Let’s solve the given expression step by step.

Problem:

The equation to solve is:

$A = -\frac{27}{4} : (-9) + \left\{ -\left(-\frac{14}{3}\right) \cdot \left(-\frac{5}{7}\right) - \left[-\frac{5}{12} : \left(\frac{1}{12} - \frac{2}{9}\right) - 3 \right] + \frac{1}{3} : (+5) \right\}$

Step 1: Simplify each term systematically.

1. $-\frac{27}{4} : (-9)$

Division becomes multiplication by the reciprocal: $-\frac{27}{4} : (-9) = -\frac{27}{4} \cdot \frac{-1}{9} = \frac{27}{36} = \frac{3}{4}$

2. Simplify the inner terms in the brackets:

$-\left(-\frac{14}{3}\right) = \frac{14}{3}$ $\frac{14}{3} \cdot \left(-\frac{5}{7}\right) = -\frac{14 \cdot 5}{3 \cdot 7} = -\frac{70}{21} = -\frac{10}{3}$

3. Handle the subtraction $\frac{1}{12} - \frac{2}{9}$:

Find the least common denominator (LCD): $\text{LCD of 12 and 9 is 36. Rewrite: } \frac{1}{12} = \frac{3}{36}, \frac{2}{9} = \frac{8}{36}.$ $\frac{1}{12} - \frac{2}{9} = \frac{3}{36} - \frac{8}{36} = -\frac{5}{36}.$

Now compute: $-\frac{5}{12} : \left(-\frac{5}{36}\right) = -\frac{5}{12} \cdot \frac{-36}{5} = \frac{36}{12} = 3.$

4. Combine with $-3$:

$3 - 3 = 0.$

5. Solve $\frac{1}{3} : (+5)$:

$\frac{1}{3} : 5 = \frac{1}{3} \cdot \frac{1}{5} = \frac{1}{15}.$

Step 2: Combine everything:

Substitute back into the original equation: $A = \frac{3}{4} + \left[ -\frac{10}{3} + 0 + \frac{1}{15} \right].$

Simplify the terms inside the brackets:

Find the LCD of 3 and 15, which is 15: $-\frac{10}{3} = -\frac{50}{15}, \quad \frac{1}{15} = \frac{1}{15}.$ $-\frac{50}{15} + \frac{1}{15} = -\frac{49}{15}.$

Add $\frac{3}{4}$ to the result:

Find the LCD of 4 and 15, which is 60: $\frac{3}{4} = \frac{45}{60}, \quad -\frac{49}{15} = -\frac{196}{60}.$ $\frac{45}{60} - \frac{196}{60} = -\frac{151}{60}.$

Final Answer:

$A = -\frac{151}{60}.$

Let me know if you'd like more clarification or further breakdown of any steps!

Related Questions:

1. How is the reciprocal of a fraction calculated, and why is it used in division?
2. Can you explain the significance of finding the least common denominator (LCD)?
3. How does distributing negative signs affect operations in a complex expression?
4. What is the strategy for combining fractions with different denominators?
5. How would you handle errors if this problem were calculated without using common denominators?

Tip:

Always break down complex expressions into smaller, manageable steps and simplify systematically to avoid mistakes!