To solve the equation [tex]\(\frac{a}{b} \div \frac{c}{d} = \frac{7}{9}\)[/tex], let's break it down step-by-step:
1. Division of fractions: When you divide by a fraction, you multiply by its reciprocal. Therefore,
[tex]\[
\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a \cdot d}{b \cdot c}
\][/tex]
This means our equation can be rewritten as:
[tex]\[
\frac{a \cdot d}{b \cdot c} = \frac{7}{9}
\][/tex]
2. To find the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], [tex]\(c\)[/tex], and [tex]\(d\)[/tex] that satisfy this equation, we need to ensure:
[tex]\[
a \cdot d = 7 \times k \quad \text{and} \quad b \cdot c = 9 \times k
\][/tex]
Here, [tex]\(k\)[/tex] is a constant that ensures the ratios fit.
3. Given the numerical solution results, no valid combinations of [tex]\(a\)[/tex], [tex]\(b\)[/tex], [tex]\(c\)[/tex], and [tex]\(d\)[/tex] within the tested range (1 to 50) satisfy these expressions. This means that there were no appropriate assignments found within those bounds to satisfy our given fraction relation exactly as [tex]\( \frac{7}{9}\)[/tex].
Therefore, within the tested range, there were no values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], [tex]\(c\)[/tex], and [tex]\(d\)[/tex] that would satisfy the equation [tex]\(\frac{a}{b} \div \frac{c}{d} = \frac{7}{9}\)[/tex].
In summary, there are no integer solutions for [tex]\(a\)[/tex], [tex]\(b\)[/tex], [tex]\(c\)[/tex], and [tex]\(d\)[/tex] in the range considered that satisfy the given equation.
