Riyaz and Ranbir are discussing rational numbers. Riyaz writes two fractions on the board:
\(a = \frac{7}{12}\) and \(b = \frac{5}{6}\)
Ranbir says that it is possible to find several rational numbers between \(a\) and \(b\) by expressing them with a common denominator. Riyaz wonders how many such numbers can be found and whether there is a condition that guarantees the existence of a given number of rational numbers between two fractions.
Ranbir says that it is possible to find several rational numbers between \(a\) and \(b\) by expressing them with a common denominator. Riyaz wonders how many such numbers can be found and whether there is a condition that guarantees the existence of a given number of rational numbers between two fractions.
Based on the above situation, answer the following questions:
1. Express \(a = \frac{7}{12}\) in the form of \(\frac{k_{1}}{m}\), where \(k_{1}\) and \(m\) are integers.
2. Express \(b = \frac{5}{6}\) in the form of \(\frac{k_{2}}{m}\), where \(k_{2}\) and \(m\) are integers.
3. Which among the following condition is necessay to find \(n\) distinct rational numbers lying
between \(a\) and \(b\) keeping an integer numerator.
between \(a\) and \(b\) keeping an integer numerator.
4. Using the same denominator \(m\), exactly how many distinct rational numbers lying between \(a\) and \(b\) keeping an integer numerator can be written.