Ian's answer is correct. I just want to expand on why he did what he did.
We'd first like to figure out how many moles of Fe 100 grams is.
You can figure that out by looking at Fe's atomic mass on the Periodic Table: 55.85. That tells you that 1 mole of Fe weighs 55.85 grams. There are 55.85 g/mol of Fe, or 1 mol of Fe/55.85 g. Use the latter to convert 100 g into mol, since that way the units will work out:
100 g × 1 mol / 55.85 g = 100 g×mol / 55.85 g = 100 mol / 55.85 = 1.79 mol
Now, let's figure out how many moles of Fe2O3 there are, since there are 1.79 moles of Fe. The reaction equation tells you that 4 moles of Fe and 3 moles of O2 will produce 2 moles of Fe2O3. The relevant part of that for us is that 4 moles of Fe will produce 2 moles of Fe2O3. In other words, there are 4 mol Fe/2 mol Fe2O3, or 2 mol Fe2O3/4 mol Fe. Use the latter to convert to moles of Fe2O3, since that way the units will work out:
1.79 mol Fe × 2 mol Fe2O3 / 4 mol Fe = (1.79(2) mol Fe × mol Fe2O3) / 4 mol Fe = 3.58 mol Fe2O3 / 4 = 0.895 mol Fe2O3
Finally, let's figure out how many grams of Fe2O3 there are if we have 0.895 moles. First, we'll need to use the atomic masses of Fe and O to find out how many moles there are per gram. We've got 2 Fe molecules and 3 O molecules, so we'll just find the atomic mass of Fe and multiply it by 2, then add that to the atomic mass of O multiplied by 3:
2(55.85 g/mol of Fe) + 3(16.00 g/mol of O) = 111.7 + 48.00 g/mol of Fe2O3 = 159.7 g/mol of Fe2O3
So, there are 159.7 g/mol of Fe2O3, or 1 mol of Fe2O3/159.7 g. Use the former to convert 0.895 mol of Fe2O3 into g, since that way the units will work out:
0.895 mol × 159.7 g / mol = 0.895(159.7) mol×g / mol = 142.9315 g
Thus, 142.93 grams of Fe2O3 are produced.
So, just to review, the steps were:
1. Figure out how many moles of Fe there are, given 100 grams of Fe
2. Figure out how many moles of Fe2O3 there are, given 1.79 moles of Fe
3. Figure out how many grams of Fe2O3 there are, given 0.895 moles of Fe2O3
That's exactly what Ian did; he just did it all in one step (and I think he made a slight error in multiplication, which is why his answer is a couple grams off):
100 g Fe x (1 mol Fe / 55.85 g Fe) x (2 mol Fe2O3 / 4 mol Fe) x (159.69 g Fe2O3 / mol Fe2O3)
100 g Fe x (conversion to mol Fe) x (conversion to mol Fe2O3) x (conversion to g Fe2O3)