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Quantitative Aptitude :: Averages Formulas

Averages Questions and Answers

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The average of the given set of quantities is calculated is dividing by dividing the sum of the quantities by their numbers.
The average is also known as mean (or arithmetic mean).

Let a1, a2, a3, ...........aN an be the N quantities and their sum be S, then

Average = Sum of quantities/Number of quantities = a1, a2, a3, ...........aN/N = S/N
If a man spends Rs. 600 in a year, we may say that his average moonthly expenditure is Rs. 600/12 = Rs. 50.
If a motorist travels 150 km in 3 hours, we may say that his average speed during the journey is 150/3 = 50 km/hr.
  • Besides arithmetic mean there are other ways in which 'mean' is defined, for example geometric mean, harmonic mean etc.
  • The average of first 'n' natural numbers is (n+1)/2
  • The average of n consecutive numbers is always the middle number.
  • The average of odd number from 1 to n is (last odd number + 1)/2 where n = natural odd number
  • The average of even number from 1 to n is (last even number + 2)/2; where n = natural even number.
  • The average of square of natural numbers till n is [(n +1)(2n + 1)]/6
  • The average of cubes of natural number till n is [[n(n + 1)]/2]2
  • The average of first n consecutive even numbers is (n + 1)
  • The average of first n consecutive odd numbers is n
 
Types of Problems Working Rules Examples
1. To find the average when the number of     quantities and their sum is given Let the number of quantities be N and their sum be S.               

Average = S/N		 If the number of quantities is 10 and their sum is 540,
Then their average = 540/10 = 54
2. To find the sum when the average and number of quantities is given. Let the number of quantities be N and thier average be X.
Average = Sum of quantities/Number of quantities
=> Sum = Average x Numbers of quantities.
i.e. S = X x N.		 If the number of quantities is 10 and their average is 54,
Then sum of quantities = 54 x 10 = 540
3. To fine the average of x + y quantities if the average of x quantities is 'a' and the average of y quantities is 'b'. Average = (xa + yb)/(x + y) If the average weight of 10 men is 50 kg, and that of 20 women is 45 kg, then the average weight of 30 persons.
= (10 x 50  + 20 x 45)/30 kg.
=1400/30 = 46 2/3 kg.
4. To find the average (or weighted mean) when a set quantities and the values or weights attached to them are given. Let x1, x2, ......., xn be the set of quantities and w1, w2, ........, wn be the weights attached to them.
Average =
W1X1 + W2X2 + ...+WnXn/W1 + W2 + .......... + Wn		 If 2, 3, 4, 5 is the number of children and 10 kg, 20 kg, 30 kg, and 40 kg is their average weight respectively, then the average weight respectively, then the average weight of all the children
(20 + 60 + 120 + 200)/(2 + 3 + 4 + 5) = 400/14 = 28 4/7
5. To find the mensure of a new quatity in place of a removed quantity, when the previous average, new average, measure of removed quantity and the number of quantities are given. Mensure of new quantity = Measure of removed quantity + (change in average x number of quantities)
(Change could be
    +ve or -ve)
 		 If the average weight of 4 men increases by 3 kg when one of them weighing 90 kg is replaced by another man then weight of the new man
= 90 + 3 x 4 = 102 kg.
6. To find the measure of a new quantity when the previous average (without the new one) and the new average (with the new one) are given along with the number of quantities. Measure of the quantity = Measure of removed quantity + (Change in average x number of quantities)
(Change could be
  +ve or -ve)		 If the average age of 20 boys of a class is 12 years and when the age of teacher is included, average becomes 14 years then age of teacher 
= 14 + 2 x 20
= 54 years
7. To find the average speed of a person if he covers a distance at x km/hour and comes back along the same route at y km/hour. Average spees = 2xy/x+y km/hour
[Note that in general Average speed = Total distance travelled /Total time taken. In this case distance travelled would be double]		 If a cyclist cycles a distance at the speed of 20 km/hour and comes back at a speed of 30 km/hour, then his average speed
= (2 x 20 x 30)/(20 + 30) = 24 km/hours.
8. To find the geometric mean (G.M) of a given set of quantities. G.M = n√X1,X2, .... Xn
where x1, x2, ..., xn are the values of a given set of n quantities.
 		 G.M of 2, 3, 4 and 5 = 4√2 x 3 x 4 x 5
                              = 4√120
9. The average of x quantities is p. The average of y of them is q. To find the average of remaining quantities. Required average 
= (px  - qy)/p-q
 		 The average of 5 quantities is 8. The average of the remaining quantities 
= (5 x 8 - 3 x 6)/2 = (40 - 18)/2 = 11