### Adding and Subtracting Rational Numbers Worksheets

We can only add or subtract rational numbers if and only if their denominators are the same. You can refer to our previous post regarding adding rational numbers worksheets by visiting this link.

Examples

Do the following operations.

\small \begin{align*} 1.\;\; \displaystyle \frac{2}{3}+\frac{4}{3}-\frac{1}{3}\end{align*}
In adding and subtracting two rational numbers having the same denominator just add or subtract the numerators and copy the common denominator. Thus,

\small \begin{align*} \quad \qquad \displaystyle \frac{2}{3}+\frac{4}{3}-\frac{1}{3}=\frac{2+4-1}{3}=\frac{5}{3}\end{align*}

\small \begin{align*} 2.\;\; &\displaystyle \frac{-8}{9}+\frac{5}{9}\\& \frac{-8}{9}+\frac{5}{9}=\frac{-8+5}{9}=\frac{-3}{9}=-\frac{1}{3} \qquad\textup{Since}\; \frac{-3}{9}\; \textup{is divisible by 3}\end{align*}
\small \begin{align*} 3.\;\; 3+0.213-1.2\end{align*}

Solution:

Since the given numbers are decimals follow the rules for adding or subtracting decimals. Thus,
\small \begin{align*} 3+0.213-1.2=3.213-1.2=2.013\end{align*}

\small \begin{align*} 4.\;\; \displaystyle \frac{2}{5}+\frac{1}{2}\end{align*}

Solution:

Since the denominators are not the same, multiply each number by the denominator of the other number. So,
\small \begin{align*} \displaystyle \frac{2}{5}\left ( \frac{2}{2} \right )+\frac{1}{2}\left ( \frac{5}{5} \right )=\frac{4}{10}+\frac{5}{10}=\frac{4+5}{10}=\frac{9}{10}\end{align*}

\small \begin{align*} 5.\;\;\displaystyle \frac{2}{3}-\frac{1}{3}\end{align*}

Solution:

In subtracting two rational numbers having the same denominator, we just need to subtract the numerators and copy the common denominator. Hence,
\small \begin{align*} \displaystyle \frac{2}{3}-\frac{1}{3}=\frac{2-1}{3}=\frac{1}{3}\end{align*}

\small \begin{align*} 6.\;\; \displaystyle \frac{4}{7}-\frac{6}{7}=\frac{4-6}{7}=\frac{-2}{7}\;\;\textup{or}\; -\frac{2}{7}\end{align*}
\small \begin{align*} 7.\;\; 0.252-0.131=0.121\end{align*}
\small \begin{align*} 8.\;\; \displaystyle \frac{3}{2}-\frac{6}{7}\end{align*}

Solution:

Same as adding rational numbers having different denominators, we need to multiply each numerator by the denominator of the other number then proceed to subtraction. Therefore,
\small \begin{align*} \displaystyle \frac{3}{2}-\frac{6}{7}=\frac{3}{2}\left ( \frac{7}{7} \right )-\frac{6}{7}\left ( \frac{2}{2}\right )=\frac{21}{4}-\frac{12}{4}=\frac{21-12}{14}=\frac{9}{14}. \end{align*}

\small \begin{align*} 9.\;\; \displaystyle \frac{2}{3}+\frac{4}{3}-\frac{7}{3} \end{align*}

Solution:

Since the three given numbers have the same denominators, we can directly do the indicated operations. Ergo,
\small \begin{align*} \displaystyle \frac{2}{3}+\frac{4}{3}-\frac{7}{3} =\frac{2+4-3}{7}=\frac{6-7}{3}=\frac{-1}{3}\; \textup{or}\; -\frac{1}{3}\end{align*}

\small \begin{align*} 10.\;\; \displaystyle \frac{3}{8}-\frac{9}{8}+\frac{4}{8} =\frac{3-9+4}{8}=\frac{-6+4}{8}=\frac{-2}{8}\;= -\frac{1}{4}\end{align*}

\small \begin{align*} 11.\;\; \displaystyle \frac{2}{3}+\frac{1}{6}-\frac{5}{2} \end{align*}

Solution:

In this example, first we need to find the LCD. Once we had the LCD, we express each number into a fraction having the LCD as the denominator. To do this, we must multiply the quotient of the LCD and the given denominator to the corresponding numerator.
\small \begin{align*} \displaystyle \frac{2}{3}+\frac{1}{6}-\frac{5}{2}&=\frac{\left ( 6\div 3 \right )\times 2}{6}+\frac{\left ( 6\div 6 \right )\times 1}{6}+\frac{\left ( 6\div 2 \right )\times 5}{6}\\&=\frac{2\times 2}{6}+\frac{1\times 1}{6}-\frac{3\times 5}{6}\\&= \frac{4}{6}+\frac{1}{6}-\frac{15}{6}\\&=\frac{4+1-15}{6}\\&=\frac{5-15}{6}=\frac{-10}{6}\div \frac{2}{2}=-\frac{5}{3}\end{align*}

\small \begin{align*}12.\;\; \displaystyle \frac{2}{5}-\frac{6}{3}-\frac{9}{15}&=\frac{16}{15}-\frac{30}{15}+\frac{9}{15}\\&=\frac{6-30+9}{15}\\&=\frac{-24+9}{15}=\frac{-15}{15}=-1\end{align*}