Question : Find the average of odd numbers from 9 to 195
Correct Answer 102
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 9 to 195
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 9 to 195 are
9, 11, 13, . . . . 195
After observing the above list of the odd numbers from 9 to 195 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 9 to 195 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 9 to 195
The First Term (a) = 9
The Common Difference (d) = 2
And the last term (ℓ) = 195
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 9 to 195
= 9 + 195/2
= 204/2 = 102
Thus, the average of the odd numbers from 9 to 195 = 102 Answer
Method (2) to find the average of the odd numbers from 9 to 195
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 9 to 195 are
9, 11, 13, . . . . 195
The odd numbers from 9 to 195 form an Arithmetic Series in which
The First Term (a) = 9
The Common Difference (d) = 2
And the last term (ℓ) = 195
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 9 to 195
195 = 9 + (n – 1) × 2
⇒ 195 = 9 + 2 n – 2
⇒ 195 = 9 – 2 + 2 n
⇒ 195 = 7 + 2 n
After transposing 7 to LHS
⇒ 195 – 7 = 2 n
⇒ 188 = 2 n
After rearranging the above expression
⇒ 2 n = 188
After transposing 2 to RHS
⇒ n = 188/2
⇒ n = 94
Thus, the number of terms of odd numbers from 9 to 195 = 94
This means 195 is the 94th term.
Finding the sum of the given odd numbers from 9 to 195
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 9 to 195
= 94/2 (9 + 195)
= 94/2 × 204
= 94 × 204/2
= 19176/2 = 9588
Thus, the sum of all terms of the given odd numbers from 9 to 195 = 9588
And, the total number of terms = 94
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 9 to 195
= 9588/94 = 102
Thus, the average of the given odd numbers from 9 to 195 = 102 Answer
Similar Questions
(1) Find the average of the first 1258 odd numbers.
(2) What is the average of the first 554 even numbers?
(3) Find the average of odd numbers from 9 to 1341
(4) Find the average of even numbers from 12 to 1230
(5) What is the average of the first 1059 even numbers?
(6) Find the average of the first 263 odd numbers.
(7) Find the average of the first 3866 even numbers.
(8) Find the average of the first 4841 even numbers.
(9) What is the average of the first 154 even numbers?
(10) Find the average of odd numbers from 15 to 967