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

1 answer

The correct response to explain why \(\frac{ad + bc}{bd}\) is a rational number 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. Closure Property: This property states that the sum, difference, and product of integers is also an integer.
  2. Numerator: \(ad + bc\) is formed by multiplying integers \(a\), \(b\), \(c\), and \(d\), and then adding the results, thereby ensuring that \(ad + bc\) is indeed an integer.
  3. Denominator: The denominator \(bd\) is the product of integers \(b\) and \(d\). Since both \(b\) and \(d\) are non-zero integers, \(bd\) is also a non-zero integer.
  4. Quotient of two integers: A rational number is defined as any number that can be expressed in the form \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q \neq 0\). Since both \(ad + bc\) and \(bd\) are integers and \(bd\) is non-zero, \(\frac{ad + bc}{bd}\) satisfies the definition of a rational number.

Thus, this confirms that \(\frac{ad + bc}{bd}\) is indeed a rational number.