1. **State the problem:** We are given the relation $$\frac{P_1 V_1}{T_1} \approx \frac{P_2 V_2}{T_2}$$ and asked to solve for $T_2$. 2. **Write the formula:** The equation relates pressures, volumes, and temperatures of a gas under two different states. 3. **Isolate $T_2$:** Multiply both sides by $T_2$ and then divide both sides by $\frac{P_1 V_1}{T_1}$ to isolate $T_2$: $$\frac{P_1 V_1}{T_1} \approx \frac{P_2 V_2}{T_2}$$ Multiply both sides by $T_2$: $$T_2 \times \frac{P_1 V_1}{T_1} \approx P_2 V_2$$ Divide both sides by $\frac{P_1 V_1}{T_1}$: $$T_2 \approx \frac{P_2 V_2}{\cancel{\frac{P_1 V_1}{T_1}}} \times \cancel{\frac{T_1}{P_1 V_1}}$$ 4. **Simplify:** Dividing by a fraction is the same as multiplying by its reciprocal: $$T_2 \approx P_2 V_2 \times \frac{T_1}{P_1 V_1}$$ 5. **Final formula:** $$\boxed{T_2 = \frac{P_2 V_2 T_1}{P_1 V_1}}$$ This means the temperature $T_2$ can be found by multiplying $P_2$, $V_2$, and $T_1$ and then dividing by the product of $P_1$ and $V_1$. This formula is useful in gas law problems where pressure, volume, and temperature change but the amount of gas remains constant.