In what proportion may three kinds of rice bought at 6,10 and 14 be mixed to produce a mixture which would earn 40% on selling it at 11.20 per kg?

Options

A. `4:1:1`
B. `3:1:1`
C. `2:1:1`
D. `2:2:1`

Answer

To solve the problem of mixing three kinds of rice bought at prices of 6, 10, and 14 rupees per kg to achieve a mixture that earns a 40% profit when sold at 11.20 rupees per kg, we can follow these steps:

### Step 1: Calculate the Cost Price of the Mixture
The selling price of the mixture is given as 11.20 rupees per kg, and we know that this selling price includes a profit of 40%.

To find the cost price (CP) of the mixture, we can use the formula:
\[ \text{Selling Price} = \text{Cost Price} + \text{Profit} \]
Here, the profit is 40% of the cost price. Therefore, we can express the selling price as:
\[ \text{Selling Price} = \text{Cost Price} \times \left(1 + \frac{40}{100}\right) = \text{Cost Price} \times 1.4 \]
Substituting the selling price:
\[ 11.20 = CP \times 1.4 \]
To find the cost price, we rearrange the equation:
\[ CP = \frac{11.20}{1.4} = 8 \text{ rupees per kg} \]

### Step 2: Set Up the Alligation Method
Now we have the cost price of the mixture (8 rupees) and the cost prices of the three types of rice:
- Rice A: 6 rupees
- Rice B: 10 rupees
- Rice C: 14 rupees

We will use the alligation method to find the ratio in which these types of rice should be mixed.

### Step 3: Calculate the Differences
Using the alligation method, we calculate the differences between the cost prices of each type of rice and the cost price of the mixture:
- For Rice A (6 rupees):
\[ \text{Difference} = 8 - 6 = 2 \]
- For Rice B (10 rupees):
\[ \text{Difference} = 10 - 8 = 2 \]
- For Rice C (14 rupees):
\[ \text{Difference} = 14 - 8 = 6 \]

### Step 4: Set Up the Ratios
Now we can set up the ratios based on the differences calculated:
- The ratio of Rice A to Rice B:
\[ \text{Ratio of A to B} = \frac{2}{2} = 1:1 \]
- The ratio of Rice A to Rice C:
\[ \text{Ratio of A to C} = \frac{6}{2} = 3:1 \]

### Step 5: Combine the Ratios
Now we need to combine the ratios we found:
- From A to B, we have 1 part of A and 1 part of B.
- From A to C, we have 3 parts of A and 1 part of C.

To combine these ratios, we can express them in terms of a common quantity for A:
- Let’s say we take 3 parts of A for the A to C ratio, which means we also take 3 parts of B (since A to B is 1:1).

Thus, the total parts will be:
- A = 3 (from A to C) + 3 (from A to B) = 6
- B = 3 (from A to B) = 3
- C = 1 (from A to C) = 1

### Final Ratio
The final ratio of Rice A, Rice B, and Rice C is:
\[ A : B : C = 4 : 1 : 1 \]

### Conclusion
The three kinds of rice should be mixed in the ratio of **4:1:1**.

---