MHB Average Problem [Need Formula]

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To find the average score of all students, first calculate the total score of the initial 36 students by multiplying their average score of 86 by 36, resulting in a total of 3096. Next, calculate the total score of the 4 additional students by multiplying their average score of 80 by 4, giving a total of 320. Combine these totals to get 3416 for all 40 students. Finally, divide this sum by the total number of students, 40, to find the overall average score, which is 85. The average value of all the students is therefore 85.
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hi all..

how to make formula for problem below :

The average math test scores of 36 students is 86. There are four students attend subsequent replications and obtain an average value of 80. Find the average value of all the students!

can you make simple?

zidan3311
 
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What do you know about the average and how it's worked out?

Could you find the total sum for the 36 students using the average given and the fact there are 36 students in the sample size?
 
You could use what's called a weighted average here. Suppose you have $n$ groups, and each group has a mean of $\overline{x_i}$ and $m_i$ members. Then the mean $\overline{x}$ of all groups combined is given by:

[box=blue]
Weighted Average

$$\overline{x}=\frac{\sum\limits_{i=1}^{n}\left(m_i\cdot\overline{x_i}\right)}{\sum\limits_{i=1}^{n}\left(m_i\right)}\tag{1}$$[/box]

In your case, you have 2 groups ($n=2$), where:

$$m_1=36,\,\overline{x_1}=86,\,m_2=4,\,\overline{x_2}=80$$

And so what do you get when you apply this data to (1)?
 
Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...

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