To determine the ratio in which the grocer must mix the two varieties of rice costing ₹128 and ₹143 per kg to get a mixture worth ₹137.75 per kg, we can use the concept of alligation.

The formula for the alligation method is:

$\text{Ratio} = \frac{C_2 - C_m}{C_m - C_1}$

Where:

• $C_1$ is the cost of the cheaper variety (₹128 per kg).
• $C_2$ is the cost of the more expensive variety (₹143 per kg).
• $C_m$ is the mean cost of the mixture (₹137.75 per kg).

Plugging in the values, we get:

$\text{Ratio} = \frac{143 - 137.75}{137.75 - 128}$

Calculate the numerator and the denominator:

$143 - 137.75 = 5.25$ $137.75 - 128 = 9.75$

So the ratio is:

$\text{Ratio} = \frac{5.25}{9.75} = \frac{21}{39} = \frac{7}{13}$

Thus, the grocer must mix the two varieties of rice in the ratio $7:13$.

Would you like further details or have any questions?

Here are some related questions:

1. How can the alligation method be applied to other commodities like fruits or spices?
2. What is the importance of understanding ratios in commercial transactions?
3. Can you explain the difference between the alligation method and weighted averages?
4. How would the ratio change if the prices of the rice varieties were different?
5. What are some practical examples of mixtures in everyday life?
6. How can one verify the accuracy of the calculated ratio in a real-world scenario?
7. What other mathematical methods can be used for similar types of problems?
8. How does the concept of ratio and proportion apply to other fields like physics or chemistry?

Tip: Always double-check the calculations, especially when dealing with ratios and proportions, to ensure accuracy.