Journey Average Speed Calculation

This solution details the calculation of average speed for a journey composed of segments traveled at different speeds.

Journey Segments Defined

The total journey is broken down into parts based on distance covered and the speed during each part:

• Part 1: Covered $ \frac{1}{4} $ of the total distance at a speed of 20 km/hr.
• Part 2: Covered $ \frac{1}{3} $ of the total distance at a speed of 30 km/hr.
• Part 3: Covered the remaining distance at a speed of 25 km/hr.

Fractions Calculation for Remaining Journey

First, determine the fraction of the journey corresponding to Part 3. The sum of all fractions must equal 1.

Fraction for Part 3 = $ 1 - (\frac{1}{4} + \frac{1}{3}) = 1 - (\frac{3}{12} + \frac{4}{12}) = 1 - \frac{7}{12} = \frac{5}{12} $.

So, Part 3 covers $ \frac{5}{12} $ of the total distance.

Time Calculation for Journey Parts

Assume the total distance is D. Calculate the time taken for each segment using the formula: Time = Distance / Speed.

• Time for Part 1 ($t_1$): $ t_1 = \frac{\frac{1}{4}D}{20 \text{ km/hr}} = \frac{D}{80} $ hours.
• Time for Part 2 ($t_2$): $ t_2 = \frac{\frac{1}{3}D}{30 \text{ km/hr}} = \frac{D}{90} $ hours.
• Time for Part 3 ($t_3$): $ t_3 = \frac{\frac{5}{12}D}{25 \text{ km/hr}} = \frac{5D}{300} = \frac{D}{60} $ hours.

Total Time for Journey

Sum the times calculated for each part to find the total time (T) for the entire journey.

$ T = t_1 + t_2 + t_3 = \frac{D}{80} + \frac{D}{90} + \frac{D}{60} $

Find the least common multiple (LCM) of 80, 90, and 60, which is 720.

$ T = \frac{9D}{720} + \frac{8D}{720} + \frac{12D}{720} = \frac{(9 + 8 + 12)D}{720} = \frac{29D}{720} $ hours.

Average Speed Formula Application

The average speed is the total distance divided by the total time.

Average Speed $ = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{D}{\frac{29D}{720}} $

$ \text{Average Speed} = \frac{720D}{29D} = \frac{720}{29} $ km/hr.

Final Average Speed Result

Convert the improper fraction $ \frac{720}{29} $ to a mixed number.

Dividing 720 by 29 gives 24 with a remainder of 24.

Thus, the average speed is $ 24 \frac{24}{29} $ km/hr.