How to calculate the weighted average?

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 $100$ in an exam but $20$ in the final test, the score of $20$ 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.

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

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

$75\times0.1+95\times0.1+85\times0.8$

The obtained weighted average is: $85$

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

$75\times0.25+95\times0.15+85\times0.6$

The obtained weighted average is: $84$

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 $80$ with a weight of $20\%$.

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

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

Calculation of the weighted average of the English grades.

$0.2\times80+0.2\times90+70\times0.6=$ weighted average $76$

An extra example to calculate the weighted average:

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

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

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

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

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

Keep in mind: if you were asked to calculate the average of the speeds (and not the weighted average speed), then the answer was $90$. 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.

Given:

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

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

Task

How many buildings have $5$ floors?

Solution

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

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

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

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

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

$47.19+4.29X=45+5X$

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

$47.19-45=5X-4.29X$

$2.19=0.71X$

We divide the equation by: $0.71$

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

Answer

The correct answer is $3$ buildings

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

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

Question

What is the class average?

Solution

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

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

$average=\frac{30\cdot75+20\cdot68+X\cdot94+(50-X)53}{100}$

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

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

Answer

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

In Mexico City, they decided to build new gardens:

In $4$ gardens they planted $47$ plants.

In $9$ gardens they planted $38$ plants.

In $Y$ gardens they planted $X$ plants.

Task

How many plants were planted in each garden on average?

Solution

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

$=\frac{47\cdot4+38\cdot9+X\times Y}{4+9+Y}$

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

Answer

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

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

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

Task

How many weights does Rebecca have that weigh$7.1$kg?

Solution

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

The number of weights that weigh $7.1$ kg - number of weights that weigh $5.2$ kg - number of weights that weigh $4.5$ kg - Number of weights = Number of weights that weigh $3.8$ kg

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

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

We multiply the equation by $17$.

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

$88.74=34.3+38+3.3X$

$88.74=72.3+3.3X$

We subtract from the equation $72.3$

$16.44=3.3X$

We divide the equation by $3.3$

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

Answer:

The number of weights that weigh $7.1$ is $5$

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

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.

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.

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.

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.