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

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

c. By the Closure Property, ad+bc
and bd
are both quotients of integers, and so ad+bcbd
is a quotient of two integers.
d. 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 determine why $\frac{ad+bc}{bd}$ is a rational number, let's analyze the provided response options based on mathematical definitions and properties.

The key points are:

1. Closure Property: The Closure Property states that when performing certain operations (like addition or multiplication) within a set (like integers), the result will remain within that set. In this case, we are using integers.

2. Rational Numbers: A number is classified as rational if it can be expressed as the quotient of two integers (i.e., $\frac{p}{q}$, where $p$ and $q$ are integers and $q\ne 0$).

Now, examining the expressions in the options, we know:

• $ad$ and $bc$ are integers because $a,b,c,d$ are integers.
• $bd$ is non-zero because both $b$ and $d$ are non-zero.

This leads us to:

• The numerator $ad+bc$ is an integer (the sum of two integers).
• The denominator $bd$ is a non-zero integer.

Thus, the fraction $\frac{ad+bc}{bd}$ is formed as a quotient of two integers—specifically, an integer over a non-zero integer.

Correct Answer Reasoning:

Among the provided options:

a. 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 correctly captures the essence of why $\frac{ad+bc}{bd}$ is a rational number, emphasizing that both the numerator and denominator are integers, which is crucial for identifying the result as rational.

Conclusion: The correct response is:

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