In the proof that ab+cd

, where a, b, c, and d are integers and b and d are non-zero, explain why ad+bcbd
is a rational number.(1 point)
Responses

a. By the Closure Property, ad+bc
and bd
are both integers, and so ad+bcbd
is a quotient of two integers.
b. By the Closure Property, A d plus b c and b d are both integers, and so the fraction with numerator A d plus b c and denominator b d is a quotient of two integers.

c. By the Closure Property, ad+bc
and bd
are both quotients of integers, and so ad+bcbd
is a quotient of two integers.
d. By the Closure Property, A d plus b c and b d are both quotients of integers, and so the fraction with numerator A d plus b c and denominator b d is a quotient of two integers.

By the Closure Property, a quotient of imaginary numbers is a rational number.
By the Closure Property, a quotient of imaginary numbers is a rational number.

By the Closure Property, ad+bcbd
reduces to a whole number, which is a subset of rational numbers.
By the Closure Property, the fraction with numerator A d plus b c and denominator b d reduces to a whole number, which is a subset of rational numbers.

1 answer

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:

  1. 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.

  2. 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.