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 divided by $11$.

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

The average of $39$ numbers is $17$.

Each number undergoes the following transformations: multiplied by $12$, then $16$ is added, then divided by $11$.

When the same operations are applied to all numbers in a set, the average changes in the same way.

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Starting with the original average of $17$:

After multiplying by $12$:

$17 \times 12$

$= 204$

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After adding $16$:

$204 + 16$

$= 220$

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After dividing by $11$:

$\dfrac{220}{11}$

$= 20$

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The transformation can be expressed as:

New Average $= \dfrac{12 \times \text{Old Average} + 16}{11}$

New Average $= \dfrac{12 \times 17 + 16}{11}$

New Average $= \dfrac{204 + 16}{11}$

New Average $= \dfrac{220}{11}$

New Average $= 20$

Therefore, the average of the new set of numbers is $20$.