Question:
Find the average of even numbers from 12 to 344
Correct Answer
178
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 344
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 344 are
12, 14, 16, . . . . 344
After observing the above list of the even numbers from 12 to 344 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 344 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 344
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 344
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 344
= 12 + 344/2
= 356/2 = 178
Thus, the average of the even numbers from 12 to 344 = 178 Answer
Method (2) to find the average of the even numbers from 12 to 344
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 344 are
12, 14, 16, . . . . 344
The even numbers from 12 to 344 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 344
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 344
344 = 12 + (n – 1) × 2
⇒ 344 = 12 + 2 n – 2
⇒ 344 = 12 – 2 + 2 n
⇒ 344 = 10 + 2 n
After transposing 10 to LHS
⇒ 344 – 10 = 2 n
⇒ 334 = 2 n
After rearranging the above expression
⇒ 2 n = 334
After transposing 2 to RHS
⇒ n = 334/2
⇒ n = 167
Thus, the number of terms of even numbers from 12 to 344 = 167
This means 344 is the 167th term.
Finding the sum of the given even numbers from 12 to 344
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 344
= 167/2 (12 + 344)
= 167/2 × 356
= 167 × 356/2
= 59452/2 = 29726
Thus, the sum of all terms of the given even numbers from 12 to 344 = 29726
And, the total number of terms = 167
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 344
= 29726/167 = 178
Thus, the average of the given even numbers from 12 to 344 = 178 Answer
Similar Questions
(1) Find the average of even numbers from 8 to 864
(2) Find the average of the first 2119 odd numbers.
(3) Find the average of even numbers from 8 to 438
(4) Find the average of the first 2956 even numbers.
(5) Find the average of the first 880 odd numbers.
(6) Find the average of even numbers from 4 to 1836
(7) What is the average of the first 56 odd numbers?
(8) Find the average of odd numbers from 15 to 139
(9) Find the average of the first 2891 odd numbers.
(10) Find the average of odd numbers from 7 to 1347