How to calculate the weighted average?

🏆Practice weighted average

Weighted Average: What Does It Really Mean?

A weighted average is an average among numbers with different weights.
Each number has its own weight and, therefore, will affect the weighted average.
Try replacing the word weight with the word importance and in this way its meaning will be better understood.
The numbers are of different importance. One number is more important and another number is less important. It does not mean a large or small number, but simply important.
When a number is more important, it has a greater weight and will have a greater effect on the weighted average.

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Test yourself on weighted average!

einstein

Norbert buys some new clothes.

When he gets home, he decides to work out how much each outfit cost him on average.

4 t-shirts = $45
2 pairs of shorts = $50
3 pairs of pants = $80
2 sweaters = $100
1 coat = $210

What answer should he come up with?

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How will we remember it?
Pay attention to the weighted word.
Remember that numbers do not have the same weight. They do not have the same importance and when calculating the weighted average you will have to take into account the weights of the numbers.
Imagine you have to calculate the average of your final grade in the subject - Spanish language.

Therefore, if you obtained 100 100 in an exam but 20 20 in the final test, the score of 20 20 will affect you much more in the final grade, since the weight of the score in the last test is higher than the weight of the score in the beginning of the year test.


Keep in mind that you must match each number with its weight according to the data of the assignment.
Multiply the number by its weight, and then add the multiplication of the second number by its weight. And so on to all the numbers for which you need to calculate the weighted average.


Examples for calculating the weighted average:

The simplest example to understand this topic is actually from a world that is familiar to you: the academic framework. As you know, throughout your math studies, you are given both exams and assessments. As is well known, exams have a greater weight on the final grade report, while assessments have a lesser weight. This is a classic case of weighted average.

Suppose these are your math grades in the first semester:

  • Equations assessment 75 75 with an approximate weight of 10% 10 \% .
  • Geometry assessment on triangles 95 95 with an approximate weight of 10% 10\%
  • A final exam on all the studied material 85 85 with an approximate weight of 80% 80\% .

The calculation of the weighted average will be done using the following formula:

75×0.1+95×0.1+85×0.8 75\times0.1+95\times0.1+85\times0.8

The obtained weighted average is: 85 85


Another example:

To illustrate the importance of each percentage in the grade, we will demonstrate another example: the same grades but with different weight percentages:

  • Equations exam 75 75 with an approximate weight of 25% 25\% .
  • Geometry exam on triangles 95 95 with an approximate weight of 15% 15\% .
  • Final exam on all the studied material 85 85 with an approximate weight of 60% 60\% .

75×0.25+95×0.15+85×0.6 75\times0.25+95\times0.15+85\times0.6

The obtained weighted average is: 84 84


Another example to calculate the weighted average:

Ivan received the following English grades in the first semester and wants to know his weighted average in the subject.

English reading comprehension exam - grade 80 80 with a weight of 20% 20\% .

English vocabulary exam - grade 90 90 with a weight of 20% 20\% .

Final semester exam - grade 70 70 with a weight of 60% 60\% .

Calculation of the weighted average of the English grades.

0.2×80+0.2×90+70×0.6= 0.2\times80+0.2\times90+70\times0.6= weighted average 76 76


An extra example to calculate the weighted average:

Miguel traveled from Madrid to Barcelona at different speeds, calculate the average travel speed (weighted average):

80 80 km/h approximately 40% 40\% of the journey

90 90 km/h approximately 20% 20\% of the journey.

100 100 km/h approximately 20% 20\% of the journey.

80×0,4+90×0,2+100×0,2= 80\times0,4+90\times0,2+100\times0,2= Miguel's average weighted speed is equal to 70 70


Keep in mind: if you were asked to calculate the average of the speeds (and not the weighted average speed), then the answer was 90 90 . Every question must be read carefully! Answering too quickly (not answering what was asked) can cause the loss of all the points for the question.

  • Turn the "problem" into a common everyday life situation.
  • As is well known, the calculation of the weighted average is based on a simple principle: each "score" / value, is calculated individually according to its weight. How do you approach a question in which you are asked to calculate a weighted average?
  • Read the question at least twice
  • Emphasize the essentials: What are you being asked to do?
  • Write down all the data from the questions in a table
  • Change the story frame to a more "friendly" everyday life situation.

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Exercises

Exercise 1

Given:

On average, each building on the street has 4.29 4.29 floors

It is known that there are two buildings with 11 11 floors, 4 4 buildings with 2 2 floors, and 5 5 buildings with 3 3 floors.

Task

How many buildings have 5 5 floors?

Solution

Average=((Number of buildings×Number of floors)+(Number of buildings×Number of floors))Total number of buildings Average=\frac{((Number~of~buildings\times Number~of~floors)+(Number~of~buildings\times Number~of~floors))}{{Total~number~of~buildings}}

We mark the number of buildings with 5 5 floors as X X

4.29=11×2+2×4+3×5+5×X2+4+5+X 4.29=\frac{11\times 2+2\times 4+3\times 5+5\times X}{2+4+5+X} ,4.29=22+8+15+5X11+X 4.29=\frac{22+8+15+5X}{11+X}

We multiply the formula by: (11+X) (11+X)

4.29(11+X)=45+5X 4.29(11+X)=45+5X

47.19+4.29X=45+5X 47.19+4.29X=45+5X

We subtract from the equation: 45 -45 and 4.29X -4.29X

47.1945=5X4.29X 47.19-45=5X-4.29X

2.19=0.71X 2.19=0.71X

We divide the equation by: 0.71 0.71

3=2.190.71=X 3=\frac{2.19}{0.71}=X

Answer

The correct answer is 3 3 buildings


Exercise 2

In the biology course class, the distribution of student results was:

  • 30% 30\% of the students scored 75 75
  • 20% 20\% scored 68 68
  • X% X\% scored…
  • The rest scored 53 53

Question

What is the class average?

Solution

Average=(....+Score×Percentage+Score×Percentage)100 Average=\frac{(....+Score\times Percentage+Score\times Percentage)}{100}

“The rest” in the question= 1003020X=50X 100-30-20-X=50-X

average=3075+2068+X94+(50X)53100 average=\frac{30\cdot75+20\cdot68+X\cdot94+(50-X)53}{100}

=2250+1360+94X+505353X100 =\frac{2250+1360+94X+50\cdot53-53X\frac{}{}}{100}

=6260+41X100=62.6+0.41X =\frac{6260+41X}{100}=62.6+0.41X

Answer

=6260+41X100=62.6+0.41X =\frac{6260+41X}{100}=62.6+0.41X


Do you know what the answer is?

Exercise 3

In Mexico City, they decided to build new gardens:

In 4 4 gardens they planted 47 47 plants.

In 9 9 gardens they planted 38 38 plants.

In Y Y gardens they planted X X plants.

Task

How many plants were planted in each garden on average?

Solution

Plants in the garden on average=(Number of plants in garden×Number of gardens+.....)Total number of gardens Plants~in~the~garden~on~average=\frac{(Number~of~plants~in~garden\times Number~of~gardens+.....)}{Total~number~of~gardens}

=474+389+X×Y4+9+Y =\frac{47\cdot4+38\cdot9+X\times Y}{4+9+Y}

=188+342+XY13+Y=530+XY13+Y =\frac{188+342+XY}{13+Y}=\frac{530+XY}{13+Y}

Answer

The correct answer is 530+XY13+Y \frac{530+XY}{13+Y}


Exercise 4

Given: Rebecca has 17 17 weights that weigh an average of 5.22 5.22 kg.

It is known that 3 3 weights weigh 4.5 4.5 kg, 4 4 weights weigh 5.2 5.2 kg and the rest weigh 7.1 7.1 kg or 3.8 3.8 kg.

Task

How many weights does Rebecca have that weigh7.1 7.1 kg?

Solution

We mark the number of weights that weigh 7.1 7.1 kg as X X.

The number of weights that weigh 7.1 7.1 kg - number of weights that weigh 5.25.2 kg - number of weights that weigh 4.5 4.5 kg - Number of weights = Number of weights that weigh 3.8 3.8 kg

Weighted average=(Weight×Number of weights+Weight×Number of weights..)Number of weights Weighted~average=\frac{(Weight\times Number~of~weights+Weight\times Number~of~weights\ldots..)}{Number~of~weights}

5.22=4.53+5.24+7.1×X+3.8×(10X)17 5.22=\frac{4.5\cdot3+5.2\cdot4+7.1\times X+3.8\times(10-X)}{17}

We multiply the equation by 17 17 .

88.74=13.5+20.8+7.1X+383.8X 88.74=13.5+20.8+7.1X+38-3.8X

88.74=34.3+38+3.3X 88.74=34.3+38+3.3X

88.74=72.3+3.3X 88.74=72.3+3.3X

We subtract from the equation 72.3 72.3

16.44=3.3X 16.44=3.3X

We divide the equation by 3.3 3.3

5=16.443.3=X 5=\frac{16.44}{3.3}=X

Answer:

The number of weights that weigh 7.1 7.1 is 5 5


Examples and exercises with solutions on how to calculate the weighted average

Exercise #1

What is Michael's score if he gets 79 on the first exam and 83 on the second, given that the weight of the first test is 30% and that of the second is 70%?

Step-by-Step Solution

To solve the weighted average, we will use the following formula:

exam 2 * weight of evaluation 2 + exam 1 * weight of the evaluation 1 = Weighted average

 

We will place the data in the formula, where the weights will be in decimal numbers:

0.3*79 + 0.7*83 = 
23.7+58.1 = 

81.8

 

Answer

81.8 81.8

Check your understanding

"Can I learn a weighted average in an online class?"

Of course! In fact, there is no subject that cannot be learned in an online class. The lesson takes place in real time, with the student and teacher connected for a private class. It is conducted via a video call so that the student can calculate the exercises and present them in front of the camera. Meanwhile, the teacher can suggest additional ways to solve them, write them on the page, and present them in front of the camera. Tips to optimize your private lesson:

  • Define in advance what topic you would like to study in the class
  • Prepare questions/exercises you would like to solve
  • Prepare in advance a notebook, a textbook, and writing materials.
  • Connect to a lesson from a quiet room and with a quality internet connection
  • Tip: at the end of the lesson, coordinate the next lesson with the tutor

How much will I need to practice until I learn how to calculate the formula?

The calculation of the weighted average is considered, on many occasions, a type of question to give away points. The difficulty is subjective and may vary from one student to another. Practice the exercises just as the teacher gives them in the classroom. If you have been successful in most of the practice, you can successfully assess the topic. If you still find some difficulty, you can perfect the topic with a teacher.  

The formula is simple to apply, and requires a basic understanding of percentages (20% which becomes 0.2) of course, competence in simple addition and multiplication exercises. Why, after all, do students fail in the calculation of the weighted average? Because they rush to answer the question without realizing what they were asked. While the question being asked is not deeply understood, the data can be calculated on the basis of a "classic average" formula.


Do you think you will be able to solve it?

How do you memorize a formula? Just practice it!

The best way to become familiar with the formula and simply "flow" with it, is to practice it. The fact that you understand the importance of the weighted average is not enough, and it is important to practice as many different exercises as possible that challenge you. Sometimes, there is a great effort to memorize the formula as a formula, but without investing time in its actual application. Keep in mind that you will need to calculate the weighted average for weights, shapes, prices, scores, etc.


For a math exam, it is not possible to study in just one day.

Calculating the weighted average does not require too much from you, but simply to focus on a specific technique. The challenge for many students is to be able to retain all the material taught throughout the semester, which sometimes proves to be not so simple a task. In this way, different gaps are created in the studied material, both in slightly more complex topics and in those that are relatively simple, such as the calculation of the weighted average. Remember that mathematics is not possible nor is it worth learning the day before the assessment, so if there are difficulties, you should study them before the upcoming exams.


Test your knowledge

Private Class - All options are open to you

There are 3 ways to attend a private class:

  • At the student's house - The teacher goes to you.
  • At the teacher's house - the students go to the tutor's home.
  • Online: both meet for a LIVE private class, each from their own home.

Choose the lesson format that suits you best, all for your success in the upcoming assessment and in next school year's math studies. Successfully!


If you are interested in this article, you might also be interested in the following articles

How to calculate the average speed?

How to calculate the area of a regular hexagon?

How to calculate percentages?

In the blog of Tutorela you will find a variety of articles about mathematics.


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