Solution:

To find the ratio at which a 25% syrup and water mixture should be mixed with a 60% syrup and water mixture to obtain a new mixture with 40% syrup, we can use the method of alligation.

Alligation:
Alligation is a mathematical technique used to find the ratio in which two or more ingredients of different strengths or concentrations should be mixed to obtain a desired strength or concentration.

Step 1: Assign Variables
Let's assume that we need x units of the 25% syrup and water mixture and y units of the 60% syrup and water mixture to obtain the desired 40% syrup mixture.

Step 2: Set Up the Alligation Diagram

We can represent the given information in the form of an alligation diagram as shown below:



The 40% syrup mixture is equidistant from the 25% and 60% syrup mixtures.

Step 3: Find the Ratio
To find the ratio of x to y, we need to find the difference between the 40% and the two given percentages (25% and 60%).

Difference between 40% and 25% = 40 - 25 = 15
Difference between 60% and 40% = 60 - 40 = 20

The ratio of x to y is given by the inverse of these differences:

Ratio of x to y = 20/15 = 4/3

This means that the 25% syrup and water mixture should be mixed with the 60% syrup and water mixture in a ratio of 4:3 to obtain the desired 40% syrup mixture.

Step 4: Calculate the Quantities
To find the quantities of the two mixtures, we need to assign a value to either x or y. Let's assume x = 4 units.

Since the ratio of x to y is 4:3, we can calculate the value of y as follows:

4 units / 4 units = 3 units / y units
y = (3 * 4 units) / 4 units
y = 3 units

Therefore, we need to mix 4 units of the 25% syrup and water mixture with 3 units of the 60% syrup and water mixture to obtain the desired 40% syrup mixture.

Conclusion:
The ratio at which a 25% syrup and water mixture should be mixed with a 60% syrup and water mixture to obtain a new mixture with 40% syrup is 4:3. This means that 4 units of the 25% syrup and water mixture should be mixed with 3 units of the 60% syrup and water mixture.