To determine which option, when multiplied with \(5 \sqrt{27}\), yields a rational number, we first simplify \(5 \sqrt{27}\):
\[ \sqrt{27} = \sqrt{9 \cdot 3} = \sqrt{9} \cdot \sqrt{3} = 3 \sqrt{3} \]
Thus, we have:
\[ 5 \sqrt{27} = 5 \cdot 3 \sqrt{3} = 15 \sqrt{3} \]
Next, we look at each option and check if multiplying it by \(15 \sqrt{3}\) results in a rational number.
Option A: \( \sqrt{12} \)
\[ 15 \sqrt{3} \cdot \sqrt{12} = 15 \sqrt{3} \cdot \sqrt{4 \cdot 3} = 15 \sqrt{3} \cdot 2 \sqrt{3} = 15 \cdot 2 \cdot 3 = 90 \]
This is rational.
Option B: \( \sqrt{9} \)
\[ 15 \sqrt{3} \cdot \sqrt{9} = 15 \sqrt{3} \cdot 3 = 45 \sqrt{3} \]
This is not rational.
Option C: \( 4 \sqrt{2} \)
\[ 15 \sqrt{3} \cdot 4 \sqrt{2} = 60 \sqrt{6} \]
This is not rational.
Option D: \( \sqrt{15} \)
\[ 15 \sqrt{3} \cdot \sqrt{15} = 15 \sqrt{3 \cdot 15} = 15 \sqrt{45} = 15 \sqrt{9 \cdot 5} = 15 \cdot 3 \sqrt{5} = 45 \sqrt{5} \]
This is not rational.
After examining all options, only Option A: \(\sqrt{12}\) results in a rational number when multiplied with \(15 \sqrt{3}\).
So the answer is:
A. \(\sqrt{12}\)