Question

Number of distinct rational numbers $x$ such that $0<x<1$ and $x=\frac{p}{q},$ where $p,q\in \{1,2,3,4,5,6\}$ is

• A. $15$
• B. $11$
• C. $12$
• D. $16$

Answer 1

The correct option is B $11$

As $0<x<1,$ we have $p<q.$
$p$ $q$

$1$	$2,3,4,5,6$
$2$	$3,4,5,6$
$3$	$4,5,6$
$4$	$5,6$
$5$	$6$

Thus, the number of rational numbers is $5+4+3+2+1=15$

When $p$ and $q$ have a common factor, we get some rational numbers, which are not different from those already counted.
Here, there are four such numbers:
$\frac{1}{2}=\frac{2}{4}=\frac{3}{6},\frac{1}{3}=\frac{2}{6},\frac{2}{3}=\frac{4}{6}.$

Therefore, the required number of rational numbers are $15-4=11$