In the proof that ab+cd

To explain why \( \frac{ad + bc}{bd} \) is a rational number, we can focus on the integer properties of the components involved.

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

**Explanation:**
- \( a \), \( b \), \( c \), and \( d \) are integers by the problem statement.
- The multiplication of integers \( a \) and \( d \) gives \( ad \), which is an integer.
- Similarly, the multiplication of integers \( b \) and \( c \) gives \( bc \), which is also an integer.
- Adding these two integer results, \( ad + bc \), yields another integer.
- \( b \) and \( d \) are non-zero integers, so their product \( bd \) is also a non-zero integer.
- Therefore, \( \frac{ad + bc}{bd} \) is a quotient of two integers (with the denominator being non-zero), which means it is a rational number.

This demonstrates that the expression is indeed rational based on the properties of integers and the definition of rational numbers.