A person covers 12 km at 3 km/hr, 18 km at 9 km/hr and 24 km at 4 km/hr. The average speed in covering the whole distance is :

Question

Accepted Answer

The correct option is 4 (4.5 km/hr):

To find the average speed for the entire journey, we must use the fundamental formula for average speed, which is:

$$	ext{Average Speed} = \frac{	ext{Total Distance Covered}}{	ext{Total Time Taken}}$$

It is crucial to remember that average speed is **not** simply the average of the individual speeds. It must be calculated using the total distance and the total time.

**Step 1: Calculate the Total Distance Covered**

The total distance is the sum of the distances covered in each segment:
$$\text{Total Distance} = 12 \text{ km} + 18 \text{ km} + 24 \text{ km} = 54 \text{ km}$$

**Step 2: Calculate the Time Taken for Each Segment**

We use the formula: $	ext{Time} = \frac{	ext{Distance}}{	ext{Speed}}$ for each part of the journey.  * **Time for the first segment (T1):**
    $$T\_1 = \frac{12 \text{ km}}{3 \text{ km/hr}} = 4 \text{ hours}$$
  * **Time for the second segment (T2):**
    $$T\_2 = \frac{18 \text{ km}}{9 \text{ km/hr}} = 2 \text{ hours}$$
  * **Time for the third segment (T3):**
    $$T\_3 = \frac{24 \text{ km}}{4 \text{ km/hr}} = 6 \text{ hours}$$**Step 3: Calculate the Total Time Taken**

The total time is the sum of the time taken for each segment:
$$\text{Total Time} = T\_1 + T\_2 + T\_3 = 4 \text{ hours} + 2 \text{ hours} + 6 \text{ hours} = 12 \text{ hours}$$

**Step 4: Calculate the Average Speed**

Now, we apply the average speed formula:
$$\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{54 \text{ km}}{12 \text{ hours}} = 4.5 \text{ km/hr}$$

Therefore, the average speed for covering the whole distance is $4.5 	ext{ km/hr}$.

The incorrect option is 1 (4 km/hr):

This result is obtained by simply dividing the total distance (54 km) by a miscalculated total time, or possibly by incorrectly taking the average of the *distances* or *speeds* without considering the time spent at each speed. For a speed of $4 \text{ km/hr}$, the total time would have to be $54 \text{ km} / 4 \text{ km/hr} = 13.5 \text{ hours}$, which is incorrect as we calculated the total time to be $12 \text{ hours}$.

The incorrect option is 2 (4.2 km/hr):

A speed of $4.2 	ext{ km/hr}$ would result in a total time of approximately $54 	ext{ km} / 4.2 	ext{ km/hr} \approx 12.86 	ext{ hours}$. This is closer to the correct average speed than $4 	ext{ km/hr}$, but it still does not match the total time of $12 	ext{ hours}$ correctly calculated from the individual segments. This answer might be derived from a slight calculation error in the total time or total distance.

The incorrect option is 3 (4.4 km/hr):

To achieve an average speed of $4.4 	ext{ km/hr}$, the total time taken for the $54 	ext{ km}$ journey would need to be $54 	ext{ km} / 4.4 	ext{ km/hr} \approx 12.27 	ext{ hours}$. This is the closest incorrect option to the actual total time of $12 	ext{ hours}$, suggesting that the student might have made a very minor arithmetic error, particularly in the calculation of the total time (e.g., calculating $T_3$ as slightly less than $6 	ext{ hours}$).

The incorrect option is 4.5 km/hr is the correct answer, based on the calculation of Total Distance / Total Time.