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, 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+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.

Answer 1

To understand why \( \frac{ad + bc}{bd} \) is a rational number, we need to look at the properties of integers and rational numbers.

1. **Understanding Rational Numbers**: A rational number is defined as a number that can be expressed as the quotient \( \frac{p}{q} \), where \( p \) and \( q \) are integers, and \( q \) is not zero.

2. **Parts of the Expression**:
- \( ad \) is the product of two integers \( a \) and \( d \), thus \( ad \) is also an integer.
- \( bc \) is the product of two integers \( b \) and \( c \), so \( bc \) is also an integer.
- Therefore, \( ad + bc \) is the sum of two integers, which is again an integer.

3. **Denominator**:
- \( bd \) is the product of \( b \) and \( d \). Since both \( b \) and \( d \) are non-zero integers, \( bd \) is also a non-zero integer.

Now, putting this together:

- The numerator, \( ad + bc \), is an integer.
- The denominator, \( bd \), is a non-zero integer.

Thus, we can conclude that \( \frac{ad + bc}{bd} \) is a quotient of two integers where the denominator is not zero, which means it is a rational number.

Given the options you provided, the most accurate response is:

**"By the Closure Property, \( ad + bc \) and \( bd \) are both integers, and so \( \frac{ad + bc}{bd} \) is a quotient of two integers."**