A grocer mixed sugar at ₹ 12 per kg with sugar at ₹ 9 per kg in a certain ratio and sold the mixture at ₹ 11 per kg to have a gain of 8th of his total investment. The ratio of two types of sugar in the mixture is

Options

A. 7 :20
B. 9 :12
C. 1 :2
D. 3 :4

Answer

To solve the problem of finding the ratio of two types of sugar mixed by the grocer, we can follow these steps:

### Step-by-Step Solution:

1. **Define Variables**:
Let the quantity of sugar costing ₹12 per kg be \( a \) kg and the quantity of sugar costing ₹9 per kg be \( b \) kg.

2. **Calculate Cost Price**:
The cost price (CP) of the mixture can be calculated as:
\[ \text{CP} = 12a + 9b \]

3. **Calculate Selling Price**:
The selling price (SP) of the mixture when sold at ₹11 per kg is:
\[ \text{SP} = 11(a + b) \]

4. **Calculate Gain**:
The gain is defined as the selling price minus the cost price:
\[ \text{Gain} = \text{SP} - \text{CP} = 11(a + b) - (12a + 9b) \]
Simplifying this gives:
\[ \text{Gain} = 11a + 11b - 12a - 9b = -a + 2b \]

5. **Set Up Gain Equation**:
According to the problem, the gain is \( \frac{1}{8} \) of the total investment (cost price):
\[ -a + 2b = \frac{1}{8}(12a + 9b) \]

6. **Clear the Fraction**:
To eliminate the fraction, multiply both sides by 8:
\[ 8(-a + 2b) = 12a + 9b \]
This simplifies to:
\[ -8a + 16b = 12a + 9b \]

7. **Rearrange the Equation**:
Rearranging gives:
\[ 16b - 9b = 12a + 8a \]
Which simplifies to:
\[ 7b = 20a \]

8. **Find the Ratio**:
Dividing both sides by \( ab \) gives:
\[ \frac{a}{b} = \frac{7}{20} \]
Therefore, the ratio of the two types of sugar in the mixture is:
\[ a : b = 7 : 20 \]

### Final Answer:
The ratio of the two types of sugar in the mixture is **7 : 20**.

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