In the proof that ab+cd

Question

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

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+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.
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GPT-4o mini · Answer

To understand why $\frac{ad + bc}{bd}$ is a rational number, we can analyze its components based on the properties of integers and rational numbers. The correct response 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. **Integer Operations**: Since $a$, $b$, $c$, and $d$ are integers, the products $ad$ and $bc$ are also integers (since the product of integers is an integer). Thus, $ad + bc$ is the sum of two integers, which is also an integer. 2. **Denominator**: The product $bd$ is the product of two integers $b$ and $d$. Since $b$ and $d$ are both non-zero integers, $bd$ is also a non-zero integer. 3. **Quotient of Integers**: The expression $\frac{ad + bc}{bd}$ takes an integer ($ad + bc$) and divides it by another integer ($bd$). The division of one integer by another non-zero integer is defined to be a rational number. Thus, $\frac{ad + bc}{bd}$ is a rational number because it is a quotient of two integers, satisfying the definition of rational numbers as numbers that can be expressed as $ \frac{p}{q} $, where $p$ and $q$ are integers and $q \neq 0$.