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Math MCQs


Question :    Find the average of odd numbers from 13 to 235


Correct Answer  124

Solution & Explanation

Solution

Method (1) to find the average of the odd numbers from 13 to 235

Shortcut Trick to find the average of the given continuous odd numbers

The odd numbers from 13 to 235 are

13, 15, 17, . . . . 235

After observing the above list of the odd numbers from 13 to 235 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 13 to 235 form an Arithmetic Series.

In the Arithmetic Series of the odd numbers from 13 to 235

The First Term (a) = 13

The Common Difference (d) = 2

And the last term (ℓ) = 235

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 13 to 235

= 13 + 235/2

= 248/2 = 124

Thus, the average of the odd numbers from 13 to 235 = 124 Answer

Method (2) to find the average of the odd numbers from 13 to 235

Finding the average of given continuous odd numbers after finding their sum

The odd numbers from 13 to 235 are

13, 15, 17, . . . . 235

The odd numbers from 13 to 235 form an Arithmetic Series in which

The First Term (a) = 13

The Common Difference (d) = 2

And the last term (ℓ) = 235

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 13 to 235

235 = 13 + (n – 1) × 2

⇒ 235 = 13 + 2 n – 2

⇒ 235 = 13 – 2 + 2 n

⇒ 235 = 11 + 2 n

After transposing 11 to LHS

⇒ 235 – 11 = 2 n

⇒ 224 = 2 n

After rearranging the above expression

⇒ 2 n = 224

After transposing 2 to RHS

⇒ n = 224/2

⇒ n = 112

Thus, the number of terms of odd numbers from 13 to 235 = 112

This means 235 is the 112th term.

Finding the sum of the given odd numbers from 13 to 235

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 13 to 235

= 112/2 (13 + 235)

= 112/2 × 248

= 112 × 248/2

= 27776/2 = 13888

Thus, the sum of all terms of the given odd numbers from 13 to 235 = 13888

And, the total number of terms = 112

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 13 to 235

= 13888/112 = 124

Thus, the average of the given odd numbers from 13 to 235 = 124 Answer


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