Question : Find the average of odd numbers from 11 to 693
Correct Answer 352
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 11 to 693
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 11 to 693 are
11, 13, 15, . . . . 693
After observing the above list of the odd numbers from 11 to 693 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 11 to 693 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 11 to 693
The First Term (a) = 11
The Common Difference (d) = 2
And the last term (ℓ) = 693
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 odd numbers from 11 to 693
= 11 + 693/2
= 704/2 = 352
Thus, the average of the odd numbers from 11 to 693 = 352 Answer
Method (2) to find the average of the odd numbers from 11 to 693
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 11 to 693 are
11, 13, 15, . . . . 693
The odd numbers from 11 to 693 form an Arithmetic Series in which
The First Term (a) = 11
The Common Difference (d) = 2
And the last term (ℓ) = 693
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 odd numbers from 11 to 693
693 = 11 + (n – 1) × 2
⇒ 693 = 11 + 2 n – 2
⇒ 693 = 11 – 2 + 2 n
⇒ 693 = 9 + 2 n
After transposing 9 to LHS
⇒ 693 – 9 = 2 n
⇒ 684 = 2 n
After rearranging the above expression
⇒ 2 n = 684
After transposing 2 to RHS
⇒ n = 684/2
⇒ n = 342
Thus, the number of terms of odd numbers from 11 to 693 = 342
This means 693 is the 342th term.
Finding the sum of the given odd numbers from 11 to 693
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 odd numbers from 11 to 693
= 342/2 (11 + 693)
= 342/2 × 704
= 342 × 704/2
= 240768/2 = 120384
Thus, the sum of all terms of the given odd numbers from 11 to 693 = 120384
And, the total number of terms = 342
Since, the average of the given numbers
= Sum of the given numbers/Total number of given numbers
Thus, the average of the given odd numbers from 11 to 693
= 120384/342 = 352
Thus, the average of the given odd numbers from 11 to 693 = 352 Answer
Similar Questions
(1) Find the average of even numbers from 10 to 2000
(2) Find the average of the first 3280 even numbers.
(3) Find the average of the first 2370 even numbers.
(4) Find the average of odd numbers from 15 to 1581
(5) Find the average of odd numbers from 9 to 729
(6) Find the average of odd numbers from 5 to 101
(7) Find the average of odd numbers from 9 to 73
(8) Find the average of the first 2410 odd numbers.
(9) Find the average of the first 3922 odd numbers.
(10) What will be the average of the first 4892 odd numbers?