	(1/6-3/2+5/2) x 4/3+4 x [2/3-(2/3 x 3/4)]

To solve the expression \((\frac{1}{6} - \frac{3}{2} + \frac{5}{2}) \times \frac{4}{3} + 4 \times \left[\frac{2}{3} - \left(\frac{2}{3} \times \frac{3}{4}\right)\right]\), we will break it down step by step.

### Step 1: Simplify the expression inside the first set of parentheses.
\[
\frac{1}{6} - \frac{3}{2} + \frac{5}{2}
\]
To simplify this, convert all fractions to have the same denominator (common denominator is 6):
\[
\frac{1}{6} - \frac{9}{6} + \frac{15}{6} = \frac{1 - 9 + 15}{6} = \frac{7}{6}
\]

### Step 2: Multiply the result by \(\frac{4}{3}\).
\[
\frac{7}{6} \times \frac{4}{3} = \frac{28}{18} = \frac{14}{9}
\]

### Step 3: Simplify the expression inside the brackets.
\[
\frac{2}{3} - \left(\frac{2}{3} \times \frac{3}{4}\right)
\]
First, calculate the product inside the brackets:
\[
\frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac{1}{2}
\]
Now subtract:
\[
\frac{2}{3} - \frac{1}{2}
\]
Find a common denominator (6):
\[
\frac{4}{6} - \frac{3}{6} = \frac{1}{6}
\]

### Step 4: Multiply the result by 4.
\[
4 \times \frac{1}{6} = \frac{4}{6} = \frac{2}{3}
\]

### Step 5: Add the results from Step 2 and Step 4.
\[
\frac{14}{9} + \frac{2}{3}
\]
Convert \(\frac{2}{3}\) to \(\frac{6}{9}\):
\[
\frac{14}{9} + \frac{6}{9} = \frac{20}{9}
\]

So, the final answer is \(\frac{20}{9}\).

Would you like further details or have any questions?

Here are 5 related questions for practice:
1. Simplify \((\frac{2}{5} - \frac{1}{3} + \frac{7}{10}) \times \frac{5}{4}\).
2. Calculate \(\frac{1}{4} \times \left(\frac{2}{3} + \frac{5}{6}\right)\).
3. Solve \((\frac{1}{8} + \frac{3}{4} - \frac{1}{2}) \times \frac{6}{5}\).
4. Simplify \(\frac{3}{7} - \left(\frac{2}{5} \times \frac{4}{7}\right)\).
5. Evaluate \(\left(\frac{1}{2} + \frac{2}{3}\right) \times 3 - \frac{5}{6}\).

**Tip:** Always remember to find a common denominator when adding or subtracting fractions!

Learn how to solve the mathematical expression (1/6 - 3/2 + 5/2) x (4/3) + 4 x [2/3 - (2/3 x 3/4)] step-by-step. This problem involves fractions and arithmetic operations, including addition, subtraction, multiplication, and division of fractions. Suitable for students in Grades 7-9.

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