In this explainer, we will learn how to subtract rational numbers, including fractions, decimals, and percentages.
There are two ways of describing the difference between integers: . We can think of this as adding the additive inverse: , or we can think of this as the difference between their displacements from 0 on a number line.
Since we move from 0 to and then remove the displacement from 0 to , we can think of this as the displacement from to .
We can use both of these ideas to consider the subtraction of rational numbers. If and are rational numbers, then we define their difference to be adding the additive inverse:
Since the negative of a rational number switches the direction of its displacement on a number line, this is equivalent to finding the difference between the displacements of and .
This time, we can note that is the displacement from to .
This gives us two ways of illustrating the difference of two rational numbers on a number line. Let us look for instance at the subtraction . We will first do this by considering this expression as . We then recall that we can add rational numbers if they have the same denominator. Since the lowest common multiple of 2 and 3 is 6, we can rewrite both rational numbers to have a denominator of 6:
We then note that , and we can add the numerators to get
This difference between and , , can be illustrated as either what is added to to reach or the number that is reached from going by in the negative direction.
We see that we can add to to get to .
We can also note that is the displacement from to .
We want to find the rational number with an equivalent displacement from 0 so that we can evaluate this difference.
To do this, we can note that when sketching on the number line, we split each integer segment into 3 sections with equal widths, and when sketching on the number line, we split each integer segment into 2 sections with equal widths. If we instead split these into 6 sections, we can note that both of these points will lie on the increments and that these increments are of width .
Finally, since the displacement from to is 1 increment of , we can conclude that .
It is worth noting that this method takes more work than using a purely numerical approach. However, this does give a useful justification of why the numerical approach works.
We can simplify this process by noting that
We can then use this process to find the difference between rational numbers in general.
Definition: Difference between Rational Numbers
For any rational numbers and , we have
For any rational numbers and , we have
It is also worth noting that we can also use the lowest common multiple of and as the common denominator when subtracting fractions. For example, we can use the above definition to evaluate as follows:
We then cancel the shared factor of 2 in the numerator and denominator to get
However, we can simplify this process by noting that the lowest common multiple of 2 and 4 is 4. Thus, we rewrite to have a denominator of 4:
Now, we evaluate the subtraction since they have a common denominator:
This is the most general way to determine the difference between rational numbers.
In our first example, we will evaluate the difference between two rational numbers given as fractions.
Example 1: Subtraction of Rational Numbers as Proper Fractions with Like Denominators
Evaluate giving the answer in its simplest form.
Answer
We first note that both rational numbers have the same denominator. We can then recall that if and are rational numbers, then
Hence,
In our next example, we will see how we can evaluate the difference of rational numbers given as mixed fractions.
Example 2: Subtraction of Rational Numbers as Mixed Numbers with Unlike Denominators
Calculate . Give your answer as a mixed number.
Answer
We could start by converting both mixed numbers into fractions; however, this is not necessary. Instead, we first note that we can evaluate the subtraction of the whole parts and fractional parts separately since
We have that , so we need to determine . To do this, we need to rewrite both fractions to have the same denominator. We can do this by noting that 4 is a divisor of 8, so the lowest common multiple of the denominators is 8. Thus,
Substituting in these values gives
We could now evaluate this expression by writing both numbers as fractions. However, we want to give our answer as a mixed number. So instead, we split 3 into as shown:
Now, we can rewrite 1 as and evaluate to get
In our next example, we will find the difference between rational numbers where one is given as a decimal.
Example 3: Finding the Difference between Rational Numbers Given in Different Forms
Evaluate , giving the answer as a fraction in its simplest form.
Answer
We could answer this question in different ways. For example, we could convert both rational numbers into decimals and then evaluate the subtraction. However, since we need to give our answer as a fraction, we will instead convert into fractions.
We first note that
We can then use the fact that 5 goes into 65 thirteen times to rewrite this as a fraction. We have
Thus,
To subtract these fractions, we want to write them with a common denominator. We can do this by noting that 5 is a factor of 20. Therefore, . We can substitute this into the expression and evaluate to get
It is also worth noting that we could answer this question by converting into the decimal value of 0.2, since , and calculating that
In our next example, we will determine the difference between two rational numbers, one of which is given as a percentage.
Example 4: Finding the Difference between Two Rational Numbers by Converting Both into Decimals
Consider that and . By converting both and into decimal form, find the value of in decimal form.
Answer
We are told to evaluate this expression by first converting both numbers into decimals. We first note that . We then recall that we can convert a percentage into a decimal by dividing by 100; we have .
Thus,
In our next example, we will evaluate an expression involving the difference of four rational numbers.
Example 5: Evaluating the Difference of Four Proper Fractions
Evaluate , giving the answer as a fraction in the simplest form.
Answer
We first note that we can evaluate this expression in any order by using the properties of the addition of rational numbers since subtracting a rational number is the same as adding its additive inverse. In particular, this means we can evaluate the sum or difference of multiple fractions by writing them to have the same denominator and then evaluating the numerators.
We can combine all of these fractions by rewriting them to have a common denominator. We see that 3 and 6 are both factors of 12, so we need to find the lowest common multiple of 8 and 12. We can calculate that this is 24. Thus, we can rewrite all fractions to have a denominator of 24 as follows:
We then rewrite this as a single fraction and evaluate as follows:
Finally, we cancel the shared factor of 3 in the numerator and denominator:
In our final example, we will evaluate an expression involving the difference of three rational numbers given in different forms.
Example 6: Evaluating the Difference of Three Rational Numbers Given in Different Forms
- Express as a fraction in its simplest form.
- Using the fraction form of , find . Express your answer as a fraction in its simplest form.
Answer
Part 1
We first recall that we can write a percentage as a fraction by dividing the numerical value by 100; this gives . We can simplify this fraction by noting that 5 is a factor of 5 and 100. Hence,
Part 2
Since we need to evaluate this expression as a fraction, we will first write all of the terms as fractions. We already found that is ; we can also show that . Thus,
Now, since the denominators are all equal, we can combine this difference into a single fraction by evaluating the numerator as follows:
Let’s finish by recapping some of the important points from this explainer.
Key Points
- We can subtract rational numbers on a number line by finding the difference in their displacements from 0.
- The difference between any two rational numbers always gives a rational number.
- We can subtract rational numbers with the same denominator by subtracting their numerators: , where , , and and .
- To subtract rational numbers with different denominators, we can rewrite both fractions to have the same denominator by finding the lowest common multiple of the denominators. In general, for any rational numbers and , we have
- We can find the difference between rational numbers in different forms by converting both rational numbers into the same form.
