Two equal glasses filled with mixture of milk and water in the proportions of 13: 12 and 17: 17 respectively are emptied into a third glass. What is the proportion of milk and water in the third glass? 

Given:

Mixture I: Milk : Water = 13 : 12

Mixture II: Milk : Water = 17 : 17

Two equal glasses are emptied into a third glass.

Calculations:

For Mixture I: Milk : Water = 13 : 12

Total parts in Mixture I = 13 + 12 = 25

For Mixture II: Milk : Water = 17 : 17

Total parts in Mixture II = 17 + 17 = 34

To make the volumes equal, we can multiply the ratio of Mixture I by the total parts of Mixture II, and the ratio of Mixture II by the total parts of Mixture I. This ensures both mixtures are scaled to an equivalent 'total parts' (25 × 34 = 850).

For Mixture I, multiply by 34 (which is the total parts of Mixture II):

MilkI : WaterI = (13 × 34) : (12 × 34)

⇒ MilkI : WaterI = 442 : 408

For Mixture II, multiply by 25 (which is the total parts of Mixture I):

MilkII : WaterII = (17 × 25) : (17 × 25)

⇒ MilkII : WaterII = 425 : 425

Now, add the milk and water quantities from both mixtures:

Total Milk = MilkI + MilkII = 442 + 425 = 867

Total Water = WaterI + WaterII = 408 + 425 = 833

Proportion of Milk : Water in the third glass = 867 : 833

∴ The proportion of milk and water in the third glass is 867 : 833.