To determine why the expression \( \frac{ad + bc}{bd} \) is a rational number, we should choose the correct response based on the properties of integers and rational numbers.
The key points are:
- \( a, b, c, d \) are integers.
- \( b \) and \( d \) are non-zero integers.
The expression \( ad + bc \) is the sum of two integers, and since integers are closed under addition, \( ad + bc \) is also an integer. The denominator \( bd \) is the product of two non-zero integers, which is also a non-zero integer.
Rational numbers are defined as the quotient of two integers (with a non-zero denominator). Thus, since both the numerator \( ad + bc \) and the denominator \( bd \) are integers (with the denominator non-zero), the result \( \frac{ad + bc}{bd} \) is indeed a rational number.
Given these points, the correct response would be:
"By the Closure Property, \( ad + bc \) and \( bd \) are both integers, and so \( \frac{ad + bc}{bd} \) is a quotient of two integers."
This choice emphasizes that both parts of the fraction are integers, leading to the conclusion that it is a rational number.
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