Average
Questions and answer based on calculation of average
Rules to calculate average
Average of given numbers is equal to the sum of given numbers divided by total number of numbers
Or, Average of given observations is equal to the sum of total observations divided by number of observations.
Rule (1) Average of given numbers 
Rule (2) Or, Average given observations 
Rule (3) Calculation of average speed
Let a person covers a distance at a speed of `x` km/h and again covers the same distance at a speed of `y` km/h.
Then average speed `=(2xy)/(x+y)` km/h
(4) Sum of n natural numbers `=(n(n+1))/2`
(5) Series of numbers in which the difference between two consecutive terms are equal is called Arithmetic series.
(a) Sum of n terms (Sn) of Arithmetic Series
`=n/2[2a+(n-1)d]`
Where, n = numbers of terms of arithmetic series
a = First terms of arithmetic series
And, d = difference between two consecutive terms, i.e. common difference of arithmetic series
(b) Sum of n terms (Sn) of Arithmetic Series
`=n/2(a+l)`
Where, n = number of terms of arithmetic series
a = first term of arithmetic series
And, `l` = last term of arithmetic series
(c) tn = a + (n – 1) d
Where, n = number of term of arithmetic series
a = first term of arithmetic series
d = common difference of arithmetic series
And, tn = nth term of Arithmetic sereis
(1) Find the average of 1, 2 and 3
Solution :
We know that,
Average of given numbers
Here sum of given numbers = 1 + 2 + 3 = 6
And total numbers of given numbers = 3
Thus, Average of given numbers `=6/3=2`
Thus, Average of given numbers = 2 Answer
(2) Find the average of 1, 2, 3, 4 and 5
Solution:
We know that average of given numbers
Sum of given numbers = 1 + 2 + 3 + 4 + 5 = 15
And, total number of given numbers = 5
Thus, Average `=15/5=3`
Thus, average of given numbers = 3 Answer
(3) Find the average of 15, 16, 17, 20 and 12
Solution :
We know that, Average of given numbers
Thus, Average of given numbers `=(15+16+17+20+12)/5`
`=80/5 = 16`
Thus, Average of given numbers = 16 Answer
(4) Find the average of first 10 natural numbers.
Solution :
We know that, Counting numbers are called natural numbers.
Thus, first 10 natural numbers are
1, 2, 3 , 4, 5, 6, 7, 8, 9, 10
Now, We know that, Average of given numbers
Thus, average `=(1+2+3+4+5+6+7+8+9+10)/10`
`=55/10 = 5.5`
Thus, average of first 10 natural numbers = 5.5 Answer
Alternate method
We know that, Average of given numbers
We know that, sum of first `n` natural numbers `=(n(n+1))/2`
Thus, sum of first 10 natural numbers
Here, n = 10
Thus, sum of first 10 natural numbers `=(10(10+1))/2`
`=110/2=55`
And total numbers = 10
Thus, average of first 10 natural numbers `=55/10=5.5`
Thus, average of first 10 natural numbers = 5.5 Answer
Shortcut method
We know that, Average of given numbers
And, sum of first `n` natural numbers `=(n(n+1))/2`
Thus, Average of first 10 natural numbers `=1/10xx(10(10+1))/2`
`=1/10xx110/2 = 5.5`
Thus, average of first 10 natural numbers = 5.5 Answer
(5) What will be the average of first 25 natural numbers?
Solution :
We know that, Counting numbers are called natural numbers.
Thus, first 25 natural numbers are
1, 2, 3, 4, . . . . . . . , 25
We know that, Average of given numbers
[Sum of first 25 natural numbers can be found by adding them. But it is easy to get their sum using formula ]
We know that, sum of first `n` natural numbers `=(n(n+1))/2`
Thus, sum of first 25 natural numbers `=(25(25+1))/2`
`=(25xx26)/2 = 325`
Here sum of first 25 natural numbers = 325
And number of given numbers = 25
Thus, average `=325/25 = 13`
∴ Average of first 25 natural numbers = 13 Answer
(6) Calculate the average of first 50 natural numbers
Solution :
We know that, counting numbers are called natural numbers.
Thus, the list of first 50 natural numbers is
1, 2, 3, 4, 5, . . . . . . 50
[Sum of first 50 natural numbers can be found by adding them. But it is easy to get the sum of first 50 natural numbers using formula ]
We know that, sum of first n natural numbers is `(n(n+1))/2`
Therefore, sum of first 50 natural `=(50(50+1))/2`
`=(50xx51)/2=`
`=2550/2` = 1275
Now, we know that average
Thus, average of given numbers `=1275/50=25.5`
Therefore, average of given numbers = 25.5 Answer
(7) Find the average of first 5 multiples of 4.
Solution
List of first 5 multiples of 4 is
4, 8, 12, 16, 20
We know that average
`=(4+8+12+16+20)/5`
`=60/5` = 12
Thus, Average of first 5 multiples of 4 = 12 Answer
(8) Find the average of first 10 multiples of 6.
Solution
List of first 10 multiples of 6
= 6 ( 1, 2, 3, . . . . , 10 )
Here numbers in bracket are natural numbers
We know that, sum of n natural numbers `= (n(n+1))/2`
Thus, sum of first 10 multiples of 6 `=6(10(10+1))/2`
= 3 (10 × 11)
⇒ sum of first 10 multiples of 6 = 330
Now, we know that average
Thus, average of first 10 multiples of 6 `=330/10 =33`
Thus, Average of first 10 multiples of 6 = 33 Answer
Alternate method
We have List of first 10 multiples of 6
= 6, 12, 18, 24, 30, 36, 42, 48, 54, 60
Thus, sum of first 10 multiples of 6
= 6 + 12 + 18 + 24 + 30 + 36 + 42 + 48 + 54 + 60
= 330
Now, we know that average
Thus, average of first 10 multiples of 6 `=330/10 =33`
Thus, Average of first 10 multiples of 6 = 33 Answer
(9) Find the average of first 20 multiples of 5
Solution :
Here, we have the List of first 20 multiples of 5
= 5 (1, 2, 3, . . . , 20)
Now, sum of first 20 multiples of 5
= 5 (1 + 2 + 3 + . . . . + 20)
`= 5xx(20(20+1))/2`
`= 5 xx(20xx21)/2`
= 5 × 10 × 21
= 1050
Now, we know that, average
Therefore, average of first 20 multiples of 5 `=1050/20`
⇒ average of first 20 multiples of 5 = 52.5 Answer
(10) Find the average of first 50 multiples of 10.
Solution :
The first 50 multiples of 10
= 10 (1, 2, 3, . . . . . . , 50)
Here numbers in bracket are natural numbers
We know that, sum of n natural numbers `= (n(n+1))/2`
Thus, sum of first 50 multiples of 10 `=10(50(50+1))/2`
= 5 × 50 × 51
Thus, sum of first 50 multiples of 10 = 12750
Now, we know that, average
Here, sum of given numbers = 12750
And, total numbers of given numbers = 50
Thus, Average `= 12750/50 =255`
Thus, Average of first 50 multiples of 10 = 255 Answer
(11) Calculate the average of first 500 multiples of 60.
Solution :
The first 500 multiples of 60
= 60 (1, 2, 3, . . . . . . , 500)
Here numbers in bracket are natural numbers
We know that, sum of n natural numbers `= (n(n+1))/2`
Thus, sum of first 500 multiples of 60 `=60(500(500+1))/2`
= 30 × 500 × 501
Thus, sum of first 500 multiples of 60 = 7515000
Now, we know that, average
Here, sum of given numbers = 7515000
And, total numbers of given numbers = 500
Thus, Average `= 7515000/500 =15030`
Thus, Average of first 500 multiples of 60 = 15030 Answer
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