Let's solve the equation step by step.

The given equation is:

\[
\frac{3}{2} \left( 4 + 12h \right) = \frac{1}{4} \left( 12 + 6h \right)
\]

### Step 1: Distribute the fractions on both sides

For the left side:

\[
\frac{3}{2} \times 4 = 6, \quad \frac{3}{2} \times 12h = 18h
\]

So the left-hand side becomes:

\[
6 + 18h
\]

For the right side:

\[
\frac{1}{4} \times 12 = 3, \quad \frac{1}{4} \times 6h = \frac{6h}{4} = \frac{3h}{2}
\]

So the right-hand side becomes:

\[
3 + \frac{3h}{2}
\]

Now, the equation is:

\[
6 + 18h = 3 + \frac{3h}{2}
\]

### Step 2: Eliminate the fraction by multiplying everything by 2

To get rid of the fraction \(\frac{3h}{2}\), multiply the entire equation by 2:

\[
2 \times \left( 6 + 18h \right) = 2 \times \left( 3 + \frac{3h}{2} \right)
\]

This simplifies to:

\[
12 + 36h = 6 + 3h
\]

### Step 3: Move all \(h\)-terms to one side

Subtract \(3h\) from both sides:

\[
12 + 36h - 3h = 6
\]

Simplifies to:

\[
12 + 33h = 6
\]

### Step 4: Isolate the \(h\)-term

Subtract 12 from both sides:

\[
33h = 6 - 12
\]

\[
33h = -6
\]

### Step 5: Solve for \(h\)

Divide both sides by 33:

\[
h = \frac{-6}{33} = -\frac{2}{11}
\]

Thus, the solution is:

\[
h = -\frac{2}{11}
\]

Would you like a detailed explanation of any step? 

Here are 5 related questions to expand your understanding:
1. How do you simplify an equation with fractions?
2. Why is it necessary to eliminate fractions by multiplying?
3. How do you check if

Learn how to solve a linear equation with fractions, step by step. This problem involves distributing fractions, eliminating fractional terms, and isolating the variable. Suitable for students in Grades 8-10.

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