OnTrack logo

Is Your Solution Possible?

When you need to solve a word problem, you should:

  1. Read the problem.
  2. Write down all the information that the problem gives you.
  3. Write down what the question is asking you to do.
  4. Use clue words in the problem to make a plan to solve the problem.
  5. Write your plan as an equation.

image of question markIn this section we will cover one of the most important steps in problem solving. Whenever you get an answer to a problem, you should think about what the answer means and whether or not it is reasonable. Ask yourself:

Based on the problem I just solved, does my answer make sense?

 

image of a man and boy playing video games
There are many factors to consider when determining whether or not an answer is reasonable. To examine the first of these factors, read the following problem and attempted solution to decide if the answer is reasonable.

 

Johnny is 24 years younger than his dad. If Johnny's dad is currently 30 years old, how old is Johnny?

Attempted Solution:

Given: Johnny's dad is 30

Question: How old is Johnny?

Plan: I should subtract 24 from Johnny's dad's age to find Johnny's age

Equation: J = 30 – 24

Calculator:

calculator showing 3 minus 24 on one line and negative 21 on the next line

Modified from: SharpEl-5120, EpiVictor, Wikimedia Commons

Answer: J = -21

  1. Is this answer reasonable?
  2. Why or why not?
  3. Can you find where the mistake happened?

The first factor to consider is whether or not the answer to the problem is physically possible.

 

 

 

 

Is Your Solution Logical?

The first question asked, "Is this possible?" Now we will ask, "Does this make sense?" There are times when even though the answer is possible, it just doesn't make any logical sense. Let's look at another attempted solution for the example from section 1.

Johnny is 24 years younger than his dad. If Johnny's dad is currently 30 years old, how old is Johnny?

Solution:

Given: Johnny's dad is 30.

Question: How old is Johnny?

Plan: I should add 24 to Johnny's dad's age to find Johnny's age.

Equation: J = 30 + 24

Calculator:

Image of calculator showing 30 plus 24 on one line and 54 as the result
Modified from: SharpEl-5120, EpiVictor, Wikimedia Commons

Answer: J = 54

  1. Is this a possible age for someone?
  2. Is this answer reasonable?
  3. Why or why not?
  4. Where is the mistake in the solution?

Checking Your Solution by Estimation

We know the answer is possible and logical. Now, we need to know if it is the answer we were expecting.

Sometimes you can make approximations and estimates to check if your answer is close to what you expected. In the next problem, Beth will make some approximations with some messy numbers so that she can do the math in her head.

Read the problem and answer the questions that follow in your notes.

image of a tv screen on the wallBeth is at a store shopping for a television. The price of the television that she wants to buy is $897.43. The mounting bracket for the television costs $101.99, and the charge for installation is $199.74. Beth does some quick mental math and decides to spread the payments out over a year–making 12 equal payments. She estimates that her monthly payment will be $100.00. Is this a reasonable estimate of the amount for each payment?

  1. What information is given in the problem?
  2. What question does the problem ask?
  3. What is your plan for solving this problem?
  4. Was Beth's answer reasonable?
  5. How did Beth do that in her head?

Rounding can be an effective way to check if your answer to a problem with messy numbers is reasonable. In the previous problem, rounding to the nearest hundred only made a difference of less than $3 in the price of each item—a very reasonable estimate for items that each cost more than $100.00. Be careful not to round too much. You want to round to the nearest number that you can so that you can accurately check your work.

Read the following situation and answer the questions that follow in your notes.

image of payment recordMr. Jones is working out a budget for household expenses. He looks at the monthly statement from his car insurance, and he sees that he pays $119.00 every month. Based on this information, he budgets $1200.00 for car insurance for the year.

  1. What was the rounded estimate that Mr. Jones used for his monthly car insurance bill?
  2. How much does Mr. Jones actually spend on car insurance in a year?
  3. Was Mr. Jones's rounding a reasonable estimate?
  4. What would a better estimate have been?

Practice

Solving Problems With a Range of Answers

Some word problems are written so that the solution is not a single number but a range of numbers. You should check to see that your answer is within the range.

Read the following problem and answer the questions below.

Woman with small child and a bucket of toysSusan babysits for several families. The different families pay her between $8 - $10 per hour. Last month she babysat for a total of 30 hours. Which of the following is a reasonable amount that she could have been paid for the month?

    A. $175.00
    B. $264.00
    C. $325.00
    D. $200.00

 

  1. What information is given in the problem?
  2. What question is the problem asking?
  3. What is your plan for solving this problem?
  4. Which pay rate would give Susan the least money?
  5. How much would she make at that rate?
  6. Which pay rate would give Susan the most money?
  7. How much would she make at that rate?
  8. What is the range of possible answers to the question?
  9. Which choice is the correct answer to the problem?