ab=a(b+c)b(b+c)=ab+acb2+bc

a+cb+c=b(a+c)b(b+c)=ab+bcb2+bc

In here , we have :

b2 + bc = b2 + bc

ab = ab

If a>b

=> ac > bc (when a,b ∈N

)

=> ab>a+cb+c

If a<b

=> ac < bc (when a,b ∈N

)

=> ab<a+cb+c

We can see :

If \(\dfrac{a}{b}< 1\)=> a < b

              => ac < ab

              => a. ( b + c ) < b. ( c + a )

              => \(\dfrac{a}{b}< \dfrac{a+c}{b+c}\)

\(\dfrac{a}{b}=\dfrac{a\left(b+c\right)}{b\left(b+c\right)}=\dfrac{ab+ac}{b^2+bc}\)

\(\dfrac{a+c}{b+c}=\dfrac{b\left(a+c\right)}{b\left(b+c\right)}=\dfrac{ab+bc}{b^2+bc}\)

In here , we have :

b² + bc = b² + bc

ab = ab

If a>b

=> ac > bc (when a,b \(\in N\))

=> \(\dfrac{a}{b}>\dfrac{a+c}{b+c}\)

If a<b

=> ac < bc (when a,b \(\in N\))

=> \(\dfrac{a}{b}< \dfrac{a+c}{b+c}\)

There aren't enough condition !