To determine why \( \frac{ad + bc}{bd} \) is a rational number, let's analyze the provided response options based on mathematical definitions and properties.
The key points are:
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Closure Property: The Closure Property states that when performing certain operations (like addition or multiplication) within a set (like integers), the result will remain within that set. In this case, we are using integers.
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Rational Numbers: A number is classified as rational if it can be expressed as the quotient of two integers (i.e., \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \)).
Now, examining the expressions in the options, we know:
- \( ad \) and \( bc \) are integers because \( a, b, c, d \) are integers.
- \( bd \) is non-zero because both \( b \) and \( d \) are non-zero.
This leads us to:
- The numerator \( ad + bc \) is an integer (the sum of two integers).
- The denominator \( bd \) is a non-zero integer.
Thus, the fraction \( \frac{ad + bc}{bd} \) is formed as a quotient of two integers—specifically, an integer over a non-zero integer.
Correct Answer Reasoning:
Among the provided options:
a. By the Closure Property, \( ad + bc \) and \( bd \) are both integers, and so \( \frac{ad + bc}{bd} \) is a quotient of two integers.
This statement correctly captures the essence of why \( \frac{ad + bc}{bd} \) is a rational number, emphasizing that both the numerator and denominator are integers, which is crucial for identifying the result as rational.
Conclusion: The correct response is:
a. By the Closure Property, \( ad + bc \) and \( bd \) are both integers, and so \( \frac{ad + bc}{bd} \) is a quotient of two integers.