In the proof that ab+cd

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.