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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page 11 of 11

Answer 1

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 \(q \neq 0\).