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A fraction is a number that can be written like p/q where p and q are integers and q ≠ 0. Such numbers are also called rational numbers.
Take any two fractions:
$$ rac{a}{b} quad ext{and} quad rac{c}{d}, qquad b
eq 0,; d
eq 0$$
To add them, make the denominators the same (use a common denominator bd):
$$ rac{a}{b} = rac{a cdot d}{b cdot d} = rac{ad}{bd}$$
$$ rac{c}{d} = rac{c cdot b}{d cdot b} = rac{cb}{bd}$$
Now add the two fractions:
$$ rac{ad}{bd} + rac{cb}{bd} = rac{ad + cb}{bd}$$
The result (dfrac{ad+cb}{bd}) is also of the form p/q (numerator and denominator are integers, denominator not zero). So the sum is a fraction (a rational number).
Example:
$$ rac{1}{3} + rac{1}{6} = rac{2}{6} + rac{1}{6} = rac{3}{6} = rac{1}{2}$$
Note: Sometimes the sum becomes a whole number, e.g.
$$ rac{1}{2} + rac{1}{2} = 1$$
A whole number is still a rational number because we can write it as a fraction with denominator 1 (here, (1 = rac{1}{1})). So the statement “The sum of two fractions is always a fraction” is true.