Numbers: Rational numbers
Addition and subtraction of fractions
Both the sum and the difference of two rational numbers is a rational number.
Addition of fractions
Let #a#, #b#, #m#, and #n# be integers with #m\ne0# and #n\ne0#.
A general formula for addition of the fractions #\frac{a}{m}# and #\frac{b}{n}# is\[\frac{a}{m}+\frac{b}{n} = \frac{a\cdot n+b\cdot m}{m\cdot n} \tiny.\]
In the case of common denominators, we have\[ \frac{a}{n}+\frac{b}{n}=\frac{a+b}{n}\tiny.\]
A common mistake is adding numerators without common denominators. For example: \[\frac{1}{2}+\frac{1}{3}\ne\frac{2}{5}\tiny.\]
A formula for the subtraction of fractions is obtained from the given formulas by replacing #c# by #-c# in the formula for addition of fractions:
\[\frac{a}{m}-\frac{b}{n} = \frac{a\cdot n-b\cdot m}{m\cdot n} \tiny.\]
The fastest method does this by
- first determining the least common multiple of the two denominators: \(\mathrm{lcm}(10,14)=\frac{10\times 14}{\mathrm{gcd}(10,14)}=\frac{140}{2}=70\),
- next adjusting the numerators accordingly: \(\frac{2}{10}=\frac{2\times 7}{10\times 7}=\frac{14}{70}\) and \(\frac{5}{14}=\frac{5\times 5}{14\times 5}=\frac{25}{70}\).
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