We start with the expression
\[ \frac{x z}{\frac{x \cdot 270}{400}}. \]
To simplify the fraction, we multiply by the reciprocal of the denominator:
\[ \frac{x z}{\frac{x \cdot 270}{400}} = x z \cdot \frac{400}{x \cdot 270}. \]
Next, we can cancel \(x\) from the numerator and denominator (assuming \(x \neq 0\)):
\[ = z \cdot \frac{400}{270}. \]
Now, we simplify the fraction \(\frac{400}{270}\):
\[ = z \cdot \frac{400 \div 10}{270 \div 10} = z \cdot \frac{40}{27}. \]
Thus, the final simplified expression is
\[ \frac{40z}{27}. \]
\(\boxed{\frac{40z}{27}}\)
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