One of the simple yet very useful topic in the section of Mathematics is Averages.
Average means the value which is representative of the various values in a set. Now, this is a very crude definition of the word. Mathematically, there are better definitions. However, this should serve ok for a beginner.
Let us say that there are 3 boys A, B and C. A scores 20 marks, B scores 15 and C scores 25. The total of all the three boys is 60. Now if B and C had also scored 20 each, then the total remains at 60.
The value '20' is a representative value of the set in general. Why does the total not change? Because the increase in B's score is offset by the same decrease in C's score. That is we took 5 from C and gave it to B.
To find the average of this set, we simply take the total of all marks and divide it by the number of students.
i.e. 60 / 3 = 20.
This is the basic concept in averages.
Average = (Total of all observations)/(number of observations)
To take another example, if 4 students score 17, 19, 21 and 23 respectively, then the total is 80 and the average is (80/4) = 20.
Isn't this the same as taking 3 from the 4th student and giving it to the first and taking 1 from the third and giving it to the second? So all the four have 20 each. i.e. Each has the same value without affecting the total.
Typically, we get problems where the weight of some boys is given. We need to find the average if one of the boys is replaced with another.
How do we solve such problems?
Let's say there are 10 boys whose average weight is 50 kg. One of the boys is replaced by another who weighs 100 kg. So, what is the new average?
If the average weight of each boy is 50, that means each boy contributes 50 kgs to the total weight. (Individual weights do not matter.) So the total weight here is 10 x 50 = 500 kg.
When we remove 1 boy, we are effectively removing 50 kg from the total. So now we have 9 boys with a total weight of (500-50) = 450 kg.
The new boy brings in 100 kgs to the total. So now, total weight = 550 and number of boys = 10. So the average weight is now 550/10 = 55 kg.
Another way of doing this is, the 100 kg boy replaces a 50 kg boy. So from 100, 50 kg go towards maintaining the original total. The excess 50 kgs is divided equally among the 10 boys. So each boy gets an excess 5 kg. So the 50 kg boys are now to be taken as 55 kg boys and hence the average is 55 kgs.
Simple!
Average means the value which is representative of the various values in a set. Now, this is a very crude definition of the word. Mathematically, there are better definitions. However, this should serve ok for a beginner.
Let us say that there are 3 boys A, B and C. A scores 20 marks, B scores 15 and C scores 25. The total of all the three boys is 60. Now if B and C had also scored 20 each, then the total remains at 60.
The value '20' is a representative value of the set in general. Why does the total not change? Because the increase in B's score is offset by the same decrease in C's score. That is we took 5 from C and gave it to B.
To find the average of this set, we simply take the total of all marks and divide it by the number of students.
i.e. 60 / 3 = 20.
This is the basic concept in averages.
Average = (Total of all observations)/(number of observations)
To take another example, if 4 students score 17, 19, 21 and 23 respectively, then the total is 80 and the average is (80/4) = 20.
Isn't this the same as taking 3 from the 4th student and giving it to the first and taking 1 from the third and giving it to the second? So all the four have 20 each. i.e. Each has the same value without affecting the total.
Typically, we get problems where the weight of some boys is given. We need to find the average if one of the boys is replaced with another.
How do we solve such problems?
Let's say there are 10 boys whose average weight is 50 kg. One of the boys is replaced by another who weighs 100 kg. So, what is the new average?
If the average weight of each boy is 50, that means each boy contributes 50 kgs to the total weight. (Individual weights do not matter.) So the total weight here is 10 x 50 = 500 kg.
When we remove 1 boy, we are effectively removing 50 kg from the total. So now we have 9 boys with a total weight of (500-50) = 450 kg.
The new boy brings in 100 kgs to the total. So now, total weight = 550 and number of boys = 10. So the average weight is now 550/10 = 55 kg.
Another way of doing this is, the 100 kg boy replaces a 50 kg boy. So from 100, 50 kg go towards maintaining the original total. The excess 50 kgs is divided equally among the 10 boys. So each boy gets an excess 5 kg. So the 50 kg boys are now to be taken as 55 kg boys and hence the average is 55 kgs.
Simple!
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