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)

Question

Responses

By the Closure Property, ad+bc and bd are both integers, and so ad+bcbd is a quotient of two integers.
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.

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.

By the Closure Property, ad+bc and bd are both quotients of integers, and so ad+bcbd is a quotient of two integers.
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.

Answer 1

The most appropriate response explaining why \( \frac{ad + bc}{bd} \) is a rational number is:

By the Closure Property, \( ad + bc \) and \( bd \) are both integers, and so the fraction with numerator \( ad + bc \) and denominator \( bd \) is a quotient of two integers.

This response correctly identifies that both the numerator and the denominator are integers, which means the entire expression \( \frac{ad + bc}{bd} \) is a rational number, as rational numbers are defined as the quotient of two integers (where the denominator is not zero).