We are tasked with finding the ratio $x : y$, where:

• $x$ kg of brand A rice (costs $12/kg) is mixed with
• $y$ kg of brand B rice (costs $8/kg),
• and the cost of the mixed rice is $10.8/kg.

Step 1: Setup the equation for the cost of the mixture.

The total cost of the rice mixture is the sum of the costs of brand A rice and brand B rice:

• The cost of $x$ kg of brand A rice is $12x$ dollars.
• The cost of $y$ kg of brand B rice is $8y$ dollars.

The total weight of the mixture is $x + y$ kg, and the total cost of the mixture is:

$\text{Cost of the mixture} = 10.8(x + y)$

Step 2: Set up the equation.

Since the total cost of the mixture is also the sum of the costs of both types of rice, we can write the equation:

$12x + 8y = 10.8(x + y)$

Step 3: Expand both sides.

Now, expand both sides of the equation:

$12x + 8y = 10.8x + 10.8y$

Step 4: Simplify the equation.

Move the terms involving $x$ to one side and the terms involving $y$ to the other side:

$12x - 10.8x = 10.8y - 8y$

Simplifying both sides:

$1.2x = 2.8y$

Step 5: Solve for $x : y$.

Now, divide both sides by 1.2 to solve for $x$ in terms of $y$:

$x = \frac{2.8y}{1.2} = \frac{28}{12}y = \frac{7}{3}y$

Thus, the ratio $x : y$ is:

$x : y = 7 : 3$

Final Answer:

The ratio $x : y$ is $\boxed{7:3}$.

Would you like any further clarification or details?

Here are some related questions to explore:

1. If the price of brand B rice were increased to $10/kg, how would the ratio $x : y$ change?
2. What if the price of brand A rice dropped to $10/kg—what would the new ratio be?
3. How would the calculation change if the cost of the mixed rice were $9.5/kg instead of $10.8/kg?
4. Can we use a similar method to solve this problem if there are more than two types of rice?
5. How would this problem change if instead of mixing, we were looking for the total cost of buying $x$ kg of brand A rice and $y$ kg of brand B rice?

Tip: When solving mixture problems, always keep the total cost per unit of the mixture in mind, and equate the total cost expression to the mixed cost for an easy solution!