Problem is about converting from one set of units to another. Start by taking what you know, and converting THAT to the units they want in the answer.

Your solution contains 800 mg in 500 mL. Same as 1600 mg in 1000 mL or 1.6 mg in each mL. So that is 1600 mcg in each mL.

You say patient weight 198 (pounds) 198 pounds is 90 kg. So we want the patient to have 12 mcg x 90 kg (1080 mcg) every minute. Multiply by 60 minutes and we find that we want the patient to have 64,800 micrograms per hour. Divide by 1000 (1,000 mcg - 1 mg) to get 64.8 mg every hour.

Now go back to the solution on hand (1,600 mcg/mL). 64,800 divided by 1,600 = 40.5. Each mL contains 1,600 mcg, so to give 64,800 mcg each hour you have to give 40.5 mL each hour.

Make sense?