`(m)/(2)-(3(m-4))/(5)=(5(3-m))/(6)`
What value o f m makes the equation above a true statements?

Options

A. `(8)/(27)`
B. `(3)/(22)`
C. `(62)/(27)`
D. `(147)/(22)`

Answer

To solve the equation \(\frac{m}{2} - \frac{3(m-4)}{5} = \frac{5(3-m)}{6}\), we will follow these steps:

### Step 1: Simplify the Left-Hand Side (LHS)
The LHS is given by:
\[ \frac{m}{2} - \frac{3(m-4)}{5} \]
To combine these fractions, we need a common denominator. The least common multiple of 2 and 5 is 10. Therefore, we can rewrite the fractions:
\[ \frac{m}{2} = \frac{5m}{10} \quad \text{and} \quad \frac{3(m-4)}{5} = \frac{6(m-4)}{10} \]
Now substituting these back into the LHS:
\[ \frac{5m}{10} - \frac{6(m-4)}{10} = \frac{5m - 6(m-4)}{10} \]

### Step 2: Distribute and Combine Like Terms
Now we distribute \(6\) in the second term:
\[ 5m - 6(m-4) = 5m - 6m + 24 = -m + 24 \]
Thus, the LHS simplifies to:
\[ \frac{-m + 24}{10} \]

### Step 3: Simplify the Right-Hand Side (RHS)
The RHS is given by:
\[ \frac{5(3-m)}{6} \]
We can leave this as is for now.

### Step 4: Set LHS Equal to RHS
Now we set the simplified LHS equal to the RHS:
\[ \frac{-m + 24}{10} = \frac{5(3-m)}{6} \]

### Step 5: Cross-Multiply
To eliminate the fractions, we can cross-multiply:
\[ 6(-m + 24) = 10 \cdot 5(3 - m) \]
This simplifies to:
\[ -6m + 144 = 150 - 50m \]

### Step 6: Rearrange the Equation
Now, we will move all terms involving \(m\) to one side and constant terms to the other:
\[ -6m + 50m = 150 - 144 \]
This simplifies to:
\[ 44m = 6 \]

### Step 7: Solve for \(m\)
Now, we divide both sides by 44:
\[ m = \frac{6}{44} = \frac{3}{22} \]

Thus, the value of \(m\) that makes the equation true is:
\[ \boxed{\frac{3}{22}} \]

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