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Let's break down the problem step-by-step.

### Given Expression:
\[
\frac{y}{4y - 48} \div \left( \frac{y}{y - 12} - \frac{y}{y + 12} \right)
\]

### Step 1: Simplify the First Fraction
The denominator of the first fraction is \(4y - 48\). We can factor out a 4:
\[
4y - 48 = 4(y - 12)
\]
So, the first fraction simplifies to:
\[
\frac{y}{4(y - 12)}
\]

### Step 2: Simplify the Second Expression
The second part is the subtraction of two fractions:
\[
\frac{y}{y - 12} - \frac{y}{y + 12}
\]

To subtract these fractions, find a common denominator:
\[
\text{Common denominator} = (y - 12)(y + 12)
\]
So, the subtraction becomes:
\[
\frac{y(y + 12) - y(y - 12)}{(y - 12)(y + 12)}
\]

### Step 3: Simplify the Numerator of the Subtraction
Expand the numerator:
\[
y(y + 12) - y(y - 12) = y^2 + 12y - (y^2 - 12y)
\]
Combine like terms:
\[
y^2 + 12y - y^2 + 12y = 24y
\]

So, the subtraction simplifies to:
\[
\frac{24y}{(y - 12)(y + 12)}
\]

### Step 4: Rewrite the Division
Now, the problem is:
\[
\frac{y}{4(y - 12)} \div \frac{24y}{(y - 12)(y + 12)}
\]

### Step 5: Multiply by the Reciprocal
To divide by a fraction, multiply by its reciprocal:
\[
\frac{y}{4(y - 12)} \times \frac{(y - 12)(y + 12)}{24y}
\]

### Step 6: Simplify the Expression
1. Cancel \(y\) in the numerator with \(y\) in the denominator:
\[
\frac{1}{4(y - 12)} \times \frac{(y - 12)(y + 12)}{24}
\]

2. Cancel \(y - 12\) in the numerator and denominator:
\[
\frac{1}{4} \times \frac{y + 12}{24}
\]

3. Multiply the fractions:
\[
\frac{1 \times (y + 12)}{4 \times 24} = \frac{y + 12}{96}
\]

### Final Answer:
\[
\frac{y + 12}{96}
\]

Would you like further details on this simplification or have any questions?

### Follow-Up Questions:
1. How do you factor common terms in polynomial expressions?
2. What is the rule for dividing fractions?
3. How do you find a common denominator for adding or subtracting fractions?
4. Why do we multiply by the reciprocal when dividing fractions?
5. How can the expression \(\frac{y}{4y - 48}\) be simplified further?

### Tip:
When dividing fractions, always remember to multiply by the reciprocal of the second fraction.

Perform the indicated operation, and simplify if possible: y/(4y - 48) ÷ (y/(y - 12) - y/(y + 12))

Learn how to simplify the complex rational expression y/(4y - 48) ÷ (y/(y - 12) - y/(y + 12)) step-by-step, covering core algebraic concepts like finding common denominators and using the reciprocal rule for division. Suitable for Grades 9-11.

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