$40$ litres of a mixture contains milk and water in the ratio of $17:3$. After the addition of some more milk to it, the ratio of milk to water in the resulting mixture becomes $7:1$. The quantity of milk added to it was:

Given total initial volume of the mixture = $40$ litres.
The initial ratio of milk to water = $17:3$.

Let the quantity of milk be $17x$ and water be $3x$.

$$17x+3x=40⟹20x=40⟹x=2$$

So, initial quantity of milk = $17\times 2=34$ litres.
Initial quantity of water = $3\times 2=6$ litres.

Let the quantity of milk added be $m$ litres.
The new ratio of milk to water becomes $7:1$.

$$\frac{34+m}{6}=\frac{7}{1}$$

$$34+m=42$$

$$m=42-34=8\text{ litres}$$

Thus, the quantity of milk added was $8$ litres.