Question : Find the average of odd numbers from 15 to 97
Correct Answer 56
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 15 to 97
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 15 to 97 are
15, 17, 19, . . . . 97
After observing the above list of the odd numbers from 15 to 97 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 15 to 97 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 15 to 97
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 97
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 97
= 15 + 97/2
= 112/2 = 56
Thus, the average of the odd numbers from 15 to 97 = 56 Answer
Method (2) to find the average of the odd numbers from 15 to 97
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 15 to 97 are
15, 17, 19, . . . . 97
The odd numbers from 15 to 97 form an Arithmetic Series in which
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 97
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 97
97 = 15 + (n – 1) × 2
⇒ 97 = 15 + 2 n – 2
⇒ 97 = 15 – 2 + 2 n
⇒ 97 = 13 + 2 n
After transposing 13 to LHS
⇒ 97 – 13 = 2 n
⇒ 84 = 2 n
After rearranging the above expression
⇒ 2 n = 84
After transposing 2 to RHS
⇒ n = 84/2
⇒ n = 42
Thus, the number of terms of odd numbers from 15 to 97 = 42
This means 97 is the 42th term.
Finding the sum of the given odd numbers from 15 to 97
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 97
= 42/2 (15 + 97)
= 42/2 × 112
= 42 × 112/2
= 4704/2 = 2352
Thus, the sum of all terms of the given odd numbers from 15 to 97 = 2352
And, the total number of terms = 42
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 97
= 2352/42 = 56
Thus, the average of the given odd numbers from 15 to 97 = 56 Answer
Similar Questions
(1) What is the average of the first 795 even numbers?
(2) Find the average of even numbers from 12 to 1584
(3) Find the average of the first 3213 even numbers.
(4) Find the average of the first 1451 odd numbers.
(5) Find the average of the first 4273 even numbers.
(6) What is the average of the first 1923 even numbers?
(7) Find the average of even numbers from 10 to 1152
(8) Find the average of even numbers from 6 to 1180
(9) Find the average of the first 2508 odd numbers.
(10) What is the average of the first 18 odd numbers?