Average Concepts and formulae with solved examples

Average concepts and formula with solved examples

Average:

The average is the ratio of the sum of the given observations and the numbers of a given observation. This is also known as the arithmetic mean.

Average = Sum of the observation / Number of observations

Properties of Average:

1.Average of a given data is less than the greatest observation and greater than the smallest observation of the given data.

2.If the observations of given data are equal, then the average will also be the same as observations.

3.If ‘0’ one of the observations of a given data, then that 0 will also be included while calculating an average.

Average speed;

Type 1.If a person covers a certain distance at a speed of A km/h and again covers the same distance at a speed of b km/h, then the average speed of the whole journey will be.

2AB /A+B

Type 2:If a person covers three equal distance at a speed of A km/h, B km/h, C km/h then the        average speed during the whole journey will be

3ABC/ AB+BC+CA

Type 3:If distance P is covered with speed x, distance Q is covered with speed y, and distance R is covered with speed z then for the whole journey.

P+Q+R+…../ (P/x +Q/y + R/z)

Type 4:If a person covers P part of his total distance with a speed of x, Q part of the total distance with speedy, and R part of the total distance with the speed of z, then

Average speed = 1/(P/x +Q/y +R/z)

Points to remember

1.Average of first n natural numbers = (n+1) /2

2.Average of first n even numbers = n+1 

3.Average of first n odd numbers = n 

4.Average of consecutive numbers = (first number + last number)/ 2 

5.Average of 1 to  n odd numbers = (Last odd number +1) / 2 

6.Average of 1 to  n even numbers = (Last even numbers +2) /2 

7.Average of squares of first n natural numbers = (n+1)(2n+1)/6 

8.Average of cubes of first n natural numbers = n(n+1)²/4 

9.Average of n multiples of any number = number * (n+1)/2

10.If Average of n1 observations is a1, the average of n2 observation is a2, and so on then 
Average of all observation is = (n1 a1 + n2 a2 + ……)/ (n1+n2)

11.If the average of m observations is a and the average of n observations is taken out of m is b then, 
Average of rest of the observation = (ma-nb)/ m-n

12.If the average of n observations is a but the average becomes b when one of the observation is eliminated or added then, 

Value of eliminated observation= N(a-b) +b

Value of added observation    = n(b-a) + b

13.If the average of total components in a group is a where the average of n components is b and the average of

remaining components is c then, 

number of remaining components  = n(a-b)/c-a

Few examples based on average:

Ex.1 If the average of m numbers is n² and that of n numbers is m², then the average of (m+n).

Sol. According to question

sum of n numbers = m*n²  =  mn²

Sum of n numbers = n*m² = nm²

(mn² + nm²)/ m+n

mn(m+n)/m+n = mn Ans.

Ex.2 The average of three numbers is 28. The first number is half of the second number and the third number is twice the second. Find the third number.

Sol. Let the first number be = x

So the second number = 2x

third number = 4x

Total = x+2x+4x = 28*3

7x = 28*3

x=12

Third number is 4x = 48 Ans

Ex.3 Out of three numbers, the first is twice the second number and half of the third number, If the average of all the three numbers is 56 the difference between the first and third numbers is.

Sol. Let the first number = 2x

Second number = x

Third number = 4x

Total of three numbers =  2x +x +4x = 56 * 3

7x = 56*3

x = 8*3

x= 24

Difference between first and third number is 

4x-2x = 2x = 48 Ans

Ex.4 There were 24 students in a class. One of them who was 18 years old left the class and his place was filled up by a new student. If the average of the class was therefore lowered by 1 month the age of the new students is.

Sol. Let the previous average be = x

Let the age of new student = y

So,

 24x - 18 + y = 24(x - 1/12)

24x - 18 + y = 24x - 24 *1/12

-18 + y = -2

y = 16 years Ans

Method 2

18 - (24*1)/12 = 16 years Ans

Ex.5 The average of six numbers is 3.95. The average of two of them is 3.4 while the average of the other two is 3.85, the average of the remaining two numbers is.

Sol. [(3.95 *6 ) -(3.4*2 +3.85*2)]/2 

[23.70 - (6.80 + 7.70)]

9.2/2 = 4.6 Ans

Ex.6 Out of 9 persons, 8 people spent Rs. 30 each for their meal. The 9th person spent Rs. 20 more than the average expenditure of all the 9 people, the total money spent by all of them was.

Sol. Let the average of nine person be x

So, total = 9x = 30*3 +x+20

8x = 260

x= 260/8 = 65/2

65/2 *9 = 292.5 Ans.

Method 2

Since 9th person spent 20 Rs. More

share of each person = 30 + 20/8 =32.5

Total expenditure = 292.5 Ans

Ex.7 In a school with 600 students the average age of boys is 12 years and that of girls is 11 years. If the average of the school is 11 years and 9 months, then the number of girls in the school.

Sol. Let the number of boys = x

So, the girls = 600 -x

According to the question

12x  + 11(600-x) = 600* 11 years 9 months

23x +6600 - 11x = 600 * 11 9⁄12

x = 7050 - 6600

x = 450

Girls = 600-450 = 150 Ans

1:2:3 Ans

Ex.8 What is the average square of the natural numbers from 1 to 41.

Sol. Average of square of the natural numbers

(n+1)(2n+1)/6 = [(41+1)(2*41 +1)]/6

42*83/6 = 3486/6 = 581 Ans

Ex.9 A train covers 50% of the journey at 30 Km/hr, 25% of the journey at 25 km.her. and the remaining at 20 km/hr. find the average speed of the train during the entire journey.

Sol. Average speed  = 100/ (A/x + b/y + c/z)

A = 50 B = 25 and C = 25 

x= 30   y= 25 and c = 20

100/(50/30 + 25/25 + 25/20)

=1200/47 Ans

Ex.10 If the mean of a,b,c is M and ab+bc+ca =0 then the mean of a²,b²,c² is.

Sol. Average of a,b,c is m

Total = (a+b+c)/3 = m

a+b+c = 3m

on squaring both sides

a²+b²+c²+ 2(ab+bc+ca) = 9m²

so the average of a²,b²,c² is

(a²+b²+c²)/3 = 9m²/3 = 3m² Ans

Ex.11 A batsman has a certain average of runs in 12 innings he scores 96 runs, thereby increasing his average by 5 runs. What is his average after the 13th inning?

Sol. Let the average of 12 innings be x

So, (12x + 96) /13

12x + 96 = 13x + 65

x = 31

the new average is x+5

31+5 = 36 Ans

Ex.12 a,b,c,d,e are five consecutive odd numbers their average is?

Sol. for odd number

the average = 1st step + (n-1)

Where n is number of step

a + (5-1) = a+4

a+4 Ans