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, 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.

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

GPT-4o mini · Answer

To explain why $ \frac{ad + bc}{bd} $ is a rational number, let's analyze the responses provided. The correct reasoning ensures that both the numerator and denominator consist of integers, and thus the overall expression is a quotient of integers. The best option is: **"By the Closure Property, $ ad + bc $ and $ bd $ are both integers, and so $ \frac{ad + bc}{bd} $ is a quotient of two integers."** Here's the reasoning: 1. **Closure Property**: This property states that the sum or product of two integers is also an integer. - $ a, b, c, $ and $ d $ are integers. Therefore, the products $ ad $ and $ bc $ are integers (since the product of integers is an integer). - Consequently, their sum $ ad + bc $ is also an integer. - Since $ b $ and $ d $ are non-zero integers, their product $ bd $ is also an integer. 2. **Quotient of Integers**: The expression $ \frac{ad + bc}{bd} $ consists of an integer in the numerator ($ ad + bc $) and a non-zero integer in the denominator ($ bd $). Therefore, the expression is a quotient of two integers. Since the quotient of two integers (where the denominator is not zero) is defined to be a rational number, it follows that $ \frac{ad + bc}{bd} $ is indeed a rational number. The other options either misstate the property or contain unnecessary complexity. Thus, in this context, the chosen response accurately captures the essence of the reasoning required.