To determine whether the quotient [tex]\(\frac{39}{139} \div \frac{9}{37}\)[/tex] represents a rational number or an irrational number, we need to understand the nature of the division of two fractions.
Step-by-step solution:
1. Identify the given fractions:
[tex]\[
\frac{39}{139} \quad \text{and} \quad \frac{9}{37}
\][/tex]
2. Recall the rule for dividing fractions:
Division of fractions can be converted to multiplication by the reciprocal of the second fraction. Hence,
[tex]\[
\frac{39}{139} \div \frac{9}{37} = \frac{39}{139} \times \frac{37}{9}
\][/tex]
3. Multiply the numerators:
[tex]\[
39 \times 37 = 1443
\][/tex]
4. Multiply the denominators:
[tex]\[
139 \times 9 = 1251
\][/tex]
5. Form the new fraction:
[tex]\[
\frac{1443}{1251}
\][/tex]
6. Determine if the fraction represents a rational number:
A rational number is defined as any number that can be expressed as the quotient [tex]\(\frac{a}{b}\)[/tex] of two integers [tex]\(a\)[/tex] and [tex]\(b\)[/tex] (where [tex]\(b \neq 0\)[/tex]). In this case, both 1443 and 1251 are integers, and the denominator (1251) is not zero.
Therefore, [tex]\(\frac{1443}{1251}\)[/tex] is a rational number.
Putting this in a structured form:
The quotient [tex]\(\frac{39}{139} \div \frac{9}{37}\)[/tex] represents a rational number. This is because the number [tex]\(\frac{39}{139}\)[/tex] is a rational number and the number [tex]\(\frac{9}{37}\)[/tex] is also a rational number. The quotient of these two fractions, [tex]\(\frac{1443}{1251}\)[/tex], is also a rational number as it can be expressed as a quotient of two integers with a non-zero denominator.
