A bus travels at 30 km/h for the first 2 h of a journey and then travels at 50 km/h for the next 2 h of the journey. Find the ratio of the speeds to the distances traveled at the two speeds.

Step-by-step solution

1. 1

   Given Data

   Speed 1: $v_1 = 30 \text{ km/h}$, Time 1: $t_1 = 2 \text{ h}$

   $$v_1 = 30 \text{ km/h}, t_1 = 2 \text{ h}$$
2. 2

   Given Data (cont.)

   Speed 2: $v_2 = 50 \text{ km/h}$, Time 2: $t_2 = 2 \text{ h}$

   $$v_2 = 50 \text{ km/h}, t_2 = 2 \text{ h}$$
3. 3

   Key Principle

   Distance, speed, and time are related by $d = v \times t$.

   $$d = v \times t$$
4. 4

   Ratio of Speeds

   Ratio of speeds: $v_1 : v_2 = 30 : 50$

   $$\frac{v_1}{v_2} = \frac{30}{50}$$
5. 5

   Simplify Speed Ratio

   The ratio of speeds is $3:5$.

   $$\frac{v_1}{v_2} = \frac{3}{5}$$
6. 6

   Calculate Distances

   Distance 1: $d_1 = v_1 \times t_1 = 30 \text{ km/h} \times 2 \text{ h} = 60 \text{ km}$

   $$d_1 = 30 \times 2 = 60 \text{ km}$$
7. 7

   Calculate Distances (cont.)

   Distance 2: $d_2 = v_2 \times t_2 = 50 \text{ km/h} \times 2 \text{ h} = 100 \text{ km}$

   $$d_2 = 50 \times 2 = 100 \text{ km}$$
8. 8

   Ratio of Distances

   Ratio of distances: $d_1 : d_2 = 60 : 100$

   $$\frac{d_1}{d_2} = \frac{60}{100}$$
9. 9

   Final Ratio

   The ratio of distances is $3:5$. Both ratios are $3:5$.

   $$\frac{d_1}{d_2} = \frac{3}{5}$$