The correct explanation for why \( \frac{ad + bc}{bd} \) is a rational number is:
By the Closure Property, \( ad + bc \) and \( bd \) are both integers, and so \( \frac{ad + bc}{bd} \) is a quotient of two integers.
Explanation:
- Integer products: Since \( a, b, c, \) and \( d \) are integers, the products \( ad \) and \( bc \) are also integers. Therefore, their sum \( ad + bc \) is an integer.
- Non-zero denominator: Since \( b \) and \( d \) are both non-zero integers, their product \( bd \) is also a non-zero integer.
- Quotient of integers: The expression \( \frac{ad + bc}{bd} \) is a fraction where both the numerator \( (ad + bc) \) and the denominator \( (bd) \) are integers, with the denominator not equal to zero.
- Rational numbers: By definition, a rational number is any number that can be expressed as the quotient of two integers (where the denominator is not zero). Hence, \( \frac{ad + bc}{bd} \) fits this definition and is thus a rational number.
The other options you've listed are either incorrect or not suitable explanations.