A person travels from place A to place B at a speed of 18 km/h and returns from B to A at a speed of 30 km/h, using the same path. What is the average speed for the entire journey?

1. Problem Description

A person travels from place A to place B at a speed of 18 km/h and returns from B to A at a speed of 30 km/h, using the same path. What is the average speed for the entire journey?

2. Solution Steps

Let the distance between A and B be $d$ km.

Time taken to travel from A to B is $t_1 = \frac{d}{18}$ hours.

Time taken to travel from B to A is $t_2 = \frac{d}{30}$ hours.

The total distance traveled is $2d$ km.

The total time taken is $t_1 + t_2 = \frac{d}{18} + \frac{d}{30}$ hours.

Average speed is defined as the total distance traveled divided by the total time taken.

$$\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}$$

$$\text{Average Speed} = \frac{2d}{\frac{d}{18} + \frac{d}{30}}$$

$$\text{Average Speed} = \frac{2d}{d(\frac{1}{18} + \frac{1}{30})}$$

$$\text{Average Speed} = \frac{2}{\frac{1}{18} + \frac{1}{30}}$$

Find a common denominator for the fractions: The least common multiple of 18 and 30 is

9

0. $$ \text{Average Speed} = \frac{2}{\frac{5}{90} + \frac{3}{90}} $$

$$\text{Average Speed} = \frac{2}{\frac{8}{90}}$$

$$\text{Average Speed} = \frac{2 \times 90}{8}$$

$$\text{Average Speed} = \frac{180}{8}$$

$$\text{Average Speed} = \frac{45}{2}$$

$$\text{Average Speed} = 22.5 \text{ km/h}$$

$$22.5 = 22\frac{1}{2}$$

3. Final Answer

The average speed for the entire journey is $22\frac{1}{2}$ km/h.