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.

Answer 1

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.