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# Formulas for Averages

** Formulas To Find Averages In Aptitude**

Average is widely used in statistics also to reduce the calculation in finding the average where the data is huge.Here We will demonstrate the application of the assumed mean to solve some aptitude questions based on averages and weighted averages.Average is the sum of observations divided by the total number of observations.The Average is also known as mean.We use concept of average in our day to day life. Formulas of Averages is very helpfull in Different Examinations.

In this Page Formulas for Averages is given which is useful to solve many problems.

**An average is defined as the sum of n different units divided by n numbers of the units.**

**Example**:- What will be the average weight of three boy, respective weight are 46,54,53?

**Solution:** \frac{\text{sum of numbers}}{\text{total numbers}}

=\frac{(46+54+53 )}{3}

=\frac{153}{3}

=51

**Example**:- A school trip by St Marry’s Academy, Meerut was organized to Appu Ghar Delhi. The total number of Girls on the trip were 160, total number of boys on the trip were 40 and total number of teachers present on the trip were 100. If they want to ride a roller coaster and all of them can not board the ride at one time. Find the average no. people boarding the ride.

**Solution: **\frac{\text{sum of numbers}}{\text{total numbers}}

= \frac{(160+40+100)}{3}

= 100

### Basic Averages Formulas:

- Mathematically, it is defined as the ratio of summation of all the numbers to the number of units present in the list.

** Average =** \mathbf{\frac{X_{1}+X_{2}+X_{3}+X_{4}+…..X_{n}}{n}}

** OR**

** Average = **\frac{\text{Sum of Observations}}{\text{Total Number of Observations}}

### Average Speed and Velocity Formula:

- Average Speed : It can be defined as total distance Travelled by a body in definite interval of time. Average speed is calculated using the below formula

**Average Speed = **\mathbf{\frac{\text{Total Distance}}{\text{Total Time}}}

**CASE 1: **

When one travels at speed ‘a’ for half the time and speed ‘b’ for other half of the time. Then, average speed is the arithmetic mean of the two speeds.

**Average Speed=** \mathbf{\frac{a+b}{2}}

**CASE 2 : **

When one travels at speed ‘a’ for half of the distance and speed ‘b’ for other half of the distance.Then, average speed is the harmonic mean of the two speeds.

**Average Speed=** \mathbf {\frac{2ab}{a+b} }

**CASE 3: **

When one travels at speed a for one-third of the distance, at speed b for another one-third of the distance and speed c for rest of the one-third of the distance

or:

**Average Speed = **\mathbf{\frac{3abc}{ab+bc+ca}}

**Average Velocity : **It can be defined as total displacement divided by total time. We calculate Average Velocity using the below formula

**Average Velocity = **\mathbf{\frac{Displacement}{Total Time}}

### Formula of Averages Related to Numbers:

- Average of ‘n’ consecutive Natural Numbers =

\mathbf{\frac{n+1}{2}} - Average of the square of consecutive n natural numbers =
\mathbf{\frac{(n+1)(2n+1)}{6}} - Average of cubes of consecutive n natural numbers =
\mathbf{\frac{n\times (n+1)^{2}}{4}} - Average of n consecutive even numbers
**=**(n+1) - Average of consecutive even numbers till n =
\mathbf{\frac{n}{2}+1} - Average of n consecutive odd numbers
**= n** - Average of consecutive odd numbers till n =
\mathbf{\frac{n+1}{2}} - Sum of 1st n even consecutive natural numbers is =
**n(n + 1)** - Sum of 1st n odd consecutive natural numbers is = \mathbf{n^{2}}

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