Calculating Average Speed for a Two-Way Journey

This problem asks us to find the average speed of a car that travels between two locations, Hayathnagar and LB Nagar, at different speeds for each part of the trip.

Understanding Average Speed

Average speed is defined as the total distance covered divided by the total time taken for the journey. It's important to remember that when the time intervals are different for different speeds, we cannot simply average the speeds arithmetically. Instead, we must use the fundamental definition:

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

Step-by-Step Calculation

Let's break down the calculation:

• Define Variables:
  • Let the distance between Hayathnagar and LB Nagar be \( d \) km.
  • Speed from Hayathnagar to LB Nagar (\( v_1 \)) = 64 km/hr.
  • Speed from LB Nagar to Hayathnagar (\( v_2 \)) = 56 km/hr.
• Calculate Time for Each Leg:
  • Time taken for the first leg (Hayathnagar to LB Nagar), \( t_1 \):

    Using the formula \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \)

    $ t_1 = \frac{d}{v_1} = \frac{d}{64} \text{ hours} $

  • Time taken for the return leg (LB Nagar to Hayathnagar), \( t_2 \):

    $ t_2 = \frac{d}{v_2} = \frac{d}{56} \text{ hours} $

• Calculate Total Distance:

  The car travels \( d \) km to LB Nagar and \( d \) km back to Hayathnagar.

  Total Distance = \( d + d = 2d \) km.

• Calculate Total Time:

  Total Time = \( t_1 + t_2 \)

  $ T = \frac{d}{64} + \frac{d}{56} \text{ hours} $

  To add these fractions, we find a common denominator for 64 and 56. The least common multiple (LCM) is 448.

  $ T = \frac{7d}{448} + \frac{8d}{448} = \frac{7d + 8d}{448} = \frac{15d}{448} \text{ hours} $

• Calculate Average Speed:

  Now, we apply the average speed formula:

  $ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{2d}{\frac{15d}{448}} $

  We can cancel out \( d \) from the numerator and denominator:

  $ \text{Average Speed} = \frac{2}{\frac{15}{448}} = 2 \times \frac{448}{15} = \frac{896}{15} \text{ km/hr} $

• Rounding the Result:

  To find the value, we perform the division:

  $ \frac{896}{15} \approx 59.7333... \text{ km/hr} $

  Rounding this to the nearest integer gives us 60 km/hr.

Conclusion

The average speed of the car for the entire journey, considering the different speeds for each leg, is approximately 60 km/hr when rounded to the nearest integer. This calculation uses the harmonic mean approach, suitable for averaging rates over equal distances.