First, we must differentiate the following two concepts to avoid confusion:

Average speed

Average velocity

At first glance, this looks like the same term, but in practice, it is not. Average speed asks you to know what is the general-classical average of the speed at which several drivers were traveling:

Example:

Ivan traveled at $70$ km/h.

Samuel at $80$ km/h.

Robert at $120$ km/h

The average velocity of all drivers by adding the speeds and dividing by $3 = 90$ km/h.

A man drives for two hours at a speed of 78 km/h, stops to get a coffee for fifteen minutes, and then continues for another hour and a half at a speed of 85 km/h.

Calculating average speed is done in a completely different way! Even before memorizing the different formulas, it's important that you understand the difference between the two terms. Remember: A lack of understanding of the question can result in the loss of all points on the exam, test, or final.

Calculating Average Speed: How Does It Work?

This type of question, by nature, includes quite a lot of data. Therefore, the first piece of advice for you is to stick to order and organization, and prepare all the data that appears in the question in an orderly table. Before you is a classic question that requires you to calculate the average speed.

Two important things:

Placing the data in a table is highly recommended in the exam! (On the quiz or on a draft).

Stoppages should also be calculated and noted (a common data point in speed questions).

Example question:

Tatiana went shopping in honor of the last day of school! She was not satisfied with going to just one mall, so she went to several different ones. First, she drove to a mall in Madrid at a speed of about 80 km/h for two hours. After the first place, she felt tired and stopped for a short time of one hour on the side of the road. After the break, she drove at a speed of about 160 km/h to the Salamanca mall for two hours. If so, what is the average speed at which Tatiana traveled?

Time

Speed

Distance

2

80

160

1

0

0

2

160

320

The formula to calculate the average speed: the entire distance Tatiana traveled, divided by the total time spent.

$160+0+320=480$

The entire distance must be divided by the total time:$2+2+1=5$

$480:5=96$ This is Tatiana's average speed.

Additional examples:

Manuel and Gastón decided to enjoy a summer vacation in Barcelona! They left Madrid at $13:00$ at a speed of about $75$ km/h. At $15:00$ they took a one-hour break. After that, they continued driving at a speed of about $90$ km/h and arrived in Barcelona at $19:00$ . What is the average speed at which Manuel and Gastón traveled?

Time

Speed

Distance

2

75

150

1

0

0

3

90

270

And now, let's calculate the average speed at which Manuel and Gastón were driving. The formula for such a calculation is to divide the distance by the total time of all the trips they made.

$150+0+270=420$ The distance must be divided by the time $3+1+2=6$ The calculation: $420:6=70$ km/h

Another example: Ramiro and Roberto decided to go to the market to buy furniture for their new home! At $10:00$ they left Pescara at $85$ km/h, and arrived in Rome at $12:00$. They walked around the market for $3$ hours and bought a new table, living room set, and buffet! On the way back home, they drove at $50$ km/h due to traffic jams, and arrived only $3$ hours later. What is the average speed at which Ramiro and Roberto were driving?

Time

Speed

Distance

2

85

170

3

0

0

3

50

50

Now, let's calculate the average speed at which Ramiro and Roberto traveled:

$170+0+150=320$ The distance should be divided by the time $2+3+3=8$ The calculation: $320:8=40$ km/h

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The answer is yes. As you know, textbooks provide you with solutions to most questions. First, you should definitely try to deal with the problem and the data it presents to you. As long as you feel that you are not close to a solution, you can definitely seek an answer in favor of learning. Note: The goal is not to check off the task, but to understand the suggested solution through the answer.

Your Mistakes Are a Gift!

So it's true, it's better to be able to find a solution and get all the points on the test. While most students know how to handle success well, dealing with mistakes is a bit more complex. Did you make a mistake in solving the question? No problem! Mistakes can be basic calculation errors, but also errors that stem from a lack of understanding of the question. Mistakes are an integral part of your learning and development process! As long as you know how to recognize your mistakes, learn from them, and grow through them, you're on the right track.

Important: The way to learn from mistakes is to understand what the mistakes are. A private math tutor will be able to diagnose the "problems" you fall into, offer you ways to deal with them, and help you turn the mistake into an opportunity for development, growth, and assimilation of the material.

Additional tips for solving average speed problems:

Break down the question into factors: times, distances, and speed

Change the "structure" to a story that is easier for you to understand

Draw the road data on a draft sheet

Do you know what the answer is?

Question 1

Gary runs at a speed of 2 meters per second for 2 minutes, then stops for a minute and runs again for 2 minutes at the same speed.

These questions mostly require comprehension, so it's important to practice the formula as much as possible. These are not complex questions, and as long as you understand what is being asked, they are almost giveaway questions. Depending on the curriculum level, the questions become more complex and involve a greater number of unknowns.

Also in private lessons: practice average speed

These questions, which require the calculation of average velocity, are initially perceived as almost threatening. Compared to the calculation of average speed, these are more challenging questions, but not impossible. Even as part of a private lesson, you can focus on solving problems. What are the important emphases for problem-solving?

Read the problem about $3$ times (yes, even within a time limit on the test).

Highlight the question's data with a marker.

In summary: What are you being asked to do?

Firstly, solving a problem of this type will take you between a quarter of an hour and about $20$ minutes. The more you practice, the shorter the times will become, which can give you a significant advantage in exams. You can study your private lesson at the teacher's house or in your own home, but also in an online lesson that will save you a lot of time!

Check your understanding

Question 1

In a relay race, three runners run one after another on a 450-meter track.

Jonathan is reviewing his cycling records from his last competition.

During the first half hour, he rode at a speed of 28 km/h.

The following two hours, he rode at a speed of 24 km/h, then 15 minutes downhill at a speed of 32 km/h, before continuing for another hour at a speed of 27 km/h.

Rodney rides a motorcycle for \( \frac{1}{3} \) of an hour over a distance of 30 km, stops to rest for \( \frac{1}{6} \) of an hour, then continues for \( \frac{1}{4} \) of an hour.

His average speed is \( 66\frac{2}{3} \) km/h.

How far does he ride in the last quarter of an hour of his trip?

A man drives for two hours at a speed of 78 km/h, stops to get a coffee for fifteen minutes, and then continues for another hour and a half at a speed of 85 km/h.

As the name suggests, it is the average of the speeds that an object travels, calculated by dividing the total displacement by the total time taken for the journey.

Example.

Julian travels from one city to another in two stages. In the first stage, he travels at a speed of $110\frac{\operatorname{km}}{h}$ for 2 hours. Then he stops to eat for an hour, and in the second stage, he travels at a speed of $80\frac{\operatorname{km}}{h}$ for $3$ hours. Calculate the average speed Julian had on the trip.

Solution:

In the first stage, he travels at a speed of $110\frac{\operatorname{km}}{h}$ for two hours, so the distance covered is:

The average speed or mean velocity is calculated as the sum of all displacements divided by the sum of all times taken on a journey, mathematically we can express this statement as follows:

Where the numerator represents the sum of displacements and the denominator the sum of all times.

How to calculate average speed from a table?

To answer this question, let's look at the following example:

Diana studies the behavior of a particle moving in a straight line, observing that it travels at a speed of $40\frac{\operatorname{km}}{h}$ for one hour. Then it accelerates to a speed of $70\frac{\operatorname{km}}{h}$ for 3 hours and finally travels at a speed of $110\frac{\operatorname{km}}{h}$ for 5 hours. What is the particle's average speed?

Let's record these speeds and times in the following table:

Time

Speed

Distance

1

40

40

3

70

210

5

110

550

So we can calculate the average speed with the table:

Instantaneous speed is the speed of an object at a specific time, this time interval is very small, meaning the time to perform this movement is extremely short (in a brief instant).

What is the difference between instantaneous speed and average speed?

As already mentioned, instantaneous speed occurs in a brief instant, in a very small amount of time, while average speed is the average of speeds that an object has over some time intervals (it is the quotient of the sum of the displacements over the sum of all the times of the movement), this interval can be much larger compared to instantaneous speed.

Check your understanding

Question 1

Gary runs at a speed of 2 meters per second for 2 minutes, then stops for a minute and runs again for 2 minutes at the same speed.

Let's first recall the formula for finding velocity:

$V=\frac{x}{t}$

X=distance t=time V=velocity

We'll input the data according to the formula:

$V=\frac{210+40+0+250}{3+1+2+2.5}$

We'll calculate accordingly and get:

$V=\frac{500}{8.5}=58.82$

The average velocity is 58.82

Answer

58.82....

Exercise #2

In a relay race, three runners run one after another on a 450-meter track.

The first runner finishes in 1.5 minutes.

The second runner finishes in 1.35 minutes.

The third runner finishes in 1.42 minutes.

What is the average speed of the relay runners?

Video Solution

Step-by-Step Solution

Answer

$5.3$ meters per second

Exercise #3

A man drives for two hours at a speed of 78 km/h, stops to get a coffee for fifteen minutes, and then continues for another hour and a half at a speed of 85 km/h.

What is his average speed?

Video Solution

Answer

$75.6$ km/h

Exercise #4

A snail crawls for 7 minutes at a speed of 4 cm per minute, rests for 3 minutes, then continues to crawl a further 30 cm in 12 minutes.

What is its average speed?

Video Solution

Answer

$2.64$ cm per minute

Exercise #5

A truck driven by George makes its journey in two parts.

In the first part, its speed is 82 km/h and it travels for 4 hours.

Then, George has a break at a petrol station for 20 minutes.

In the second part, George travels at a speed of 70 km/h for 3 hours.