Let's solve the expression step by step:

The expression is:

\[
\frac{\left( \frac{1}{6} - \frac{2}{5} \times \frac{15}{4} \right)}{\left( 2 \times \frac{1}{3} - \frac{1}{4} \right)}
\]

### Step 1: Simplify the numerator

The numerator is:

\[
\frac{1}{6} - \left( \frac{2}{5} \times \frac{15}{4} \right)
\]

First, simplify \( \frac{2}{5} \times \frac{15}{4} \):

\[
\frac{2 \times 15}{5 \times 4} = \frac{30}{20} = \frac{3}{2}
\]

Now subtract \( \frac{3}{2} \) from \( \frac{1}{6} \):

\[
\frac{1}{6} - \frac{3}{2}
\]

To subtract, find a common denominator (6 in this case):

\[
\frac{1}{6} = \frac{1}{6}, \quad \frac{3}{2} = \frac{9}{6}
\]

\[
\frac{1}{6} - \frac{9}{6} = \frac{-8}{6} = \frac{-4}{3}
\]

### Step 2: Simplify the denominator

The denominator is:

\[
2 \times \frac{1}{3} - \frac{1}{4}
\]

First, simplify \( 2 \times \frac{1}{3} \):

\[
2 \times \frac{1}{3} = \frac{2}{3}
\]

Now subtract \( \frac{1}{4} \) from \( \frac{2}{3} \):

Find a common denominator (12 in this case):

\[
\frac{2}{3} = \frac{8}{12}, \quad \frac{1}{4} = \frac{3}{12}
\]

\[
\frac{8}{12} - \frac{3}{12} = \frac{5}{12}
\]

### Step 3: Divide the numerator by the denominator

Now we have:

\[
\frac{\frac{-4}{3}}{\frac{5}{12}}
\]

Dividing fractions is the same as multiplying by the reciprocal:

\[
\frac{-4}{3} \times \frac{12}{5} = \frac{-4 \times 12}{3 \times 5} = \frac{-48}{15}
\]

Simplify the fraction:

\[
\frac{-48}{15} = \frac{-16}{5}
\]

### Final Answer:

\[
\boxed{\frac{-16}{5}}
\]

Do you want more details on any step or have further questions?

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Here are five related questions you could explore:

1. How do you find the reciprocal of a fraction?
2. Can you explain how to subtract fractions with different denominators?
3. How do you simplify a fraction after multiplication?
4. What is the method for dividing one fraction by another?
5. How can you simplify improper fractions into mixed numbers?

**Tip:** Always ensure that you simplify fractions as early as possible in a calculation to make the arithmetic easier.

Discover how to solve the fraction expression (1/6 - 2/5 * 15/4) ÷ (2 * 1/3 - 1/4) with a detailed step-by-step solution. This problem covers key concepts such as multiplying, subtracting, and dividing fractions, making it perfect for students in Grades 6-8.

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