We are given the following units of measure;
[tex]1\text{ Atomic mass}=8.3\times10^{-24}Carat[/tex]To determine the equivalent as 1 carat, we would divide the quantity of 1 atomic mass in carat by the value in carat.
This is shown below;
[tex]\begin{gathered} 1\text{ AM}=8.3\times10^{-24}Carat \\ \frac{1}{8.3\times10^{-24}}=\frac{8.3\times10^{-24}}{8.3\times10^{-24}}\text{Carat} \\ \frac{1}{8.3\times\frac{1}{10^{24}}}=1\text{Carat} \end{gathered}[/tex]We shall refine the left side of the equation and we'll have;
[tex]\begin{gathered} \frac{1}{1}\div\frac{8.3}{10^{24}} \\ =\frac{1}{1}\times\frac{10^{24}}{8.3} \\ =1.204819277\times10^{23} \end{gathered}[/tex]Therefore;
[tex]1\text{Carat}=1.2\times10^{23}[/tex]For the second part:
[tex]1AM=1.66\times10^{-21}mg[/tex]We now divide the quantity in atomic mass by the quantity in milligrams and that gives us;
[tex]\begin{gathered} \frac{1}{1.66\times10^{-21}}=\frac{1.66\times10^{-21}}{1.66\times10^{-21}}mg \\ \frac{1}{1.66\times\frac{1}{10^{21}}}=1mg \end{gathered}[/tex]We will refine the left side and we'll have;
[tex]\begin{gathered} \frac{1}{1}\div\frac{1.66}{10^{21}}=1mg \\ 1\times\frac{10^{21}}{1.66}=1mg \\ 6.0240963855\times10^{20}=1mg \end{gathered}[/tex]ANSWER:
1 Carat equals;
[tex]\begin{gathered} 1.2\times10^{23} \\ \text{Option B} \end{gathered}[/tex]1 milligram equals;
[tex]\begin{gathered} 6.02\times10^{20} \\ \text{Option D} \end{gathered}[/tex]