To determine whether the difference of the fractions [tex]\(\frac{14}{29} - \frac{21}{113}\)[/tex] represents a rational or irrational number, let's break down the problem step-by-step.
1. Identify the type of numbers:
- The fraction [tex]\(\frac{14}{29}\)[/tex] is a rational number because it can be expressed as the ratio of two integers, 14 and 29.
- Similarly, the fraction [tex]\(\frac{21}{113}\)[/tex] is also a rational number as it can be expressed as the ratio of two integers, 21 and 113.
2. Difference of rational numbers:
- A fundamental property of rational numbers is that the difference between two rational numbers is always a rational number. This is due to the fact that the set of rational numbers is closed under subtraction. If you subtract one rational number from another, the result will still be a rational number.
Given these points, we conclude the following:
- The number [tex]\(\frac{14}{29}\)[/tex] is a rational number.
- The number [tex]\(\frac{21}{113}\)[/tex] is a rational number.
- The difference [tex]\(\frac{14}{29} - \frac{21}{113}\)[/tex] is a rational number.
Thus, filling in the blanks:
The difference [tex]\(\frac{14}{29} - \frac{21}{113}\)[/tex] represents a rational number. This is because the number [tex]\(\frac{14}{29}\)[/tex] is rational and the number [tex]\(\frac{21}{113}\)[/tex] is rational. The difference of rational numbers is always rational.
