Simplify for m : `((2m+1/(2m))(3m-3/m)+3/(2m^2))/((m+1/m)(2m-1/(2m))+1/(2m^2))=39/19`

Options

A. 2
B. 4
C. 0
D. 3

Answer

To simplify the equation for \( m \):

\[ \frac{\left( \frac{2m + 1}{2m} \cdot \frac{3m - 3}{m} + \frac{3}{2m^2} \right)}{\left( \frac{m + 1}{m} \cdot \frac{2m - 1}{2m} + \frac{1}{2m^2} \right)} = \frac{39}{19} \]

### Step 1: Simplify the Numerator
The numerator is:

\[ \frac{2m + 1}{2m} \cdot \frac{3m - 3}{m} + \frac{3}{2m^2} \]

First, simplify \( \frac{2m + 1}{2m} \cdot \frac{3m - 3}{m} \):

1. \( (2m + 1)(3m - 3) = 6m^2 - 6m + 3m - 3 = 6m^2 - 3m - 3 \)
2. Divide by \( 2m \cdot m = 2m^2 \):
\[ \frac{6m^2 - 3m - 3}{2m^2} = 3 - \frac{3}{2m} - \frac{3}{2m^2} \]

Now add \( \frac{3}{2m^2} \):
\[ 3 - \frac{3}{2m} \]

### Step 2: Simplify the Denominator
The denominator is:

\[ \frac{m + 1}{m} \cdot \frac{2m - 1}{2m} + \frac{1}{2m^2} \]

1. Simplify \( \frac{m + 1}{m} \cdot \frac{2m - 1}{2m} \):
\[ \frac{(m + 1)(2m - 1)}{2m^2} = \frac{2m^2 - m + 2m - 1}{2m^2} = \frac{2m^2 + m - 1}{2m^2} \]
2. Add \( \frac{1}{2m^2} \):
\[ \frac{2m^2 + m - 1 + 1}{2m^2} = \frac{2m^2 + m}{2m^2} = 1 + \frac{m}{2m^2} = 1 + \frac{1}{2m} \]

### Step 3: Set Up the Equation
Now we have:

\[ \frac{3 - \frac{3}{2m}}{1 + \frac{1}{2m}} = \frac{39}{19} \]

### Step 4: Cross Multiply
Cross multiplying gives:

\[ 19(3 - \frac{3}{2m}) = 39(1 + \frac{1}{2m}) \]

Expanding both sides:

\[ 57 - \frac{57}{2m} = 39 + \frac{39}{2m} \]

### Step 5: Combine Like Terms
Combine terms involving \( m \):

\[ 57 - 39 = \frac{57}{2m} + \frac{39}{2m} \]
\[ 18 = \frac{96}{2m} \implies 18 = \frac{48}{m} \]

### Step 6: Solve for \( m \)
Cross multiplying gives:

\[ 18m = 48 \implies m = \frac{48}{18} = \frac{8}{3} \]

### Step 7: Final Answer
The value of \( m \) is:

\[ m = \frac{8}{3} \]