1.02 Add and subtract rational numbers

When adding or subtracting rational numbers, it is helpful to write the numbers in the same form like we did when comparing rational numbers.

Let's review some of the key points for adding and subtracting with different types of rational numbers.

• Fractions: make sure that there is a common denominator. Once the denominators are equal, we add the numerators and the denominator remains the same.

  \displaystyle \frac{1}{3} + \frac{1}{4}	\displaystyle =	\displaystyle \frac{⬚}{12} + \frac{⬚}{12}	The least common denominator between 3 and 4 is 12.
  	\displaystyle =	\displaystyle \frac{1}{3}\cdot \frac{4}{4} + \frac{1}{4} \cdot \frac{3}{3}	Multiply to create common denominator
  	\displaystyle =	\displaystyle \frac{4}{12} + \frac{3}{12}	Evaluate the multiplication
  	\displaystyle =	\displaystyle \frac{7}{12}	Add the fractions and keep common denominator

• Mixed numbers: it is usually helpful to change any mixed numbers into improper fractions and then follow the same steps for adding fractions.

  \displaystyle 1\frac{2}{3} + 2\frac{1}{4}	\displaystyle =	\displaystyle \frac{5}{3} + \frac{9}{4}	Convert mixed numbers to improper fractions
  	\displaystyle =	\displaystyle \frac{5}{3}\cdot \frac{4}{4} + \frac{9}{4} \cdot \frac{3}{3}	Multiply to get least common denominator of 12.
  	\displaystyle =	\displaystyle \frac{20}{12} + \frac{27}{12}	Evaluate the multiplication
  	\displaystyle =	\displaystyle \frac{47}{12}	Add the fractions and keep common denominator
  	\displaystyle =	\displaystyle 3 \frac{11}{12}	Convert back to a mixed number

• Decimals: we can use a vertical algorithm to line up the decimals and their place values. Pay careful attention to whether a whole is gained during additon or lost during subtraction.

\quad\quad\,\,\text{Tens }\,\text{ Ones }. \text{ Tenths}\\ \begin{array}{c} & & & & & & 8 & . & 2&&&\\ + & & & & & & 3 & . & 5 & &\\ \hline & & & & 1 & & 1 & . & 7\\ \hline \end{array}

Using the number line or zero pairs may be helpful when working positive and negative rational numbers.

For example, adding -\dfrac{7}{10} to \dfrac{4}{10} is the same as subtracting \dfrac{7}{10}

On the number line, this is moving \dfrac{7}{10} units to the left of the \dfrac{4}{10}

This gives us: \dfrac{4}{10}+ \left(-\dfrac{7}{10}\right) = -\dfrac{3}{10}

It is helpful to understand how to interpret adjacent signs with rational numbers:

• Adding a positive integer means we move to the right, and simplify the problem to addition:2 + (+3) \quad = \quad 2 + 3
• Adding a negative decimal means we move to the left, and simplify the problem to subtraction:4.5 + (-1.5) \quad = \quad 4.5 - 1.5
• Subtracting a positive fraction means we move to the left, and simplify the problem to subtraction:\frac{5}{6} - \left(+\frac{1}{6}\right) \quad = \quad \frac{5}{6} - \frac{1}{6}
• Subtracting a negative mixed number means we move to the right, and simplify the problem to addition:3\frac{1}{4} - \left(-\frac{3}{4}\right) \quad = \quad 3\frac{1}{4} + \frac{3}{4}

Examples