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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}{b}# and #\frac{c}{d}# is\[\frac{a}{b}+\frac{c}{d} = \frac{a\cdot d+b\cdot c}{b\cdot d} \tiny.\]
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:
\[\frac{a}{b}-\frac{c}{d} = \frac{a\cdot d-b\cdot c}{b\cdot d} \tiny.\]
The fastest method does this by
- first determining the least common multiple of the two denominators: \(\mathrm{lcm}(5,13)=\frac{5\times 13}{\mathrm{gcd}(5,13)}=\frac{65}{1}=65\),
- next adjusting the numerators accordingly: \(\frac{3}{5}=\frac{3\times 13}{5\times 13}=\frac{39}{65}\) and \(\frac{3}{13}=\frac{3\times 5}{13\times 5}=\frac{15}{65}\).
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