1.2: Addition of Fractions
Lily and Howard ordered a pizza that was cut into 8 slices. Lily ate 3 slices and Howard ate 4 slices. What fraction of the pizza did each person eat? What fraction of the pizza did they eat all together?
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Khan Academy Adding and Subtracting Fractions
Guidance
\begin{align*}\frac{2}{5}+\frac{1}{5}=?\end{align*}
The problem above can be represented using fraction strips.
\begin{align*}\boxed{\frac{2}{5} + \frac{1}{5} = \frac{2+1}{5} = \frac{3}{5}}\end{align*}
To add fractions, the fractions must have the same bottom numbers (denominators). Both fractions have a denominator of 5. The answer is the result of adding the top numbers (numerators). The numbers in the numerator are 1 and 2. The sum of 1 and 2 is 3. This sum is written in the numerator over the denominator of 5. Therefore \begin{align*}\frac{2}{5}+\frac{1}{5}=\frac{3}{5}\end{align*}.
A number line can also be used to show the addition of fractions, as you will explore in Example C.
The sum of two fractions will sometimes result in an answer that is an improper fraction. An improper fraction is a fraction which has a larger numerator than denominator. This answer can be written as a mixed number. A mixed number is a number made up of a whole number and a fraction.
In order to add fractions that have different denominators, the fractions must be expressed as equivalent fractions with a LCD. The sum of the numerators can be written over the common denominator.
Example A
\begin{align*}\frac{3}{7}+\frac{2}{7}=?\end{align*}
\begin{align*}\boxed{\frac{3}{7} + \frac{2}{7} = \frac{3+2}{7} = \frac{5}{7}}\end{align*}
Example B
Louise is taking inventory of the amount of water in the water coolers located in the school. She estimates that one cooler is \begin{align*}\frac{2}{3}\end{align*} full and the other is \begin{align*}\frac{1}{4}\end{align*} full. What single fraction could Louise use to represent the amount of water of the two coolers together?
Use fraction strips to represent each fraction.
\begin{align*}\frac{2}{3}\end{align*} and \begin{align*}\frac{8}{12}\end{align*} are equivalent fractions. \begin{align*}\frac{2}{3} \left(\frac{4}{4}\right)=\frac{8}{12}\end{align*}.
\begin{align*}\frac{1}{4}\end{align*} and \begin{align*}\frac{3}{12}\end{align*} are equivalent fractions. \begin{align*}\frac{1}{4} \left(\frac{3}{3}\right)=\frac{3}{12}\end{align*}.
The two green pieces will be replaced with eight purple pieces and the one blue piece will be replaced with three purple pieces.
The amount of water in the two coolers can be represented by the single fraction \begin{align*}\frac{11}{12}\end{align*}.
\begin{align*} \frac{2}{3}+\frac{1}{4}=\frac{8}{12}+\frac{3}{12}=\frac{11}{12}\end{align*}
The denominator of 12 is the LCD (least common denominator) of \begin{align*}\frac{2}{3}\end{align*} and \begin{align*}\frac{1}{4}\end{align*} because it is the LCM (least common multiple) of the denominators 3 and 4.
Example C
What is \begin{align*}2\frac{3}{4}+\frac{1}{2}\end{align*}?
The number line is labeled in intervals of 4 which indicates that each interval represents \begin{align*}\frac{1}{4}\end{align*}. From zero, move to the number 2 plus 3 more intervals to the right. Mark the location. This represents \begin{align*}2 \frac{3}{4}\end{align*}.
From here, move to the right \begin{align*}\frac{1}{2}\end{align*} or \begin{align*}\frac{1}{2}\end{align*} of 4, which is 2 intervals.
The sum of \begin{align*}2 \frac{3}{4}\end{align*} is \begin{align*}\frac{1}{2}\end{align*} is \begin{align*}3\frac{1}{4}\end{align*}.
Concept Problem Revisited
Lily ate \begin{align*}\frac{3}{8}\end{align*} of the pizza because she ate 3 out of the 8 slices. Howard ate \begin{align*}\frac{4}{8}\end{align*} (or \begin{align*}\frac{1}{2}\end{align*}) of the pizza. Together they ate 7 slices which is \begin{align*}\frac{7}{8}\end{align*} of the pizza.
Vocabulary
 Denominator
 The denominator of a fraction is the number on the bottom that indicates the total number of equal parts in the whole or the group. \begin{align*}\frac{5}{8}\end{align*} has denominator 8.
 Fraction
 A fraction is any rational number that is not an integer.
 Improper Fraction
 An improper fraction is a fraction in which the numerator is larger than the denominator.
 \begin{align*}\frac{8}{3}\end{align*} is an improper fraction.
 LCD
 The least common denominator is the lowest common multiple of the denominators of two or more fractions. The least common denominator of \begin{align*}\frac{3}{4}\end{align*} and \begin{align*}\frac{1}{5}\end{align*} is 20.
 LCM
 The least common multiple is the lowest common multiple that two or more numbers share. The least common multiple of 6 and 5 is 30.
 Mixed Number
 A mixed number is a number made up of a whole number and a fraction such as \begin{align*}4\frac{3}{5}\end{align*}.
 Numerator
 The numerator of a fraction is the number on top that is the number of equal parts being considered in the whole or the group. \begin{align*}\frac{5}{8}\end{align*} has numerator 5.
Guided Practice
1. Use a model to answer the problem \begin{align*}\frac{1}{2}+\frac{1}{6}=?\end{align*}
2. Use a number line to determine the answer to the problem \begin{align*}\frac{3}{4}+\frac{1}{2}\end{align*}.
3. Determine the answer to \begin{align*}\frac{1}{6}+\frac{3}{4}=?\end{align*} and \begin{align*}\frac{2}{5}+\frac{2}{3}=?\end{align*} by using the rules for adding fractions.
Answers:
1.
Use fraction strips to represent each fraction.
\begin{align*}\frac{2}{3}\end{align*} and \begin{align*}\frac{3}{6}\end{align*} are equivalent fractions. \begin{align*}\frac{1}{2} \left(\frac{3}{3}\right)=\frac{3}{6}\end{align*}.
The two fractions now have the same denominator of 6.
The one yellow strip can be replaced with three green strips and the one orange strip can be replaced with one green strip.
\begin{align*}& \frac{1}{2}+\frac{1}{6}=\frac{3}{6}+\frac{1}{6}= \frac{4}{6}\end{align*}
2. \begin{align*}\frac{3}{4}+\frac{1}{2}\end{align*}
Use a \begin{align*}\frac{1}{4}\end{align*} number line. The number line is labeled in intervals of 4. Place the starting point at \begin{align*}\frac{3}{4}\end{align*}.
From this point, move to the right a total of 2 intervals. \begin{align*}\frac{1}{2}\end{align*} of \begin{align*}4=2\end{align*}. An equivalent fraction for \begin{align*}\frac{1}{2}\end{align*} is \begin{align*}\frac{2}{4} \cdot \frac{1}{2} \left(\frac{2}{2}\right)=\frac{2}{4}\end{align*}. The point where you stop is the sum of \begin{align*}\frac{3}{4}+\frac{1}{2}\end{align*}.
\begin{align*}& \frac{3}{4}+\frac{1}{2}=\frac{3}{4}+\frac{2}{4}=\frac{5}{4}\end{align*}
On the number line you stopped at the point \begin{align*}1 \frac{1}{4}\end{align*}. This is equal to \begin{align*}\frac{5}{4}\end{align*}.
3. \begin{align*}\frac{1}{6}+\frac{3}{4}=?\end{align*}
The least common multiple of 6 and 4 is 12. This means that both fractions must have a common denominator of 12 before they can be added.
\begin{align*}\frac{1}{6} \left(\frac{2}{2}\right)=\frac{2}{12}\end{align*}
\begin{align*}\frac{3}{4} \left(\frac{3}{3}\right)=\frac{9}{12}\end{align*}
\begin{align*}& \frac{1}{6}+\frac{3}{4}=\frac{2}{12}+\frac{9}{12}=\frac{11}{12}\end{align*}
\begin{align*}\frac{2}{5}+\frac{2}{3}=?\end{align*}
The least common multiple of 5 and 3 is 15. This means that both fractions must have a common denominator of 15 before they can be added.
\begin{align*}\frac{2}{5} \left(\frac{3}{3}\right)=\frac{6}{15}\end{align*}
\begin{align*}\frac{2}{3} \left(\frac{5}{5}\right)=\frac{10}{15}\end{align*}
\begin{align*}& \frac{2}{5}+\frac{2}{3}=\frac{6}{15}+\frac{10}{15}=\frac{16}{15}=1 \frac{1}{15}\end{align*}
\begin{align*}\frac{16}{15}\end{align*} is an improper fraction. An improper fraction is one which has a larger numerator than denominator. \begin{align*}\frac{15}{15}=1\end{align*} plus there is \begin{align*}\frac{1}{15}\end{align*} left over. This can be written as a whole number and a fraction \begin{align*}1 \frac{1}{15}\end{align*}. This representation is called a mixed number.
Practice
Complete the following addition problems using any method.
 \begin{align*}\frac{1}{4}+\frac{5}{8}\end{align*}
 \begin{align*}\frac{2}{5}+\frac{1}{3}\end{align*}
 \begin{align*}\frac{2}{9}+\frac{2}{3}\end{align*}
 \begin{align*}\frac{3}{7}+\frac{2}{3}\end{align*}
 \begin{align*}\frac{7}{10}+\frac{1}{5}\end{align*}
 \begin{align*}\frac{2}{3}+\frac{1}{2}\end{align*}
 \begin{align*}\frac{2}{5}+\frac{3}{10}\end{align*}
 \begin{align*}\frac{5}{9}+\frac{2}{3}\end{align*}
 \begin{align*}\frac{3}{8}+\frac{3}{4}\end{align*}
 \begin{align*}\frac{3}{5}+\frac{3}{10}\end{align*}
 \begin{align*}\frac{7}{11}+\frac{1}{2}\end{align*}
 \begin{align*}\frac{7}{8}+\frac{5}{12}\end{align*}
 \begin{align*}\frac{3}{4}+\frac{5}{6}\end{align*}
 \begin{align*}\frac{5}{6}+\frac{2}{5}\end{align*}
 \begin{align*}\frac{4}{5}+\frac{3}{4}\end{align*}
For each of the following questions, write an addition statement and find the result. Express all answers as either proper fraction or mixed numbers.
 Karen used \begin{align*}\frac{5}{8} \ cups\end{align*} of flour to make cookies. Jenny used \begin{align*}\frac{15}{16} \ cups\end{align*} of flour to make a cake. How much flour did they use altogether?
 Lauren used \begin{align*}\frac{3}{4} \ cup\end{align*} of milk, \begin{align*}1 \frac{1}{3} \ cups\end{align*} of flour and \begin{align*}\frac{3}{8} \ cup\end{align*} of oil to make pancakes. How many cups of ingredients did she use in total?
 Write two fractions with different denominators whose sum is \begin{align*}\frac{5}{6}\end{align*}. Use fraction strips to model your answer.
 Allan’s cat ate \begin{align*}2 \frac{2}{3} \ cans\end{align*} of food in one week and \begin{align*}3 \frac{1}{4} \ cans\end{align*} the next week. How many cans of food did the cat eat in two weeks?
 Amanda and Justin each solved the same problem.

 Amanda’s Solution:


 \begin{align*}& \frac{1}{6}+\frac{3}{4}\\ & \frac{2}{12}+\frac{9}{12}\\ & =\frac{11}{24}\end{align*}


 Justin’s Solution:


 \begin{align*}& \frac{1}{6}+\frac{3}{4}\\ & \frac{2}{12}+\frac{9}{12}\\ & =\frac{11}{12}\end{align*}


 Who is correct? What would you tell the person who has the wrong answer?
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Image Attributions
Here you will learn to add real numbers using different representations. You will learn to add fractions by using appropriate models and by using the number line. These methods will lead to the formation of rules for adding fractions.
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