3.10: Adding Common Fractions

Vocabulary review:

[latex]4+5=9[/latex]

  • 4 and 5 are addends.
  • 9 is the sum.

[latex]\dfrac{3}{7}[/latex]

  • 3 is the numerator.
  • 7 is the denominator.

Like fractions are fractions that have the same denominator.

Example: [latex]\tfrac{1}{4}[/latex],  [latex]\tfrac{2}{4}[/latex],  [latex]\tfrac{3}{4}[/latex],  [latex]\tfrac{4}{4}[/latex],  etc.

Adding and subtracting fractions has some different rules from multiplying and dividing.

Two circles. Circles are divided in three with shading of one of the three.
Figure 3.10.1

There are two cakes that are left over (see Figure 3.10.1). There is 1 piece of each cake left. If you were to put all the pieces left onto one plate, how much cake would you have?

Shade in your answer in Figure 3.10.2.

 

Visual representation of equation about two cakes. Three circles. Circles are divided in three with shading of one of the three for the items to add together.
Figure 3.10.2

If you made your plate like Figure 3.10.3, then you are right!

A circle divided into three parts with two-thirds shaded in.
Figure 3.10.3

Try the example in Figure 3.10.4.

 

Image of equation using three rectangle objects. One rectangle divided in four, shaded in one segment. Addition between. Second rectangle divided in four segments, shaded in first and third segments. Equals sign. And rectangle with three four segments. All unshaded.
Figure 3.10.4

Figure 3.10.5 shows the answer.

 

Image of equation using three rectangle objects. One rectangle divided in four, shaded in one segment. Addition between. Second rectangle divided in four segments, shaded in first and third segments. Equals sign. And rectangle with three shaded segments, one blank.
Figure 3.10.5

What you are doing is adding two like fractions.

  • You are moving pieces of fractions that are the same size into one whole shape. The pieces do not change size, so the denominator must stay the same size.
  • When adding two fractions, your answer is a fraction.

Look back at the two examples (Figure 3.10.1 and Figure 3.10.4).

When you add fractions, does the denominator or the numerator stay the same?

Common fractions must have the same denominator when you add them together. Add the numerators and keep the denominators the same.

Look at the next two examples (Figure 3.10.6 and Figure 3.10.7):

 

Image of equation using three rectangle objects. One rectangle divided in four, shaded in one top segment. Addition between. Second rectangle divided in four segments, shaded in two bottom segments. Equals sign. Sum shown as rectangle with three shaded segments, one blank.
Figure 3.10.6

[latex]\dfrac{1}{4} + \dfrac{2}{4} = \dfrac{3}{4}[/latex]

 

Image of equation using four rectangle objects for fractions and their sum. One rectangle divided in four, shaded in one left segment. Addition between. Second rectangle divided in four segments, shaded in two left segments. Third rectangle divided in four segments, shaded in one left segments. Equals sign. Sum shown as rectangle with three shaded segments, one blank.
Figure 3.10.7

[latex]\dfrac{1}{5}+ \dfrac{2}{5}+ \dfrac{1}{5} = \dfrac{4}{5}[/latex]

Exercise 1

Try a few for yourself.

    1. Visual representation of the equation to solve.

      [latex]\dfrac{2}{9} + \dfrac{3}{9} = \dfrac{ }{9}[/latex]

  1. Visual representation of the equation to solve.

    [latex]\dfrac{2}{4} + \dfrac{1}{4} = \dfrac{ }{4}[/latex]

  2. Visual representation of the equation to solve.

    [latex]\dfrac{1}{3} + \dfrac{1}{3} = \dfrac{ }{3}[/latex]

  3. Visual representation of the equation to solve.

    [latex]\dfrac{3}{6} + \dfrac{3}{6} = \dfrac{ }{6}[/latex]

  4. Visual representation of the equation to solve.
    [latex]\dfrac{3}{8} + \dfrac{4}{8} = \dfrac{ }{8}[/latex]

Exercise 1 Answers

  1. [latex]\dfrac{5}{9}[/latex]
  2. [latex]\dfrac{3}{4}[/latex]
  3. [latex]\dfrac{2}{3}[/latex]
  4. [latex]\dfrac{5}{6}[/latex]
  5. [latex]\dfrac{7}{8}[/latex]

Exercise 2

Now, find the answers to these additions without diagrams.

  1. [latex]\dfrac{2}{4} + \dfrac{1}{4} = \dfrac{ }{4}[/latex]
  2. [latex]\dfrac{1}{3} + \dfrac{1}{3} = \dfrac{ }{3}[/latex]
  3. [latex]\dfrac{1}{5} + \dfrac{1}{5} = \dfrac{ }{5}[/latex]
  4. [latex]\dfrac{2}{11} + \dfrac{7}{11} = \dfrac{ }{11}[/latex]

Exercise 2 Answers

  1. [latex]\dfrac{3}{4}[/latex]
  2. [latex]\dfrac{2}{3}[/latex]
  3. [latex]\dfrac{2}{5}[/latex]
  4. [latex]\dfrac{9}{11}[/latex]

Exercise 3

Add these common fractions.

  1. [latex]\dfrac{1}{5} + \dfrac{2}{5} =[/latex]
  2. [latex]\dfrac{3}{6} + \dfrac{2}{6} =[/latex]
  3. [latex]\dfrac{3}{7} + \dfrac{2}{7} =[/latex]
  4. [latex]\dfrac{3}{10} + \dfrac{6}{10} =[/latex]
  5. [latex]\dfrac{14}{20} + \dfrac{3}{20} =[/latex]
  6. [latex]\dfrac{7}{37} + \dfrac{19}{37} =[/latex]

Exercise 3 Answers

  1. [latex]\dfrac{3}{5}[/latex]
  2. [latex]\dfrac{5}{6}[/latex]
  3. [latex]\dfrac{5}{7}[/latex]
  4. [latex]\dfrac{9}{10}[/latex]
  5. [latex]\dfrac{17}{20}[/latex]
  6. [latex]\dfrac{26}{37}[/latex]

Sometimes, the sum of a fraction will need to be reduced (take a look at this example to remind yourself how to do this).

Example A

[latex]\dfrac{2}{8} + \dfrac{2}{8} = \dfrac{4}{8}\rightarrow\dfrac{÷ 4}{÷ 4} = \dfrac{1}{2}[/latex]

Example B

[latex]\dfrac{3}{4} + \dfrac{3}{4} = \dfrac{6}{4}\rightarrow\dfrac{6}{4} \dfrac{÷ 2}{÷ 2} = \dfrac{3}{2} = 1\dfrac{1}{2}[/latex]

Exercise 4

Find the sums to the following additions. Make sure your answer is in the lowest terms.

  1. [latex]\dfrac{1}{4} + \dfrac{1}{4} =[/latex]
  2. [latex]\dfrac{1}{3} + \dfrac{1}{3} =[/latex]
  3. [latex]\dfrac{3}{10} + \dfrac{2}{10} =[/latex]
  4. [latex]\dfrac{7}{25} + \dfrac{8}{25} =[/latex]
  5. [latex]\dfrac{3}{5} + \dfrac{1}{5} =[/latex]
  6. [latex]\dfrac{9}{27} + \dfrac{12}{27} =[/latex]

Exercise 4 Answers

  1. [latex]\dfrac{1}{2}[/latex]
  2. [latex]\dfrac{2}{3}[/latex]
  3. [latex]\dfrac{1}{2}[/latex]
  4. [latex]\dfrac{3}{5}[/latex]
  5. [latex]\dfrac{4}{5}[/latex]
  6. [latex]\dfrac{7}{9}[/latex]

So far all your answers have been less than one (a proper fraction). Sometimes adding fractions can result in more than one whole.

Look at this example in Figure 3.10.8:

 

A rectangle with two out of four parts shaded in is added to a rectangle with three out of four parts shaded in. The result is one fully shaded rectangle and one rectangle with one out of four parts shaded in.
Figure 3.10.8

[latex]\dfrac{2}{4} + \dfrac{3}{4}=\dfrac{4}{4}\text{and}\dfrac{1}{4}[/latex]   (or [latex]\left(\dfrac{5}{4}\right)[/latex])

There are not enough parts in the first square to hold all your shaded parts, so you need to draw a second square to hold the extra shaded parts.

You would also have to convert this answer from an improper fraction to a mixed number:

[latex]\dfrac{5}{4} = 1\dfrac{1}{4}[/latex]

Exercise 5

Try these additions. Remember to always reduce!

  1. Visual represenation of the equation to solve.
    [latex]\dfrac{4}{6}+\dfrac{5}{6} =[/latex]
  2. Visual representation of the equation to solve.
    [latex]\dfrac{6}{8}+\dfrac{3}{8} =[/latex]
  3. Visual representation of the equation to solve.
    [latex]\dfrac{3}{4}+\dfrac{3}{4}=[/latex]
  4. Visual representation of the equation to solve.
    [latex]\dfrac{8}{9}+\dfrac{4}{9} =[/latex]
  5. Visual representation of the equation to solve.
    [latex]\dfrac{3}{5}+\dfrac{4}{5} =[/latex]

Exercise 5 Answers

  1. [latex]1\dfrac{1}{2}[/latex]
  2. [latex]1\dfrac{1}{8}[/latex]
  3. [latex]1\dfrac{1}{2}[/latex]
  4. [latex]1\dfrac{1}{3}[/latex]
  5. [latex]1\dfrac{2}{5}[/latex]

Example C

Sometimes, you will have to add 3 or more fractions together (see Figure 3.10.9).

 

Visual representation of the equation to solve.
Figure 3.10.9

[latex]\dfrac{2}{3} + \dfrac{1}{3} + \dfrac{2}{3} = \dfrac{5}{3} = 1\dfrac{2}{3}[/latex]

Example D

Visual representation of the equation to solve.
Figure 3.10.10

[latex]\dfrac{1}{4} + \dfrac{2}{4} + \dfrac{1}{4} + \dfrac{3}{4} = \dfrac{7}{4}[/latex]

Exercise 6

Add these common fractions. Be sure your answers are in the lowest terms.

Example:     [latex]\dfrac{2}{3} + \dfrac{1}{3} = \dfrac{3}{3} = 1[/latex]

 

  1. [latex]\dfrac{7}{10} + \dfrac{3}{10} =[/latex]
  2. [latex]\dfrac{3}{5} + \dfrac{2}{5} =[/latex]
  3. [latex]\begin{array}{rr}&\dfrac{3}{4}\\+&\dfrac{1}{4}\\ \hline&\end{array}[/latex]
  4. [latex]\begin{array}{rr}&\dfrac{5}{6}\\+&\dfrac{5}{6}\\ \hline&\end{array}[/latex]
  5. [latex]\begin{array}{rr}&\dfrac{4}{8}\\+&\dfrac{3}{8}\\ \hline&\end{array}[/latex]
  6. [latex]\begin{array}{rr}&\dfrac{1}{8}\\+&\dfrac{3}{8}\\ \hline&\end{array}[/latex]
  7. [latex]\begin{array}{rr}&\dfrac{2}{5}\\&\dfrac{3}{5}\\+&\dfrac{3}{5}\\ \hline&\end{array}[/latex]
  8. [latex]\begin{array}{rr}&\dfrac{3}{6}\\&\dfrac{1}{6}\\ +&\dfrac{1}{6}\\ \hline&\end{array}[/latex]

Exercise 6 Answers

  1. 1
  2. 1
  3. 1
  4. [latex]1\dfrac{2}{3}[/latex]
  5. [latex]\dfrac{7}{8}[/latex]
  6. [latex]\dfrac{1}{2}[/latex]
  7. [latex]1\dfrac{3}{5}[/latex]
  8. [latex]\dfrac{5}{6}[/latex]

Adding Mixed Numbers

To add mixed numbers:

  • Be sure the denominators are the same.
  • Add the common fractions.
  • Add the whole numbers.
  • Simplify the common fraction.

Example E

[latex]\begin{array}{rr}&3\dfrac{1}{8}\\+&2\dfrac{3}{8}\\ \hline\end{array}[/latex]

[latex]5\dfrac{4}{8}[/latex] = [latex]5\dfrac{1}{2}[/latex]

[latex]\dfrac{4}{8}[/latex] = [latex]\dfrac{4}{8}\left(\dfrac{÷4}{÷4}\right )[/latex] = [latex]\dfrac{1}{2}[/latex]

Example F

[latex]\begin{array}{rr}&12\dfrac{1}{3}\\+&6\dfrac{1}{3}\\\hline&18\dfrac{2}{3}\end{array}[/latex]

Exercise 7

Add the following numbers. Reduce the answers to the lowest terms.

Example:     [latex]\begin{array}{rr}&6\dfrac{1}{12}\\+&8 \dfrac{5}{12}\\ \hline \end{array}[/latex]     [latex]14 \dfrac{6}{12} = 14 \dfrac{1}{2}[/latex]

 

  1. [latex]\begin{array}{rr}&22\dfrac{1}{6}\\+&14\dfrac{6}{12}\\ \hline\end{array}[/latex]
  2. [latex]\begin{array}{rr}&8\dfrac{1}{4}\\+&3\dfrac{1}{4}\\ \hline\end{array}[/latex]
  3. [latex]\begin{array}{rr}&18\dfrac{1}{2}\\+&10\\ \hline\end{array}[/latex]
  4. [latex]\begin{array}{rr}&4\dfrac{1}{10}\\+&\dfrac{3}{10}\\ \hline\end{array}[/latex]

Exercise 7 Answers

  1. [latex]36\dfrac{1}{3}[/latex]
  2. [latex]11\dfrac{1}{2}[/latex]
  3. [latex]28\dfrac{1}{2}[/latex]
  4. [latex]4\dfrac{2}{5}[/latex]

Exercise 8

Add these numbers. Give your answers in the lowest terms.

Example:     [latex]\begin{array}{ll}&6\dfrac{4}{5}\\+&3\dfrac{2}{5}\\ \hline\end{array}[/latex]     [latex]9\dfrac{6}{5}=10\dfrac{1}{5}[/latex]

 

  1. [latex]\begin{array}{ll}&9\dfrac{1}{3}\\+&2\dfrac{2}{3}\\ \hline\end{array}[/latex]
  2. [latex]\begin{array}{ll}&3\dfrac{3}{8}\\+&12\dfrac{7}{8}\\ \hline\end{array}[/latex]
  3. [latex]\begin{array}{ll}&100\dfrac{7}{10}\\+&50\dfrac{5}{10}\\ \hline\end{array}[/latex]
  4. [latex]\begin{array}{ll}&3\dfrac{4}{7}\\+&6\dfrac{5}{7}\\ \hline\end{array}[/latex]
  5. [latex]\begin{array}{ll}&8\dfrac{4}{5}\\+&\dfrac{4}{5}\\ \hline\end{array}[/latex]

Exercise 8 Answers

  1. 12
  2. [latex]16\dfrac{1}{4}[/latex]
  3. [latex]151\dfrac{1}{5}[/latex]
  4. [latex]10\dfrac{2}{7}[/latex]
  5. [latex]12\dfrac{3}{5}[/latex]

The next question is:

What happens when two fractions in an addition (the addends) do not have the same denominator? If the addends do not have a common denominator, you will need to find equivalent fractions to make the addends have a common denominator.

Read on to find out how!

Multiples & Least Common Multiples (LCM)

When you learned the multiplication tables you learned the multiples of each number. Multiples are the answers when you multiply a whole number by 1, 2, 3, 4, 5, 6, 7, and so on.

Table 3.10.1 shows some multiples of 2 and 6.

Table 3.10.1: Multiples of 2 and 6
Multiplied By Multiples of 2 Multiples of 6
1 2 6
2 4 12
3 6 18
4 8 24
5 10 30
6 12 36
7 14 42
8 16 48
9 18 54
10 20 60
11 22 66
12 24 72

You can keep going as high as you want.

The multiples of 2 are 2, 4, 6, 8, 10, 12, 14, and so on.
The multiples of 6 are 6, 12, 18, 24, 30, 36, and so on.

Exercise 9

Create a chart of the first ten multiples of each number. This chart may be useful to you later.

Example: Multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18, 20

 

  1. 3
  2. 4
  3. 5
  4. 9
  5. 10
  6. 11
  7. 12

Exercise 9 Answers

Table 3.10.2: Exercise 9 Answers
Number × 2 × 3 × 4 × 5 × 6 × 7 × 8 × 9
 × 10
2 4 6 8 10 12 14 16 18 20
3 6 9 12 15 18 21 24 27 30
4 8 12 16 20 24 28 32 36 40
5 10 15 20 25 30 35 40 45 50
9 18 27 36 45 54 63 72 81 90
10 20 30 40 50 60 70 80 90 100
11 22 33 44 55 66 77 88 99 110
12 24 36 48 60 72 84 96 108 120
This is a quick method to find the least common multiple (LCM):
  • Least means smallest.
  • Common means shared.
  • Multiple means the answers when you multiply by 1, 2, 3, etc.

Example G

What is the least common multiple (LCM) of 3 and 5?

Multiples:

  • Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30
  • Multiples of 5: 5, 10, 15, 20, 25, 30…

The least common multiple of 3 and 5 is 15.

Example H

What is the LCM of 3 and 4?

Multiples:

  • Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30…
  • Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32 ….

The least common multiple of 3 and 4 is 12.

Example I

What is the LCM of 4 and 8?

Multiples:

  • Multiples of 4: 4, 8, 12, 16, 20…
  • Multiples of 8:  8, 16, 24, 32, 40…

The least common multiple of 4 and 8 is 8.

Hint: Always check to see if the larger number is a multiple of the smaller number. If it is, then the larger number is the Least Common Multiple (LCM).
  • LCM of 3 and 6 is 6
  • LCM of 2 and 4 is 4
  • LCM of 5 and 15 is 15

Exercise 10

Find the least common multiple (LCM) of these pairs of numbers.

Use your chart from Exercise 9 to help you. You may want to add the multiples of other numbers to that chart.

  1. 3,6
  2. 2,5
  3. 12, 3
  4. 6, 12
  5. 5, 4
  6. 4, 8
  7. 8, 16
  8. 4, 7
  9. 25, 5
  10. 2, 9
  11. 6, 10
  12. 8, 12

Exercise 10 Answers

  1. 6
  2. 10
  3. 12
  4. 12
  5. 20
  6. 8
  7. 16
  8. 28
  9. 25
  10. 18
  11. 30
  12. 24

Now that you know how to find an LCM, you can apply this knowledge to adding and subtracting fractions.

Least Common Denominator (LCD)

To find the Least Common Denominator (LCD) of common fractions, find the least common multiple of the denominators.

Example J

What is the least common denominator of [latex]\dfrac{1}{2}[/latex] and [latex]\dfrac{3}{4}[/latex]?

  • The denominators are 2 and 4.
  • The least common multiple of 2 and 4 is 4.

So, the least common denominator (LCD) for [latex]\dfrac{1}{2}[/latex] and [latex]\dfrac{3}{4}[/latex] is 4.

Example K

What is the LCD of [latex]\dfrac{3}{4}[/latex] and [latex]\dfrac{2}{3}[/latex]?

  • The denominators are 4 and 3.
  • The least common multiple of 4 and 3 is 12.

So, the least common denominator for [latex]\dfrac{3}{4}[/latex] and [latex]\dfrac{2}{3}[/latex] is 12.

Exercise 11

Find the Least Common Denominator (LCD) for these pairs of fractions.

Example:     [latex]\dfrac{5}{8}[/latex], [latex]\dfrac{2}{3}[/latex]
Denominators: 8, 3     LCD: 24

 

  1. [latex]\dfrac{1}{5}[/latex], [latex]\dfrac{1}{10}[/latex]
  2. [latex]\dfrac{1}{3}[/latex], [latex]\dfrac{3}{4}[/latex]
  3. [latex]\dfrac{2}{3}[/latex], [latex]\dfrac{1}{5}[/latex]
  4. [latex]\dfrac{5}{8}[/latex], [latex]\dfrac{1}{16}[/latex]

Exercise 11 Answers (only least common denominator is given)

  1. 10
  2. 12
  3. 15
  4. 16

You know how to find the least common denominator (LCD). The next step is to make equivalent fractions using the LCD.

Step 1: Find the least common denominator.

[latex]\begin{array}{rr}&\dfrac{3}{4}\\+&\dfrac{1}{3}\\ \hline\end{array}[/latex]

LCD of 4 and 3 is 12.

Step 2: Write an = sign after each fraction, followed by the common denominator.

[latex]\begin{array}{rrrr}&\dfrac{3}{4}=\dfrac{ }{12}\\+&\dfrac{1}{3} = \dfrac{ }{12}\\ \hline\end{array}[/latex]

Step 3: Rename the fractions as equivalent fractions with the LCD.

[latex]\dfrac{3}{4} = \dfrac{ }{12}[/latex]

4 times what = 12?

[latex]4\times 3 = 12[/latex]

If the denominator was multiplied by 3, the numerator must be multiplied by 3.

[latex]\dfrac{3}{4} \dfrac{×3}{×3} = \dfrac{9}{12}[/latex]

Now rename the other fraction.

[latex]\dfrac{1}{3} = \dfrac{ }{12}[/latex]

3 times what = 12?

[latex]3\times 4 = 12[/latex]

If this denominator was multiplied by 4, this numerator must be multiplied by 4.

[latex]\dfrac{1}{3} \dfrac{×4}{×4} = \dfrac{4}{12}[/latex]

Now, rename the other fraction.

Step 4: The question now looks like this and can be added.

[latex]\begin{array}{rrrr}&\dfrac{3}{4}&=&\dfrac{9}{12}\\ +&\dfrac{1}{3} &= &\dfrac{4}{12}\\ \hline \\ & \dfrac{13}{12}&=& 1\dfrac{1}{12}\end{array}[/latex]

Example L

[latex]\dfrac{1}{4} + \dfrac{3}{8}[/latex] =

Step 1 & 2: Find the least common denominator.

[latex]\begin{array}{ll}&\dfrac{1}{4} = \dfrac{ }{8}\\+&\dfrac{3}{8}= \dfrac{ }{8}\\ \hline\end{array}[/latex]

Step 3: Rename as equivalent fractions.

[latex]\begin{array}{ll}&\dfrac{1}{4}\left(\dfrac{\times2}{\times2}\right) = \dfrac{2}{8}\\+&\dfrac{3}{8}\left(\dfrac{\times1}{\times1}\right)= \dfrac{3}{8}\\ \hline\end{array}[/latex]

Step 4: Add and simplify the answer.

[latex]\begin{array}{lll}&\dfrac{1}{4}\left(\dfrac{×2}{×2}\right) = &\dfrac{2}{8}\\+&\dfrac{3}{8}\left(\dfrac{×1}{×1}\right)= &\dfrac{3}{8}\\ \hline&&\dfrac{5}{6}\end{array}[/latex]

Exercise 12

Add these common fractions. Express the answer in the lowest terms.

Examples:

  • [latex]\begin{array}{rr}&\dfrac{1}{2}\left(\dfrac{×4}{×4}\right) =&\dfrac{4}{8}\\+&\dfrac{3}{8}\left(\dfrac{×1}{×1}\right)= &\dfrac{3}{8}\\ \hline&&\dfrac{7}{8}\end{array}[/latex]
  • [latex]\begin{array}{rr}&\dfrac{1}{4}\left(\dfrac{×2}{×2}\right) =&\dfrac{2}{8}\\+&\dfrac{3}{8}\left(\dfrac{×1}{×1}\right)= &\dfrac{3}{8}\\ \hline&&\dfrac{5}{8}\end{array}[/latex]

 

  1. [latex]\begin{array}{rr}&\dfrac{1}{5}\\+&\dfrac{1}{10}\\ \hline&\end{array}[/latex]
  2. [latex]\begin{array}{rr}&\dfrac{5}{16}\\+&\dfrac{1}{4}\\ \hline&\end{array}[/latex]
  3. [latex]\begin{array}{rr}&\dfrac{1}{3}\\+&\dfrac{7}{12}\\ \hline&\end{array}[/latex]
  4. [latex]\begin{array}{rr}&\dfrac{2}{3}\\+&\dfrac{1}{6}\\ \hline&\end{array}[/latex]
  5. [latex]\begin{array}{rr}&\dfrac{3}{10}\\ +&\dfrac{2}{5}\\ \hline&\end{array}[/latex]
  6. [latex]\begin{array}{rr}&\dfrac{1}{12}\\ +&\dfrac{1}{4}\\ \hline&\end{array}[/latex]

Exercise 12 Answers

  1. [latex]\dfrac{3}{10}[/latex]
  2. [latex]\dfrac{9}{16}[/latex]
  3. [latex]\dfrac{11}{12}[/latex]
  4. [latex]\dfrac{5}{6}[/latex]
  5. [latex]\dfrac{7}{10}[/latex]
  6. [latex]\dfrac{1}{3}[/latex]

Exercise 13

More practice. Do only as many as you think you need.

Examples:

  • [latex]\begin{array}{rrr}&\dfrac{2}{3}\left(\dfrac{×4}{×4}\right) &=\dfrac{8}{12}\\&\dfrac{1}{2}\left(\dfrac{×6}{×6}\right)&=\dfrac{6}{12}\\+&\dfrac{3}{4}\left(\dfrac{×3}{×3}\right)& = \dfrac{9}{12}\\ \hline&&\dfrac{23}{12}&=1\dfrac{11}{12}\end{array}[/latex]
  • [latex]\begin{array}{rrr}&\dfrac{5}{24}\left(\dfrac{×1}{×1}\right)& =\dfrac{5}{24}\\&\dfrac{1}{3}\left(\dfrac{×8}{×8}\right)&= \dfrac{8}{24}\\+&\dfrac{3}{8}\left(\dfrac{×3}{×3}\right)&= \dfrac{9}{24}\\ \hline&&\dfrac{22}{24}&=1\dfrac{11}{12}\end{array}[/latex]

 

  1. [latex]\begin{array}{rr}&\dfrac{5}{12}\\&\dfrac{5}{6}\\+&\dfrac{3}{4}\\ \hline\end{array}[/latex]
  2. [latex]\begin{array}{rr}&\dfrac{3}{10}\\&\dfrac{3}{4}\\+&\dfrac{4}{5}\\ \hline\end{array}[/latex]
  3. [latex]\begin{array}{rr}&\dfrac{1}{2}\\&\dfrac{2}{5}\\+&\dfrac{7}{10}\\ \hline\end{array}[/latex]
  4. [latex]\begin{array}{rr}&\dfrac{5}{6}\\&\dfrac{3}{4}\\+&\dfrac{1}{3}\\ \hline\end{array}[/latex]
  5. [latex]\begin{array}{rr}&\dfrac{7}{16}\\+&\dfrac{3}{4}\\ \hline\end{array}[/latex]
  6. [latex]\begin{array}{rr}&\dfrac{4}{5}\\+&\dfrac{1}{3}\\ \hline\end{array}[/latex]

Exercise 13 Answers

  1. 2
  2. [latex]1\dfrac{17}{20}[/latex]
  3. [latex]1\dfrac{3}{5}[/latex]
  4. [latex]1\dfrac{1}{12}[/latex]
  5. [latex]1\dfrac{3}{16}[/latex]
  6. [latex]1\dfrac{2}{15}[/latex]

Addition questions are often written with the fractions side by side instead of one fraction above the other. For example:

[latex]\dfrac{2}{3} + \dfrac{5}{8} =[/latex]

You may solve as shown in this example or rewrite the question with the fractions one above the other.

[latex]\dfrac{2}{3} + \dfrac{5}{8} = \dfrac{2}{3}\dfrac{×8}{×8} + \dfrac{5}{8}\dfrac{×3}{×3}= \dfrac{16}{24}+ \dfrac{15}{24}=\dfrac{31}{24}= 1 \dfrac{7}{24}[/latex]

or

[latex]\begin{array}{rrr}&\dfrac{2}{3}\left(\dfrac{×8}{×8}\right) &=\dfrac{16}{24}\\&\dfrac{5}{8}\left(\dfrac{×3}{×3}\right)&=\dfrac{15}{24}\\ \hline&&\dfrac{31}{24}&=1\dfrac{7}{24}\end{array}[/latex]

 

Exercise 14

Find the sum. Do enough questions to be confident in your skill.

Example:     [latex]\begin{array}{rr} \\ \dfrac{1}{2} + \dfrac{1}{6} = &\\ \dfrac{1}{2}\left( \dfrac{\times 3}{\times 3}\right) + \dfrac{1}{6} = &\\ \dfrac{3}{6} + \dfrac{1}{6} = &\dfrac{4}{6} = \dfrac{2}{3} \end{array}[/latex]

 

  1. [latex]\dfrac{1}{4} + \dfrac{7}{8} =[/latex]
  2. [latex]\dfrac{1}{5} + \dfrac{3}{5} =[/latex]
  3. [latex]\dfrac{1}{12}+\dfrac{2}{3} =[/latex]
  4. [latex]\dfrac{1}{3} + \dfrac{2}{3} =[/latex]
  5. [latex]\dfrac{1}{6} + \dfrac{3}{8} =[/latex]
  6. [latex]\dfrac{3}{4} + \dfrac{1}{2} =[/latex]
  7. [latex]\dfrac{1}{3} + \dfrac{5}{8} =[/latex]

Exercise 14 Answers

  1. [latex]1\dfrac{1}{8}[/latex]
  2. [latex]\dfrac{4}{5}[/latex]
  3. [latex]\dfrac{3}{4}[/latex]
  4. 1
  5. [latex]\dfrac{13}{24}[/latex]
  6. [latex]1\dfrac{1}{4}[/latex]
  7. [latex]\dfrac{23}{14}[/latex]

You already know how to add mixed numbers which have the same (like) denominators.

To add mixed numbers with different denominators, you must:

  1. Find the least common denominator (LCD) for the fractions.
  2. Rename the fractions as equivalent fractions using the LCD
  3. Be sure to bring the whole number across the equal sign when you rename.
  4. Add the fractions.
  5. Add the whole numbers.
  6. Simplify the answer.
  7. Remember that if the sum of the fractions is an improper fraction, you must rename it as a mixed number that is added to the whole number in your answer.

Example M

[latex]\begin{array}{rr}&3\dfrac{3}{4}\left(\dfrac{\times5}{\times5}\right)=3\dfrac{15}{20} \\ +&6\dfrac{1}{5}\left(\dfrac{\times 4}{\times 4}\right)=6\dfrac{4}{20} \\ \hline \\ & =9\dfrac{19}{20}\end{array}[/latex]

Example N

[latex]\begin{array}{rr}&3\dfrac{3}{4}\left(\dfrac{\times3}{\times3}\right)=3\dfrac{3}{12} \\ &8\dfrac{2}{3}\left(\dfrac{\times 4}{\times 4}\right)=8\dfrac{8}{12} \\ +& 2 \dfrac{1}{2} \left(\dfrac{\times 6}{\times 6}\right) = 2 \dfrac{6}{12} \\ \hline \\ & =13\dfrac{17}{12}\end{array}[/latex]

[latex]\dfrac{17}{12}[/latex] is an improper fraction so we simplify it:

[latex]\dfrac{17}{12} = 1 \dfrac{5}{12}[/latex]

Therefore, the answer becomes:

[latex]13 \dfrac{17}{12} = 13 + 1 \dfrac{5}{12} = 14 \dfrac{5}{12}[/latex]

Exercise 15

Add. Express the sums in the lowest terms.

Example:     [latex]\begin{array}{rrrrr}&1\dfrac{3}{8}\left(\dfrac{\times1}{\times1}\right)&=&1\dfrac{3}{8}&\\+&1\dfrac{1}{4}\left(\dfrac{\times2}{\times2}\right)&=&1\dfrac{2}{8}&\\\hline&&&2\dfrac{5}{8} \end{array}[/latex]

 

  1. [latex]\begin{array}{rr}&3\dfrac{1}{5}\\+&2\dfrac{3}{10}\\ \hline&\end{array}[/latex]
  2. [latex]\begin{array}{rr}&6\dfrac{2}{15}\\+&1\dfrac{3}{10}\\ \hline&\end{array}[/latex]
  3. [latex]\begin{array}{rr}&8\dfrac{1}{4}\\+&4\dfrac{1}{3}\\ \hline&\end{array}[/latex]
  4. [latex]\begin{array}{rr}&5\dfrac{2}{3}\\+&6\dfrac{1}{4}\\ \hline&\end{array}[/latex]
  5. [latex]\begin{array}{rr}&116\dfrac{5}{8}\\+&9\dfrac{1}{24}\\ \hline&\end{array}[/latex]

Exercise 15 Answers

  1. [latex]5\dfrac{1}{2}[/latex]
  2. [latex]7\dfrac{13}{30}[/latex]
  3. [latex]12\dfrac{7}{12}[/latex]
  4. [latex]11\dfrac{11}{12}[/latex]
  5. [latex]125\dfrac{2}{3}[/latex]

Exercise 16

Add. Express the sums in the lowest terms.

Example:      [latex]\begin{array}{rrrrr}&4\dfrac{1}{2}\left(\dfrac{\times6}{\times6}\right)&=&4\dfrac{6}{12}&\\+&2\dfrac{1}{3}\left(\dfrac{\times4}{\times4}\right)&=&2\dfrac{4}{12}&\\\hline&&&6\dfrac{10}{12}&=6\dfrac{5}{6}\end{array}[/latex]

 

  1. [latex]\begin{array}{rr}&3\dfrac{2}{3}\\+&1\dfrac{1}{2}\\ \hline&\end{array}[/latex]
  2. [latex]\begin{array}{rr}&6\dfrac{1}{2}\\+&4\dfrac{1}{4}\\ \hline&\end{array}[/latex]
  3. [latex]\begin{array}{rr}&2\dfrac{1}{8}\\+&4\dfrac{3}{16}\\ \hline&\end{array}[/latex]
  4. [latex]\begin{array}{rr}&2\dfrac{1}{5}\\+&3\dfrac{2}{3}\\ \hline&\end{array}[/latex]
  5. [latex]\begin{array}{rr}&3\dfrac{3}{8}\\&2\dfrac{3}{4}\\+&1\dfrac{1}{2}\\\hline&\end{array}[/latex]
  6. [latex]\begin{array}{rr}&4\dfrac{3}{4}\\&2\dfrac{1}{5}\\+&4\dfrac{1}{2}\\ \hline&\end{array}[/latex]

Exercise 16 Answers

  1. [latex]5\dfrac{1}{6}[/latex]
  2. [latex]10\dfrac{3}{4}[/latex]
  3. [latex]6\dfrac{5}{16}[/latex]
  4. [latex]12\dfrac{7}{15}[/latex]
  5. [latex]7\dfrac{5}{8}[/latex]
  6. [latex]11\dfrac{9}{20}[/latex]

Problems Using Addition of Common Fractions

Exercise 17

Solve the word problems.

  1. The bathroom shelf is crowded with hand lotion bottles, each with a little lotion left inside. Everyone always likes to try the new bottle before the old one is emptied! One bottle is [latex]\dfrac{1}{3}[/latex] full, another is [latex]\dfrac{1}{4}[/latex] full, one is only [latex]\dfrac{1}{8}[/latex] full and one is still [latex]\dfrac{1}{2}[/latex] full. How much lotion is in the bottles altogether?
  2. Sometimes, Joan thinks she will go crazy when she packs the lunches for her family. Little Sarah has decided she only wants [latex]\dfrac{3}{4}[/latex] of a sandwich, Megan wants [latex]\dfrac{1}{4}[/latex] of a sandwich, Joan’s husband takes [latex]1\dfrac{1}{2}[/latex] sandwiches, and their son, who does heavy work, takes 3 sandwiches! How many sandwiches does Joan make?
  3. Dave paid the babysitter for the week. The sitter worked [latex]3\dfrac{3}{4}[/latex] hours on Monday, [latex]4\dfrac{1}{4}[/latex] hours on Tuesday and [latex]6\dfrac{1}{2}[/latex] hours on Friday. How many hours did the babysitter work looking after Dave’s children that week?
  4. Quite a lot of watermelon was left after the watermelon-eating contest: [latex]1\dfrac{1}{2}[/latex] watermelons on one table, [latex]2\dfrac{3}{4}[/latex] of a watermelon on another table and [latex]\dfrac{5}{8}[/latex] of a watermelon on the third table. The organizers want to know exactly how much was left over so they will not buy so much next year. Calculate the amount of watermelon left over.
  5. Jeanette has a novel to read for English. She read [latex]\dfrac{1}{2}[/latex] of the book on the weekend and only had time to read [latex]\dfrac{1}{8}[/latex] of the book on Monday and another [latex]\dfrac{1}{4}[/latex] on Wednesday. How much of the book has she read?
  6. Dion walks around this route each day for exercise. How far does he walk each day? Is this a perimeter or area question?Rectangle. 1 half kilometres by 1 and 2 thirds kilometres.
  7. How many metres of baseboard are needed for a rectangular room [latex]4\dfrac{1}{2}[/latex] m by [latex]3\dfrac{1}{5}[/latex] m? Deduct 1 m for the doorway. (TIP: Draw a picture)
  8. Sana is going to frame a large piece of art with a wooden frame. The art piece is [latex]1\dfrac{1}{10}[/latex] m by [latex]\dfrac{3}{5}[/latex] m. How much framing material should she buy?
  9. Find the perimeter of the following figure.
  10. Find the perimeter of a picture frame if one side is 12 1/10 cm and the other side measures 14 1/5 cm.
  11. Find the perimeter of this triangle.

Exercise 17 Answers

  1. 1 5/24 bottles total
  2. 5 1/2 sandwiches
  3. 14 1/2 hours
  4. 4 7/8 watermelons
  5. 7/8 of the book
  6. He walks 4 1/3 km each day, perimeter
  7. 14 2/5 m of material
  8. 3 2/5 m of material
  9. 15 2/3 cm
  10. 52 3/5 cm k) 17 11/24 cm

3.10: Practice Questions

  1. Add and express the answers in the lowest terms.
    1. [latex]\begin{array}{rr}&\dfrac{1}{4}\\+&\dfrac{3}{4}\\ \hline&\end{array}[/latex]
    2. [latex]\begin{array}{rr}&1\dfrac{3}{5}\\+&3\dfrac{4}{5}\\ \hline&\end{array}[/latex]
    3. [latex]\begin{array}{rr}&\dfrac{3}{8}\\+&\dfrac{3}{4}\\ \hline&\end{array}[/latex]
    4. [latex]\begin{array}{rr}&2\dfrac{1}{6}\\+&3\dfrac{5}{12}\\ \hline&\end{array}[/latex]
    5. [latex]\begin{array}{rr}&6\dfrac{3}{4}\\+&2\dfrac{1}{2}\\ \hline&\end{array}[/latex]
    6. [latex]\begin{array}{rr}&6\dfrac{7}{8}\\+&9\dfrac{1}{3}\\ \hline&\end{array}[/latex]

     

  2. Solve the following word problems.
    1. The flight from Vancouver to Sandspit took [latex]1\dfrac{1}{4}[/latex] hours. The wait in Sandspit was [latex]1\dfrac{1}{2}[/latex] hours, and the flight from there to Ketchikan, Alaska was [latex]\dfrac{3}{4}[/latex] of an hour. How long did it take to make the trip from Vancouver, BC to Ketchikan, Alaska?
    2. Dave built [latex]\dfrac{1}{8}[/latex] of the fence around his house on Monday, [latex]\dfrac{1}{4}[/latex] of it on Tuesday and another [latex]\dfrac{1}{4}[/latex] on Wednesday. How much of the fence has he built?
    3. John bought snacks in bulk for the class party. His items weighed [latex]\dfrac{2}{5}[/latex] kg of chips, [latex]\dfrac{3}{5}[/latex] kg of peanuts, [latex]\dfrac{1}{2}[/latex] kg of cheese and [latex]1\dfrac{1}{4}[/latex] kg of fresh veggies. How much did all his snacks weigh?
    4. Clarence is making a frame for his favourite photo. The frame needs to be [latex]\dfrac{1}{8}[/latex] m by [latex]\dfrac{5}{6}[/latex] m. How much material should he buy?

3.10: Practice Answers

  1. Add and express the answers in the lowest terms.
    1. 1
    2. [latex]5\dfrac{2}{5}[/latex]
    3. [latex]1\dfrac{1}{8}[/latex]
    4. [latex]5\dfrac{7}{12}[/latex]
    5. [latex]9\dfrac{1}{4}[/latex]
    6. [latex]16\dfrac{5}{24}[/latex]

     

  2. Solve the following word problems.
    1. [latex]3 \dfrac{1}{2}[/latex] hr
    2. [latex]\dfrac{5}{8}[/latex] of the fence
    3. [latex]2 \dfrac{3}{4}[/latex] kg of food
    4. [latex]1 \dfrac{11}{12}[/latex] m of material

Attribution

All figures in this chapter are from Topic A: Adding Common Fractions in Adult Literacy Fundamental Mathematics: Book 5 – 2nd Edition by Liz Girard, Wendy Tagami, and Leanne Caillier-Smith, via BCcampus.

This chapter has been adapted from Topic A: Adding Common Fractions in Adult Literacy Fundamental Mathematics: Book 5 – 2nd Edition (BCcampus) by Liz Girard, Wendy Tagami, and Leanne Caillier-Smith (2023), which is under a CC BY 4.0 license.

License

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Intermediate PreAlgebra: Building Success Copyright © 2024 by Kim Tamblyn, TRU Open Press is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

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