Question : Find the average of even numbers from 12 to 194
Correct Answer 103
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 194
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 194 are
12, 14, 16, . . . . 194
After observing the above list of the even numbers from 12 to 194 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 194 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 194
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 194
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 even numbers from 12 to 194
= 12 + 194/2
= 206/2 = 103
Thus, the average of the even numbers from 12 to 194 = 103 Answer
Method (2) to find the average of the even numbers from 12 to 194
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 194 are
12, 14, 16, . . . . 194
The even numbers from 12 to 194 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 194
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 even numbers from 12 to 194
194 = 12 + (n – 1) × 2
⇒ 194 = 12 + 2 n – 2
⇒ 194 = 12 – 2 + 2 n
⇒ 194 = 10 + 2 n
After transposing 10 to LHS
⇒ 194 – 10 = 2 n
⇒ 184 = 2 n
After rearranging the above expression
⇒ 2 n = 184
After transposing 2 to RHS
⇒ n = 184/2
⇒ n = 92
Thus, the number of terms of even numbers from 12 to 194 = 92
This means 194 is the 92th term.
Finding the sum of the given even numbers from 12 to 194
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 even numbers from 12 to 194
= 92/2 (12 + 194)
= 92/2 × 206
= 92 × 206/2
= 18952/2 = 9476
Thus, the sum of all terms of the given even numbers from 12 to 194 = 9476
And, the total number of terms = 92
Since, the average of the given numbers
= Sum of the given numbers/Total number of given numbers
Thus, the average of the given even numbers from 12 to 194
= 9476/92 = 103
Thus, the average of the given even numbers from 12 to 194 = 103 Answer
Similar Questions
(1) Find the average of even numbers from 12 to 78
(2) Find the average of the first 3354 even numbers.
(3) Find the average of the first 1479 odd numbers.
(4) Find the average of the first 536 odd numbers.
(5) Find the average of even numbers from 12 to 496
(6) Find the average of the first 3358 odd numbers.
(7) What will be the average of the first 4311 odd numbers?
(8) Find the average of the first 2468 even numbers.
(9) What will be the average of the first 4252 odd numbers?
(10) Find the average of the first 2995 even numbers.