Question:
Find the average of even numbers from 12 to 552
Correct Answer
282
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 552
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 552 are
12, 14, 16, . . . . 552
After observing the above list of the even numbers from 12 to 552 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 552 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 552
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 552
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 552
= 12 + 552/2
= 564/2 = 282
Thus, the average of the even numbers from 12 to 552 = 282 Answer
Method (2) to find the average of the even numbers from 12 to 552
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 552 are
12, 14, 16, . . . . 552
The even numbers from 12 to 552 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 552
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 552
552 = 12 + (n – 1) × 2
⇒ 552 = 12 + 2 n – 2
⇒ 552 = 12 – 2 + 2 n
⇒ 552 = 10 + 2 n
After transposing 10 to LHS
⇒ 552 – 10 = 2 n
⇒ 542 = 2 n
After rearranging the above expression
⇒ 2 n = 542
After transposing 2 to RHS
⇒ n = 542/2
⇒ n = 271
Thus, the number of terms of even numbers from 12 to 552 = 271
This means 552 is the 271th term.
Finding the sum of the given even numbers from 12 to 552
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 552
= 271/2 (12 + 552)
= 271/2 × 564
= 271 × 564/2
= 152844/2 = 76422
Thus, the sum of all terms of the given even numbers from 12 to 552 = 76422
And, the total number of terms = 271
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 552
= 76422/271 = 282
Thus, the average of the given even numbers from 12 to 552 = 282 Answer
Similar Questions
(1) Find the average of odd numbers from 13 to 1433
(2) What is the average of the first 139 even numbers?
(3) Find the average of the first 3535 even numbers.
(4) Find the average of the first 2750 even numbers.
(5) Find the average of even numbers from 8 to 336
(6) Find the average of odd numbers from 15 to 1139
(7) Find the average of odd numbers from 9 to 69
(8) Find the average of even numbers from 4 to 1036
(9) Find the average of odd numbers from 11 to 535
(10) Find the average of even numbers from 10 to 346