Step 1: Understand atomic mass of elements with isotopes.
Most elements exist as a mixture of isotopes. The observed atomic mass is a weighted average of the masses of all isotopes, where the weight is the fractional (percentage) abundance of each isotope.
Step 2: Write the weighted average formula.
\[ \bar{A} = \frac{\sum_i (A_i \times \% abundance_i)}{100} \] where $ A_i $ is the mass number of each isotope and $ \% abundance_i $ is its percentage.
Step 3: List the given isotopes of iron.
$ ^{54}Fe $: 10% abundance, mass 54. $ ^{56}Fe $: 85% abundance, mass 56. $ ^{57}Fe $: 5% abundance, mass 57. Note that $ 10 + 85 + 5 = 100\% $, confirming completeness.
Step 4: Compute the weighted sum.
\[ \text{Numerator} = (54 \times 10) + (56 \times 85) + (57 \times 5) \] \[ = 540 + 4760 + 285 = 5585 \]
Step 5: Divide by 100 to get the average atomic mass.
\[ \bar{A} = \frac{5585}{100} = 55.85 \] This value closely matches the standard atomic mass of iron listed in periodic tables (55.845), confirming our calculation is correct.
Step 6: Final answer.
\[ \boxed{55.85} \]