Decide whether the quotient represents a rational number or an irrational number. Explain how you know without simplifying.

[tex]\[ \frac{39}{139} \div \frac{9}{37} \][/tex]

The quotient [tex]\(\frac{39}{139} \div \frac{9}{37}\)[/tex] represents [tex]\(\square\)[/tex]. This is because the number [tex]\(\frac{39}{139}\)[/tex] is [tex]\(\square\)[/tex] and the number [tex]\(\frac{9}{37}\)[/tex] is [tex]\(\square\)[/tex]. The quotient of [tex]\(\square\)[/tex] is [tex]\(\square\)[/tex].



Answer :

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.