IF the price of three types of rice are 480 576 and 696 per quintal, then find the ratio in which these types of rices should be mixed, so that the resulant mixture cost 564 per quintal ?

Options

A. 0.92440972222222
B. 1.1764699074074
C. 0.5884375
D. `11:77:7`

Answer

To find the ratio in which three types of rice priced at 480, 576, and 696 per quintal should be mixed to achieve a resultant mixture cost of 564 per quintal, we can use the method of alligation. Here’s a step-by-step solution:

### Step 1: Identify the Prices
The prices of the three types of rice are:
- Rice A: 480 per quintal
- Rice B: 576 per quintal
- Rice C: 696 per quintal

### Step 2: Identify the Average Price
The average price of the mixture is given as 564 per quintal.

### Step 3: Set Up the Alligation
We will use the alligation method to find the ratio. The formula for alligation is:

\[ \text{Ratio} = \frac{\text{Price of higher} - \text{Average Price}}{\text{Average Price} - \text{Price of lower}} \]

### Step 4: Calculate the Differences
1. **For Rice A (480) and Rice B (576)**:
- Higher price (B) = 576
- Lower price (A) = 480
- Average price = 564

\[ \text{Difference} = 576 - 564 = 12 \quad (for Rice B) \]
\[ \text{Difference} = 564 - 480 = 84 \quad (for Rice A) \]

2. **For Rice A (480) and Rice C (696)**:
- Higher price (C) = 696
- Lower price (A) = 480
- Average price = 564

\[ \text{Difference} = 696 - 564 = 132 \quad (for Rice C) \]
\[ \text{Difference} = 564 - 480 = 84 \quad (for Rice A) \]

### Step 5: Set Up the Ratios
1. **Ratio of Rice A to Rice B**:
\[ \text{Ratio} = \frac{12}{84} = \frac{1}{7} \]

2. **Ratio of Rice A to Rice C**:
\[ \text{Ratio} = \frac{132}{84} = \frac{11}{7} \]

### Step 6: Combine the Ratios
Now we have:
- Ratio of Rice A to Rice B = 1:7
- Ratio of Rice A to Rice C = 11:7

To combine these ratios, we need to equalize the quantities of Rice A in both ratios.

### Step 7: Equalizing the Ratios
Let’s equalize the quantities of Rice A:
- From Rice A to Rice B, we have 1 part of Rice A.
- From Rice A to Rice C, we have 11 parts of Rice A.

To equalize, we can multiply the first ratio by 11 and the second ratio by 1:

- Rice A : Rice B = 1 * 11 : 7 * 11 = 11 : 77
- Rice A : Rice C = 11 * 1 : 7 * 1 = 11 : 7

### Step 8: Final Ratio
Now we can write the final ratio of Rice A : Rice B : Rice C:
- Rice A = 11 parts
- Rice B = 77 parts
- Rice C = 7 parts

Thus, the final ratio of Rice A : Rice B : Rice C is:
\[ \text{Final Ratio} = 11 : 77 : 7 \]

### Conclusion
The ratio in which the three types of rice should be mixed is **11 : 77 : 7**.

---