Let's break this down step-by-step:

Expression:

$\frac{28}{15} \times (0.50)^2 \times 3 + \left( \frac{8}{15} - \frac{79}{60} \right) \div \frac{27}{47}$

Step 1: Calculate $(0.50)^2$

$(0.50)^2 = 0.25$

Step 2: Multiply $\frac{28}{15} \times 0.25 \times 3$

Start by multiplying: $\frac{28}{15} \times 0.25 = \frac{28 \times 0.25}{15} = \frac{7}{15}$ Then multiply by 3: $\frac{7}{15} \times 3 = \frac{21}{15} = 1.4$

Step 3: Simplify the second part $\frac{8}{15} - \frac{79}{60}$

To subtract these fractions, first find a common denominator. The least common denominator (LCD) of 15 and 60 is 60.

$\frac{8}{15} = \frac{32}{60}$ $\frac{8}{15} - \frac{79}{60} = \frac{32}{60} - \frac{79}{60} = \frac{32 - 79}{60} = \frac{-47}{60}$

Step 4: Divide $\frac{-47}{60} \div \frac{27}{47}$

Dividing by a fraction is the same as multiplying by its reciprocal: $\frac{-47}{60} \times \frac{47}{27} = \frac{-47 \times 47}{60 \times 27} = \frac{-2209}{1620}$

Step 5: Simplify $\frac{-2209}{1620}$

This fraction is already in its simplest form.

Step 6: Add the two parts together

Now we add the results from Step 2 and Step 5: $1.4 + \frac{-2209}{1620}$

To add these, convert $1.4$ into a fraction: $1.4 = \frac{7}{5}$

Now, we need a common denominator for $\frac{7}{5}$ and $\frac{-2209}{1620}$. The least common denominator is 1620.

$\frac{7}{5} = \frac{7 \times 324}{5 \times 324} = \frac{2268}{1620}$ $1.4 + \frac{-2209}{1620} = \frac{2268}{1620} + \frac{-2209}{1620} = \frac{2268 - 2209}{1620} = \frac{59}{1620}$

Final Answer:

$\frac{59}{1620}$

You can simplify this fraction further if needed, but it's already in a manageable form.

Would you like further explanation or have any questions about the steps?

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Here are 5 related questions to deepen your understanding:

1. How do you subtract fractions with different denominators?
2. What’s the easiest method to divide fractions?
3. How do you multiply fractions with decimal numbers involved?
4. How do you convert a decimal into a fraction and vice versa?
5. What is the concept of the least common denominator in fraction operations?

Tip: When working with fractions, always make sure to simplify intermediate steps and check for common denominators when adding or subtracting fractions.