This problem asks us to calculate the average speed of Smith who travels along the four sides of a square at different speeds. To find the average speed, we need to determine the total distance traveled and the total time taken for the entire journey.

Understanding Average Speed Calculation

The formula for average speed is:

\( \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \)

In this scenario, the journey is along the four sides of a square. A square has four sides of equal length.

Step-by-Step Calculation

Let's assume the length of each side of the square is \(d\) kilometers. Since there are four sides, the total distance traveled is \(4d\).

Smith travels along each side at a different speed:

• Side 1: 20 km/hr
• Side 2: 10 km/hr
• Side 3: 40 km/hr
• Side 4: 30 km/hr

We can calculate the time taken to travel each side using the formula: \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \).

• Time for Side 1 (\(t_1\)): \( t_1 = \frac{d}{20} \) hours
• Time for Side 2 (\(t_2\)): \( t_2 = \frac{d}{10} \) hours
• Time for Side 3 (\(t_3\)): \( t_3 = \frac{d}{40} \) hours
• Time for Side 4 (\(t_4\)): \( t_4 = \frac{d}{30} \) hours

The total time taken for the entire journey is the sum of the time taken for each side:

\( \text{Total Time} = t_1 + t_2 + t_3 + t_4 \)

\( \text{Total Time} = \frac{d}{20} + \frac{d}{10} + \frac{d}{40} + \frac{d}{30} \)

To add these fractions, we need to find a common denominator. The least common multiple (LCM) of 20, 10, 40, and 30 is 120.

• \( \frac{d}{20} = \frac{d \times 6}{20 \times 6} = \frac{6d}{120} \)
• \( \frac{d}{10} = \frac{d \times 12}{10 \times 12} = \frac{12d}{120} \)
• \( \frac{d}{40} = \frac{d \times 3}{40 \times 3} = \frac{3d}{120} \)
• \( \frac{d}{30} = \frac{d \times 4}{30 \times 4} = \frac{4d}{120} \)

Now, add the fractions:

\( \text{Total Time} = \frac{6d}{120} + \frac{12d}{120} + \frac{3d}{120} + \frac{4d}{120} \)

\( \text{Total Time} = \frac{(6 + 12 + 3 + 4)d}{120} = \frac{25d}{120} \)

Simplifying the fraction \( \frac{25}{120} \) by dividing both numerator and denominator by their greatest common divisor, 5:

\( \frac{25}{120} = \frac{25 \div 5}{120 \div 5} = \frac{5}{24} \)

So, Total Time \( = \frac{5d}{24} \) hours.

Now we can calculate the average speed using the total distance and total time:

\( \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{4d}{\frac{5d}{24}} \)

To divide by a fraction, we multiply by its reciprocal:

\( \text{Average Speed} = 4d \times \frac{24}{5d} \)

The \(d\) in the numerator and denominator cancel out:

\( \text{Average Speed} = \frac{4 \times 24}{5} = \frac{96}{5} \)

Now, convert the fraction to a decimal:

\( \text{Average Speed} = \frac{96}{5} = 19.2 \)

Therefore, the average speed of Smith is 19.2 km/hr.

Revision Table: Average Speed Concepts

Additional Information: Harmonic Mean and Average Speed

When the distance covered in different segments is the same, the average speed is the harmonic mean of the speeds for those segments. For \(n\) segments of equal distance \(d\) with speeds \(v_1, v_2, \dots, v_n\), the total distance is \(nd\) and the total time is \( \frac{d}{v_1} + \frac{d}{v_2} + \dots + \frac{d}{v_n} \). The average speed is:

\( \text{Average Speed} = \frac{nd}{\frac{d}{v_1} + \frac{d}{v_2} + \dots + \frac{d}{v_n}} = \frac{n}{\frac{1}{v_1} + \frac{1}{v_2} + \dots + \frac{1}{v_n}} \)

This is the formula for the harmonic mean of the speeds. In this problem, \(n=4\) and the speeds are 20, 10, 40, and 30 km/hr. The average speed is:

\( \text{Average Speed} = \frac{4}{\frac{1}{20} + \frac{1}{10} + \frac{1}{40} + \frac{1}{30}} \)

The denominator is \( \frac{6}{120} + \frac{12}{120} + \frac{3}{120} + \frac{4}{120} = \frac{25}{120} = \frac{5}{24} \).

\( \text{Average Speed} = \frac{4}{\frac{5}{24}} = 4 \times \frac{24}{5} = \frac{96}{5} = 19.2 \) km/hr.

This confirms the result obtained by directly calculating total distance and total time.