In what proportion must a grocer mix sugar at Rs.12 per kg and Rs.7 per kg so as to make a mixture worth Rs. 8 per kg.?

Options

A. `7:12`
B. `1:4`
C. `2:3`
D. `12:7`

Answer

To solve the problem of mixing sugar priced at Rs. 12 per kg and Rs. 7 per kg to achieve a mixture worth Rs. 8 per kg, we can use the method of allegation. Here’s a step-by-step solution:

### Step 1: Identify the Prices
We have two types of sugar:
- Sugar A: Rs. 12 per kg
- Sugar B: Rs. 7 per kg
- Mixture: Rs. 8 per kg

### Step 2: Set Up the Allegation
We will arrange the prices in the following manner:

```
Sugar A (Rs. 12) Mixture (Rs. 8) Sugar B (Rs. 7)
```

### Step 3: Calculate the Differences
Now, we calculate the differences between the prices of the sugars and the price of the mixture:

- Difference between Sugar A and the Mixture:
\[ 12 - 8 = 4 \]

- Difference between Sugar B and the Mixture:
\[ 8 - 7 = 1 \]

### Step 4: Write the Ratios
The ratio of the quantities of Sugar A to Sugar B is given by the differences calculated above. Therefore, the ratio is:
\[ \text{Ratio of Sugar A to Sugar B} = \text{Difference from Sugar B} : \text{Difference from Sugar A} = 1 : 4 \]

### Conclusion
The grocer must mix the sugar at Rs. 12 per kg and Rs. 7 per kg in the ratio of **1:4**.

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