Introduction:
The problem involves a barrel containing a mixture of wine and water in a specific ratio. A certain fraction of the mixture is to be drawn off and substituted by water to achieve a new ratio of wine and water in the resultant mixture.

Given:
Wine and water are in the ratio 3:1 in the original mixture.

To find:
The fraction of mixture to be drawn off and substituted by water to achieve a new ratio of wine and water in the resultant mixture as 1:1.

Solution:
Let's assume that the initial volume of the mixture in the barrel is 4 units. Then, the volume of wine and water in the mixture would be 3 units and 1 unit, respectively.

Step 1:
Let's assume that x units of the mixture is to be drawn off from the barrel.

Step 2:
In x units of the mixture, the amount of wine would be (3/4)x units, and the amount of water would be (1/4)x units.

Step 3:
After removing x units of the mixture from the barrel, the volume of wine and water in the barrel would be 3 units - (3/4)x units and 1 unit - (1/4)x units, respectively.

Step 4:
Let's assume that y units of water is to be added to the barrel after removing x units of the mixture.

Step 5:
After adding y units of water to the barrel, the volume of water in the barrel would be 1 unit - (1/4)x + y units.

Step 6:
To achieve a 1:1 ratio of wine and water in the resultant mixture, the volume of wine in the barrel should be equal to the volume of water in the barrel.

Step 7:
Equating the volume of wine and water in the barrel, we get:
3 - (3/4)x = 1 - (1/4)x + y

Step 8:
Solving the above equation for y, we get:
y = (1/2)x

Conclusion:
Therefore, to achieve a new ratio of wine and water in the resultant mixture as 1:1, we need to draw off 1/2 of the mixture in the barrel and replace it with water.