Calculating Average Speed with Equal Distances

The question asks for the average speed when Ram travels three equal distances at different speeds: 15 km/hr, 10 km/hr, and 12 km/hr. The total distance travelled is given as 180 km, which means each of the three equal distances is \( \frac{180 \text{ km}}{3} = 60 \text{ km} \).

Understanding Average Speed

Average speed is defined as the total distance travelled divided by the total time taken. When the speed varies over different parts of the journey, we cannot simply take the arithmetic average of the speeds unless the time taken for each part is equal.

In this case, the distances are equal, but the speeds are different, meaning the time taken for each segment will be different. Therefore, we must calculate the total time taken for the entire journey.

Method 1: Using Total Distance and Total Time

Since the total distance is 180 km and it's divided into three equal parts, each part is 60 km.

• Distance 1 = 60 km at 15 km/hr
• Distance 2 = 60 km at 10 km/hr
• Distance 3 = 60 km at 12 km/hr

Now, calculate the time taken for each part:

• Time for Part 1: \( t_1 = \frac{\text{Distance}_1}{\text{Speed}_1} = \frac{60 \text{ km}}{15 \text{ km/hr}} = 4 \text{ hours} \)
• Time for Part 2: \( t_2 = \frac{\text{Distance}_2}{\text{Speed}_2} = \frac{60 \text{ km}}{10 \text{ km/hr}} = 6 \text{ hours} \)
• Time for Part 3: \( t_3 = \frac{\text{Distance}_3}{\text{Speed}_3} = \frac{60 \text{ km}}{12 \text{ km/hr}} = 5 \text{ hours} \)

Total time taken = \( t_1 + t_2 + t_3 = 4 + 6 + 5 = 15 \text{ hours} \).

Total distance travelled = 180 km.

Average Speed = \( \frac{\text{Total Distance}}{\text{Total Time}} = \frac{180 \text{ km}}{15 \text{ hours}} = 12 \text{ km/hr} \).

Method 2: Using Harmonic Mean Formula for Equal Distances

When equal distances are travelled at different speeds, the average speed is the harmonic mean of the speeds. For three equal distances travelled at speeds \(v_1, v_2, v_3\), the average speed \(V_{avg}\) is given by the formula:

\( V_{avg} = \frac{3}{\frac{1}{v_1} + \frac{1}{v_2} + \frac{1}{v_3}} \)

Given speeds are \(v_1 = 15\) km/hr, \(v_2 = 10\) km/hr, and \(v_3 = 12\) km/hr.

Substitute the values into the formula:

\( V_{avg} = \frac{3}{\frac{1}{15} + \frac{1}{10} + \frac{1}{12}} \)

Find a common denominator for 15, 10, and 12, which is 60.

\( \frac{1}{15} = \frac{4}{60} \)

\( \frac{1}{10} = \frac{6}{60} \)

\( \frac{1}{12} = \frac{5}{60} \)

Now, sum the fractions in the denominator:

\( \frac{1}{15} + \frac{1}{10} + \frac{1}{12} = \frac{4}{60} + \frac{6}{60} + \frac{5}{60} = \frac{4+6+5}{60} = \frac{15}{60} = \frac{1}{4} \)

Substitute this sum back into the average speed formula:

\( V_{avg} = \frac{3}{\frac{1}{4}} = 3 \times 4 = 12 \text{ km/hr} \)

Both methods yield the same average speed of 12 km/hr.

Summary of Calculations

The average speed of Ram for the entire journey is 12 km/hr.

Revision Table: Key Concepts

Additional Information on Average Speed Problems

Problems involving speed, distance, and time often require careful consideration of whether distance or time is constant across different parts of the journey.

• When an object travels equal distances at different speeds, the average speed is calculated using the harmonic mean. This is because the slower speeds impact the total time significantly more than faster speeds over the same distance.
• When an object travels for equal durations of time at different speeds, the average speed is simply the arithmetic mean of the speeds. In this scenario, the total distance is the sum of (speed * time) for each segment.
• If neither distance nor time is equal, you must calculate the time taken for each segment (Time = Distance / Speed), sum up the total distance and total time, and then divide total distance by total time to find the average speed.