Fraction Multiplication Simplification

Calculate the value of the expression:

$ \frac{32}{46} \times \frac{94}{576} \times \frac{184}{282} $

Simplify by canceling common factors between numerators and denominators.

Stepwise Calculation

1. Simplify individual fractions by dividing by common factors:

   • $ \frac{32}{46} = \frac{16}{23} $ (Divide by 2)
   • $ \frac{94}{576} = \frac{47}{288} $ (Divide by 2)
   • $ \frac{184}{282} = \frac{92}{141} $ (Divide by 2)

   The expression becomes:

   $ \frac{16}{23} \times \frac{47}{288} \times \frac{92}{141} $

2. Identify factors for cross-cancellation:

   • $ 288 = 16 \times 18 $
   • $ 92 = 23 \times 4 $
   • $ 141 = 47 \times 3 $

   Substitute these into the expression:

   $ \frac{16}{23} \times \frac{47}{16 \times 18} \times \frac{23 \times 4}{47 \times 3} $

3. Cancel the common factors ($16$, $23$, $47$) from numerators and denominators:

   $ \frac{\cancel{16}}{\cancel{23}} \times \frac{\cancel{47}}{\cancel{16} \times 18} \times \frac{\cancel{23} \times 4}{\cancel{47} \times 3} $

   This leaves:

   $ \frac{1}{1} \times \frac{1}{18} \times \frac{4}{3} $

4. Multiply the remaining terms:

   $ \frac{1 \times 1 \times 4}{1 \times 18 \times 3} = \frac{4}{54} $

5. Simplify the final fraction by dividing the numerator and denominator by 2:

   $ \frac{4}{54} = \frac{2}{27} $

The value of the expression is $ \frac{2}{27} $.