The average of 39 numbers is 17. Each number is multiplied by 12 and then 16 is added to each number. Finally, each number thus formed is divide by 11.

Determine the average of the new sets of numbers thus formed?

20

The correct answer is Option (4) → 20

Step 1: Let the numbers be $x_1, x_2, \dots, x_{39}$

Given:

$\text{Average of 39 numbers} = 17 ⇒\frac{\sum x_i}{39} = 17 ⇒\sum x_i = 39 \times 17 = 663$

Step 2: Apply the operations

Each number is transformed as:

$y_i = \frac{12x_i + 16}{11}$

Step 3: Average of new numbers

$\text{New average} = \frac{\sum y_i}{39} = \frac{\sum \frac{12x_i + 16}{11}}{39} = \frac{1}{39} \cdot \frac{1}{11} \sum (12x_i + 16)$

$= \frac{1}{429} \sum (12x_i + 16) \quad (\text{since } 39 \times 11 = 429)$

$\sum (12x_i + 16) = 12 \sum x_i + 39 \cdot 16 = 12 \cdot 663 + 624 = 7956 + 624 = 8580$

$\text{New average} = \frac{8580}{429} = 20$