The sum of two rational numbers is rational because of the way rational numbers are defined. A rational number is any number that can be expressed as the quotient of two integers, i.e., in the form $\frac{a}{b}$, where $a$ and $b$ are integers, and $b \neq 0$.

Let's break it down step by step:

Given:

Two rational numbers $\frac{a}{b}$ and $\frac{c}{d}$, where $a, b, c, d$ are integers and $b, d \neq 0$.

Step 1: Adding the two rational numbers:

$\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$ In this step, we find a common denominator (by multiplying the denominators $b$ and $d$), and the numerators are adjusted accordingly.

Step 2: Simplification:

$\frac{ad + bc}{bd}$ Here, the numerator is $ad + bc$, and the denominator is $bd$. Since $a, b, c, d$ are integers, both $ad + bc$ (the sum of two products of integers) and $bd$ (the product of two integers) are also integers.

Step 3: Conclusion:

Since the result can be written as the quotient of two integers, and the denominator $bd$ is not zero (because neither $b$ nor $d$ is zero), the result is a rational number.

Thus, the sum of two rational numbers is rational.

Do you want further clarification on any part of this explanation?

Here are 5 related questions to explore this concept further:

1. What happens when you subtract two rational numbers?
2. How do you multiply two rational numbers, and why is the result rational?
3. What is the sum of a rational and an irrational number, and why is it irrational?
4. Can a rational number ever have a repeating decimal form?
5. How do rational and irrational numbers differ in terms of their decimal expansions?

Tip: When adding or subtracting fractions, always remember to find the least common denominator to simplify calculations.