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Addition of Rational Numbers

Adding fractions

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Addition of Rational Numbers

Suppose you drank \begin{align*}\frac{7}{8}\end{align*} of a bottle of juice on Monday, \begin{align*}\frac{2}{5}\end{align*} of a bottle on Tuesday, and \begin{align*}\frac{3}{2}\end{align*} of a bottle on Wednesday. Can you identify which of the fractions of a bottle of juice are improper, and can you write the fractions with a common denominator? Also, can you add these fractions of a bottle of juice together? 

Addition of Rational Numbers

Recall that rational numbers are numbers that can be written in the form of \begin{align*}\frac{a}{b}\end{align*} where \begin{align*}a \text{ and } b\end{align*} are integers. Some decimals and all fractions are rational numbers. To add rational numbers, you need to know how to add both.  

Adding Decimals

When adding decimals, you can treat them like whole numbers. The major difference is that you must align the decimals so that you are only adding numbers in the same place value. 

Let's add the following decimals:

  1. \begin{align*}87.296+48.6\end{align*}

First, align the decimals:

\begin{align*} 87.296 & \\ \underline{+48.6\phantom{00}}\end{align*}

The decimal points must be kept directly under each other as well as the digits must be kept in the same place value in line with each other. To ensure that the digits are aligned, add zeros to 48.6 to represent the hundredths and thousandths place shown in 87.296.

\begin{align*} 87.296 & \\ \underline{+48.6{\color{blue}00}} & \end{align*}

Add the numbers.

\begin{align*} 87.296 & \\ \underline{+ 48.6{\color{blue}00}} & \\ 135.896 & \end{align*}

The answer is 135.896.

  1. \begin{align*}(97.38)+(-45.17)\end{align*}

    The first step is to write the problem with the decimals aligned. The two decimal numbers that are being added have opposite signs. When this happens, subtract the numbers and use the sign of the larger number in the answer.

    \begin{align*} 97.38 & \\ \underline{- 45.17} & \\ \quad 52.21 & \end{align*}

    The larger decimal number is 97.38 and it has a positive sign. This means that the sign of the answer will also be a positive value.

  2. The first step is to write the problem with the decimals aligned. The two decimal numbers that are being added have the same signs. When this happens, add the numbers and use the sign of the numbers in the answer.

    \begin{align*} -168.8\phantom{000} & \\ \underline{+-217.4536} & \end{align*}

    To ensure that the digits are aligned correctly, add zeros to 168.8. Add the numbers.

    \begin{align*} -168.8{\color{blue}000} & \\ \underline{+-217.4536} & \end{align*}

    Add the numbers.

    \begin{align*} -168.8{\color{blue}000} & \\ \underline{+-217.4536} & \\ -386.2536 & \end{align*}

    The decimal numbers being added both had negative signs. This means that the sign of the answer is also a negative value.

Adding Fractions

To add fractions, we must first remember how to rewrite mixed numbers as improper fractions. Begin by multiplying the denominator of the mixed number to the whole value. Then, add the numerator to this product. This value is the numerator of the improper fraction. The denominator is the original.

Let's use the steps above and write \begin{align*}11 \frac{2}{3}\end{align*} as an improper fraction:

\begin{align*}3 \times 11 = 33 + 2 = 35\end{align*}. This is the numerator of the improper fraction.

\begin{align*}11 \frac{2}{3} = \frac{35}{3}\end{align*}

Addition with the Same Denominator

Now that we know how to rewrite a mixed number as an improper fraction, we can begin to add rational numbers. There is one thing to remember when finding the sum or difference of rational numbers: The denominators must be equivalent.

The Addition Property of Fractions: For all real numbers \begin{align*}a, \ b,\end{align*} and \begin{align*}c, \ \frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}.\end{align*}

Let's add the following sets of fractions (rational numbers):

  1. \begin{align*}\frac{1}{3}+\frac{2}{3}\end{align*}

Since the denominators are the same, we can go ahead and add the numerators:


  1. \begin{align*}4\frac{1}{7}+\frac{2}{7}\end{align*}

\begin{align*}4\frac{1}{7}\end{align*} is a mixed fraction, and reads "4 and one seventh." This means we can think of it as \begin{align*}4+\frac{1}{7}\end{align*}. Since the other fraction also has 7 as a denominator, we can add the two fractions:


This mixed fraction can be turned into an improper fraction as follows:

\begin{align*}4\times 7=28\end{align*}


Common Denominators

In order to add two fractions, they must have a common denominator. This means that they must have the same number in the denominator. If two fractions to be added do not have common denominators, either one or both of the fractions can be changed so that they do have common denominators. In general, when two fractions have different denominators, use the pattern below.

\begin{align*}\frac{a}{b}+\frac{c}{d}=\frac{a}{b}\cdot \frac{d}{d}+\frac{c}{d}\cdot \frac{b}{b}=\frac{ad}{bd}+\frac{bc}{bd}=\frac{ad+bc}{bd}\end{align*}

Now, let's find common denominators for the follow sums, and then add them.

  1.  \begin{align*}\frac{2}{11}+\frac{1}{3}\end{align*} 

Follow the pattern; that is, multiply the top and bottom of each fraction by the denominator of the other fraction:

\begin{align*}\frac{2}{11}+\frac{1}{3}=\frac{2}{11}\cdot \frac{3}{3}+\frac{1}{3}\cdot \frac{11}{11}=\frac{6}{33}+\frac{11}{33}=\frac{6+11}{33}=\frac{17}{33}\end{align*}

  1. \begin{align*}\frac{1}{5}+\frac{3}{10}\end{align*}

In this problem, our denominator, 10, is a multiple of the other denominator, 5. There is no need to change the fraction with a denominator of 10. Simply multiply the top and bottom of the first fraction in order to make its denominator 10:

\begin{align*}\frac{1}{5}+\frac{3}{10}=\frac{1}{5}\cdot \frac{2}{2}+\frac{3}{10}=\frac{2}{10}+\frac{3}{10}=\frac{2+3}{10}=\frac{5}{10}=\frac{1}{2}\end{align*}

Algebraic Properties of Addition

There are properties of addition that apply to all real numbers and so they apply to all rational numbers as well. Let's review them here:

The Commutative Property of Addition: For all real numbers \begin{align*}a\end{align*},and \begin{align*}b\end{align*}, \begin{align*}a + b = b + a\end{align*}.

To commute means to change locations, so the Commutative Property of Addition allows you to rearrange the objects in an addition problem.

The Associative Property of Addition: For all real numbers \begin{align*}a, \ b,\end{align*} and \begin{align*}c\end{align*}, \begin{align*}(a + b)+ c = a + (b + c).\end{align*}

To associate means to group together, so the Associative Property of Addition allows you to regroup the objects in an addition problem.

The Identity Property of Addition: For any real number \begin{align*}a, \ a + 0 = a.\end{align*}

Another way you sometimes see a rational number is as a decimal number, such as 2.5, 30.01, or 2.9999. We will practice some of the above properties on rational numbers in their different forms.

Let's show that the following properties are true by showing that the equations are equal:

  1. Commutative Property: \begin{align*} 2.5 + 3.5 =3.5 + 2.5 \end{align*}

We will check each side separately to see if they equal the same thing.

\begin{align*} 2.5 + 3.5 =6\end{align*}

\begin{align*}3.5 + 2.5=6 \end{align*}

So we conclude that the equality is satisfied.

  1. Associate Property: \begin{align*} \frac{1}{9}+\left(\frac{2}{9}+\frac{5}{9}\right)=\left(\frac{1}{9}+\frac{2}{9}\right)+\frac{5}{9}\end{align*}

We check each side of the equation here as well.

\begin{align*} \frac{1}{9}+\left(\frac{2}{9}+\frac{5}{9}\right)=\frac{1}{9}+\frac{7}{9}=\frac{8}{9}\end{align*}


So we conclude that the equality is satisfied.


Example 1

Earlier, you were told that you drank \begin{align*}\frac{7}{8}\end{align*} of a bottle of juice on Monday, \begin{align*}\frac{2}{5}\end{align*} of a bottle on Tuesday, and \begin{align*}\frac{3}{2}\end{align*} of a bottle on Wednesday. Which of the fractions of a bottle of juice are improper? What is the common denominator? Can you add these fractions of a bottle of juice together to determine how much of a bottle you drank this week?

\begin{align*}\frac{3}{2}\end{align*} is the only fraction of a bottle of juice that is improper because it is the only fraction in which the numerator is greater than the denominator. The three denominators that we have are 8, 4, and 2. Using the pattern, multiply the first two denominators together and you get 40. Notice that 2 is a factor of 40 so you can multiply \begin{align*}\frac{3}{2}\end{align*} by 20 to get a common denominator. Therefore the common denominator is 40. 

Now, add the fractions:

\begin{align*}\frac{7}{8} + \frac{2}{5} + \frac{3}{2}=\frac{7}{8}\cdot \frac{5}{5}+\frac{2}{5}\cdot \frac{8}{8}+\frac{3}{2}\cdot \frac{20}{20} =\frac{35}{40} + \frac{16}{40} + \frac{60}{40}=\frac{111}{40}\end{align*} 

\begin{align*}\frac{111}{40}\end{align*} cannot be simplified. Therefore, you drank \begin{align*}\frac{111}{40}\end{align*} or \begin{align*}2\frac{31}{40}\end{align*} bottles of juice this week. 

Example 2

Perform the addition of rational numbers and give your final answer as an mixed fraction.


We can break up the mixed fractions:


Using the Commutative Property we can rearrange and simplify by adding integers:


Now we just need to add the fractions. Since they do not have common denominators, we have to give them common denominators. The denominators do not share any factors, so we need to multiply them by each other:

\begin{align*}\frac{1}{3}+\frac{3}{4}=\frac{1\times 4}{3\times 4}+\frac{3\times 3}{4\times 3}=\frac{4}{12}+\frac{9}{12}=\frac{4+9}{12}=\frac{13}{12}.\end{align*}

Now we know what the sum of the fractions is:


Since our answer needs to be a mixed fraction, we will turn the improper fraction into a mixed fraction. Since 12 goes into 13 one time with a remainder of 1, we get:


The answer is:



Find the sum. Write the answer in its simplest form.

  1. \begin{align*}\frac{3}{7} + \frac{2}{7}\end{align*}
  2. \begin{align*}\frac{3}{10} + \frac{1}{5}\end{align*}
  3. \begin{align*}\frac{5}{16} + \frac{5}{12}\end{align*}
  4. \begin{align*}\frac{3}{8} + \frac{9}{16}\end{align*}
  5. \begin{align*}\frac{8}{25} + \frac{7}{10}\end{align*}
  6. \begin{align*}\frac{1}{6} + \frac{1}{4}\end{align*}
  7. \begin{align*}\frac{7}{15} + \frac{2}{9}\end{align*}
  8. \begin{align*}\frac{5}{19} + \frac{2}{27}\end{align*}
  9. \begin{align*}-2.6 + 11.19\end{align*}
  10. \begin{align*}-8 + 13\end{align*}
  11. \begin{align*}-7.1 + (-5.63)\end{align*}
  12. \begin{align*}9.99 + (-0.01)\end{align*}
  13. \begin{align*}4 \frac{7}{8} + 1\frac{1}{2}\end{align*}
  14. \begin{align*}-3 \frac{1}{3} + \left (-2 \frac{3}{4} \right )\end{align*}

In 15–17, which property of addition does each situation involve?

  1. Whichever order your groceries are scanned at the store, the total will be the same.
  2. Suppose you go buy a DVD for $8.00, another for $29.99, and a third for $14.99. You can add \begin{align*}(8 + 29.99) + 14.99\end{align*} or you can add \begin{align*}8 + (29.99 + 14.99)\end{align*} to obtain the total.
  3. Nadia, Peter, and Ian are pooling their money to buy a gallon of ice cream. Nadia is the oldest and gets the greatest allowance. She contributes half of the cost. Ian is next oldest and contributes one third of the cost. Peter, the youngest, gets the smallest allowance and contributes one fourth of the cost. They figure that this will be enough money. When they get to the check-out, they realize that they forgot about sales tax and worry there will not be enough money. Amazingly, they have exactly the right amount of money. What fraction of the cost of the ice cream was added as tax?

In 18–24, evaluate each expression

  1. \begin{align*}9.1 + 5.8\end{align*}
  2. \begin{align*}5.8+(\text{-}5.8)\end{align*}
  3. \begin{align*}\text{-}5.8+4.12\end{align*}
  4. \begin{align*}\text{-}23.14+\text{-}5.8\end{align*}
  5. \begin{align*}7.86+(\text{-}5.8)\end{align*}
  6. \begin{align*}\text{-}5.8+3.5\end{align*}
  7. \begin{align*}\text{-}5.8+5.8\end{align*}

Review (Answers)

To see the Review answers, open this PDF file and look for section 2.4. 

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Addition Property of Fractions For all real numbers a, \ b, and c, \ \frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}.
Additive Identity Property The sum of any number and zero is the number itself.
Associative Property The associative property states that you can change the groupings of numbers being added or multiplied without changing the sum. For example: (2+3) + 4 = 2 + (3+4), and (2 X 3) X 4 = 2 X (3 X 4).
Commutative Property The commutative property states that the order in which two numbers are added or multiplied does not affect the sum or product. For example a+b=b+a \text{ and\,} (a)(b)=(b)(a).
Equivalent Fractions Equivalent fractions are fractions that can each be simplified to the same fraction. An equivalent fraction is created by multiplying both the numerator and denominator of the original fraction by the same number.
multiplicative identity property The product of any number and one is the number itself.
simplest form The simplest form of a fraction has no common factors in the numerator and the denominator. The simplest form of 3/6 is 1/2.

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