Question:
Find the average of odd numbers from 7 to 299
Correct Answer
153
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 7 to 299
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 7 to 299 are
7, 9, 11, . . . . 299
After observing the above list of the odd numbers from 7 to 299 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 7 to 299 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 7 to 299
The First Term (a) = 7
The Common Difference (d) = 2
And the last term (ℓ) = 299
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 7 to 299
= 7 + 299/2
= 306/2 = 153
Thus, the average of the odd numbers from 7 to 299 = 153 Answer
Method (2) to find the average of the odd numbers from 7 to 299
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 7 to 299 are
7, 9, 11, . . . . 299
The odd numbers from 7 to 299 form an Arithmetic Series in which
The First Term (a) = 7
The Common Difference (d) = 2
And the last term (ℓ) = 299
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 7 to 299
299 = 7 + (n – 1) × 2
⇒ 299 = 7 + 2 n – 2
⇒ 299 = 7 – 2 + 2 n
⇒ 299 = 5 + 2 n
After transposing 5 to LHS
⇒ 299 – 5 = 2 n
⇒ 294 = 2 n
After rearranging the above expression
⇒ 2 n = 294
After transposing 2 to RHS
⇒ n = 294/2
⇒ n = 147
Thus, the number of terms of odd numbers from 7 to 299 = 147
This means 299 is the 147th term.
Finding the sum of the given odd numbers from 7 to 299
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 7 to 299
= 147/2 (7 + 299)
= 147/2 × 306
= 147 × 306/2
= 44982/2 = 22491
Thus, the sum of all terms of the given odd numbers from 7 to 299 = 22491
And, the total number of terms = 147
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 7 to 299
= 22491/147 = 153
Thus, the average of the given odd numbers from 7 to 299 = 153 Answer
Similar Questions
(1) Find the average of even numbers from 12 to 1792
(2) Find the average of even numbers from 10 to 1636
(3) Find the average of even numbers from 8 to 938
(4) Find the average of even numbers from 8 to 1032
(5) Find the average of even numbers from 10 to 252
(6) What is the average of the first 160 odd numbers?
(7) Find the average of odd numbers from 13 to 765
(8) Find the average of even numbers from 10 to 970
(9) Find the average of even numbers from 4 to 1088
(10) Find the average of odd numbers from 13 to 187