Question:
Find the average of even numbers from 12 to 780
Correct Answer
396
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 780
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 780 are
12, 14, 16, . . . . 780
After observing the above list of the even numbers from 12 to 780 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 780 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 780
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 780
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 12 to 780
= 12 + 780/2
= 792/2 = 396
Thus, the average of the even numbers from 12 to 780 = 396 Answer
Method (2) to find the average of the even numbers from 12 to 780
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 780 are
12, 14, 16, . . . . 780
The even numbers from 12 to 780 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 780
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 12 to 780
780 = 12 + (n – 1) × 2
⇒ 780 = 12 + 2 n – 2
⇒ 780 = 12 – 2 + 2 n
⇒ 780 = 10 + 2 n
After transposing 10 to LHS
⇒ 780 – 10 = 2 n
⇒ 770 = 2 n
After rearranging the above expression
⇒ 2 n = 770
After transposing 2 to RHS
⇒ n = 770/2
⇒ n = 385
Thus, the number of terms of even numbers from 12 to 780 = 385
This means 780 is the 385th term.
Finding the sum of the given even numbers from 12 to 780
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 12 to 780
= 385/2 (12 + 780)
= 385/2 × 792
= 385 × 792/2
= 304920/2 = 152460
Thus, the sum of all terms of the given even numbers from 12 to 780 = 152460
And, the total number of terms = 385
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 12 to 780
= 152460/385 = 396
Thus, the average of the given even numbers from 12 to 780 = 396 Answer
Similar Questions
(1) Find the average of odd numbers from 13 to 1379
(2) Find the average of the first 2964 even numbers.
(3) Find the average of the first 1865 odd numbers.
(4) Find the average of the first 1657 odd numbers.
(5) Find the average of the first 1404 odd numbers.
(6) What is the average of the first 1506 even numbers?
(7) Find the average of even numbers from 10 to 806
(8) Find the average of odd numbers from 11 to 1117
(9) Find the average of odd numbers from 13 to 787
(10) Find the average of even numbers from 6 to 1658