Question : Find the average of even numbers from 10 to 318
Correct Answer 164
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 10 to 318
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 10 to 318 are
10, 12, 14, . . . . 318
After observing the above list of the even numbers from 10 to 318 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 10 to 318 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 10 to 318
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 318
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 318
= 10 + 318/2
= 328/2 = 164
Thus, the average of the even numbers from 10 to 318 = 164 Answer
Method (2) to find the average of the even numbers from 10 to 318
Finding the average of given continuous even numbers after finding their sum
The even numbers from 10 to 318 are
10, 12, 14, . . . . 318
The even numbers from 10 to 318 form an Arithmetic Series in which
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 318
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 318
318 = 10 + (n – 1) × 2
⇒ 318 = 10 + 2 n – 2
⇒ 318 = 10 – 2 + 2 n
⇒ 318 = 8 + 2 n
After transposing 8 to LHS
⇒ 318 – 8 = 2 n
⇒ 310 = 2 n
After rearranging the above expression
⇒ 2 n = 310
After transposing 2 to RHS
⇒ n = 310/2
⇒ n = 155
Thus, the number of terms of even numbers from 10 to 318 = 155
This means 318 is the 155th term.
Finding the sum of the given even numbers from 10 to 318
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 318
= 155/2 (10 + 318)
= 155/2 × 328
= 155 × 328/2
= 50840/2 = 25420
Thus, the sum of all terms of the given even numbers from 10 to 318 = 25420
And, the total number of terms = 155
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 318
= 25420/155 = 164
Thus, the average of the given even numbers from 10 to 318 = 164 Answer
Similar Questions
(1) Find the average of odd numbers from 13 to 175
(2) Find the average of odd numbers from 15 to 631
(3) Find the average of the first 700 odd numbers.
(4) Find the average of even numbers from 8 to 1240
(5) Find the average of the first 1734 odd numbers.
(6) Find the average of the first 3895 odd numbers.
(7) Find the average of even numbers from 4 to 1046
(8) Find the average of the first 1047 odd numbers.
(9) What is the average of the first 601 even numbers?
(10) Find the average of odd numbers from 15 to 973