n the proof that ab+cd

Question

n the proof that  ab+cd
 , 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+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, 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.

GPT-4o mini · Answer

The correct response to explain why $\frac{ad + bc}{bd}$ is a rational number would be: **By the Closure Property, $ad + bc$ and $bd$ are both integers, and so $\frac{ad + bc}{bd}$ is a quotient of two integers.** ### Explanation: 1. **Closure Property:** This property states that the sum, difference, and product of integers is also an integer. 2. **Numerator:** $ad + bc$ is formed by multiplying integers $a$, $b$, $c$, and $d$, and then adding the results, thereby ensuring that $ad + bc$ is indeed an integer. 3. **Denominator:** The denominator $bd$ is the product of integers $b$ and $d$. Since both $b$ and $d$ are non-zero integers, $bd$ is also a non-zero integer. 4. **Quotient of two integers:** A rational number is defined as any number that can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. Since both $ad + bc$ and $bd$ are integers and $bd$ is non-zero, $\frac{ad + bc}{bd}$ satisfies the definition of a rational number. Thus, this confirms that $\frac{ad + bc}{bd}$ is indeed a rational number.