to school each day. If each bus can fit 52 students, how many buses are needed?
Read and understand the question. This question is looking for the number of buses needed for a certain number of students.
Make a plan. Divide the total number of students by the number of students that can fit on a bus. Because the numbers are not easily divided, make the numbers simpler by rounding. Then, a reasonable solution can be found much more easily.
Carry out the plan. Instead of using 294 students, use a value of 300. Since each bus can fit 52 students, round this value to 50. Divide 300 by 50 to get 6 buses.
Check your answer. Check the solution by multiplying 6 buses by 50, which is 300. Since there are only 294 students who need to ride the bus and each bus can actually fit 52 students, there will be extra seats left. The solution is checking.
Be careful when using rounding to make simpler, more compatible numbers. As in the question, the number of students was rounded up slightly and the number of students who would fit on each bus was rounded down. This way, the number of buses needed was not underestimated. Always check a solution to make sure it is reasonable.
Here is another example of making a problem simpler. In the question that follows, you are asked to find the sum of many numbers. Notice how finding the sum of a few numbers can lead to an easier way to find a solution.
What is the sum of the first 19 natural numbers?
Read and understand the question. This question is asking for the total when the first 19 whole numbers beginning with 1 are added together.
Make a plan. This problem seems difficult when you are trying to add all 19 values. Begin by making the problem simpler by adding just a few numbers, and then look for a pattern.
Carry out the plan. You need to add the natural numbers from 1 to 19. This would appear as 1 + 2 + 3 + 4 + … + 18 + 19. However, instead of adding the numbers in order, begin by adding the first and last numbers: 1 + 19 = 20. Continue this pattern using the next numbers: 2 + 18 = 20, 3 + 17 = 20, 4 + 16 = 20, and so on until you reach 9 + 11 = 20. The value 10 in the list will not have a paired value. Therefore, there are nine sums of 20, plus the number 10: 9 × 20 + 10 = 180 + 10 = 190. The sum is 190.
Check your work. One way to check your work is to add the numbers in the list by hand or with a calculator. Each of these methods results in a sum of 190. The solution is checking.
Guess and Check
Guess and check, also known as trial and error or guess and test, is another important strategy. This is the method that many people choose when they cannot come up with any other approach to solve a problem. Although guess and check can be used on just about any question, the problems in this lesson lend themselves to this strategy.
The method of guess and check is exactly what it says: You guess an answer and then check to see if it works. It is important to note, however, if you get a correct answer on the first guess that you should show at least three trials each time this strategy is used. You should always show at least three guesses, two incorrect and, of course, the correct answer.
Read through the following sample to see an example of how to use the strategy of guess and check.
Gail has 13 coins for a total of $1.00. If she only has dimes and nickels, how many of each coin does she have?
Read and understand the question. This question is looking for the number of dimes and nickels that Gail has. She has a total of 13 coins that add to exactly $1.00.
Make a plan. Use the strategy of guess and check to solve this problem. Begin with a guess as to the number of nickels and number of dimes, and be sure that the number of coins adds to 13. Then, check to see if the total amount of money adds to $1.00.
Carry out the plan. Start your guesses with 6 dimes and 7 nickels. Six dimes is equal to $0.60 and 7 nickels is equal to $0.35. This is a total of $0.60 + $0.35 = $0.95. This guess is too low.
For the next guess, try a greater amount of dimes. Try 8 dimes and 5 nickels. This is a total of $0.80 + $0.25 = $1.05. This guess is too high.
For the next guess, try 7 dimes and 6 nickels. Seven dimes is equal to $0.70 and 6 nickels is equal to $0.30. This is a total of $0.70 + $0.30 = $1.00. This is the correct answer.
Check your work. Be sure that the solution meets all of the facts in the question. Gail had only nickels and dimes for a total of 13 coins. Seven dimes ($0.70) and six nickels ($0.30) is a total of 13 coins, which add to $1.00. This answer is checking.
It is important to show at least three trials when you are using the strategy of guess and check. Show one guess greater than the right answer, one guess less than the right answer, and the correct answer. Always show all work to prove the solution works, and remember, it may take more than three trials to find the correct answer.
Find practice problems and solutions for these concepts at Strategies to Simplify Word Problems Practice Questions.
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