Question : Find the average of odd numbers from 11 to 555.
Correct Answer 283
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 11 to 555
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 11 to 555 are
11, 13, 15, 17, . . . . 555
The odd numbers from 11 to 555 form an Arithmetic Series
The First Term (a) = 11
The Common Difference (d) = 2
And the last term (ℓ) = 555
The average of 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 odd numbers from 11 and 555
= 11 + 555/2
= 566/2 = 283
Thus, the average of odd numbers from 11 and 555 = 283 Answer
Method (2) to find the average of odd numbers from 11 to 555
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 11 to 555 are
11, 13, 15, 17, . . . . 555
The odd numbers from 11 to 555 form an Arithmetic Series
The First Term (a) = 11
The Common Difference (d) = 2
And the last term (ℓ) = 555
The Average of given numbers
= Sum of the given numbers/Total number of given numbers
Thus, to find the average of 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
an = a + (n – 1) d
a = First term
d = Common difference
n = number of terms
Where, an = nth term
Thus, for the given series of odd numbers from 11 to 555
555 = 11 + (n – 1) × 2
⇒ 555 = 11 + 2 n – 2
⇒ 555 = 11 – 2 + 2 n
⇒ 555 = 9 + 2 n
After transposing 9 to LHS
⇒ 555 – 9 = 2 n
⇒ 546 = 2 n
After rearranging the above expression
⇒ 2 n = 546
After transposing 2 to RHS
⇒ n = 546/2
⇒ n = 273
Thus, the number of terms of odd numbers from 11 to 555 = 273
This means 555 is the 273th term.
Finding the sum of given odd numbers from 11 to 555
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 given odd numbers from 11 to 555
= 273/2 (11 + 555)
= 273/2 × 566
= 273 × 566/2
= 154518/2 =77259
Thus, the sum of all terms of the given odd numbers from 11 to 555 = 77259
And, the total number of terms = 273
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 555
= 77259/273 = 283
Thus, the average of the given odd numbers from 11 to 555 = 283 Answer
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