A bus made a roundtrip journey from Madison towards Chicago

Solution

Given:
• A bus made a roundtrip journey from Madison to Chicago
• Distance between Madison and Chicago = 240 km
• In the onward journey, the total journey time was divided into 3 equal blocks
• The bus increased its speed by 10 kph after every stoppage

o Speed in the 1st time block = 50 kph
o Speed in the 2nd time block = 60 kph
o Speed in the 3rd (final) time block = 70 kph

• Halt time in Chicago = 7 hrs 44 minutes = \(\frac{116}{15}\) hrs
• In the return journey, the total journey was divided into 2 equal distance blocks

o Speed in the 1st distance block = 45 kph
o Speed in the 2nd distance block = 75 kph

To find:
• The value of the average speed of the bus in the whole journey is what amount less than the sum of the average speeds of the onward and return journeys

Approach and Working:
• In the onward journey, let us assume each time block takes t hrs

o Therefore, we can say 50t + 60t + 70t = 240
Or, 180t = 240
Or, t = \(\frac{240}{180}\) hrs = \(\frac{4}{3}\) hrs

• Hence, the total time taken in the onward journey = 3t = \(3 * \frac{4}{3}\) hrs = 4 hrs
• Therefore, the average speed in the onward journey = \(\frac{240}{4}\) kph = 60 kph
[as the time taken in each block is same, we could have also calculated the average speed by taking the arithmetic mean of the individual speeds, i.e. \(\frac{(50 + 60 + 70)}{3}\) = 60 mph]


• In the return journey, because the total distance is equally divided into 2 parts, we can say the average speed is = \(\frac{(2 * 45 * 75)}{(45 + 75)}\) = 56.25 kph
• Also, the time taken in the return journey = (\(\frac{120}{45} + \frac{120}{75}\)) hrs = (\(\frac{8}{3} + \frac{8}{5}\)) hrs = \(\frac{64}{15}\) hrs


• Now, if we consider the whole journey,
The total time taken = (\(4 + \frac{116}{15} + \frac{64}{15}\)) hrs = 16 hrs

o Total distance travelled = (240 + 240) km = 480 km

• Therefore, the average speed of the whole journey = \(\frac{480}{16}\) kph = 30 kph

• So, the amount by which overall average speed is less than the sum of the individual average speeds = (60 + 56.25) – 30 kph = 116.25 – 30 kph = 86.25 kph

Hence, the correct answer is option B.

Answer: B