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

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 π‘Žπ‘, where π‘Ž, π‘βˆˆβ„€ and 𝑏≠0. There are several different ways to visualize rational numbers. For example, since we are dividing π‘Ž by an integer 𝑏, we are dividing the value of π‘Ž into 𝑏 equal parts. We can represent this in many different ways. One way is to consider the position of π‘Ž on a number line.

To represent π‘Žπ‘ on this number line, we want to split the line segment between 0 and π‘Ž into 𝑏 equal sections. For example, if 𝑏 is 6, we split this into 6 sections of equal length and the first increment will have a value of π‘Ž6 as shown.

This is equivalent to dividing 1 into |𝑏| sections and then taking |π‘Ž| 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.

Let’s see an example of adding 37 and 27 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 17 so that we can represent 27 and 37, as shown.

The second increment will be 27 and the third will be 37. To add these numbers together, we need to add their displacements from 0 together. We note that 27 is 2 increments of 17 away from 0 in the positive direction and 37 is 3 increments of 17 away from 0 in the positive direction. Hence, when we add these together, we will get 2+3=5 increments of 17 away from 0 in the positive direction.

Therefore, 37+27 is 5 increments of 17 from 0 in the positive direction. We have shown that 37+27=57.

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 π‘Žπ‘+𝑐𝑏 for integers π‘Ž, 𝑏, and 𝑐 with 𝑏≠0, we know that π‘Žπ‘ is π‘Ž increments of 1𝑏 from 0, 𝑐𝑏 is 𝑐 increments of 1𝑏 from 0, and their sum will be π‘Ž+𝑐 increments of 1𝑏 from 0. Hence, π‘Žπ‘+𝑐𝑏=π‘Ž+𝑐𝑏.

This result is also true when subtracting rational numbers. This gives us the following result:

Definition: Adding Rational Numbers with the Same Denominator

If π‘Ž, 𝑏, and 𝑐 are integers with 𝑏≠0, then π‘Žπ‘+𝑐𝑏=π‘Ž+𝑐𝑏.

Let’s 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 310+110. Give your answer in its simplest form.

Answer

We first note that these rational numbers have the same denominator. We then recall that π‘Žπ‘+𝑐𝑏=π‘Ž+𝑐𝑏, so we just need to add the numerators. This gives 310+110=3+110=410.

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 4=2Γ—2 and 10=2Γ—5, so they both share a factor of 2. Canceling this shared factor, we get 410=2Γ—22Γ—5=25.

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 825+ο€Όβˆ’435. Give your answer as a mixed number.

Answer

Let’s start by rewriting both mixed numbers as fractions. We have 825=8Γ—5+25=425,βˆ’435=βˆ’4Γ—5+35=βˆ’235.

We first note that these rational numbers have the same denominator. We then recall that π‘Žπ‘+𝑐𝑏=π‘Ž+𝑐𝑏, so we just need to add the numerators. This gives 825+ο€Όβˆ’435=425+ο€Όβˆ’235=42+(βˆ’23)5=195.

We can then convert this into a mixed number by noting that 19=5Γ—3+4. Hence, 195=345.

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 12+23. We first represent each number on a number line by splitting the number line into increments of 12 and 13 as shown.

To add these together, we add their displacements. So, we want the number that is 1 increment of 12 and 2 increments of 13 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 12 and 23 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 12=1Γ—32Γ—3=36,23=2Γ—23Γ—2=46.

We then have that 12+23=36+46=3+46=76.

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.

Let’s 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 βˆ’310+=25.

Answer

We first note that each increment on the number line represents 110, since they are equally spaced and the first increment after 0 is labeled as 110. We can then recall that we can add numbers on a number line by adding their displacements from 0. Therefore, when we add βˆ’310 and the missing value together, we will end at 25 on the number line.

In other words, the equation βˆ’310+=25 is asking us how far we need to move from βˆ’310 to get to 25. 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 110, 710 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.

Let’s 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 π‘₯=15 and 𝑦=16%. By converting π‘₯ and 𝑦 to decimal form, find the value of π‘₯+𝑦 approximated to the nearest two decimals.

Answer

We are asked to find this sum by first converting both π‘₯ and 𝑦 into decimals. First, we can note that 15=210=0.2. 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 16%=16100=0.16.

Hence, 15+16%=0.2+0.16=0.36.

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 14+13+34+ο€Όβˆ’26 giving the answer in its simplest form.

Answer

There are two ways to evaluate this expression. The easiest way is to note that βˆ’26 can be simplified since the numerator and denominator share a factor of 2. Hence, βˆ’26=βˆ’2Γ—13Γ—2=βˆ’13.

Substituting this into the expression gives 14+13+34+ο€Όβˆ’26=14+13+34+ο€Όβˆ’13.

We can then add the fractions with equal denominators by adding the numerators. We have 14+34=1+34=44,13+ο€Όβˆ’13=1βˆ’13=03.

We know that 03=0 and 44=1, so 14+13+34+ο€Όβˆ’26=0+1=1.

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 4=2 and 6=2Γ—3, 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 LCM(3,4,6)=2Γ—3=12.

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 14+13+34+ο€Όβˆ’26=1Γ—34Γ—3+1Γ—43Γ—4+3Γ—34Γ—3+ο€Όβˆ’2Γ—26Γ—2=312+412+912+ο€Όβˆ’412=3+4+9βˆ’412=1212=1.

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 35+0.7 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 0.7=710. 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 35 to have a denominator of 10 by multiplying both the numerator and denominator by 2 to get 35=3Γ—25Γ—2=610.

We can then substitute these into the expression and evaluate to get 35+0.7=610+710=1310.

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 18+(βˆ’0.1)+45+15+(βˆ’0.3), 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, let’s first evaluate the decimal part of this expression. We have (βˆ’0.1)+(βˆ’0.3)=βˆ’0.1βˆ’0.3=βˆ’0.4.

Thus, 18+(βˆ’0.1)+45+15+(βˆ’0.3)=18+45+15+(βˆ’0.4).

We can then convert βˆ’0.4 into a rational number as follows: βˆ’0.4=βˆ’410=βˆ’2Γ—25Γ—2=βˆ’25.

So, 18+45+15+(βˆ’0.4)=18+45+15+ο€Όβˆ’25.

We can simplify this by adding the fractions with same denominator. This gives us 18+45+15+ο€Όβˆ’25=18+4+1+(βˆ’2)5=18+35.

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 8=2Γ—2Γ—2, 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 LCM(5,8)=2Γ—5=40.

We now rewrite each fraction to have a denominator of 40 as follows: 18+35=1Γ—58Γ—5+3Γ—85Γ—8=540+2440.

Finally, we can add rational numbers with the same denominator by just adding their numerators. This gives us 540+2440=5+2440=2940.

Let’s 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.

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