# Adding a different constant to numerator and denominator

Suppose that $a$ is less than $b$ , $c$ is less than $d$.

What is the relation between $\dfrac{a}{b}$ and $\dfrac{a+c}{b+d}$? Is $\dfrac{a}{b}$ less than, greater than or equal to $\dfrac{a+c}{b+d}$?

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Any possibility can occur. Take $a=1$ and $b=2$. Then consider $c=1, d=2$ (equality); $c=1, d=3$ (greater than ); and $c=4,d=5$ (less than). –  David Mitra Dec 16 '11 at 10:00
Note that if $b$ and $d$ have the same sign, then $$\frac{a}{b}-\frac{a+c}{b+d}=\frac{ad-bc}{b(b+d)}$$ and $$\frac{a+c}{b+d}-\frac{c}{d}=\frac{ad-bc}{d(b+d)}$$ also have the same sign.
Therefore, if $b$ and $d$ have the same sign, then $\dfrac{a+c}{b+d}$ is between $\dfrac{a}{b}$ and $\dfrac{c}{d}$.
Comment: As Srivatsan points out, if $b$ and $d$ are both positive, $$\frac{a}{b}\lesseqqgtr\frac{a+c}{b+d}\text{ if }\frac{a}{b}\lesseqqgtr\frac{c}{d}$$
My impression is that the OP might be most interested in the case when all the numbers are positive. In that case, it will be nice to point out explicitly that $\frac{a+c}{b+d}$ will be larger than $\frac{a}{b}$ if and only if $\frac{c}{d} > \frac{a}{b}$. –  Srivatsan Dec 16 '11 at 10:43