In the proof that ab+cd

Question

, 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

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+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.

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.

Answer 1

To explain why \( \frac{ad + bc}{bd} \) is a rational number, we can refer to the properties of integers and rational numbers.

The correct reasoning is:

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 is accurate because:

1. \( a, b, c, \) and \( d \) are integers, and the product of any two integers is also an integer (Closure Property of Integers).
2. \( bd \) is the product of \( b \) and \( d \), which are both non-zero integers, so \( bd \) is also an integer and not zero.
3. \( ad + bc \) is a sum of two products of integers (and thus also an integer).
4. Therefore, \( \frac{ad + bc}{bd} \) is a valid expression for a rational number since it is the quotient of two integers, with a non-zero denominator, which satisfies the definition of a rational number.