Q.In a periodic table the average atomic mass of magnesium is given as 24.312 u. The average value is based on their relative natural abundance on earth. The three isotopes and their masses are $^{24}_{12}\text{Mg}$ (23.98504u), $^{25}_{12}\text{Mg}$ (24.98584u) and $^{26}_{12}\text{Mg}$ (25.98259u). The natural abundance of $^{24}_{12}\text{Mg}$ is 78.99% by mass. Calculate the abundances of other two isotopes.

$24.312 = (0.7899)(23.98504) + x(24.98584) + y(25.98259)$

$(0.7899)(23.98504) = 18.9458\ \text{u}$

$24.312 - 18.9458 = 5.3662\ \text{u} = x(24.98584) + y(25.98259)$

$5.3662 = x(24.98584) + (0.2101-x)(25.98259)$

$5.3662 = (0.2101)(25.98259) + x[24.98584 - 25.98259]$

$5.3662 = 5.45894 - 0.99675\,x$

$0.99675\,x = 5.45894 - 5.3662 = 0.09274$

$x = 0.09303 \implies x \approx 9.30\%$

$y = 0.2101 - 0.09303 = 0.11707 \implies y \approx 11.71\%$