# Quantitative Aptitude :: Average

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**Average Important Formulas**

The result obtained by adding several quantities together and then dividing this total by the number of quantities is called Average. The main term of average is equal sharing of a value among all, where it may share persons or things. We obtain the average of a number using the formula that is the sum of observations divides by Number of observations.
**1. Average:**

Formula to find Average = \(\frac{Sum of observations}{Number of observations}\)

**2. Average Speed Formula:**

** What is the formula for average speed?** Suppose a man covers a certain distance at

**x**

**kmph**

**and an equal distance at**

**y**

**kmph**

**.**

Then, the average speed during the whole journey will be \(\left(\frac{2xy}{x+y}\right)\)kmph.

__Average Methods shortcut tricks__

- If both the time taken are equal i.e
**t1 = t2 = t**,then, \(\frac{t1 + t2}{2}\) - The average of odd numbers from1 to n is = \(\frac{[Last \ odd \ no. + 1]}{2}\).
- The average of even numbers from1 to n is = \(\frac{[Last \ even \ no. + 2]}{2}\).
- The Average of any number of quantities is sum of their quantities by the number of quantities (n) => Average = \(\frac{Sum of quantities}{n}\).
- If there are two types of items say A and B , A has m number of sub items and B has n number of sum items then the average of A and B is \(\frac{A \times m + B \times n}{m+n}\).
- If a vehicle travels from one place to another at a speed of a kmph but returns at the speed of b kmph then its average speed during the whole journey is \(\left(\frac{2ab}{a + b}\right)\) kmph.
- Out of three numbers, first number is x times of the second number and y times of the third number. If the average of all the three numbers is z then the first number is \(\frac{3xyz}{xy + x + y}\)
- The average age of a group of N student is 'X' years. If M students joins, the average age of the group increases by 'Y' years, then the average age of the new students = \(x + \left(1 + \frac{N}{M}\right)\times Y \) years.
- The average age of a group of N student is 'X' years. If M students joins, the average age of the group decreased by 'Y' years, then the average age of the new students = \(x - \left(1 + \frac{N}{M}\right)\times Y \) years.
- The average age of a group of N student is 'X' years. If M student (Rahul) join the group, the average age of the group increases by 'Y' years, then the age of the new student (Rahul) is = \(x + \left(1 - \frac{N}{M}\right)\times Y \) years.
- The average age of a group of N student is 'X' years. If M student (Ram) left the group, the average age of the group decreased by 'Y' years, then the age of the new student (Ram) was = \(x - \left(1 - \frac{N}{M}\right)\times Y \) years.
- In a group of N persons whose average age is increased by 'Y' years when a person of 'X' years is replaced by a new man. Then the age of new comer is = \(\left(X + N \times Y\right)\) years.