To subtract the fractions \( \frac{5}{9} - \frac{5}{12} \), we need a common denominator. The least common multiple (LCM) of 9 and 12 is 36.
Next, we convert each fraction to have this common denominator:
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For \( \frac{5}{9} \): \[ \frac{5}{9} \times \frac{4}{4} = \frac{20}{36} \]
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For \( \frac{5}{12} \): \[ \frac{5}{12} \times \frac{3}{3} = \frac{15}{36} \]
Now we can subtract the two fractions: \[ \frac{20}{36} - \frac{15}{36} = \frac{20 - 15}{36} = \frac{5}{36} \]
Thus, the answer in simplest form is: \[ \frac{5}{36} \]