In the proof that a/b+c/d

Question

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

A) By the Closure Property, ad+bc/bd
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.

B) 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.

C) By the Closure Property, ad+bc
and bd
are both integers, and so ad+bc/bd
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.

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

Answer 1

The correct response is:

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

Explanation: The operation of addition (ad + bc) yields an integer because the sum of integers is an integer. Additionally, the product (bd) is also an integer since the product of two integers is an integer. Since both the numerator (ad + bc) and the denominator (bd) are integers and the denominator is non-zero (as b and d are non-zero), the expression ad + bc / bd represents a quotient of two integers, which is the definition of a rational number. Thus, ad + bc / bd is indeed a rational number.