n the proof that ab+cd

Question

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, 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 quotient of imaginary numbers is a rational number.

GPT-4o mini · Answer

The correct explanation for why $ \frac{ad + bc}{bd} $ is a rational number 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. **Integer products:** Since $ a, b, c, $ and $ d $ are integers, the products $ ad $ and $ bc $ are also integers. Therefore, their sum $ ad + bc $ is an integer. 2. **Non-zero denominator:** Since $ b $ and $ d $ are both non-zero integers, their product $ bd $ is also a non-zero integer. 3. **Quotient of integers:** The expression $ \frac{ad + bc}{bd} $ is a fraction where both the numerator $ (ad + bc) $ and the denominator $ (bd) $ are integers, with the denominator not equal to zero. 4. **Rational numbers:** By definition, a rational number is any number that can be expressed as the quotient of two integers (where the denominator is not zero). Hence, $ \frac{ad + bc}{bd} $ fits this definition and is thus a rational number. The other options you've listed are either incorrect or not suitable explanations.