Let N₀ be the initial number of atoms of element X. After time t, the number of atoms of X remaining is N_t. The number of atoms of stable element Y formed is N_y = N₀ - N_t. The half-life of X is T = 2 hours. The formula for radioactive decay is N_t = N₀ (1/2)^(t/T). The problem states the ratio of atoms is 1:7. This typically refers to the ratio of the remaining radioactive element (X) to the stable product (Y), i.e., N_t : N_y = 1 : 7. So, N_t / N_y = 1/7. Substituting N_y = N₀ - N_t, we get N_t / (N₀ - N_t) = 1/7. 7 N_t = N₀ - N_t 8 N_t = N₀ N_t / N₀ = 1/8. Now substitute this into the decay formula: 1/8 = (1/2)^(t/2). Since 1/8 = (1/2)³, we have: (1/2)³ = (1/2)^(t/2). Equating the exponents, 3 = t/2. Therefore, t = 6 hours.