The equilibrium constant for the reaction is given as \( K = 10^{-1} \). The number of moles of \( A \) initially is 1 mole. After adding 0.5 moles, we now have 1.5 moles of \( A \). The equilibrium concentration is shifted, and the new ratio of concentrations will be calculated using the new equilibrium condition. After the second equilibrium is established, the value of \( \left[\frac{A}{B}\right] \) is \( 10^{2} \).