Home > Pre-Algebra > Fractions > Adding Fractions and Mixed Numbers

Adding Fractions and Mixed Numbers

Introduction

In order to add fractions together, we find the highest common factor of the denominator, and add the equivalent numerators together.

Terms

Fraction - Fractions are used when using numbers to express parts of whole.
Least/Lowest Common Multiple - The lowest multiple of two or more numbers.
Equivalent Fraction - Equivalent fractions are fractions that have the same value.

Lesson

When adding fractions together, we have to make sure the denominators are the same. So if we are given 15+25 rac{1}{5} + rac{2}{5}, in this case the denominators are the same, so we simply add the two numerators together: 15+25=35dfrac{1}{5} + dfrac{2}{5} = dfrac{3}{5}

When the denominators are different, we ‘make’ them the same by finding their lowest common multiple. So in the case of 23+27 rac{2}{3} + rac{2}{7}, the LCM of 3 and 7 is 21. So we need to make the denominator of each fraction equal to 21.

23=x21=1421dfrac{2}{3} = dfrac{x}{21} = dfrac{14}{21}

27=x21=621dfrac{2}{7} = dfrac{x}{21} = dfrac{6}{21}

These fractions now have the same denominators, so we simply add the numerators

1421+921=2021dfrac{14}{21} + dfrac{9}{21} = dfrac{20}{21}

So we know that 23+27=2021dfrac{2}{3} + dfrac{2}{7} = dfrac{20}{21}

When adding mixed fractions, we follow the same procedure but we need to convert the mixed fraction to an improper fraction first.

So if we are given 213+122 rac{1}{3} + rac{1}{2}, we convert 2132 rac{1}{3} to the improper fraction 73 rac{7}{3}. Then we ‘make’ the denominators the same by finding the LCM, so we end up with:

73=146dfrac{7}{3} = dfrac{14}{6}

12=36dfrac{1}{2} = dfrac{3}{6}

Since the denominators are now the same, we add the numerators together:

146+36=176=256dfrac{14}{6} + dfrac{3}{6} = dfrac{17}{6} = 2 dfrac{5}{6}

Examples

23+12\dfrac{2}{3} + \dfrac{1}{2}
Since these fractions have different denominators, we need to find the least common multiple of the denominators
The least common multiple of 2 and 3 is 6, so we need to multiply to make each of the denominators = 6
23∗22=46\dfrac{2}{3} * \dfrac{2}{2} = \dfrac{4}{6}
12∗33=36\dfrac{1}{2} * \dfrac{3}{3} = \dfrac{3}{6}
Since these fractions have the same denominator, we can just add the numerators
46+36=76\dfrac{4}{6} + \dfrac{3}{6} = \dfrac{7}{6}
Because 76\dfrac{7}{6} is an improper fraction (the numerator is greater than the denominator), we need to convert it to a mixed number
76=116\dfrac{7}{6} = 1\dfrac{1}{6}

23+23\dfrac{2}{3} + \dfrac{2}{3}
Since these fractions have the same denominator, we can just add the numerators
23+23=43\dfrac{2}{3} + \dfrac{2}{3} = \dfrac{4}{3}
Because 43\dfrac{4}{3} is an improper fraction (the numerator is greater than the denominator), we need to convert it to a mixed number
43=113\dfrac{4}{3} = 1\dfrac{1}{3}

16+16\dfrac{1}{6} + \dfrac{1}{6}
Since these fractions have the same denominator, we can just add the numerators
16+16=26\dfrac{1}{6} + \dfrac{1}{6} = \dfrac{2}{6}
26\dfrac{2}{6} can be reduced, since 22 is a factor of both 22 and 66:
26÷22=13\dfrac{2}{6} \div \dfrac{2}{2} = \dfrac{1}{3}
The fraction is now in lowest terms

316+2163\dfrac{1}{6} + 2\dfrac{1}{6}
316+2163\dfrac{1}{6} + 2\dfrac{1}{6}
3+2=53 + 2 = 5
Since these fractions have the same denominator, we can just add the numerators
16+16=26\dfrac{1}{6} + \dfrac{1}{6} = \dfrac{2}{6}
26\dfrac{2}{6} can be reduced, since 22 is a factor of both 22 and 66:
26÷22=13\dfrac{2}{6} \div \dfrac{2}{2} = \dfrac{1}{3}
The fraction is now in lowest terms
5+13=5135 + \dfrac{1}{3} = 5\dfrac{1}{3}

Adding Fractions and Mixed Numbers Worksheets (PDF)

Adding Fractions Worksheet 1

Download PDF

Adding Fractions Worksheet 2

Download PDF

Adding Fractions Worksheet 3

Download PDF

67