Lesson Video: Addition of Rational Numbers Mathematics • 7th Grade

Video Transcript

In this video, we will learn how to add rational numbers, including fractions, decimals, and percentages. We will begin by recalling the definition of a rational number.

We know that a rational number is a quotient of integers and can therefore be written in the form 𝑎 over 𝑏, where 𝑎 and 𝑏 are integers and 𝑏 is nonzero. There are several ways to visualize rational numbers. For example, since we’re dividing 𝑎 by an integer 𝑏, we are dividing the value of 𝑎 into 𝑏 equal parts. One way of representing this is to consider the position of 𝑎 on a number line. To represent 𝑎 over 𝑏 on this number line, we want to split the line segment between zero and 𝑎 into 𝑏 equal sections. For example, if 𝑏 is six, we split this into six sections of equal lengths as shown. And the first increment will have a value of 𝑎 over six.

We can use a number line like this to add rational numbers. And to do so, we first recall that to add two numbers on a number line, we add their displacements from zero. Let’s consider an example of adding three-sevenths and two-sevenths using a number line. We split the line segment between zero and one into seven equal sections, where each increment will represent one-seventh. We can therefore represent two-sevenths and three-sevenths as shown.

To add these numbers together, we need to add their displacements from zero together. Since two plus three is equal to five, we are now five increments of one-seventh away from zero in the positive direction. This means that three-sevenths plus two-sevenths is equal to five-sevenths. We note that we are just adding the numerators of the two rational numbers. This is true when adding any two fractions where the denominators are equal and leads us to the general rule that if 𝑎, 𝑏, and 𝑐 are integers with 𝑏 not equal to zero, then 𝑎 over 𝑏 plus 𝑐 over 𝑏 is equal to 𝑎 plus 𝑐 over 𝑏.

We will now consider a couple of examples, where we can use this rule.

Find the value of three-tenths plus one-tenth. Give your answer in its simplest form.

We begin by noting that we have two rational numbers with the same denominator. Since the denominator is nonzero, we can use the rule 𝑎 over 𝑏 plus 𝑐 over 𝑏 is equal to 𝑎 plus 𝑐 over 𝑏. When adding two fractions with the same denominator, we simply add their numerators. This means that three over 10 plus one over 10 is equal to three plus one over 10. And this simplifies to four over 10 or four-tenths.

We are asked to give our answer in its simplest form, so we need to check whether the numerator and denominator have any common factors apart from one. As both four and 10 are even, we can divide the numerator and denominator by two. The fraction four-tenths simplifies to two-fifths. We can therefore conclude that the value of three-tenths plus one-tenth in its simplest form is two-fifths.

In our next example, we will add two mixed numbers with the same denominator.

Calculate eight and two-fifths plus negative four and three-fifths. Give your answer as a mixed number.

In this question, we will begin by rewriting the two mixed numbers as improper or top-heavy fractions. We recall that the mixed number 𝑎 and 𝑏 over 𝑐 is equal to 𝑎 𝑐 plus 𝑏 over 𝑐. This means that eight and two-fifths is equal to eight multiplied by five plus two over five. The numerator of our improper fraction is equal to the whole number multiplied by the denominator plus the numerator of the mixed number. Eight and two-fifths is therefore equal to 42 over five or forty-two fifths.

We can repeat this process for the mixed number negative four and three-fifths. We begin by converting the mixed number four and three-fifths into an improper fraction. This is equal to twenty-three fifths. And as such, negative four and three-fifths is equal to negative twenty-three fifths.

We are now in a position where we can add the two improper fractions. Since the denominators are the same, we simply add the numerators, giving us 42 plus negative 23 over five. This is equal to 19 over five or nineteen-fifths. And our final step is to convert this back into a mixed number. Dividing 19 by five gives us three remainder four. And as such, nineteen-fifths is equivalent to three and four-fifths. We can therefore conclude that this is the value of eight and two-fifths plus negative four and three-fifths.

At this stage, it is worth noting that there is no reason to assume that the denominators of the fractions we wish to add will be equal. We can still evaluate these expressions by using a number line or finding a common denominator.

Let’s now consider an example of this type.

Evaluate one-half plus two-thirds.

In order to answer this question, we will use a number line together with our knowledge of equivalent fractions. By drawing a number line from zero to two, we’ll firstly mark on the fractions one-half and two-thirds. To do this, we split each integer into two and three equal sections, respectively. Next, we recall that in order to add the fractions, we add their displacements from zero. This can be represented on a number line as shown.

However, there is a small problem; if we try to find this using our number line, we see that this point is not marked. As such, we can find the value of this point by using our knowledge of adding fractions with equal denominators and our knowledge of equivalent fractions. The fraction one-half is equivalent to three-sixths, as we can multiply the numerator and denominator by three. In the same way, two-thirds is equivalent to four-sixths. This time, we multiply the numerator and denominator by two.

We can therefore rewrite one-half plus two-thirds as three-sixths plus four-sixths. And recalling that when adding fractions with the same denominator, we simply add the numerators, this becomes three plus four over six, which is equal to seven over six or seven-sixths. On our number line, instead of splitting each integer segment into two and three sections, we could instead split it into six sections. This again shows that we are adding three-sixths and four-sixths, giving us an answer of seven-sixths. We can therefore conclude that one-half plus two-thirds is equal to seven-sixths, which can also be written as the mixed number one and one-sixth.

We will now move on to two further examples where our rational numbers are given as decimals and percentages as well as fractions.

Consider that 𝑥 equals one-fifth and 𝑦 is equal to 16 percent. By converting 𝑥 and 𝑦 to decimal form, find the value of 𝑥 plus 𝑦 approximated to the nearest two decimals.

In this question, we’ll begin by converting the values of 𝑥 and 𝑦 into decimal form. Let’s consider the fraction one-fifth. By multiplying both the numerator and denominator by two, we see that one-fifth is equivalent to two-tenths. Using our knowledge of place value, this can be written as 0.2. And our value of 𝑥 is therefore equal to 0.2.

Next, we recall that we can convert a percentage to a decimal by dividing by 100. 16 percent is equal to 16 divided by 100 or sixteen hundredths, which, written in decimal form, is 0.16. We now have values of both 𝑥 and 𝑦 in decimal form. And as such, we can find the value of 𝑥 plus 𝑦 by adding 0.2 and 0.16. This is equal to 0.36, which is already written to two decimal places. Adding the fraction one-fifth and 16 percent gives us 0.36. Whilst it is not required in this question, we could also give our answer as 36 percent or 36 over 100. This fraction could also be given in its simplified form of nine over 25 or nine twenty-fifths.

We will now consider one final example of this type.

Evaluate three-fifths plus 0.7, giving the answer in its simplest form.

In this question, we need to evaluate the sum of two rational numbers: one that is written as a fraction and one as a decimal. Using our knowledge of place value, we know that 0.7 is equal to seven-tenths. We therefore need to add the fractions three-fifths and seven-tenths. We know that in order to add fractions, we need a common denominator. So we therefore need to find an equivalent fraction to three-fifths.

Since five multiplied by two is 10, we need to multiply the numerator by two. Three-fifths is equivalent to six-tenths. This means that in order to evaluate three-fifths plus 0.7, we can simply add six-tenths and seven-tenths. As the denominators are now the same, we simply add the numerators, giving us a final answer of thirteen-tenths. It is worth noting that this could also be written as the mixed number one and three-tenths, as 13 divided by 10 is one remainder three. This can also be written in the decimal form 1.3.

We will now finish this video by summarizing the key points. We saw in this video that 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 also saw that we can add rational numbers given in different forms such as mixed numbers, decimals, and percentages by first converting all of the numbers into the same form. Finally, we also saw that we can add rational numbers by using the increments between them on a number line.