# Interpreting Differences of Rational Numbers

Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

Rational numbers are numbers that can be written as a fraction. We will look at how to find the difference between rational numbers, and how to interpret these differences both abstractly and literally based on a given scenario.

## Difference of Rational Numbers

Suppose that Susan goes out for a walk and walks 2/3 of a mile. At the same time, Susan's friend Meghan goes out for a walk and walks 5/4 of a mile. Both Susan's and Meghan's distances are examples of rational numbers. A rational number is a number that can be written in the form a/b, where a and b are integers. Basically, a rational number is a number that can be written as a fraction, like 2/3 or 5/4.

Later in the day, Susan and Meghan meet up and are talking about how far each of them walked today. They want to know how much farther Meghan walked than Susan. In other words, they want to know the difference between 2/3 and 5/4.

In mathematics, when we want to find the difference between two numbers, we simply need to subtract the smaller number from the larger number. When we are dealing with the difference between two rational numbers, as we are in this example, we use the following steps to subtract the smaller number from the larger number:

1. Get a common denominator by multiplying the two denominators together. Then multiply the numerators of each number by the denominator of the other number: a/b - c/d = ad/bd - bc/bd.
2. Subtract the numerators, and keep the common denominator: a/b - c/d = ad/bd - bc/bd = (ad - bc) / bd.
3. Simplify the resulting fraction.

As it turns out, subtracting rational numbers, a/b - c/d is the same as adding -c/d to a/b (this is called adding the additive inverse). That is,

• a/b - c/d = a/b + (-c/d)

If we simplify this, we will end up with the same formula as our steps gave us, so let's give this formula a go with our example.

We see that 5/4 - 2/3 = 7/12. Therefore, Meghan walked 7/12 of a mile farther than Susan. That's not too hard!

## Interpreting the Difference Between Rational Numbers

As we said, in general, when we are trying to find the difference between two numbers, we are talking about subtracting the smaller number from the larger number. When we do this, we are finding out 'how much' is in-between the two numbers. This is equal to the distance between the two numbers on a number line, or the absolute value of their difference. This interpretation of the difference between numbers is the same for the interpretation of the difference between rational numbers.

As we just saw in Susan and Meghan's example, when we are finding the difference between two rational numbers, we are finding 'how much' is in-between the two numbers. Of course, in this particular case the literal interpretation of the difference between the two numbers is 'how much' farther Meghan walked than Susan, or the number of miles between the two distances. The abstract interpretation is the distance between the two numbers on a number line or 'how much' is in-between the two numbers.

We have that the abstract interpretation of the difference between rational numbers can be made into a literal interpretation when we are dealing with a real-world application, and those literal interpretations all depend on the context of the problem. Let's look at a couple of examples.

## Examples

Imagine that you are making some oatmeal raisin cookies. The recipe calls for 3/4 teaspoons cinnamon and 1/4 teaspoons nutmeg. If we subtract 1/4 from 3/4, we get 1/2.

We know that the abstract interpretation of the difference between 3/4 and 1/4 is that there is 1/2 between the two numbers, or the distance between 1/4 and 3/4 on a number line is 1/2. In the context of this problem, the literal interpretation is that the difference between 3/4 tsp. and 1/4 tsp. is 1/2 tsp., and it is 'how much' more cinnamon there is than nutmeg in the cookies.

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