Question

In the proof that a/b+c/d
, 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

By the Closure Property, a/d+b/c
and b/d
are both quotients of 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 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+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.

By the Closure Property, a/d+b/c
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.

By the Closure Property, a quotient of imaginary numbers is a rational number.

Answer 1

By the Closure Property, ad and bc are integers, so ad + bc is an integer, and bd is an integer (and bd ≠ 0). Thus (ad + bc)/(bd) is a quotient of two integers with nonzero denominator, hence a rational number.