Question : Find the average of even numbers from 10 to 174
Correct Answer 92
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 10 to 174
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 10 to 174 are
10, 12, 14, . . . . 174
After observing the above list of the even numbers from 10 to 174 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 10 to 174 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 10 to 174
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 174
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 10 to 174
= 10 + 174/2
= 184/2 = 92
Thus, the average of the even numbers from 10 to 174 = 92 Answer
Method (2) to find the average of the even numbers from 10 to 174
Finding the average of given continuous even numbers after finding their sum
The even numbers from 10 to 174 are
10, 12, 14, . . . . 174
The even numbers from 10 to 174 form an Arithmetic Series in which
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 174
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 10 to 174
174 = 10 + (n – 1) × 2
⇒ 174 = 10 + 2 n – 2
⇒ 174 = 10 – 2 + 2 n
⇒ 174 = 8 + 2 n
After transposing 8 to LHS
⇒ 174 – 8 = 2 n
⇒ 166 = 2 n
After rearranging the above expression
⇒ 2 n = 166
After transposing 2 to RHS
⇒ n = 166/2
⇒ n = 83
Thus, the number of terms of even numbers from 10 to 174 = 83
This means 174 is the 83th term.
Finding the sum of the given even numbers from 10 to 174
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 10 to 174
= 83/2 (10 + 174)
= 83/2 × 184
= 83 × 184/2
= 15272/2 = 7636
Thus, the sum of all terms of the given even numbers from 10 to 174 = 7636
And, the total number of terms = 83
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 10 to 174
= 7636/83 = 92
Thus, the average of the given even numbers from 10 to 174 = 92 Answer
Similar Questions
(1) Find the average of even numbers from 8 to 1336
(2) Find the average of the first 2805 odd numbers.
(3) What is the average of the first 190 odd numbers?
(4) Find the average of odd numbers from 11 to 1261
(5) Find the average of the first 4980 even numbers.
(6) What is the average of the first 1558 even numbers?
(7) Find the average of the first 2532 odd numbers.
(8) Find the average of the first 2858 odd numbers.
(9) Find the average of the first 2826 even numbers.
(10) Find the average of the first 4791 even numbers.