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------------------------------------------------ Simplify the expression:

[tex]
\[
\begin{array}{l}
\frac{3 \times 6}{4 \times 6} + \frac{5 + 4}{6 + 4} = \frac{18}{24} + \frac{9}{10} \\
= \frac{3}{4} + \frac{9}{10} \\
= \frac{15}{20} + \frac{18}{20} \\
= \frac{33}{20} = 1\frac{13}{20}
\end{array}
\]
[/tex]

Note: Ensure all steps in calculations are shown and verified for accuracy.

Sure, let's go through the steps to solve the problem and understand how the result was obtained.

1. Calculate Each Fraction Separately:
- The first term is calculated as [tex]\(\frac{3 \times 6}{4 \times 6}\)[/tex].
- Simplifying that, we have [tex]\(\frac{18}{24}\)[/tex], which is equal to 0.75.

2. Calculate the Second Term:
- The second term is [tex]\(\frac{5+4}{6+4}\)[/tex].
- Simplifying, we get [tex]\(\frac{9}{10}\)[/tex], which is equal to 0.9.

3. Add the Two Terms Together:
- The combined value of the two terms is 0.75 + 0.9, which equals 1.65.

4. Combine Fractions and Simplify:
- Combining the fractions [tex]\(\frac{18}{24} + \frac{20}{24}\)[/tex], we focus on a common denominator.
- This gives us [tex]\(\frac{38}{24}\)[/tex], which simplifies to approximately 1.58333.

5. Express the Result as a Mixed Number:
- The next step is to express the total as a mixed number.
- This results in [tex]\(1 + \frac{14}{24}\)[/tex].
- The fractional part, [tex]\(\frac{14}{24}\)[/tex], simplifies to approximately 0.58333.

6. Simplifying Further:
- Finally, when you divide [tex]\(\frac{14}{24}\)[/tex] by 2, you get approximately 0.29167.

That's how we reach the solution with the series of calculations we performed!