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, 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, a quotient of imaginary numbers is a rational number.

Answer 1

The correct response to explain why $\frac{ad+bc}{bd}$ is a rational number 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 is valid because:

• $a,b,c,$ and $d$ are all integers, and since the product and sum of integers are also integers, both $ad$ and $bc$ are integers.
• Since $b$ and $d$ are non-zero, $bd$ is also a non-zero integer.
• Therefore, $\frac{ad+bc}{bd}$ is a quotient of integers where the denominator is not zero, confirming that it is a rational number.