Add Fractions (mixed numbers)
The most difficult part of adding fractions is finding the common denominator. Even when you add mixed numbers, the hardest part is finding the common denominators. When you add fractions with mixed numbers, focus first on adding the fractions. Then add the whole numbers (it's just like regular addition, start with the digits to the right, and move to the left). Then, you just have to simplify and combine the fraction and whole number.
Example:
| $$\eqalign{2\dfrac{1}{6}+1\dfrac{3}{8}\\ \text{Write it vertically }&\left\{ \begin{array}{rcl}2 \dfrac{1}{6}&\\ +1\dfrac{3}{8}& \\\hline \qquad \qquad\\ \qquad \end{array}\right. \\\text{Find the common denominator: 24 }&\left\{ \begin{array}{rcl} 2\dfrac{1}{6}&=&&\dfrac{?}{24}\\+1\dfrac{3}{8}&=&&\dfrac{?}{24}\\\hline \qquad \qquad \\ \qquad \end{array}\right.\\\begin{array}{rcl}\text{Multiply each fraction by a fractional version of 1 }\\\text{that will make each denominator equal 24 }\end{array}&\left\{ \begin{array}{rcl}2\dfrac{1}{6}&\times&&\dfrac{4}{4}=&&&\dfrac{4}{24}\\+1\dfrac{3}{8}&\times&&\dfrac{3}{3}=&&&\dfrac{9}{24}\\ \hline \qquad \qquad \end{array}\right. \\\text{Add fractions and add whole numbers }&=\begin{array}&\quad && \;\; 3 &&& \quad &&&&\;\dfrac{13}{24}\end {array}}$$ |
In this case, the problem is done. $2\dfrac{1}{6}+1\dfrac{3}{8}=3\dfrac{13}{24}$
These problems get a little more complicated when the answer to the fraction side of the problem is an improper fraction. In those cases, students need to turn the improper fraction into a mixed number and then combine it with the whole number.
Example:
| $$\eqalign{1\dfrac{2}{3}+1\dfrac{5}{6}\\ \text{Write it vertically }&\left\{ \begin{array}{rcl}1 \dfrac{2}{3}&\\ +1\dfrac{5}{6}& \\\hline \qquad \qquad\\ \qquad \end{array}\right. \\\text{Find the common denominator: 6 }&\left\{ \begin{array}{rcl} 1\dfrac{2}{3}&=&&\dfrac{?}{6}\\+1\dfrac{5}{6}&=&&\dfrac{?}{6}\\\hline \qquad \qquad \\ \qquad \end{array}\right.\\\begin{array}{rcl}\text{Multiply each fraction by a fractional version of 1 }\\\text{that will make each denominator equal 6 }\end{array}&\left\{ \begin{array}{rcl}1\dfrac{2}{3}&\times&&\dfrac{2}{2}=&&&\dfrac{4}{6}\\+1\dfrac{5}{6}&\times&&\dfrac{1}{1}=&&&\dfrac{5}{6}\\ \hline \qquad \qquad \end{array}\right.\\\text{Add fractions and add whole numbers }&=\begin{array}&\quad && \;\; 2 &&& \quad &&&&\;\dfrac{7}{6}\end {array}}$$ |
In this case, you have a whole number and an improper fraction, so you have to convert the improper fraction into a mixed number and then combine it with the whole number (for more on converting improper fractions to mixed numbers see lesson Convert Improper Fractions to Mixed Numbers):
$$\eqalign{1\dfrac{2}{3}+1\dfrac{5}{6}&=2\dfrac{7}{6}\\&=2 + \dfrac{7}{6}\\&=2+1+\dfrac{1}{6}\\&=3\dfrac{1}{6}}$$
Overall, add mixed numbers the same way you add regular numbers; start from the right and move left. But, rather than carrying, find the fractional sum and the whole number sum and then combine them at the end.
Practice Problems:
Fraction Addition (Unlike Denominators, Mixed Numbers)
Find the sum. Simplify all answers completely. Change improper fractions to mixed numbers.
1. $4\dfrac{9}{10}+8\dfrac{1}{22}=$
2. $6\dfrac{6}{28}+8\dfrac{3}{4}=$
3. $4\dfrac{10}{50}+5\dfrac{14}{25}=$
4. $2\dfrac{8}{24}+6\dfrac{2}{3}=$
5. $4\dfrac{7}{14}+7\dfrac{6}{7}=$
6. $2\dfrac{3}{27}+9\dfrac{5}{9}=$
7. $3\dfrac{1}{4}+5\dfrac{1}{22}=$
8. $6\dfrac{20}{46}+9\dfrac{4}{23}=$
9. $5\dfrac{2}{3}+5\dfrac{1}{39}=$
10. $1\dfrac{16}{47}+5\dfrac{19}{94}=$
11. $1\dfrac{3}{11}+5\dfrac{20}{77}=$
12. $6\dfrac{3}{6}+8\dfrac{3}{4}=$