In this explainer, we will learn how to add rational numbers, including fractions, decimals, and percentages.
We first recall that a rational number is one that is the quotient of integers, so all rational numbers can be written in the form , where , and . There are several different ways to visualize rational numbers. For example, since we are dividing by an integer , we are dividing the value of into equal parts. We can represent this in many different ways. One way is to consider the position of on a number line.
To represent on this number line, we want to split the line segment between 0 and into equal sections. For example, if is 6, we split this into 6 sections of equal length and the first increment will have a value of as shown.
This is equivalent to dividing 1 into sections and then taking of these sections in the direction dictated by the sign of the rational number.
We can use a number line to add rational numbers. To do this, we first recall that to add two numbers on a number line, we add their displacements from 0.
Letβs see an example of adding and together by using a number line. First, we need to split the line segment between 0 and 1 into 7 equal sections. Each increment will represent so that we can represent and , as shown.
The second increment will be and the third will be . To add these numbers together, we need to add their displacements from 0 together. We note that is 2 increments of away from 0 in the positive direction and is 3 increments of away from 0 in the positive direction. Hence, when we add these together, we will get increments of away from 0 in the positive direction.
Therefore, is 5 increments of from 0 in the positive direction. We have shown that .
We can note that we are just adding the numerators of the two rational numbers, and this result is true in general, provided the denominators are equal; we can show this using the exact same argument. To evaluate for integers , , and with , we know that is increments of from 0, is increments of from 0, and their sum will be increments of from 0. Hence,
This result is also true when subtracting rational numbers. This gives us the following result:
Definition: Adding Rational Numbers with the Same Denominator
If , , and are integers with , then
Letβs now see some examples of applying this result to find the sum of two rational numbers with the same denominator.
Example 1: Adding Rational Numbers with Equal Denominators
Find the value of . Give your answer in its simplest form.
Answer
We first note that these rational numbers have the same denominator. We then recall that , so we just need to add the numerators. This gives
We might stop here; however, we are told to give our answer in its simplest form. To do this, we need to check if there are any common divisors in the numerator and denominator. We see that and , so they both share a factor of 2. Canceling this shared factor, we get
We can then note that 2 and 5 share no common factors, so this is the simplest form of this fraction.
Example 2: Adding Two Mixed Numbers with Different Signs Giving the Answer as a Mixed Number
Calculate . Give your answer as a mixed number.
Answer
Letβs start by rewriting both mixed numbers as fractions. We have
We first note that these rational numbers have the same denominator. We then recall that , so we just need to add the numerators. This gives
We can then convert this into a mixed number by noting that . Hence,
It is worth noting that there is no reason to assume that the denominators of the fractions we want to add or subtract will be equal. However, we can still evaluate these expressions by using a number line. For example, consider . We first represent each number on a number line by splitting the number line into increments of and as shown.
To add these together, we add their displacements. So, we want the number that is 1 increment of and 2 increments of from 0 in the positive direction. However, there is a small problem: if we try to find this using our number line, we see that this point is not marked.
We can find the value of this point by using our knowledge of adding fractions with equal denominators and our knowledge of equivalent fractions. Since we can add fractions with equal denominators, we could try and rewrite and to have the same denominator. To do this, we note that both 2 and 3 are factors of 6. In fact, 6 is the lowest common multiple of 2 and 3. We can then rewrite
We then have that
On our number line, this is the same as saying that when we partition each integer segment into 2 sections and 3 sections, we could instead partition into 6 sections and both rational numbers would be represented as shown.
We follow this same process for the sum of any rational numbers. We need to find the lowest common multiple of their denominators so that we can rewrite each fraction to have the same denominator, then we can add the fractions by adding their numerators. Finally, we can check if we can simplify the result by looking for shared factors in the numerator and denominator.
Letβs now see an example of completing an equation involving the sum of rational numbers using a number line.
Example 3: Completing Addition Equations of Fractions with Unlike Denominators Using Number Lines
Use the drawn number line to complete .
Answer
We first note that each increment on the number line represents , since they are equally spaced and the first increment after 0 is labeled as . We can then recall that we can add numbers on a number line by adding their displacements from 0. Therefore, when we add and the missing value together, we will end at on the number line.
In other words, the equation is asking us how far we need to move from to get to . We can answer this by using the given number line.
We see that we need to move 7 increments to the right, which is the positive direction.
Since each increment is , is the missing value.
Thus far, we have focused on adding rational numbers given as fractions. However, this is not the only way of representing rational numbers, nor is it the only way we can add these numbers. For example, we recall that we can represent rational numbers as decimals; this means we can add rational numbers together by first converting them into decimals and then adding the decimal values together.
Letβs see an example of applying this method to find the sum of two given rational numbers in different forms.
Example 4: Adding Rational Numbers Given in Different Forms
Consider that and . By converting and to decimal form, find the value of approximated to the nearest two decimals.
Answer
We are asked to find this sum by first converting both and into decimals. First, we can note that . Second, we recall that to write a percentage as a decimal, we divide the value by 100, which is the same as moving the decimal point two spaces to the left. We have
Hence,
In our next example, we will find the sum of multiple rational numbers of different signs.
Example 5: Evaluating Numerical Expressions Involving the Addition of Rational Numbers
Evaluate giving the answer in its simplest form.
Answer
There are two ways to evaluate this expression. The easiest way is to note that can be simplified since the numerator and denominator share a factor of 2. Hence,
Substituting this into the expression gives
We can then add the fractions with equal denominators by adding the numerators. We have
We know that and , so
However, it is not always possible to combine fractions in this way; we may need to write them with a common denominator. To do this, we first need to find the lowest common multiple of all the denominators. We will do this by factoring each denominator into primes. We have and , and 3 is prime. Taking the product of all the different prime factors (or their highest powers) of the three numbers gives their lowest common multiple
Therefore, we want to rewrite all of these fractions with a denominator of 12. We do this by multiplying both the numerator and denominator by the same value to get
In our next example, we find the sum of a fraction and decimal value giving our answer as a fraction in its simplest form.
Example 6: Adding Fractions to Decimals and Simplifying the Answer
Evaluate giving the answer as a fraction in its simplest form.
Answer
We can evaluate the sum of two rational numbers by first converting them into fractions with equal denominators. To do this, we note that . To write these fractions with the same denominator, we first need to find the lowest common multiple of the two denominators. Since 5 is a factor of 10, we must have that 10 is their lowest common multiple.
We can then rewrite to have a denominator of 10 by multiplying both the numerator and denominator by 2 to get
We can then substitute these into the expression and evaluate to get
Since 13 and 10 have no common factors greater than 1, we cannot simplify this any further.
In our final example, we will evaluate an expression involving the sum of multiple fractions and decimals giving our answer as a fraction in its simplest form.
Example 7: Simplifying an Expression Involving Fractions and Decimals
Find the value of , giving your answer as a fraction.
Answer
We recall that we can evaluate the sum of rational numbers by first converting them into fractions with equal denominators. To do this, letβs first evaluate the decimal part of this expression. We have
Thus,
We can then convert into a rational number as follows:
So,
We can simplify this by adding the fractions with same denominator. This gives us
We now want to rewrite the fractions in the expression so that they have the same denominators. To do this, we first need to find the lowest common multiple of all of the denominators. We can find this by factoring each of the denominators into primes. We have , and 5 is prime. We then find the product of all of the different prime factors (or their highest powers) of the two numbers. This gives
We now rewrite each fraction to have a denominator of 40 as follows:
Finally, we can add rational numbers with the same denominator by just adding their numerators. This gives us
Letβs finish by recapping some of the important points from this explainer.
Key Points
- We can add rational numbers with the same denominator by adding their numerators.
- We can add rational numbers with different denominators by first converting both fractions to have the same denominator. We do this by finding the lowest common multiple of both denominators and then multiplying both the numerator and denominator of each fraction by the same number such that the denominators are both equal to the lowest common multiple.
- We can add rational numbers given in different forms (such as mixed fractions, decimals, and percentages) by first converting all of the numbers into the same form.
- We can add rational numbers by using the increments between them on a number line.
