Question

In the proof that a/b + c/d where a,b,c, and d are integers and b and d are non zero

Answer 1

To prove that the expression \(\frac{a}{b} + \frac{c}{d}\) where \(a\), \(b\), \(c\), and \(d\) are integers and \(b\) and \(d\) are non-zero results in a rational number, you can follow these steps:

1. **Establish the Expression**: We start with the expression

\[
\frac{a}{b} + \frac{c}{d}
\]

2. **Find a Common Denominator**: The common denominator of \(b\) and \(d\) is \(bd\). We can rewrite each fraction with this common denominator:

\[
\frac{a}{b} = \frac{a \cdot d}{b \cdot d}
\]
and
\[
\frac{c}{d} = \frac{c \cdot b}{d \cdot b}
\]

Thus, we can write the sum as:

\[
\frac{ad}{bd} + \frac{cb}{bd} = \frac{ad + cb}{bd}
\]

3. **Combine the Fractions**: Now we combine the fractions:

\[
\frac{ad + cb}{bd}
\]

4. **Check the Numerator and Denominator**: Here \(ad + cb\) is an integer because it is the sum of two integers. The denominator \(bd\) is also an integer, and since both \(b\) and \(d\) are non-zero, \(bd \neq 0\).

5. **Conclude Rationality**: Since the result \(\frac{ad + cb}{bd}\) is a fraction where both the numerator and the denominator are integers and the denominator is non-zero, the result is a rational number.

Thus, we conclude that:

\[
\frac{a}{b} + \frac{c}{d} \text{ is a rational number.}
\]