# Using Reasonableness to Solve Math Problems

Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

When solving math problems, checking your possible solution for reasonableness will help you determine whether your possible solution is a real one or if you have done something wrong. Learn how in this lesson.

## Reasonableness

When you have a math problem that you are working on, and you have worked out the problem and think you have found the solution, a good way to help you make sure is to use reasonableness. In math terms, reasonableness means to verify the answer you have found by either estimating or plugging in your answer to check to see if it works.

It might feel like you are doing double work, but if your life depended on you getting an A on a math test, then making sure that your answers are correct by using reasonableness will be a lifesaver. And you never know, getting an A on that math test can mean the difference between getting that job you always wanted or getting a job you really hate.

Let's say you are working on the problem 235 * 4 - 10. You have worked out the problem, and you have an answer of 93. Is this the correct answer? Well, one way to use reasonableness is to estimate the answer to see if your answer is an outrageous answer or not. So, looking at your problem, you see that you are multiplying a two hundred something number by 4 and then subtracting 10 from it. Estimating, you can say that your answer should be somewhat more than 800. Looking at your answer of 93, you see that your answer is nowhere near 800! Your answer then is an outrageous answer and therefore not a correct answer. You will need to rework your problem.

You can use estimation to check for reasonableness when you have math problems with all numbers. When you do use estimation, it won't tell you whether you actually have THE correct answer, but it will tell you if you are close and your answer is probably right.

Yet another way to use reasonableness to verify your answer is to plug in your answer to see if it is correct. This method to check for reasonableness is best used for algebraic problems or any other problem where you can plug in your answer to check if the resulting mathematical equation is correct. This method will tell you that you have found THE correct answer!

For example, say you are working on this problem:

Find x when y = 3.

21x -7y = 21

You go through the process of solving for x, and you get x = 2. Is this a correct answer? You can use reasonableness to check by plugging in the value for x that you found. Let's see what happens.

Plugging in 2 for x, you get this:

21 (2) - 7 (3) = 21

42 - 21 = 21

21 = 21

Ah ha! That's correct. That means that x = 2 is the correct answer! You answered correctly!

## Example

Let's look at another example.

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