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The average of thirteen numbers is 47. The average of the first three numbers is 39 and that of the next seven numbers is 49. The 11th number is two times the 12th number and the 12th number is 3 less than the 13th number. What is the average of 11th and 13th numbers?

The average of thirteen numbers is 47. The average of the first three numbers is 39 and that of the next seven numbers is 49. The 11th number is two times the 12th number and the 12th number is 3 less than the 13th number. What is the average of 11th and 13th numbers? Correct Answer 57

The average of thirteen numbers is 47

Sum of the thirteen numbers is 47 × 13 = 611

The average of first three number = 39

Sum of the first three number = 39 × 3 = 117

The average of next seven number = 49

Sum of next seven number = 49 × 7 = 343

Sum of last the three number = 611 – 117 – 343 = 151

Let 12th number be x

11th number = 2x

13th number = x + 3

According to the question

⇒ 2x + x + x + 3 = 151

⇒ 4x + 3 = 151

⇒ 4x = 151 – 3

⇒ 4x = 148

⇒ x = 37

11th number = 2x = 2 × 37 = 74

13th number = x + 3 = 37 + 3 = 40

Average of 11th and 13th number = (74 + 40)/2 = 114/2 = 57

Short Trick ∶

Average of 13 number = 47

Average of first three numbers = 39

Deviation of first three numbers = 3 × (47 – 39) = 3 × 8 = 24

Average of next 7 numbers = 49

Deviation of next 7 numbers = 7 × (47 – 49) = 7 × (-2) = (-14)

Sum of 11th, 12th and 13th numbers = 47 × 3 + 24 – 14 = 141 + 10 = 151

Let 12th number be x

11th number = 2x

13th number = x + 3

According to the question

⇒ 2x + x + x + 3 = 151

⇒ 4x + 3 = 151

⇒ 4x = 151 – 3

⇒ x = 37

11th number is two times of x and 13th number is 3 more than x so sum of extra number = 37 + 3 = 40

So, average number = 37 + 40/2 = 37 + 20 = 57

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