Solve for m: 150 + 75m + m/8 - m/3 = (900 - 5m/2)/12

To solve this problem, we'll follow these steps:

• Step 1: Simplify the right-hand side.
• Step 2: Work with fractions on the left-hand side.
• Step 3: Solve for $m$.

Let's work through each step:

Step 1: Simplify the right-hand side.
The right side of the equation is $\left( 900 - \frac{5m}{2} \right) \cdot \frac{1}{12}$. Distribute $\frac{1}{12}$ across the terms inside the parentheses:

$= 900 \cdot \frac{1}{12} - \frac{5m}{2} \cdot \frac{1}{12}$

$= \frac{900}{12} - \frac{5m}{24}$

$= 75 - \frac{5m}{24}$

So, the simplified equation becomes:

$150 + 75m + \frac{m}{8} - \frac{m}{3} = 75 - \frac{5m}{24}$

Step 2: Combine and simplify terms.
We will first find a common denominator for the fractions on the left side. The least common multiple of the denominators 8, 3, and 24 is 24. Convert each fraction to have this common denominator:

$\frac{m}{8} = \frac{3m}{24}$ and $\frac{m}{3} = \frac{8m}{24}$.

Rewrite the left-hand side:

$150 + 75m + \frac{3m}{24} - \frac{8m}{24}$

Combine the like terms:

$150 + 75m + \left(\frac{3m}{24} - \frac{8m}{24}\right)$

$= 150 + 75m - \frac{5m}{24}$

The equation becomes:

$150 + 75m - \frac{5m}{24} = 75 - \frac{5m}{24}$

Now add $\frac{5m}{24}$ to both sides to eliminate the fraction:

$150 + 75m = 75$

Step 3: Solve for $m$.
Subtract 150 from both sides:

$75m = 75 - 150$

$75m = -75$

Divide both sides by 75:

$m = -1$

Therefore, the solution to the problem is $m = -1$.