Question:
Find the average of even numbers from 12 to 632
Correct Answer
322
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 632
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 632 are
12, 14, 16, . . . . 632
After observing the above list of the even numbers from 12 to 632 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 632 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 632
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 632
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 632
= 12 + 632/2
= 644/2 = 322
Thus, the average of the even numbers from 12 to 632 = 322 Answer
Method (2) to find the average of the even numbers from 12 to 632
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 632 are
12, 14, 16, . . . . 632
The even numbers from 12 to 632 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 632
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 632
632 = 12 + (n – 1) × 2
⇒ 632 = 12 + 2 n – 2
⇒ 632 = 12 – 2 + 2 n
⇒ 632 = 10 + 2 n
After transposing 10 to LHS
⇒ 632 – 10 = 2 n
⇒ 622 = 2 n
After rearranging the above expression
⇒ 2 n = 622
After transposing 2 to RHS
⇒ n = 622/2
⇒ n = 311
Thus, the number of terms of even numbers from 12 to 632 = 311
This means 632 is the 311th term.
Finding the sum of the given even numbers from 12 to 632
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 632
= 311/2 (12 + 632)
= 311/2 × 644
= 311 × 644/2
= 200284/2 = 100142
Thus, the sum of all terms of the given even numbers from 12 to 632 = 100142
And, the total number of terms = 311
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 632
= 100142/311 = 322
Thus, the average of the given even numbers from 12 to 632 = 322 Answer
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