During a round-trip journey where a person walks to a location at 5 km/h and returns at 3 km/h, which statement about average speed is correct?

The correct option is: Average speed will be less than the arithmetic mean of 5 km/h and 3 km/h.

Step 1: Understand the Definition of Average Speed
Average speed for any total journey is defined as total distance traveled divided by total time taken:

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

Step 2: Calculate the Average Speed for a Round Trip
Let the distance to the location be _d km.
The person travels distance _d at speed _{v1 = 5} km/h and returns the same distance _d at speed _{v2 = 3} km/h.

Time taken for the forward journey:

$t_1=\frac{d}{5}$

Time taken for the return journey:

$t_2=\frac{d}{3}$

Total distance = _{d + d = 2d} km.
Total time taken:

$t_{\text{total}}=\frac{d}{5}+\frac{d}{3}=\frac{3d+5d}{15}=\frac{8d}{15}$

Now, calculate the average speed (which is the harmonic mean of the two speeds):

$\text{Average Speed}=\frac{2d}{\frac{8d}{15}}=\frac{2\times 15}{8}=\frac{30}{8}=3.75\text{ km/h}$

Step 3: Calculate the Arithmetic Mean of the Speeds
The arithmetic mean of 5 km/h and 3 km/h is:

$\text{Arithmetic Mean}=\frac{5+3}{2}=4\text{ km/h}$

Step 4: Compare Average Speed and Arithmetic Mean
Comparing the calculated values:

$3.75\text{ km/h}<4\text{ km/h}$

Thus, the average speed (3.75 km/h) will be less than the arithmetic mean (4 km/h) of 5 km/h and 3 km/h.