The dosage of quinine when a 145−lb adult takes a 200.−mg tablet is equivalent to μg drug per kg of body weight.

Basically, all you have to do to solve this problem is convert units and calculate for their ratio. Convert pounds to kilograms and milligrams to micrograms.

Pounds→kilograms
2.2 lbs → 1 kg
∴, 145 lbs * 1 kg/2.2 lbs = 65.91 kg

milligrams → micrograms
1 mg → 1000 μg
∴, 200 mg * 1000 μg/1 mg = 200,000 μg

The ratio is, therefore,
200,000 μg/65.91 kg = 3,034.44 μg/kg