A vessel is filled with liquid, 3 parts of which are water and 5 parts of syrup. How much of the mixture must be drawn off and replaced with water so that the mixture may be half water and half syrup?

To solve the problem, let's define the prices and quantities of the different varieties of tea being mixed.

Given:

• Price of the first variety of tea = Rs. 135/kg
• Price of the second variety of tea = Rs. 126/kg
• Ratio of the first, second, and third varieties = 1 : 1 : 2

Let the price of the third variety of tea be $x$ Rs./kg.

The total mixture is worth Rs. 153/kg. Since the ratio of the tea varieties is 1:1:2, we can assume the quantities of each type of tea in the mixture to be:

• Quantity of the first variety = 1 unit
• Quantity of the second variety = 1 unit
• Quantity of the third variety = 2 units

The total cost of the mixture can be expressed as the weighted average cost of the individual varieties based on their respective quantities.

Let's compute the cost per unit for the mixture:

The total quantity of the mixture is:

The average price per kg of the mixture is given as Rs. 153. Therefore, we set up the equation:

First, simplify the numerator:

Next, substitute this back into the equation and solve for $x$:

Multiply both sides by 4 to clear the denominator:

Now, isolate $x$:

Thus, the price of the third variety of tea is Rs. 175.50 per kg.