Two most common errors you should avoid when you see an average speed calculation

**Mistake #1**

The most common mistakes in average speed problems occur when the question asks you to calculate the average speed of an object moving at two different speeds for different parts of the journey. Students are generally tempted to calculate the arithmetic mean (average) of the two speeds and select a corresponding answer. This averaging approach is wrong.

*Incorrect way*

$\text {Average speed} = \text {Arithmetic mean of speeds} = \frac {v_1 + v_2}{2}$

*Correct Way*

$\text {Average Speed} =\frac {\text{ Total distance }}{\text {Total Time taken}}$

**Mistake #2**

Another problem that happens Quite often is with the questions where we have different units of measurement in one question. For example, if a Truck takes 60 minutes to travel 60 miles, and then takes 6 hours to travel 60 miles, you should calculate the average speed of the truck for the entire journey expressing 60 minutes as 1 hour or by converting 6 hours to 360 minutes. Either way, you should never calculate time, distance, or speed using different units of measurement for the different parts of the journey. In this problem, if you had wrongly used two different units of time for calculating average speed, you would obtain the wrong answers even with the correct formula.

**There are Shortcuts for average speed calculation**

There are two situations in which you can save time when dealing with average speed questions:

-When the distances are equal

*Average speed*= $ \frac {2(ab)}{(a+b)}$

Here a and b are the speed in two parts. This formula is particularly useful for round trips where the outbound and return speeds are different

This can be also written as

$<v> = \frac {2}{\frac {1}{a} + \frac {1}{b} }$

Which is the harmonic mean of the speed

-When times are equal*Average Speed*= $ \frac {(a+b)}{2}$

Similar to distances, if two legs of a trip take equal time but are traveled at different speeds, the average speed can be found as the simple average of the two speeds.

This is called the arithmetic mean of the speeds

**Important points**

- When distances are unequal, calculate the total distance and total time separately, then use the basic formula. There’s no simple shortcut, but breaking the journey into segments and summing their distances and times keeps it manageable.
- Always define what is constant (distance, time, or speed) and what varies in your specific problem. This approach helps in selecting the most appropriate shortcut or method.

**Example 1**

A man moves from Home to Shop at speed 10 km/hr and then comes back through same path with speed 20 km/hr. What is the average speed of the Journey**Solution**

Incorrect way will be

Average Speed = $ \frac {10 + 20}{2} = 15$ km/hr

Correct Way will be

*Average speed*= $ \frac {2(ab)}{(a+b)} = \frac { 2 \times 10 \times 20}{10 + 20} =13.33$ km/hr

## Solved Questions

**Question 1**

A person drives 60 km at 30 km/h and returns along the same route at 60 km/h. What is their average speed for the entire trip?

**Solution:**

Using the shortcut for journeys with two segments at different speeds over equal distances:

$$

\text{Average Speed} = \frac{2 \times ab}{\a + b} = \frac{2 \times 30 \times 60}{30 + 60} = \frac{3600}{90} = 40 \text{ km/h}

$$

**Question 2**

A person drives 100 km at 50 km/h, then 200 km at 100 km/h. What is their average speed for the entire trip?

**Solution:**

First, calculate total distance and total time separately, then use the basic formula.

- Total Distance = 100 km + 200 km = 300 km
- Time for first part = 100 km / 50 km/h = 2 hours
- Time for second part = 200 km / 100 km/h = 2 hours
- Total Time = 2 hours + 2 hours = 4 hours

$$

\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{300 \text{ km}}{4 \text{ h}} = 75 \text{ km/h}

$$

**Question 3**

A cyclist goes to a destination 45 km away at a speed of 15 km/h and returns at a speed of 30 km/h. What is the average speed for the entire trip?

**Solution:**

This is a round trip with equal distances at different speeds, so we use the formula for equal distances:

$$

\text{Average Speed} = \frac{2 \times ab}{a + b} = \frac{2 \times 15 \times 30}{15 + 30} = \frac{900}{45} = 20 \text{ km/h}

$$

**Question 4**

A trip consists of three segments: 60 km at 60 km/h, 40 km at 80 km/h, and 100 km at 50 km/h. What is the average speed for the entire trip?

**Solution**

Calculate total distance and total time separately:

- Total Distance = 60 km + 40 km + 100 km = 200 km
- Time for first segment = 60 km / 60 km/h = 1 hour
- Time for second segment = 40 km / 80 km/h = 0.5 hours
- Time for third segment = 100 km / 50 km/h = 2 hours
- Total Time = 1 + 0.5 + 2 = 3.5 hours

$$

\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{200 \text{ km}}{3.5 \text{ h}} \approx 57.14 \text{ km/h}

$$

**Question 5**

A car travels 300 km at an average speed of 75 km/h. However, the car stops for a total of 1 hour during the trip. What is the effective average speed for the entire duration including the stop?

**Solution:**

First, calculate the driving time without stops:

- Driving Time = 300 km / 75 km/h = 4 hours
- Total Time with stop = 4 hours + 1 hour = 5 hours

$$

\text{Effective Average Speed} = \frac{\text{Total Distance}}{\text{Total Time including stop}} = \frac{300 \text{ km}}{5 \text{ h}} = 60 \text{ km/h}

$$

**Question 6**

A person travels for 1 hour at 30 km/h and another hour at 60 km/h. What is their average speed?

**Solution**

**Method 1**

To find the average speed when a person travels for equal time intervals at different speeds, we can simply take the arithmetic mean of the speeds or calculate using same distance ,speed formula

Given:

- Speed for the first hour: 30 km/h
- Speed for the second hour: 60 km/h
- Time for each interval: 1 hour

The total distance traveled can be calculated as the sum of distances for each interval, and the total time is the sum of times for each interval.

- Total Distance: The distance traveled in the first hour at 30 km/h is (30 \times 1 = 30) km, and the distance traveled in the second hour at 60 km/h is (60 \times 1 = 60) km. Thus, the total distance is (30 + 60 = 90) km.
- Total Time: The total time spent traveling is (1 + 1 = 2) hours.
- Average Speed: The average speed is the total distance divided by the total time:

$$

\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{90 \text{ km}}{2 \text{ h}} = 45 \text{ km/h}

$$

Therefore, the average speed over the entire journey is 45 km/h.

**Method II**

As time is equal , we can find the arithmetic mean

$\text{Average Speed} = \frac {30 + 60}{2} = 45 km/h.$