To explain why \( \frac{ad + bc}{bd} \) is a rational number where \( a, b, c, \) and \( d \) are integers and \( b \) and \( d \) are non-zero, we should analyze the options provided and identify the correct reasoning based on properties of integers and rational numbers.
The correct explanation for why \( \frac{ad + bc}{bd} \) is a rational number is:
Option 2: By the Closure Property, \( ad + bc \) and \( bd \) are both quotients of integers, and so the fraction with numerator \( ad + bc \) and denominator \( bd \) is a quotient of two integers.
Reasoning:
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Definition of Rational Numbers: A rational number is defined as a number that can be expressed as \( \frac{p}{q} \) where \( p \) and \( q \) are integers and \( q \neq 0 \).
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Closure Property: The Closure Property for integers states that the sum, difference, and product of any two integers result in another integer. Thus:
- Since \( a, b, c, \) and \( d \) are integers, their products \( ad \) and \( bc \) are also integers.
- Additionally, the sum \( ad + bc \) is an integer because the sum of integers is an integer.
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Denominator \( bd \): The product \( bd \) is also an integer (product of two integers) and since both \( b \) and \( d \) are non-zero, \( bd \neq 0 \).
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Formation of a Rational Number: Since both \( ad + bc \) (the numerator) and \( bd \) (the denominator) are integers with the denominator not equal to zero, the fraction \( \frac{ad + bc}{bd} \) is indeed a rational number.
Therefore, Option 2 correctly explains how \( ad + bc \) and \( bd \) being integers leads to \( \frac{ad + bc}{bd} \) being a rational number.