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)
1) By the Closure Property, ad+bc and bd are both quotients of integers, and so ad+bcbd is a quotient of two integers.
2) 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.
3) 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.
4) 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 explain why \( \frac{ad + bc}{bd} \) is a rational number where \( a, b, c, \) and \( d \) are integers and \( b \) and \( d \) are non-zero, we should analyze the options provided and identify the correct reasoning based on properties of integers and rational numbers.

The correct explanation for why \( \frac{ad + bc}{bd} \) is a rational number is:

**Option 2**: By the Closure Property, \( ad + bc \) and \( bd \) are both quotients of integers, and so the fraction with numerator \( ad + bc \) and denominator \( bd \) is a quotient of two integers.

### Reasoning:
1. **Definition of Rational Numbers**: A rational number is defined as a number that can be expressed as \( \frac{p}{q} \) where \( p \) and \( q \) are integers and \( q \neq 0 \).

2. **Closure Property**: The Closure Property for integers states that the sum, difference, and product of any two integers result in another integer. Thus:
- Since \( a, b, c, \) and \( d \) are integers, their products \( ad \) and \( bc \) are also integers.
- Additionally, the sum \( ad + bc \) is an integer because the sum of integers is an integer.

3. **Denominator \( bd \)**: The product \( bd \) is also an integer (product of two integers) and since both \( b \) and \( d \) are non-zero, \( bd \neq 0 \).

4. **Formation of a Rational Number**: Since both \( ad + bc \) (the numerator) and \( bd \) (the denominator) are integers with the denominator not equal to zero, the fraction \( \frac{ad + bc}{bd} \) is indeed a rational number.

Therefore, **Option 2** correctly explains how \( ad + bc \) and \( bd \) being integers leads to \( \frac{ad + bc}{bd} \) being a rational number.