Average problem tricks and shortcuts | average math concept.

Averages math concept :

The word 'average' is common to everyone.
Data collected are always necessary for every project.

Definition :

" The sum of all observations divided by total number of observations is known as average also know as arithmetic mean".
Or
" When objects are divided into equal parts, the number representing the same part is called average."

For example

- Average marks of students.

- Average height of students.

- Average speed of the bike.

- Average profit of the year.

etc.

Properties of Average :

1. The average of the numbers is also called the mean of the numbers.
2. The sequential numbers are equal to the numbers in the mean.
3. The average of sequential even or odd numbers is equal to the mean number.

Some important formulae :
1. Average = ( The sum of all the given numbers ) / Total number.
2. The sum of all the given numbers = Average × Total number.
3. Total number = Average × The sum of all the given numbers.
4. Sequential average of odd numbers = Total odd number
5. Sequential average of even numbers = Total even number + One.

Objective of an Average :

- The main object in calculating an average is to obtain a single figure, which will represent the whole data.

For example :

If we consider the ages of all the children in standard VI we will find that some are 11 years old, some 12 and some 13. That is, the age of Std VI children is around 12 years. In mathematical language, we say that the average age of Std. VI children is 12 years.

- We often need to compare one set of figures with another.

Types of Averages :

I. Arithmetic Mean

II. Median

III. Mode

Examples :

1. In six cricket matches, Virat scored 52, 74, 80, 40, 102 and 78 runs respectively. What is his average score?

Ans. : The sum of his scores in all matches

= 52 + 74 + 80 + 40 + 102 + 78

= 426

Total number of matches = 6

Therefore Average score = 426 / 6 = 71.

2. The daily income of a vegetable vender on four days was Rs. 200, Rs. 300, Rs. 430 and Rs. 270 what was his average daily income?

Ans. : The sum of his income in all days

= 200 + 300 + 430 + 270

= 1200

Total number of days = 4

Therefore Average income = 1200 / 4 = 300.

3. What is the average of the first 4 prime numbers ?
Ans. :
First five prime numbers : 2, 3, 5, 7.
Average = ( 2 + 3 + 5 + 7 ) / 4
                = 17 / 4
                = 4.25

4.What is the average of 1 + 2 + 3 + ----+ 10.
Or. 1 + 2 + 3 + - - - + 10 = ?
Ans. :
These are sequential numbers.
Therefore, Average = ( First number + Last number ) / 2
Average = ( 1 + 10 ) / 2 = 5.5

5. What is the average of first 5 odd numbers ?
Or. 1 + 3 + 5 + 7 + 9 = ?
Ans. : Sequential average of odd numbers = Total odd number
Therefore,
Average of first 5 odd numbers = 5.

6. What is the average of first 5 even numbers ?
Or. 2 + 4 + 6 + 8 + 10 = ?
Ans. : Sequential average of even numbers = Total even number + 1
Therefore,
Average of first 5 even number = 5 + 1 = 6

7. The average of the 5 numbers is 40, the average of the first 4 numbers is 35, which is the 5th number?
Ans. :
Total number × Average = The sum of the all the given numbers.
5 × 40 = 200.
4 × 35 = 140.
Therefore, 5th number = 200 - 140 = 60.

8. There are 30 students in a class and their average age is 15 years. If the teacher comes to the class average increases by 1, then how old is the teacher.
Ans. :
Shortcut Method :
In a given example, if the average only increases by 1, then,
Teacher Age = The sum of the all numbers.
                       = 30 + 15 + 1
                       = 46 years old.


 Related Links:

1. Divisibility Rules :

2. Number System math :

3. Ratio and Proportion :

4. Percentage tricks and shortcuts :

5. Profit and loss math tricks :

6. Average problems tricks and shortcuts :

7. lcm and gcd questions and answers :