In the proof that ab+cd

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+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, ad+bcbd
 reduces to a whole number, which is a subset of rational numbers.

GPT-4o mini · Answer

To explain why $ \frac{ad + bc}{bd} $ is a rational number given that $ a, b, c, $ and $ d $ are integers with $ b $ and $ d $ being non-zero, we can use the Closure Property of integers. The relevant 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.** ### Explanation: 1. **Integers and Closure Property**: The integers are closed under addition and multiplication. This means that when you add or multiply integers, you will always get an integer. 2. **Numerator**: The term $ ad + bc $ is formed by multiplying integers $ a $ and $ d $ to get $ ad $, and multiplying integers $ b $ and $ c $ to get $ bc $. Since both $ ad $ and $ bc $ are integers, their sum $ ad + bc $ is also an integer. 3. **Denominator**: The term $ bd $ is the product of two integers $ b $ and $ d $, which also results in an integer. 4. **Quotient of Integers**: Since both the numerator $ ad + bc $ and the denominator $ bd $ are integers, and since $ bd $ is non-zero (because both $ b $ and $ d $ are non-zero), we can conclude that $ \frac{ad + bc}{bd} $ is the quotient of two integers, which makes it a rational number. Therefore, $ \frac{ad + bc}{bd} $ is a rational number.