The ratio of the quantities of sugar, in which sugar costing Rs 20 per kg and Rs 15 per kg,should be mixed so that there will be neither loss nor gain on selling the mixed sugar at the rate of 18 per kg, is

Options

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

Answer

To solve the problem of mixing two types of sugar (one costing Rs 20 per kg and the other costing Rs 15 per kg) so that there is neither a loss nor a gain when selling the mixture at Rs 18 per kg, we can use the concept of weighted averages.

### Step-by-Step Solution:

1. **Identify the Costs and Selling Price:**
- Cost of sugar A (Rs 20 per kg) = C1
- Cost of sugar B (Rs 15 per kg) = C2
- Selling price of the mixture = SP = Rs 18 per kg

2. **Set Up the Equation for No Profit, No Loss:**
The average cost of the mixture should equal the selling price. If we let the quantities of sugar A and B be x kg and y kg respectively, the total cost of the mixture can be expressed as:
\[ \text{Total Cost} = 20x + 15y \]
The total quantity of the mixture is:
\[ \text{Total Quantity} = x + y \]
Therefore, the average cost per kg of the mixture is:
\[ \text{Average Cost} = \frac{20x + 15y}{x + y} \]

3. **Set the Average Cost Equal to Selling Price:**
To ensure there is neither profit nor loss, we set the average cost equal to the selling price:
\[ \frac{20x + 15y}{x + y} = 18 \]

4. **Cross-Multiply to Eliminate the Fraction:**
Cross-multiplying gives us:
\[ 20x + 15y = 18(x + y) \]

5. **Expand the Right Side:**
Expanding the right side:
\[ 20x + 15y = 18x + 18y \]

6. **Rearrange the Equation:**
Rearranging the equation to isolate terms involving x and y:
\[ 20x - 18x = 18y - 15y \]
Simplifying this gives:
\[ 2x = 3y \]

7. **Find the Ratio of x to y:**
Dividing both sides by y and rearranging gives:
\[ \frac{x}{y} = \frac{3}{2} \]
Therefore, the ratio of the quantities of sugar A to sugar B is:
\[ x:y = 3:2 \]

### Final Answer:
The ratio of the quantities of sugar costing Rs 20 per kg and Rs 15 per kg to be mixed is **3:2**.