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
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
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
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
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 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
a2 + b2 + c2 + 2(ab+bc+ca) = 9m2
(a2 + b2 + c2) / 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
(12x + 96) / 13
12x + 96 = 13x + 65
x = 31
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