Question:
Find the average of even numbers from 12 to 324
Correct Answer
168
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 324
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 324 are
12, 14, 16, . . . . 324
After observing the above list of the even numbers from 12 to 324 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 324 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 324
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 324
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 324
= 12 + 324/2
= 336/2 = 168
Thus, the average of the even numbers from 12 to 324 = 168 Answer
Method (2) to find the average of the even numbers from 12 to 324
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 324 are
12, 14, 16, . . . . 324
The even numbers from 12 to 324 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 324
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 324
324 = 12 + (n – 1) × 2
⇒ 324 = 12 + 2 n – 2
⇒ 324 = 12 – 2 + 2 n
⇒ 324 = 10 + 2 n
After transposing 10 to LHS
⇒ 324 – 10 = 2 n
⇒ 314 = 2 n
After rearranging the above expression
⇒ 2 n = 314
After transposing 2 to RHS
⇒ n = 314/2
⇒ n = 157
Thus, the number of terms of even numbers from 12 to 324 = 157
This means 324 is the 157th term.
Finding the sum of the given even numbers from 12 to 324
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 324
= 157/2 (12 + 324)
= 157/2 × 336
= 157 × 336/2
= 52752/2 = 26376
Thus, the sum of all terms of the given even numbers from 12 to 324 = 26376
And, the total number of terms = 157
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 324
= 26376/157 = 168
Thus, the average of the given even numbers from 12 to 324 = 168 Answer
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