“pounds per kilogram” can be your first conversion, so do you multiply by 2.2 lb/kg or divide? Well, you want to get rid of the lb unit, so you want a fraction with lb on the bottom and kg on the top. That means you have to multiply by 1 kg/2.2 lb. Then the lb factors cancel and you’re left with a number followed by kg.

You can then step down all at once to milligrams or do it in stages. There are 1000 grams/kg; do you want to multiply or divide? You have kg on top, so you want kg in the denominator of the fraction. You multiply by 1000 grams/kg.

Then the kg cancel, leaving grams.

1 gram is 1000 milligrams — do you multiply by 1 gram/1000 mg or by 1000 mg/1 gram? You want the grams to cancel, so grams has to be on the bottom. Multiply by 1000 mg/g, and you have your answer in milligrams.

Dimensional analysis doesn’t solve problems — it is a check to make sure you’re going about the problem correctly to get the units you need. It’s easy, say in this problem, to divide by 1000 at some point rather than multiply. Checking the units can help prevent that from happening.

RE:
How do you convert 242 lbs into mg using dimension analysis?

242 lbs x (453.6 g/1 lb) x (1000 mg/1 g)

just multiply the top three numbers

*although this is the correct way to do it, it would be better if you understood the concept so that you can use it yourself in other problems without asking for help