Solve the Fractions Puzzle: Find X in 3/11 - 8/12x + 1/3x = 1/2 + 2/4 - 22/24x

To solve this problem, let's break down the equation step-by-step:

Start with the original equation:

$\frac{3}{11} - \frac{8}{12}x + \frac{1}{3}x = \frac{1}{2} + \frac{2}{4} - \frac{22}{24}x$

Step 1: Simplify the fractions where possible.

• $\frac{8}{12} = \frac{2}{3}$: Simplifying $\frac{8}{12}x$ gives us:$-\frac{2}{3}x$
• $\frac{2}{4} = \frac{1}{2}$
• $\frac{22}{24}$ simplifies to $\frac{11}{12}$: So, $-\frac{22}{24}x$ becomes $-\frac{11}{12}x$

The equation now looks like this:

$\frac{3}{11} - \frac{2}{3}x + \frac{1}{3}x = \frac{1}{2} + \frac{1}{2} - \frac{11}{12}x$

Step 2: Combine like terms.

• $-\frac{2}{3}x + \frac{1}{3}x = -\frac{1}{3}x$
• $\frac{1}{2} + \frac{1}{2} = 1$

So, the equation simplifies to:

$\frac{3}{11} - \frac{1}{3}x = 1 - \frac{11}{12}x$

Step 3: Move all terms involving $x$ to one side:

Add $\frac{1}{3}x$ to both sides:

$\frac{3}{11} = 1 - \frac{11}{12}x + \frac{1}{3}x$

Combine the terms with $x$:

• $-\frac{11}{12}x + \frac{1}{3}x = -\frac{11}{12}x + \frac{4}{12}x = -\frac{7}{12}x$

Thus, the equation is:

$\frac{3}{11} = 1 - \frac{7}{12}x$

Step 4: Isolate $x$:

Subtract 1 from both sides:

$\frac{3}{11} - 1 = -\frac{7}{12}x$

Convert $-1$ to a fraction with a common denominator to the left side:$\frac{3}{11} - \frac{11}{11} = \frac{3 - 11}{11} = -\frac{8}{11}$

Now the equation is:

$-\frac{8}{11} = -\frac{7}{12}x$

Multiply both sides by the reciprocal of $-\frac{7}{12}$ to solve for $x$:

$x = -\frac{8}{11} \cdot -\frac{12}{7} = \frac{8 \cdot 12}{11 \cdot 7} = \frac{96}{77}$

Thus, the solution to the equation is $\boxed{\frac{96}{77}}$.