Answer:
[tex]\dfrac{ad+bc}{bd}[/tex]
Step-by-step explanation:
Let [tex]\frac{a}{b}[/tex] and [tex]\frac{c}{d}[/tex] be two rational numbers, where b and d are not zero and a, b, c and d are integers.
1. Given:
[tex]\dfrac{a}{b}+\dfrac{c}{d}[/tex]
2. Multiply to get a common denominator :
[tex]\dfrac{a}{b}+\dfrac{c}{d}=\dfrac{ad}{bd}+\dfrac{cb}{db}[/tex]
3. Simplify:
[tex]\dfrac{a}{b}+\dfrac{c}{d}=\dfrac{ad}{bd}+\dfrac{cb}{db}=\dfrac{ad+bc}{bd}[/tex]
4. Since [tex]b\neq 0,\ d\neq 0,[/tex] then [tex]bd\neq 0.[/tex]
If [tex]a,b,c,d[/tex] are integers, then [tex]bd, ad,bc, ad+bc[/tex] are integers too. So the fraction
[tex]\dfrac{ad+bc}{bd}[/tex]
is a rational number
