Question

In the proof that ab+cd
𝑎
𝑏
+
𝑐
𝑑
, where a, b, c, and d are integers and b and d are non-zero, explain why ad+bcbd
𝑎
𝑑
+
𝑏
𝑐
𝑏
𝑑
is a rational number.(1 point)
Responses

By the Closure Property, ad+bcbd
𝑎
𝑑
+
𝑏
𝑐
𝑏
𝑑
reduces to a whole number, which is a subset of rational numbers.
By the Clo

Answer 1

To explain why \( \frac{ad + bc}{bd} \) is a rational number, we can follow this reasoning:

The expression \( \frac{ad + bc}{bd} \) consists of two parts:

1. The numerator: \( ad + bc \)
2. The denominator: \( bd \)

### Step 1: Integers in the Numerator
- Since \( a \), \( b \), \( c \), and \( d \) are all integers, the products \( ad \) and \( bc \) are also integers because the product of integers is an integer.
- The sum of two integers \( ad + bc \) is also an integer.

### Step 2: Non-zero Denominator
- The denominator \( bd \) is a product of integers \( b \) and \( d \). Given that both \( b \) and \( d \) are non-zero, the product \( bd \) is also non-zero.

### Step 3: Rational Number Definition
- A rational number is defined as a number that can be expressed as the quotient of two integers, where the denominator is not zero.
- In this case, \( ad + bc \) is an integer (i.e., the numerator) and \( bd \) is a non-zero integer (i.e., the denominator), which fits the definition of a rational number.

### Conclusion
Therefore, \( \frac{ad + bc}{bd} \) is a rational number because it can be expressed as the ratio of two integers, where the denominator is non-zero.

Thus, the correct response to the question is:

**By the Closure Property, \( \frac{ad + bc}{bd} \) results in a whole number (in the numerator) divided by a non-zero integer (in the denominator), which is a rational number.**