Question : Find the average of odd numbers from 15 to 373
Correct Answer 194
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 15 to 373
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 15 to 373 are
15, 17, 19, . . . . 373
After observing the above list of the odd numbers from 15 to 373 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 15 to 373 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 15 to 373
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 373
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 15 to 373
= 15 + 373/2
= 388/2 = 194
Thus, the average of the odd numbers from 15 to 373 = 194 Answer
Method (2) to find the average of the odd numbers from 15 to 373
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 15 to 373 are
15, 17, 19, . . . . 373
The odd numbers from 15 to 373 form an Arithmetic Series in which
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 373
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 15 to 373
373 = 15 + (n – 1) × 2
⇒ 373 = 15 + 2 n – 2
⇒ 373 = 15 – 2 + 2 n
⇒ 373 = 13 + 2 n
After transposing 13 to LHS
⇒ 373 – 13 = 2 n
⇒ 360 = 2 n
After rearranging the above expression
⇒ 2 n = 360
After transposing 2 to RHS
⇒ n = 360/2
⇒ n = 180
Thus, the number of terms of odd numbers from 15 to 373 = 180
This means 373 is the 180th term.
Finding the sum of the given odd numbers from 15 to 373
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 15 to 373
= 180/2 (15 + 373)
= 180/2 × 388
= 180 × 388/2
= 69840/2 = 34920
Thus, the sum of all terms of the given odd numbers from 15 to 373 = 34920
And, the total number of terms = 180
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 15 to 373
= 34920/180 = 194
Thus, the average of the given odd numbers from 15 to 373 = 194 Answer
Similar Questions
(1) Find the average of odd numbers from 13 to 683
(2) Find the average of even numbers from 10 to 920
(3) Find the average of odd numbers from 5 to 365
(4) Find the average of odd numbers from 3 to 713
(5) What is the average of the first 371 even numbers?
(6) Find the average of odd numbers from 9 to 1177
(7) What is the average of the first 1347 even numbers?
(8) What will be the average of the first 4434 odd numbers?
(9) Find the average of odd numbers from 3 to 663
(10) Find the average of odd numbers from 9 to 721