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To determine which of the equations is an identity, we need to analyze each equation and simplify both sides. An equation is an **identity** if it is true for all values of \( x \). Let's go through each option:

### Option A:
\[
-5 \left( \frac{x}{15} - 16 \right) - 30 = 50 - \frac{1}{3}x
\]
Distribute the \(-5\):
\[
-5 \cdot \frac{x}{15} + 5 \cdot 16 - 30 = 50 - \frac{1}{3}x
\]
\[
- \frac{5x}{15} + 80 - 30 = 50 - \frac{1}{3}x
\]
Simplify:
\[
-\frac{x}{3} + 50 = 50 - \frac{1}{3}x
\]
Both sides are the same, so **Option A** is an identity.

### Option B:
\[
3 \left( \frac{x}{2} + 16 \right) - 16 = \frac{3}{2}x
\]
Distribute the 3:
\[
3 \cdot \frac{x}{2} + 3 \cdot 16 - 16 = \frac{3}{2}x
\]
\[
\frac{3x}{2} + 48 - 16 = \frac{3}{2}x
\]
Simplify:
\[
\frac{3x}{2} + 32 = \frac{3}{2}x
\]
The constants don't match, so **Option B** is **not** an identity.

### Option C:
\[
-4(3 - 2x) = -12 - 8x
\]
Distribute the \(-4\):
\[
-12 + 8x = -12 - 8x
\]
This simplifies to:
\[
8x - 12 = -12 - 8x
\]
Add 12 to both sides:
\[
8x = -8x
\]
This simplifies to \( 16x = 0 \), which is only true when \( x = 0 \). So, **Option C** is **not** an identity.

### Option D:
\[
36 \left( \frac{3}{4}x - 2 \right) + 72 = -72 + 27x
\]
Distribute the 36:
\[
36 \cdot \frac{3}{4}x - 36 \cdot 2 + 72 = -72 + 27x
\]
\[
27x - 72 + 72 = -72 + 27x
\]
Simplify:
\[
27x = -72 + 27x
\]
Subtract \( 27x \) from both sides:
\[
0 = -72
\]
This is false, so **Option D** is **not** an identity.

### Option E:
\[
\frac{9x}{15} + 27 = \frac{9x}{15} + \frac{27}{15}
\]
Simplify the fractions:
\[
\frac{3x}{5} + 27 = \frac{3x}{5} + \frac{9}{5}
\]
The constants \( 27 \) and \( \frac{9}{5} \) do not match, so **Option E** is **not** an identity.

### Conclusion:
The correct answer is **Option A**.

Would you like a detailed breakdown of any of the steps? 

Here are 5 related questions:
1. What is the general definition of an identity in algebra?
2. How do you solve equations involving fractions?
3. What are common methods for checking if an equation is an identity?
4. How do you simplify complex expressions with variables and constants?
5. What is the difference between an equation that is an identity and one that has specific solutions?

**Tip:** When simplifying equations, always check if both sides reduce to identical expressions to identify potential identities.

Which equation is an identity? Choose the correct answer from the options.

Analyze the given equations to identify which one is an identity. Detailed explanation covers key algebraic concepts like distributive property, simplifying equations, and verifying identities. Suitable for high school students (Grades 9-12).

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