If `1(2)/(3)+ ( 2)/( 7 ) xx ( x)/( 7) = 1( 1)/( 4) xx ( 2)/( 3) +( 1)/( 6) ` then find the value of x.

Options

A. 0.006
B. `(1)/( 6)`
C. 0.6
D. 6

Answer

To solve the equation \( 1\frac{2}{3} + \frac{2}{7} \cdot \frac{x}{7} = 1\frac{1}{4} \cdot \frac{2}{3} + \frac{1}{6} \), we will follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
Convert \( 1\frac{2}{3} \) and \( 1\frac{1}{4} \) to improper fractions.

\[ 1\frac{2}{3} = \frac{3 \cdot 1 + 2}{3} = \frac{5}{3} \]
\[ 1\frac{1}{4} = \frac{4 \cdot 1 + 1}{4} = \frac{5}{4} \]

### Step 2: Rewrite the Equation
Now rewrite the equation with the improper fractions:

\[ \frac{5}{3} + \frac{2}{7} \cdot \frac{x}{7} = \frac{5}{4} \cdot \frac{2}{3} + \frac{1}{6} \]

### Step 3: Simplify the Right Side
Calculate \( \frac{5}{4} \cdot \frac{2}{3} \):

\[ \frac{5 \cdot 2}{4 \cdot 3} = \frac{10}{12} = \frac{5}{6} \]

Now rewrite the equation:

\[ \frac{5}{3} + \frac{2x}{49} = \frac{5}{6} + \frac{1}{6} \]

### Step 4: Combine the Right Side
Combine \( \frac{5}{6} + \frac{1}{6} \):

\[ \frac{5}{6} + \frac{1}{6} = \frac{6}{6} = 1 \]

Now the equation looks like this:

\[ \frac{5}{3} + \frac{2x}{49} = 1 \]

### Step 5: Isolate the \( x \) Term
Subtract \( \frac{5}{3} \) from both sides:

\[ \frac{2x}{49} = 1 - \frac{5}{3} \]

### Step 6: Find a Common Denominator
To subtract \( 1 - \frac{5}{3} \), convert \( 1 \) to a fraction with a denominator of \( 3 \):

\[ 1 = \frac{3}{3} \]
\[ \frac{3}{3} - \frac{5}{3} = \frac{3 - 5}{3} = \frac{-2}{3} \]

### Step 7: Substitute Back
Now substitute back into the equation:

\[ \frac{2x}{49} = \frac{-2}{3} \]

### Step 8: Cross Multiply
Cross multiply to solve for \( x \):

\[ 2x \cdot 3 = -2 \cdot 49 \]
\[ 6x = -98 \]

### Step 9: Solve for \( x \)
Now divide both sides by \( 6 \):

\[ x = \frac{-98}{6} = \frac{-49}{3} \]

### Final Answer
Thus, the value of \( x \) is:

\[ \boxed{\frac{-49}{3}} \]