Nested Fraction Evaluation: Solve for a:(b:c):(b·c)?

To solve this problem, we'll follow these steps:

• Interpret the mathematical notation correctly.
• Simplify the expression step-by-step.
• Compare the result to the provided answer choices.

Let's begin by interpreting the symbols in the expression $(a:(b:c)):(b\cdot c)$. The colon ":" signifies division. Thus, we can rewrite the expression as:

$\left(\frac{a}{\left(\frac{b}{c}\right)}\right) : (b \cdot c)$

Now, simplify the innermost expression $\frac{b}{c}$, and express the inverse as:

$\frac{a}{\frac{b}{c}} = a \cdot \frac{c}{b}$

This simplifies to:

$\frac{a \cdot c}{b}$

Substitute this result back into the original expression to get:

$\frac{\left(\frac{a \cdot c}{b}\right)}{b \cdot c}$

This further simplifies to:

$\frac{a \cdot c}{b} \cdot \frac{1}{b \cdot c}$

Canceling $c$ and rearranging the expression, we get:

$\frac{a}{b^2}$

Finally, let's verify which answer choice this computation corresponds to. Comparing our result to the provided answer choices:

• Choice 4: $\frac{a}{b^2}$ is correct.

Therefore, the solution to this problem is $\frac{a}{b^2}$.