Question

One-fourth of a certain journey was covered at the speed of 20 km/hr, one-third at 30 km/hr and the rest at the speed of 25 km/hr. The average speed (in km/hr) for the entire journey is :

• A. $29\frac{2}{29}$
• B. $24\frac{24}{29}$
• C. $42\frac{5}{29}$
• D. $31\frac{3}{29}$

Hint:

To avoid working with fractions, assume a total distance that is a common multiple of the denominators.
For denominators $4$, $3$, and $12$, let the total distance be $1200 \text{ km}$.
- Part 1: $300 \text{ km}$ at $20 \text{ km/hr} \rightarrow T_1 = 15 \text{ hours}$.
- Part 2: $400 \text{ km}$ at $30 \text{ km/hr} \rightarrow T_2 = 13.33 \text{ hours}$.
- Part 3: $500 \text{ km}$ at $25 \text{ km/hr} \rightarrow T_3 = 20 \text{ hours}$.
Total time $= 15 + \frac{40}{3} + 20 = 35 + \frac{40}{3} = \frac{145}{3} \text{ hours}$.
Average speed $= \frac{1200}{\frac{145}{3}} = \frac{3600}{145} = \frac{720}{29} = 24\frac{24}{29} \text{ km/hr}$.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
This problem is based on the concept of average speed.
Average speed is not simply the arithmetic mean of the individual speeds.
Instead, it is defined as the total distance traveled divided by the total time taken to cover that distance.
The journey is divided into three parts with different fractional distances and different speeds.

Step 2: Key Formula or Approach:
Let the total distance of the journey be $d$.
The formula for average speed ($v_{\text{avg}}$) is:
\[ v_{\text{avg}} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{d}{t_1 + t_2 + t_3} \] where $t_1, t_2, t_3$ are the time intervals taken to complete each of the three stages of the journey.
For each stage, time is calculated as:
\[ t = \frac{\text{Distance covered in that stage}}{\text{Speed of that stage}} \]

Step 3: Detailed Explanation:

• Define Distance Fractions and Speeds:
Let $d$ be the total distance of the journey.
- Part 1: Distance $d_1 = \frac{d}{4}$ covered at speed $v_1 = 20 \text{ km/hr}$.
- Part 2: Distance $d_2 = \frac{d}{3}$ covered at speed $v_2 = 30 \text{ km/hr}$.
- Part 3: The remaining distance $d_3$ is:
\[ d_3 = d - \left(\frac{d}{4} + \frac{d}{3}\right) = d \left(1 - \frac{7}{12}\right) = \frac{5d}{12} \] This remaining distance is covered at speed $v_3 = 25 \text{ km/hr}$.

• Calculate the Time taken for each part:
- Time for Part 1:
\[ t_1 = \frac{d/4}{20} = \frac{d}{80} \text{ hours} \]
- Time for Part 2:
\[ t_2 = \frac{d/3}{30} = \frac{d}{90} \text{ hours} \]
- Time for Part 3:
\[ t_3 = \frac{5d/12}{25} = \frac{5d}{12 \times 25} = \frac{d}{60} \text{ hours} \]

• Calculate Total Time ($T$):
\[ T = t_1 + t_2 + t_3 = d \left( \frac{1}{80} + \frac{1}{90} + \frac{1}{60} \right) \]
Find the Least Common Multiple (LCM) of $80, 90,$ and $60$, which is $720$:
\[ T = d \left( \frac{9 + 8 + 12}{720} \right) = d \left( \frac{29}{720} \right) \]

• Calculate Average Speed:
\[ v_{\text{avg}} = \frac{d}{T} = \frac{d}{d \left(\frac{29}{720}\right)} = \frac{720}{29} \text{ km/hr} \]
Convert $\frac{720}{29}$ into a mixed fraction:
\[ 720 = 29 \times 24 + 24 \]
Therefore:
\[ v_{\text{avg}} = 24\frac{24}{29} \text{ km/hr} \]

Step 4: Final Answer:
The average speed for the entire journey is $24\frac{24}{29} \text{ km/hr}$.
Thus, the correct option is (B).