If `(2)/(3) p-2 (1)/(2) = 3 (1)/(2)`, then the value of p is _______.

Options

A. `-9`
B. `6`
C. `9`
D. 0

Answer

To solve the equation \( \frac{2}{3} p - \frac{2}{1} = \frac{7}{2} \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ \frac{2}{3} p - \frac{2}{1} = \frac{7}{2} \]

### Step 2: Convert mixed numbers to improper fractions
Convert \( \frac{2}{1} \) and \( \frac{7}{2} \) into improper fractions:
\[ \frac{2}{1} = 2 = \frac{4}{2} \]
\[ \frac{7}{2} \text{ remains the same.} \]

### Step 3: Add \( \frac{4}{2} \) to both sides
To isolate \( \frac{2}{3} p \), we add \( \frac{4}{2} \) to both sides:
\[ \frac{2}{3} p = \frac{7}{2} + \frac{4}{2} \]

### Step 4: Combine the fractions on the right side
Now we add the fractions:
\[ \frac{2}{3} p = \frac{7 + 4}{2} = \frac{11}{2} \]

### Step 5: Multiply both sides by the reciprocal of \( \frac{2}{3} \)
To solve for \( p \), we multiply both sides by the reciprocal of \( \frac{2}{3} \), which is \( \frac{3}{2} \):
\[ p = \frac{11}{2} \times \frac{3}{2} \]

### Step 6: Multiply the fractions
Now we multiply:
\[ p = \frac{11 \times 3}{2 \times 2} = \frac{33}{4} \]

### Step 7: Convert to a mixed number (if necessary)
Convert \( \frac{33}{4} \) to a mixed number:
\[ 33 \div 4 = 8 \quad \text{(remainder 1)} \]
So, \( \frac{33}{4} = 8 \frac{1}{4} \).

### Final Answer
Thus, the value of \( p \) is:
\[ p = 8.25 \text{ or } 8 \frac{1}{4} \]

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