To solve this problem, we'll use the concept of weighted averages. Let's denote the quantities of the three varieties of rice (costing 18, 25, and 28 Rs per kilo) as $x$, $y$, and $z$ respectively.

Step 1: Set Up the Equation

We want to mix the three varieties such that the resulting mixture costs 26 Rs per kilo. The equation representing this condition is:

$\frac{18x + 25y + 28z}{x + y + z} = 26$

Step 2: Multiply Both Sides by $x + y + z$

This simplifies to:

$18x + 25y + 28z = 26(x + y + z)$

Expanding the right-hand side:

$18x + 25y + 28z = 26x + 26y + 26z$

Step 3: Rearrange the Equation

Now, rearrange the terms to group like terms together:

$18x + 25y + 28z - 26x - 26y - 26z = 0$

Simplifying further:

$-8x - y + 2z = 0$

Step 4: Express the Equation in Terms of a Ratio

Rearrange the equation to express one variable in terms of the others:

$y = 8x + 2z$

Step 5: Determine the Ratio

To find a possible ratio, choose convenient values for $x$ and $z$ and solve for $y$. Let's assume $x = 1$ and $z = 1$:

$y = 8(1) + 2(1) = 8 + 2 = 10$

So, one possible ratio is $x : y : z = 1 : 10 : 1$.

Final Answer

A possible ratio in which the three varieties of rice could be mixed to obtain a mixture costing 26 Rs per kilo is 1 : 10 : 1.

Would you like more details or have any questions?

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Here are five related questions to expand your understanding:

1. What if the resultant price was 24 Rs per kilo? How would the ratio change?
2. What is the ratio if the third variety costs 30 Rs instead of 28 Rs per kilo?
3. How much of each variety should be mixed to get 100 kg of the mixture at 26 Rs per kilo?
4. What is the cost per kilo if the three varieties are mixed in the ratio 2:3:5?
5. Can you find another ratio that would give a 26 Rs per kilo mixture?

Tip: When working with weighted averages, it's often helpful to consider extreme cases (like setting one of the variables to 1) to simplify the calculation of ratios.