Let's break down the given expression step by step:

The expression is:

$\left( \frac{10 \cdot \sqrt{1.44}}{3} + \frac{24 \cdot \sqrt{0.25}}{7} \right) : \left( \frac{12}{7} + \frac{\sqrt{144}}{9} \right)$

Step 1: Simplifying the square roots

1. $\sqrt{1.44} = 1.2$
2. $\sqrt{0.25} = 0.5$
3. $\sqrt{144} = 12$

Step 2: Substitute the simplified square roots into the expression

$\left( \frac{10 \cdot 1.2}{3} + \frac{24 \cdot 0.5}{7} \right) : \left( \frac{12}{7} + \frac{12}{9} \right)$

Step 3: Simplifying the individual terms

Numerator:

$\frac{10 \cdot 1.2}{3} = \frac{12}{3} = 4$ $\frac{24 \cdot 0.5}{7} = \frac{12}{7}$

Thus, the numerator becomes:

$4 + \frac{12}{7}$

To combine these terms, express $4$ as $\frac{28}{7}$:

$4 + \frac{12}{7} = \frac{28}{7} + \frac{12}{7} = \frac{40}{7}$

Denominator:

$\frac{12}{7} + \frac{12}{9}$

We need to find a common denominator. The least common denominator (LCD) of 7 and 9 is 63. Now, rewrite each fraction:

$\frac{12}{7} = \frac{12 \cdot 9}{63} = \frac{108}{63}$ $\frac{12}{9} = \frac{12 \cdot 7}{63} = \frac{84}{63}$

Thus, the denominator becomes:

$\frac{108}{63} + \frac{84}{63} = \frac{192}{63}$

Step 4: Dividing the numerator by the denominator

Now, divide the simplified numerator by the denominator:

$\frac{\frac{40}{7}}{\frac{192}{63}} = \frac{40}{7} \times \frac{63}{192} = \frac{40 \cdot 63}{7 \cdot 192}$

Simplify:

$\frac{40 \cdot 63}{7 \cdot 192} = \frac{2520}{1344}$

Simplify the fraction by dividing both the numerator and denominator by their greatest common divisor (GCD), which is 48:

$\frac{2520}{1344} = \frac{2520 \div 48}{1344 \div 48} = \frac{52.5}{28}$

Thus, the final simplified result is:

$\frac{52.5}{28} \approx 1.875$

Final Answer:

$\boxed{1.875}$

Would you like further details or have any questions? Here are some related questions you can explore:

1. How do you simplify fractions involving square roots?
2. What are the common denominators for fractions with different denominators?
3. How do you simplify square roots in expressions?
4. How do you divide fractions?
5. What is the importance of finding the greatest common divisor when simplifying fractions?

Tip: Always try to simplify expressions step-by-step, breaking down complex operations into smaller, manageable parts!