To determine the average atomic mass of an element with isotopes [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex], one must use the concept of the weighted average. This involves multiplying the mass of each isotope by its relative abundance (expressed as a percentage) and then summing these values. Each percentage must be converted from a percent to a fraction (by dividing by 100) before performing the multiplication.
Given the choices:
Choice A:
[tex]\[
\frac{\text{(mass of } A + \text{ mass of } B + \text{ mass of } C)}{3}
\][/tex]
This simply averages the masses of the isotopes without considering their relative abundances. Hence, it is not correct.
Choice B:
[tex]\[
\frac{[(\text{ mass of } A) \times (\% \text{ of } A) + (\text{ mass of } B) \times (\% \text{ of } B) + (\text{ mass of } C) \times (\% \text{ of } C)]}{3}
\][/tex]
This formula incorrectly divides the sum of the weighted masses by 3, which is not how weighted averages are calculated.
Choice C:
[tex]\[
\frac{\text{(mass of } A)}{(\% \text{ of } A)} + \frac{\text{(mass of B)}}{(\% \text{ of } B)} + \frac{\text{(mass of } C)}{(\% \text{ of } C)}
\][/tex]
This formula reverses the intended operation by dividing the mass of each isotope by their respective percentages. This does not produce a correct weighted average.
Choice D:
[tex]\[
(\text{mass of } A) \times (\% \text{ of } A) + (\text{mass of } B) \times (\% \text{ of } B) + (\text{mass of } C) \times (\% \text{ of } C)
\][/tex]
This choice calculates the average atomic mass correctly by taking the weighted sum of the masses of the isotopes multiplied by their relative abundances.
Therefore, the correct choice is:
[tex]\[
\boxed{D}
\][/tex]
