Addition and subtraction of fractions

Calculating with numbers: Calculating with fractions

Addition and subtraction of fractions

Addition and subtraction of fractions with the same denominator

Adding and subtracting two fractions with the same denominator is easy: the numerators are added or subtracted while the denominator is left unchanged. if possible, the answer is simplified.

Example

\[\begin{aligned}\tfrac{5}{14} +\tfrac{3}{14}&=\tfrac{5+3}{14}=\tfrac{8}{14}=\tfrac{4}{7} \\[0.2cm] \tfrac{5}{14} -\tfrac{3}{14}&=\tfrac{5-3}{14}=\tfrac{2}{14}=\tfrac{1}{7}\end{aligned}\]

Addition and subtraction of fractions with the different denominator

Adding and subtracting two fractions with the different denominators can be done via the following algorithm:

rewrite the rational numbers as fractions with the same denominator;
add or subtract these fractions;
if possible, simplify the answer.

Example

\[\begin{aligned}\tfrac{3}{10} +\tfrac{2}{15}&=\tfrac{\phantom{1}3\times 15}{10\times 15}+\tfrac{10\times 2\phantom{1}}{10\times 15}\\&=\tfrac{45}{150}+\tfrac{20}{150}=\tfrac{65}{150}=\tfrac{13}{30} \\[0.2cm] \tfrac{3}{10} -\tfrac{2}{15}&=\tfrac{\phantom{1}3\times 3}{10\times 3}-\tfrac{\phantom{1}2\times 2}{15\times 2}\\&=\tfrac{9}{30}-\tfrac{4}{30}=\tfrac{5}{30}=\tfrac{1}{6}\end{aligned}\]

Explanation of the first example The first example of adding fractions with different denominators illustrates a method that can always be applied to rewrite rational numbers as fractions with the same denominator, namely working with the product of the denominators. For two fractions \(\tfrac{a}{b}\) and \(\tfrac{c}{d}\), one multiplies the numerator and denominator of the first fraction by \(d\), and the numerator and denominator of the second fraction by \(b\). In this way one gets: \[\frac{a}{b}\pm\frac{c}{d}=\frac{a\times d}{b\times d}\pm\frac{b\times c}{b\times d}=\frac{a\times d\pm b\times c}{b\times d}\] The fraction obtained in this way is not always in its simplest form: sometimes the numerator and denominator of the result have a common divisor. Then it is neat to divide the numerator and denominator by this common divisor.

Explanation of the second example The second example of subtracting fractions with different denominators illustrates a method to work with fractions with the smallest common denominator, namely the use of the least common multiple (lcm) of the denominators. In this way one gets: \[\frac{a}{b}\pm\frac{c}{d}=\frac{a\times e}{b\times e}\pm\frac{c\times f}{d\times f}=\frac{a\times e\pm c\times f}{k}\] met \[k=\mathrm{lcm}(b,d),\quad e=\frac{k}{b},\quad f=\frac{k}{d}\]

The examples below with two or three fractions illustrate the computational work.

Calculate \(\frac{9}{10}+\frac{4}{13}\) and simplify the answer as much as possible.

The least common multiple of the denominators \(10\) and \(13\) is equal to \(130\).
We use it to rewrite the fractions as fractions with a common denominator. \[\frac{9}{10}=\frac{9\times 13}{10\times 13}=\frac{117}{130}\quad\text{and}\quad\frac{4}{13}=\frac{4\times 10}{13\times 10}=\frac{40}{130}\] Then the computation is easy: \[\begin{aligned}\frac{9}{10}+\frac{4}{13} &=\frac{117}{130}+\frac{40}{130} &\blue{\text{construction of fractions with a common denominator}} \\ \\ &=\frac{117+40}{130} &\blue{\text{collection of numerators}} \\ \\ &={{157}\over{130}}&\blue{\text{calculation and sometimes simplification}} \end{aligned}\]

New example

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Addition and Subtraction (22:32)