Question : Find the average of even numbers from 12 to 358
Correct Answer 185
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 358
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 358 are
12, 14, 16, . . . . 358
After observing the above list of the even numbers from 12 to 358 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 358 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 358
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 358
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 358
= 12 + 358/2
= 370/2 = 185
Thus, the average of the even numbers from 12 to 358 = 185 Answer
Method (2) to find the average of the even numbers from 12 to 358
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 358 are
12, 14, 16, . . . . 358
The even numbers from 12 to 358 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 358
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 358
358 = 12 + (n – 1) × 2
⇒ 358 = 12 + 2 n – 2
⇒ 358 = 12 – 2 + 2 n
⇒ 358 = 10 + 2 n
After transposing 10 to LHS
⇒ 358 – 10 = 2 n
⇒ 348 = 2 n
After rearranging the above expression
⇒ 2 n = 348
After transposing 2 to RHS
⇒ n = 348/2
⇒ n = 174
Thus, the number of terms of even numbers from 12 to 358 = 174
This means 358 is the 174th term.
Finding the sum of the given even numbers from 12 to 358
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 358
= 174/2 (12 + 358)
= 174/2 × 370
= 174 × 370/2
= 64380/2 = 32190
Thus, the sum of all terms of the given even numbers from 12 to 358 = 32190
And, the total number of terms = 174
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 358
= 32190/174 = 185
Thus, the average of the given even numbers from 12 to 358 = 185 Answer
Similar Questions
(1) Find the average of even numbers from 12 to 726
(2) Find the average of even numbers from 10 to 1894
(3) Find the average of the first 3335 odd numbers.
(4) Find the average of the first 3972 even numbers.
(5) Find the average of odd numbers from 7 to 1201
(6) What is the average of the first 1345 even numbers?
(7) Find the average of odd numbers from 5 to 455
(8) Find the average of odd numbers from 5 to 631
(9) Find the average of the first 4729 even numbers.
(10) What will be the average of the first 4409 odd numbers?