Question

How many positive rational numbers less than $1$ are there such that the sum of the numerator and denominator does not exceed $10$?

Answer 1

The positive rational numbers less than $1$ whose the sum of the numerator and denominator does not exceed $10$ are as follows:

$\frac{1}{9}=0.11,$ $\frac{1}{8}=0.12,$ $\frac{1}{7}=0.14,$ $\frac{1}{6}=0.16,$ $\frac{1}{5}=0.2,$ $\frac{1}{4}=0.25,$ $\frac{1}{3}=0.33,$ $\frac{1}{2}=0.5$

$\frac{2}{7}=0.28,$ $\frac{2}{5}=0.4,$ $\frac{2}{3}=0.66,$ $\frac{3}{7}=0.42,$ $\frac{3}{5}=0.6,$ $\frac{3}{4}=0.75,$ $\frac{4}{5}=0.8$

Hence, the above $15$ positive rational numbers are the numbers that are less than $1$ and the sum of the numerator and denominator does not exceed $10$.