An 84-litre mixture has milk and water in the ratio 5:2. If 21 litres of the mixture is removed and 12 litres of pure milk is added, what is the new ratio of milk to water?

The correct option is 19:6.

Let us solve the problem step-by-step.

Step 1: Understand the initial mixture
Total volume of the initial mixture = 84 litres.
Ratio of milk to water = 5 : 2.
Total parts in the ratio = 5 + 2 = 7 parts.

Step 2: Calculate the quantities of milk and water removed
When 21 litres of the mixture is removed, the milk and water in the removed portion will also be in the ratio 5 : 2.
Amount of milk removed = $\frac{5}{7}\times 21=15\text{ litres}$
Amount of water removed = $\frac{2}{7}\times 21=6\text{ litres}$

Step 3: Calculate the remaining mixture volume
Remaining volume of the mixture = 84 - 21 = 63 litres.

Alternatively, we can directly find the remaining quantities of milk and water in the 63 litres of mixture:
Remaining milk = $\frac{5}{7}\times 63=45\text{ litres}$
Remaining water = $\frac{2}{7}\times 63=18\text{ litres}$

Step 4: Add pure milk to the mixture
Now, 12 litres of pure milk is added to the remaining mixture.
New quantity of milk = 45 + 12 = 57 litres.
New quantity of water remains unchanged = 18 litres.

Step 5: Find the new ratio of milk to water
New Ratio = Milk : Water
$\text{New Ratio}=57:18$
Dividing both terms by their greatest common divisor (3):
$\frac{57}{3}:\frac{18}{3}=19:6$

Therefore, the new ratio of milk to water is 19:6.