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Bot GPT 4-o mini 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.