If `1 (2)/(3) div (2)/(7) xx (x)/(7) = 1 (1)/(4) xx (2)/(3) div (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 \( 1 \frac{2}{3} \div \frac{2}{7} \times \frac{x}{7} = 1 \frac{1}{4} \times \frac{2}{3} \div \frac{1}{6} \), we will follow these steps:

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

- For \( 1 \frac{2}{3} \):
\[ 1 \frac{2}{3} = \frac{3 \times 1 + 2}{3} = \frac{5}{3} \]

- For \( 1 \frac{1}{4} \):
\[ 1 \frac{1}{4} = \frac{4 \times 1 + 1}{4} = \frac{5}{4} \]

### Step 2: Rewrite the Equation
Now, substitute the improper fractions back into the equation:
\[ \frac{5}{3} \div \frac{2}{7} \times \frac{x}{7} = \frac{5}{4} \times \frac{2}{3} \div \frac{1}{6} \]

### Step 3: Simplify the Left Side
To simplify the left side, we can rewrite the division as multiplication by the reciprocal:
\[ \frac{5}{3} \times \frac{7}{2} \times \frac{x}{7} \]
The \( 7 \) in the numerator and denominator cancels out:
\[ \frac{5}{3} \times \frac{x}{2} \]

### Step 4: Simplify the Right Side
Now simplify the right side:
\[ \frac{5}{4} \times \frac{2}{3} \div \frac{1}{6} = \frac{5}{4} \times \frac{2}{3} \times 6 \]
This can be simplified as:
\[ \frac{5 \times 2 \times 6}{4 \times 3} = \frac{60}{12} = 5 \]

### Step 5: Set Up the Equation
Now we have:
\[ \frac{5x}{6} = 5 \]

### Step 6: Solve for \( x \)
To isolate \( x \), multiply both sides by \( 6 \):
\[ 5x = 30 \]
Now divide by \( 5 \):
\[ x = 6 \]

### Final Answer
Thus, the value of \( x \) is \( 6 \).

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