Question:
Find the average of even numbers from 12 to 670
Correct Answer
341
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 670
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 670 are
12, 14, 16, . . . . 670
After observing the above list of the even numbers from 12 to 670 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 670 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 670
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 670
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 670
= 12 + 670/2
= 682/2 = 341
Thus, the average of the even numbers from 12 to 670 = 341 Answer
Method (2) to find the average of the even numbers from 12 to 670
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 670 are
12, 14, 16, . . . . 670
The even numbers from 12 to 670 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 670
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 670
670 = 12 + (n – 1) × 2
⇒ 670 = 12 + 2 n – 2
⇒ 670 = 12 – 2 + 2 n
⇒ 670 = 10 + 2 n
After transposing 10 to LHS
⇒ 670 – 10 = 2 n
⇒ 660 = 2 n
After rearranging the above expression
⇒ 2 n = 660
After transposing 2 to RHS
⇒ n = 660/2
⇒ n = 330
Thus, the number of terms of even numbers from 12 to 670 = 330
This means 670 is the 330th term.
Finding the sum of the given even numbers from 12 to 670
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 670
= 330/2 (12 + 670)
= 330/2 × 682
= 330 × 682/2
= 225060/2 = 112530
Thus, the sum of all terms of the given even numbers from 12 to 670 = 112530
And, the total number of terms = 330
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 670
= 112530/330 = 341
Thus, the average of the given even numbers from 12 to 670 = 341 Answer
Similar Questions
(1) Find the average of even numbers from 4 to 898
(2) What is the average of the first 1183 even numbers?
(3) Find the average of odd numbers from 11 to 261
(4) Find the average of odd numbers from 15 to 673
(5) Find the average of odd numbers from 13 to 473
(6) Find the average of even numbers from 10 to 372
(7) What will be the average of the first 4170 odd numbers?
(8) Find the average of the first 1279 odd numbers.
(9) Find the average of the first 2878 odd numbers.
(10) What will be the average of the first 4082 odd numbers?