**Shortcuts in Quantitative Aptitude for Competitive Exams – Average**

Shortcuts in Quantitative AptitudeReasoningEnglish

**Average**

**AVERAGE**

‘Average ’ is a very simple but effective way of representing an entire group by a single value.

To calculate the sum of quantities, they should be in the same unit,

**Shortcut Approach
**

If X is the average of x1, x2, x3, …xn then

- The average of x
_{1}+a, x_{2}+a, x_{3}+ a ……….. +x_{n}+a is X + a. - The average of x
_{1}-a, x_{2}-a, x_{3}– a ……….. x_{n}-a is X -a - The average ofax
_{1, }ax_{2, }ax_{3, ……… }ax_{n }is aX, provied a ≠ 0 - The average of , provided a ≠ 0

**Average of a group consisting two different groups when their averages are known :**

Let Group A contains m quantities and their average is a and Group B contains n quantities and their average is b, then average of group

**WEIGHTED AVERAGE**

If we have two or more groups of members whose individual averages are known, then combined average of all the members of all the groups is known as weighted average. Thus if there are k groups having member of number n_{1, }n_{2, }n3_{, ……… }n_{k }with averages A_{1, }A_{2, }A3_{, ……… }A_{k} respectively then weighted average.

**Shortcut Approach**

If, in a group, one or more new quantities are added or excluded, then the new quantity or sum of added or excluded quantities = [Change in no. of quantities x original average] ± [change in average x final no. of quantities].

Take +ve sign if quantities added and take -ve sign if quantities removed.

**AVERAGE SPEED IF EQUAL DISTANCES ARE TRAVELLED BY TWO DIFFERENT SPEEDS**

If a car travels at a speed S_{1}, from A to B and at a speed S_{2} from B to A. Then

The above formula can be found out as follows: If distance between A and B is d, then

**AVERAGE SPEED IF EQUAL DISTANCES ARE TRAVELLED BY THREE DIFFERENT SPEEDS**

Average speed

Where x, y and 2 are these different speeds.

**REMEMBER**

** Average of first n natural numbers **=

** Average of first n consecutive x 2 even numbers = (n + 1)**

** Average of first n consecutive x 2 odd numbers = n**

** If n is odd: The average of n consecutive numbers, consecutive even numbers or consecutive odd numbers is always the middle number.**

** If n is even: The average of n consecutive numbers, consecutive even numbers or consecutive odd numbers is always the average of the middle two numbers.**

** The average of squares of first n consecutive even number is**

** The average of squares of consecutive even numbers till n is**

** The average of square of consecutive odd numbers till n is**

** If the average of n consecutive numbers is m, then the difference between the smallest and the largest number is 2(n – 1).**

** If a person or a motor car covers three equal distances at the speed of x km/h, y km/h and z km/h, respectively, then for the entire journey average speed of the person or motor car is .**

**Shortcut Approach**

- If average of n observations is a but the average becomes b when one observation is eliminated, then Value of eliminated observation = n (a – b) + b
- If average of n observations is a but the average becomes b when a new observation is added, then Value of added observation = n (b – a) + b. We have n observations out of which some observations (a
_{1, }a_{2, }a_{3,…………}) are replaced by some other new observations in this way, if the average increases or ‘ decreases by b, then

**NOTE :** In this formula, the signs of ‘+ ’ and ‘ — ’ depend upon the increment or decrement in the average.

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