Question:
Find the average of first 30 multiples of 11.
Correct Answer
170.5
Solution And Explanation
Solution
Shortcut method/Trick
Formula to find the average of the first n multiples of a number p
= p + (n – 1) p/2
Here to find the average of the first 30 multiples of 11.
Here, p = 11 and n = 30
Therefore, Average = 11 + (30 – 1) 11/2
= 11 + 29 × 5.5
= 11 + 159.5
= 170.5
Thus, the average of the first 30 multiples of 1 = 170.5 Answer
Alternate method to find the average of the first 30 multiples of 11
Step (1) Find the sum of the first 30 multiples of 11
Step (2)
Divide the sum of the first 30 multiples of 11 by 30 to find the average.Solution
The first 30 multiples of 11 are
11, 22, 33, . . . .330
These numbers form an Arithmetic Series because the differences between two consecutive numbers are equal.
Here, the first term "a" = 11
And, common difference (d) = 11
And, number of terms = 30
Sum of "n" terms of a Arithmetic Series
Sn = n/2 [2a + (n – 1)d]
Therefore, S30 = 30/2 [2 × 11 + (30 – 1)11]
= 15 [22 + (29 × 11)]
= 15 [22 + 319]
= 15 × 341
= 5115
Average = Sum of given obeservations/Number of observations
Therefore, the Average of the first 30 multiles of 11
= 5115/30 = 170.5
Therefore, the Average of the first 30 multiles of 11 = 170.5 Answer
Second Alternate method to find the average of the first 30 multiples of 11.
The first 30 multiples of 11 are
11, 22, 33, . . . .330
Therefore, the sum of first 30 multiples of 11
= 11 + 22 + . . . + 330
= 11(1 + 2 + 3 + . . . + 30)
= 11 × [30(30 + 1)]/2
[∵ Sum of first n natural numbers = n(n + 1)/2]
= 11 × (30 × 31)/2
= 11 × 930/2
= 11 × 465
= 5115
Average = Sum of given obeservations/Number of observations
Therefore, the Average of the first 30 multiles of 11
= 5115/30 = 170.5
Therefore, the Average of the first 30 multiles of 11 = 170.5 Answer
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