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, 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.

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.

Answer 1

In the proof that \( \frac{ab + cd}{bd} \) is a rational number, the explanation for why \( \frac{ad + bc}{bd} \) (where \( a, b, c, d \) are integers and \( b, d \) are non-zero) is a rational number is captured well by the following response:

By the Closure Property, \( ad + bc \) and \( bd \) are both integers, and so \( \frac{ad + bc}{bd} \) is a quotient of two integers.

This response correctly notes that:

1. Since \( a, b, c, d \) are integers, the products \( ad \) and \( bc \) are also integers.
2. The sum \( ad + bc \) is an integer because the set of integers is closed under addition.
3. The product \( bd \) is a non-zero integer because both \( b \) and \( d \) are non-zero integers.
4. Therefore, since both the numerator \( ad + bc \) and the denominator \( bd \) are integers (with the denominator non-zero), the expression \( \frac{ad + bc}{bd} \) represents a quotient of two integers, which is defined to be a rational number.

The other options either confuse the property being applied (e.g., talking about imaginary numbers which aren't relevant here) or misstate the conclusion, so this response clearly and accurately addresses the proof's reasoning.