Question : Find the average of even numbers from 10 to 1150
Correct Answer 580
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 10 to 1150
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 10 to 1150 are
10, 12, 14, . . . . 1150
After observing the above list of the even numbers from 10 to 1150 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 10 to 1150 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 10 to 1150
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 1150
The average of the numbers forming an Arithmetic Series
= The first term (a) + The last term (ℓ)/2
⇒ The average of numbers forming an Arithmetic Series = a + ℓ/2
Thus, the average of the even numbers from 10 to 1150
= 10 + 1150/2
= 1160/2 = 580
Thus, the average of the even numbers from 10 to 1150 = 580 Answer
Method (2) to find the average of the even numbers from 10 to 1150
Finding the average of given continuous even numbers after finding their sum
The even numbers from 10 to 1150 are
10, 12, 14, . . . . 1150
The even numbers from 10 to 1150 form an Arithmetic Series in which
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 1150
The Average of the given numbers
= Sum of the given numbers/Total number of given numbers
Thus, to find the average of the given numbers, first, we need to find their sum and the total number of given numbers
Finding the number of terms
For an Arithmetic Series, the nth term
an = a + (n – 1) d
Where
a = First term
d = Common difference
n = number of terms
an = nth term
Thus, for the given series of the even numbers from 10 to 1150
1150 = 10 + (n – 1) × 2
⇒ 1150 = 10 + 2 n – 2
⇒ 1150 = 10 – 2 + 2 n
⇒ 1150 = 8 + 2 n
After transposing 8 to LHS
⇒ 1150 – 8 = 2 n
⇒ 1142 = 2 n
After rearranging the above expression
⇒ 2 n = 1142
After transposing 2 to RHS
⇒ n = 1142/2
⇒ n = 571
Thus, the number of terms of even numbers from 10 to 1150 = 571
This means 1150 is the 571th term.
Finding the sum of the given even numbers from 10 to 1150
The sum of all terms (S) in an Arithmetic Series
= n/2 (a + ℓ)
Where, n = number of terms
a = First term
And, ℓ = Last term
Thus, the sum of all terms (S) of the given even numbers from 10 to 1150
= 571/2 (10 + 1150)
= 571/2 × 1160
= 571 × 1160/2
= 662360/2 = 331180
Thus, the sum of all terms of the given even numbers from 10 to 1150 = 331180
And, the total number of terms = 571
Since, the average of the given numbers
= Sum of the given numbers/Total number of given numbers
Thus, the average of the given even numbers from 10 to 1150
= 331180/571 = 580
Thus, the average of the given even numbers from 10 to 1150 = 580 Answer
Similar Questions
(1) Find the average of odd numbers from 13 to 611
(2) Find the average of the first 754 odd numbers.
(3) Find the average of odd numbers from 9 to 1243
(4) Find the average of even numbers from 12 to 918
(5) What is the average of the first 80 odd numbers?
(6) Find the average of the first 3940 even numbers.
(7) Find the average of odd numbers from 9 to 71
(8) Find the average of the first 2232 even numbers.
(9) Find the average of the first 3141 odd numbers.
(10) Find the average of even numbers from 6 to 1882