1. **State the problem:** We need to find the ratio of the speeds of a bus and a train given their distances traveled and times taken. 2. **Formula for speed:** Speed is calculated as $$\text{Speed} = \frac{\text{Distance}}{\text{Time}}$$ 3. **Calculate the speed of the bus:** $$\text{Speed}_{bus} = \frac{126}{3} = 42 \text{ km/hr}$$ 4. **Calculate the speed of the train:** $$\text{Speed}_{train} = \frac{315}{5} = 63 \text{ km/hr}$$ 5. **Find the ratio of their speeds:** $$\text{Ratio} = \frac{\text{Speed}_{bus}}{\text{Speed}_{train}} = \frac{42}{63}$$ 6. **Simplify the ratio:** $$\frac{\cancel{42}^{6 \times 7}}{\cancel{63}^{9 \times 7}} = \frac{6}{9}$$ 7. **Further simplify:** $$\frac{\cancel{6}^{2 \times 3}}{\cancel{9}^{3 \times 3}} = \frac{2}{3}$$ **Final answer:** The ratio of the speeds of the bus to the train is $2:3$.