Question : Find the average of even numbers from 12 to 700
Correct Answer 356
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 700
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 700 are
12, 14, 16, . . . . 700
After observing the above list of the even numbers from 12 to 700 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 700 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 700
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 700
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 700
= 12 + 700/2
= 712/2 = 356
Thus, the average of the even numbers from 12 to 700 = 356 Answer
Method (2) to find the average of the even numbers from 12 to 700
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 700 are
12, 14, 16, . . . . 700
The even numbers from 12 to 700 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 700
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 700
700 = 12 + (n – 1) × 2
⇒ 700 = 12 + 2 n – 2
⇒ 700 = 12 – 2 + 2 n
⇒ 700 = 10 + 2 n
After transposing 10 to LHS
⇒ 700 – 10 = 2 n
⇒ 690 = 2 n
After rearranging the above expression
⇒ 2 n = 690
After transposing 2 to RHS
⇒ n = 690/2
⇒ n = 345
Thus, the number of terms of even numbers from 12 to 700 = 345
This means 700 is the 345th term.
Finding the sum of the given even numbers from 12 to 700
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 700
= 345/2 (12 + 700)
= 345/2 × 712
= 345 × 712/2
= 245640/2 = 122820
Thus, the sum of all terms of the given even numbers from 12 to 700 = 122820
And, the total number of terms = 345
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 700
= 122820/345 = 356
Thus, the average of the given even numbers from 12 to 700 = 356 Answer
Similar Questions
(1) Find the average of odd numbers from 5 to 691
(2) Find the average of odd numbers from 11 to 69
(3) Find the average of the first 2324 odd numbers.
(4) Find the average of odd numbers from 15 to 861
(5) Find the average of the first 2798 odd numbers.
(6) Find the average of odd numbers from 15 to 1375
(7) Find the average of the first 1944 odd numbers.
(8) What is the average of the first 809 even numbers?
(9) Find the average of even numbers from 4 to 380
(10) What is the average of the first 968 even numbers?