Question

If $p,q$ and $r$ are positive real number, then the quantity $\frac{(p+r)}{(q+r)}$ is

• A. $>\frac{p}{q}$ if $p>q$
• B. $=\frac{p}{q}$ if $p>q$
• C. $>\frac{p}{q}$ if $p<q$
• D. $<\frac{p}{q}$ if $p>q$

Answer 1

The correct option is A $>\frac{p}{q}$ if $p>q$

Given:$p,q,r$ are positive real number

Case 1: If $p>q$

then $\frac{p}{q}>1$

$\Rightarrow p+r>q+r$

$\frac{p+r}{q+r}>1$

Let us assign some values

Let $p=2,q=1,r=3$

i) $\frac{p+r}{q+r}=\frac{5}{4}=1.25$

ii) $\frac{p}{q}=\frac{2}{1}=2$

$\Rightarrow \frac{p}{q}>\frac{p+r}{q+r}$ ...... $\left(1\right)$

Case 2: if $p<q$

$\frac{p}{q}<1$

$\Rightarrow p+r<q+r$

Let us assign some value

$p=1,q=2,r=3$

i)$\frac{p}{q}=\frac{1}{2}=0.5$

ii)$\frac{p+r}{q+r}=\frac{1+3}{2+3}=\frac{4}{5}=0.8$

$\Rightarrow \frac{p}{q}<\frac{p+r}{q+r}$ ...... $\left(2\right)$

From equation $\left(1\right)$ and $\left(2\right),$ we get

for $p>q$ ,$\frac{p}{q}<\frac{p+r}{q+r}$