Question : Find the average of even numbers from 12 to 472
Correct Answer 242
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 472
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 472 are
12, 14, 16, . . . . 472
After observing the above list of the even numbers from 12 to 472 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 472 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 472
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 472
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 472
= 12 + 472/2
= 484/2 = 242
Thus, the average of the even numbers from 12 to 472 = 242 Answer
Method (2) to find the average of the even numbers from 12 to 472
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 472 are
12, 14, 16, . . . . 472
The even numbers from 12 to 472 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 472
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 472
472 = 12 + (n – 1) × 2
⇒ 472 = 12 + 2 n – 2
⇒ 472 = 12 – 2 + 2 n
⇒ 472 = 10 + 2 n
After transposing 10 to LHS
⇒ 472 – 10 = 2 n
⇒ 462 = 2 n
After rearranging the above expression
⇒ 2 n = 462
After transposing 2 to RHS
⇒ n = 462/2
⇒ n = 231
Thus, the number of terms of even numbers from 12 to 472 = 231
This means 472 is the 231th term.
Finding the sum of the given even numbers from 12 to 472
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 472
= 231/2 (12 + 472)
= 231/2 × 484
= 231 × 484/2
= 111804/2 = 55902
Thus, the sum of all terms of the given even numbers from 12 to 472 = 55902
And, the total number of terms = 231
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 472
= 55902/231 = 242
Thus, the average of the given even numbers from 12 to 472 = 242 Answer
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