To determine if the quotient of [tex]\(\frac{10}{31} \div \frac{35}{127}\)[/tex] is a rational number or an irrational number, let's examine the division of these two fractions in detail.
Firstly, dividing by a fraction is equivalent to multiplying by its reciprocal. Therefore, we can rewrite the given expression as follows:
[tex]\[
\frac{10}{31} \div \frac{35}{127} = \frac{10}{31} \times \frac{127}{35}
\][/tex]
Next, let's multiply the fractions:
[tex]\[
\frac{10}{31} \times \frac{127}{35}
\][/tex]
To perform the multiplication, we multiply the numerators together and the denominators together:
[tex]\[
= \frac{10 \times 127}{31 \times 35}
\][/tex]
This yields:
[tex]\[
= \frac{1270}{1085}
\][/tex]
Now, let's consider whether [tex]\(\frac{1270}{1085}\)[/tex] is a rational number. By definition, a rational number is any number that can be expressed as the quotient of two integers, where the numerator and denominator are both integers and the denominator is not zero.
Here, both the numerator (1270) and the denominator (1085) are integers. Since the denominator is not zero, this quotient [tex]\(\frac{1270}{1085}\)[/tex] is indeed a rational number.
Therefore, the quotient of [tex]\(\frac{10}{31} \div \frac{35}{127}\)[/tex] represents a rational number. Based on the observed result:
[tex]\[
\frac{10}{31} \div \frac{35}{127} = 1.1705069124423961, \quad \text{and it's rational}.
\][/tex]
Conclusively, the quotient is a rational number.
