Solving the Fraction Proportion

The problem asks us to find the value of \(x\) in the given proportion:

\(4/9 : x = 2/3 : 16/81\)

This proportion can be rewritten as an equation involving fractions:

\(\frac{4/9}{x} = \frac{2/3}{16/81}\)

Step-by-Step Calculation for x

To find \(x\), we can rearrange the equation. We want to isolate \(x\).

1. Rewrite the equation to solve for \(x\). Multiply both sides by \(x\) and by \(\frac{16}{81}\), then divide by \(\frac{2}{3}\):

   \(x = \frac{4/9}{\null} \times \frac{16/81}{2/3}\)

   Or, more directly from the cross-multiplication or standard proportion solving method:

   \(x = \frac{4/9 \times 16/81}{2/3}\)

2. First, calculate the product in the numerator:

   \(\frac{4}{9} \times \frac{16}{81} = \frac{4 \times 16}{9 \times 81} = \frac{64}{729}\)

3. Now, divide this result by \(2/3\). Dividing by a fraction is the same as multiplying by its reciprocal:

   \(x = \frac{64/729}{2/3} = \frac{64}{729} \times \frac{3}{2}\)

4. Perform the multiplication:

   \(x = \frac{64 \times 3}{729 \times 2} = \frac{192}{1458}\)

5. Simplify the fraction. Both numerator and denominator are divisible by common factors. We can simplify before multiplying as well:

   \(x = \frac{64}{729} \times \frac{3}{2} = \frac{64 \div 2}{729 \div 3} \times \frac{3 \div 3}{2 \div 2} = \frac{32}{243} \times \frac{1}{1}\)

   \(x = \frac{32}{243}\)

Thus, the value of \(x\) is \(32/243\).