Question : Find the average of even numbers from 10 to 500
Correct Answer 255
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 10 to 500
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 10 to 500 are
10, 12, 14, . . . . 500
After observing the above list of the even numbers from 10 to 500 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 10 to 500 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 10 to 500
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 500
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 10 to 500
= 10 + 500/2
= 510/2 = 255
Thus, the average of the even numbers from 10 to 500 = 255 Answer
Method (2) to find the average of the even numbers from 10 to 500
Finding the average of given continuous even numbers after finding their sum
The even numbers from 10 to 500 are
10, 12, 14, . . . . 500
The even numbers from 10 to 500 form an Arithmetic Series in which
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 500
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 10 to 500
500 = 10 + (n – 1) × 2
⇒ 500 = 10 + 2 n – 2
⇒ 500 = 10 – 2 + 2 n
⇒ 500 = 8 + 2 n
After transposing 8 to LHS
⇒ 500 – 8 = 2 n
⇒ 492 = 2 n
After rearranging the above expression
⇒ 2 n = 492
After transposing 2 to RHS
⇒ n = 492/2
⇒ n = 246
Thus, the number of terms of even numbers from 10 to 500 = 246
This means 500 is the 246th term.
Finding the sum of the given even numbers from 10 to 500
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 10 to 500
= 246/2 (10 + 500)
= 246/2 × 510
= 246 × 510/2
= 125460/2 = 62730
Thus, the sum of all terms of the given even numbers from 10 to 500 = 62730
And, the total number of terms = 246
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 10 to 500
= 62730/246 = 255
Thus, the average of the given even numbers from 10 to 500 = 255 Answer
Similar Questions
(1) Find the average of odd numbers from 7 to 1099
(2) What is the average of the first 995 even numbers?
(3) Find the average of odd numbers from 7 to 1375
(4) Find the average of odd numbers from 15 to 719
(5) What is the average of the first 341 even numbers?
(6) Find the average of odd numbers from 11 to 609
(7) Find the average of the first 2882 odd numbers.
(8) Find the average of odd numbers from 3 to 309
(9) Find the average of the first 3277 even numbers.
(10) Find the average of odd numbers from 3 to 337