To explain why \( \frac{ad + bc}{bd} \) is a rational number when \( a, b, c, \) and \( d \) are integers and \( b \) and \( d \) are non-zero, we can break down the reasoning as follows:
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Identify the Components: In the expression \( \frac{ad + bc}{bd} \), we have:
- \( ad \): This is the product of two integers (since \( a \) and \( d \) are both integers), and thus \( ad \) is an integer.
- \( bc \): This is also the product of two integers (since \( b \) and \( c \) are integers), and thus \( bc \) is an integer.
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Sum of Integers: The numerator of our fraction is \( ad + bc \). Since both \( ad \) and \( bc \) are integers, and the sum of two integers is also an integer, \( ad + bc \) is an integer.
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Denominator: The denominator \( bd \) is the product of two non-zero integers \( b \) and \( d \). Since both \( b \) and \( d \) are non-zero, \( bd \) is also a non-zero integer.
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Quotient of Integers: The fraction \( \frac{ad + bc}{bd} \) is then a quotient of two integers, where the numerator \( ad + bc \) is an integer and the denominator \( bd \) is a non-zero integer.
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Definition of Rational Numbers: By definition, a number is rational if it can be expressed as the quotient of two integers where the denominator is not zero. Therefore, \( \frac{ad + bc}{bd} \) fits this definition because both the numerator and denominator are integers, and the denominator is non-zero.
Thus, \( \frac{ad + bc}{bd} \) is a rational number.
In summary, the correct explanation aligns with the principles of integer arithmetic and the definition of rational numbers, confirming that \( \frac{ad + bc}{bd} \) is indeed a rational number.
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