2.2: Addition of Rational Numbers
A football team gains 11 yards on one play then loses 5 yards on another play and loses 2 yards on the third play. What is the total yardage loss or gain?
A loss can be expressed as a negative integer. A gain can be expressed as a positive integer. To find the net gain or loss, the individual values must be added together. Therefore, the sum is \begin{align*}11 + (-5) + (-2) = 4\end{align*}. The team has a net gain of 4 yards.
Addition can also be shown using a number line. If you need to add \begin{align*}2 + 3\end{align*}, start by making a point at the value of 2 and move three integers to the right. The ending value represents the sum of the values.
Example 1: Find the sum of \begin{align*}-2 + 3\end{align*} using a number line.
Solution: Begin by making a point at –2 and moving three units to the right. The final value is 1, so \begin{align*}-2 + 3 = 1.\end{align*}
When the value that is being added is positive, we jump to the right. If the value is negative, we jump to the left (in a negative direction).
Example 2: Find the sum of \begin{align*}2 - 3\end{align*} using a number line.
Solution: Begin by making a point at 2. The expression represents subtraction, so we will count three jumps to the left.
The solution is: \begin{align*}2 - 3 = -1\end{align*}
Algebraic Properties of Addition
In Lesson 2.1, you learned the Additive Inverse Property. This property states that the sum of a number and its opposite is zero. Algebra has many other properties that help you manipulate and organize information.
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*}
This property allows you to use the fact that the sum of any number and zero is the original value.
Example 3: Simplify the following using the properties of addition:
a) \begin{align*}9 + ( 1 + 22)\end{align*}
b) \begin{align*}4,211 + 0\end{align*}
Solution:
a) It is easier to regroup \begin{align*}9 + 1\end{align*}, so by applying the Associative Property of Addition, \begin{align*}(9 + 1) + 22 = 10 + 22 = 32.\end{align*}
b) The Additive Identity Property states the sum of a number and zero is itself; therefore, \begin{align*}4,211 + 0 = 4,211.\end{align*}
Nadia and Peter are building sand castles on the beach. Nadia built a castle two feet tall, stopped for ice-cream, and then added one more foot to her castle. Peter built a castle one foot tall before stopping for a sandwich. After his sandwich, he built up his castle by two more feet. Whose castle is the taller?
Nadia’s castle is \begin{align*}(2 + 1)\end{align*} feet tall. Peter’s castle is \begin{align*}(1 + 2)\end{align*} feet tall. According to the Commutative Property of Addition, the two castles are the same height.
Adding Rational Numbers
To add rational numbers, 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. Add the numerator to this product. This value is the numerator of the improper fraction. The denominator is the original.
Example 4: Write \begin{align*}11 \frac{2}{3}\end{align*} as an improper fraction:
Solution: \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*}
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*}
Watch this video for further explanation on adding fractions with unlike denominators. This video shows how to add fractions using a visual model.
Practice Set
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Basic Algebra: Addition of Rational Numbers (7:40)
In exercises 1 and 2, write the sum represented by the moves on the number line.
Find the sum. Write the answer in its simplest form.
- \begin{align*}\frac{3}{7} + \frac{2}{7}\end{align*}
- \begin{align*}\frac{3}{10} + \frac{1}{5}\end{align*}
- \begin{align*}\frac{5}{16} + \frac{5}{12}\end{align*}
- \begin{align*}\frac{3}{8} + \frac{9}{16}\end{align*}
- \begin{align*}\frac{8}{25} + \frac{7}{10}\end{align*}
- \begin{align*}\frac{1}{6} + \frac{1}{4}\end{align*}
- \begin{align*}\frac{7}{15} + \frac{2}{9}\end{align*}
- \begin{align*}\frac{5}{19} + \frac{2}{27}\end{align*}
- \begin{align*}-2.6 + 11.19\end{align*}
- \begin{align*}-8 + 13\end{align*}
- \begin{align*}-7.1 + (-5.63)\end{align*}
- \begin{align*}9.99 + (-0.01)\end{align*}
- \begin{align*}4 \frac{7}{8} + 1\frac{1}{2}\end{align*}
- \begin{align*}-3 \frac{1}{3} + \left (-2 \frac{3}{4} \right )\end{align*}
In 17 – 20, which property of addition does each situation involve?
- Whichever order your groceries are scanned at the store, the total will be the same.
- 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.
- Shari’s age minus the negative of Jerry’s age equals the sum of the two ages.
- Kerri has 16 apples and has added zero additional apples. Her current total is 16 apples.
- 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?
- A blue whale dives 160 feet below the surface then rises 8 feet. Write the addition problem and find the sum.
- The temperature in Chicago, Illinois one morning was \begin{align*}-8^\circ F\end{align*}. Over the next six hours the temperature rose 25 degrees Fahrenheit. What was the new temperature?
In 24 – 30, evaluate each expression for \begin{align*}v = 5.8.\end{align*}
- \begin{align*}9.1 + v\end{align*}
- \begin{align*}v+(-v)\end{align*}
- \begin{align*}-v+4.12\end{align*}
- \begin{align*}-23.14+ -v\end{align*}
- \begin{align*}7.86+(-v)\end{align*}
- \begin{align*}-v+3.5\end{align*}
- \begin{align*}-v+v\end{align*}
Mixed Review
- Find the opposite of –72.
- Evaluate \begin{align*}|16-29+78-114|.\end{align*}
- What is the domain and range of the following: \begin{align*}\left \{(2,-1),(3,0),(4,6),(1,-3) \right \}?\end{align*}
- Write a rule for the table:
Volume (in cubic inches) | Mass (in grams) |
---|---|
1 | 20.1 |
2 | 40.2 |
3 | 60.3 |
4 | 80.4 |