Shortcuts in Quantitative Aptitude for Competitive Exams – Average

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 x₁+a, x₂+a, x₃+ a ……….. +x_n+a is X + a.
• The average of x₁-a, x₂-a, x₃– 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 S₁, from A to B and at a speed S₂ 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.