I have 0.009 moles of CaCO3 and I needed to find how many moles of CO3 are in the CaCO3. I am just making sure I am doing this correctly.

0.009 mol CaCO3 x 60.01/100.09= 0.54g CO3

0.54g CO3/ 100.09g CaCO3 = 0.005 mol CO3

Nope.You made two mistakes; i.e., one math and one concept.

math first. You correctly converted 0.009 mol CaCO3 to g CO3. The math error is in converting 0.54 g CO3 to mols CO3. mols = g/molar mass; therefore, 0.54/60.01 = 0.009 mols CO3 which is the correct answer.
Now you see the concept error. Converting mol CaCO3 to mols CO3 is as simple as noting that there is 1 mol CO3 in 1 mol CaCO3 which makes 0.009 mol CaCO3 = 0.009 mol CO3. How many mols C in 0.009 mol CaCO3? 0.009 mol C of course. How many mols O atoms? That is 0.009 mols CaCO3 x (3 mol O atoms/1 mol CaCO3) = 3*0.009 = ?

Oh ok I thought I had to divide by the molecular weight of CaCO3 to find the moles of CO3.

So I take the 0.54g CO3 and divide that by 60.01g which gives me 0.009 mol CO3 in 0.009 mol CaCO3?

Yes, if you want to do it the long way (and hard way). The easier way is to look at it and see there is 1 mol CO3 in 1 mol CaCO3.

To find the number of moles of CO3 in 0.009 moles of CaCO3, you can use the molar ratio between CaCO3 and CO3.

The molar ratio between CaCO3 and CO3 is 1:1, as there is one CO3 ion in each formula unit of CaCO3.

So, you can multiply the number of moles of CaCO3 (0.009 mol) by the molar ratio (1 mol CO3 / 1 mol CaCO3) to find the number of moles of CO3:

0.009 mol CaCO3 x 1 mol CO3 / 1 mol CaCO3 = 0.009 mol CO3

Therefore, there are 0.009 moles of CO3 in 0.009 moles of CaCO3.

However, if you want to convert this value to grams, you can use the molar mass of CO3, which is 60.01 g/mol, and calculate as follows:

0.009 mol CO3 x 60.01 g CO3 / 1 mol CO3 = 0.54 g CO3

So, there are 0.54 grams of CO3 in 0.009 moles of CaCO3.

Your calculations are correct.