The average of 24 numbers is 56. The average of the first 10 numbers is 71.7 and that of the next 11 number is 42. Then next three numbers (i.e 22nd, 23rd, and 24th) are in the ratio 1/2 : 1/3 : 5/12 What is the average of the 22nd and 24th numbers?

The average of 24 numbers is 56. The average of the first 10 numbers is 71.7 and that of the next 11 number is 42. Then next three numbers (i.e 22nd, 23rd, and 24th) are in the ratio 1/2 : 1/3 : 5/12 What is the average of the 22nd and 24th numbers? Correct Answer 60.5

Average of 24 numbers = 56

Sum of 24 numbers = 56 × 24 = 1344

Average of first 10 numbers = 71.7

Sum of first 10 numbers = 71.7 × 10 = 717

Average of next 11 numbers = 42

Sum of next 11 numbers = 42 × 11 = 462

Sum of 22^nd, 23^rd and 24^th = 1344 – 717 – 462 = 165

Ratio of 22^nd, 23^rd and 24^th = 1/2 : 1/3 : 5/12 = 12/2 : 12/3 : (12 × 5)/12 = 6 : 4 : 5

Ratio of 22^nd, 23^rd and 24^th = 6x : 4x : 5x

6x + 4x + 5x = 165

⇒ 15x = 165

⇒ x = 11

22^nd number = 6x = 6 × 11 = 66

24^th number = 5x = 5 × 11 = 55

∴ Average of 22^nd and 24^th number = (66 + 55)/2 = 121/2 = 60.5

Short Trick:

Average of 24 numbers = 56

Deviation on first 10 numbers = (56 – 71.7) × 10 = –157

Deviation on next 11 numbers = (56 – 42) 11 = 154

Sum of 22^nd, 23^rd and 24^th = 56 × 3 – 157 + 154 = 168 – 3 = 165

Ratio of 22^nd, 23^rd and 24^th = 1/2 : 1/3 : 5/12 = 12/2 : 12/3 : (12 × 5)/12 = 6 : 4 : 5

Average of 22^nd and 24^th number = (6 + 5)/2 = 5.5

6 + 4 + 5 = 15 unit

⇒ 15 unit = 165

⇒ 1 unit = 11