A man travels a distance of 420 km by train which moves at the speed of 75 km/h and returns back by car at the speed of 50 km/h. Find his average speed for the whole journey.
60 km/h
Calculating Average Speed for a Round Journey
The question asks us to find the average speed of a man for his entire journey, which involves travelling from one point to another by train and returning to the starting point by car. The key to calculating average speed is to use the formula:
$$\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}$$
We are given the distance for one leg of the journey and the speeds for both legs. Let's break down the calculation step-by-step.
Step 1: Determine the Total Distance Travelled
The man travels a distance of 420 km by train to reach a destination and then returns back by car. This means the distance covered in the return journey is also 420 km.
- Distance one way (Train): 420 km
- Distance return way (Car): 420 km
- Total Distance = Distance (Train) + Distance (Car)
- Total Distance = 420 km + 420 km = 840 km
Step 2: Calculate the Time Taken for Each Leg of the Journey
We can calculate the time taken for each part of the journey using the formula:
$$\text{Time} = \frac{\text{Distance}}{\text{Speed}}$$
- Time taken by Train:
- Speed of Train = 75 km/h
- Distance by Train = 420 km
- Time by Train = $\frac{420 \text{ km}}{75 \text{ km/h}}$
- Time by Train = $\frac{420}{75}$ hours
Let's simplify the fraction for time taken by train:
$$\frac{420}{75} = \frac{420 \div 15}{75 \div 15} = \frac{28}{5} \text{ hours}$$
Or, as a decimal: $\frac{28}{5} = 5.6$ hours.
- Time taken by Car:
- Speed of Car = 50 km/h
- Distance by Car = 420 km
- Time by Car = $\frac{420 \text{ km}}{50 \text{ km/h}}$
- Time by Car = $\frac{420}{50}$ hours
Let's simplify the fraction for time taken by car:
$$\frac{420}{50} = \frac{420 \div 10}{50 \div 10} = \frac{42}{5} \text{ hours}$$
Or, as a decimal: $\frac{42}{5} = 8.4$ hours.
Step 3: Calculate the Total Time Taken for the Whole Journey
The total time is the sum of the time taken for the journey by train and the journey by car.
- Total Time = Time by Train + Time by Car
- Total Time = $\frac{28}{5} \text{ hours} + \frac{42}{5} \text{ hours}$
- Total Time = $\frac{28 + 42}{5} \text{ hours} = \frac{70}{5} \text{ hours}$
- Total Time = 14 hours
Step 4: Calculate the Average Speed for the Whole Journey
Now we have the total distance and the total time. We can use the average speed formula.
- Total Distance = 840 km
- Total Time = 14 hours
- Average Speed = $\frac{\text{Total Distance}}{\text{Total Time}}$
- Average Speed = $\frac{840 \text{ km}}{14 \text{ hours}}$
Let's calculate the division:
$$\frac{840}{14} = \frac{84 \times 10}{14} = 6 \times 10 = 60 \text{ km/h}$$
So, the average speed for the whole journey is 60 km/h.
| Leg of Journey | Distance (km) | Speed (km/h) | Time (hours) |
|---|---|---|---|
| Outward (Train) | 420 | 75 | $\frac{420}{75} = \frac{28}{5}$ |
| Return (Car) | 420 | 50 | $\frac{420}{50} = \frac{42}{5}$ |
| Total | $420 + 420 = 840$ | - | $\frac{28}{5} + \frac{42}{5} = \frac{70}{5} = 14$ |
| Average Speed = $\frac{\text{Total Distance}}{\text{Total Time}} = \frac{840}{14} = 60$ km/h | |||
The average speed for the entire journey is 60 km/h.
Revision Table: Key Concepts for Speed, Distance, Time
| Concept | Formula | Units (Standard) |
|---|---|---|
| Speed | $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$ | m/s, km/h, mph |
| Distance | $\text{Distance} = \text{Speed} \times \text{Time}$ | m, km, miles |
| Time | $\text{Time} = \frac{\text{Distance}}{\text{Speed}}$ | s, hours, minutes |
| Average Speed (General) | $\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}$ | m/s, km/h, mph |
Additional Information: Average Speed Shortcut for Equal Distances
When a journey is divided into two equal distances, travelled at different speeds, say $v_1$ and $v_2$, there is a useful shortcut formula for the average speed. This is the harmonic mean of the two speeds.
If a person travels a distance 'd' at speed $v_1$ and returns the same distance 'd' at speed $v_2$, the total distance is $d+d=2d$.
The time taken for the first part is $t_1 = \frac{d}{v_1}$.
The time taken for the second part is $t_2 = \frac{d}{v_2}$.
The total time is $T = t_1 + t_2 = \frac{d}{v_1} + \frac{d}{v_2} = d \left(\frac{1}{v_1} + \frac{1}{v_2}\right) = d \left(\frac{v_2 + v_1}{v_1 v_2}\right)$.
The average speed is $\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{2d}{d \left(\frac{v_1 + v_2}{v_1 v_2}\right)} = \frac{2}{\left(\frac{v_1 + v_2}{v_1 v_2}\right)} = \frac{2 v_1 v_2}{v_1 + v_2}$.
In this problem, $v_1 = 75$ km/h and $v_2 = 50$ km/h. We can use this formula to verify our answer:
$$\text{Average Speed} = \frac{2 \times 75 \times 50}{75 + 50} = \frac{2 \times 75 \times 50}{125}$$
$$= \frac{2 \times 75 \times 50}{125} = \frac{2 \times (3 \times 25) \times (2 \times 25)}{5 \times 25} = \frac{2 \times 3 \times 25 \times 2 \times 25}{5 \times 25}$$
Cancel one 25 from numerator and denominator:
$$= \frac{2 \times 3 \times 2 \times 25}{5} = \frac{12 \times 25}{5}$$
Cancel 5 from numerator and denominator:
$$= 12 \times 5 = 60 \text{ km/h}$$
This shortcut confirms our detailed calculation. This formula is very useful for problems involving average speed over equal distances.
Important Questions from Average Speed
A car covers the first 41 km of its journey in 45 min and covers the remaining 23 km in 35 min. What is the average speed (in m/sec) of the car?
Akhil drives a car from his home to office at an average speed of 50 km/h and reaches office 10 minutes early. But one day due to some problem with the car, he could drive at an average speed of 30 km/h only and reached office 10 minutes late. How far is his office from home ?
During a flight of 900 km, an aircraft was slowed down due to bad weather. Its average speed for the trip was reduced by 300 km/h and the time of flight increased by 30 min. The original duration of the flight was :
If a man travels from A to B at a speed of 50 km/h and returns by increasing his speed by 40%, then find his average speed (to 2 decimal places) for both the trips.
A train runs at a speed of 90 km/h in the first 10 minutes and 60 km/h in next 25 minutes and 15 km/h in last 4 minutes. What is the average speed of the train?
