Price of two types of rice is 20 and 16 kg. A shopkeeper mixed them in the ratio of n : (n+1) where n is an integer and sold it at 17.8 kg. What is the minimum value of n so that he is in loss ?

Options

A. 3
B. 7
C. 6
D. 5

Answer

To solve the problem step by step, we will use the method of allegation to determine the minimum value of \( n \) such that the shopkeeper incurs a loss.

### Step 1: Understand the Prices
The prices of the two types of rice are:
- Type 1: \( 20 \) rupees per kg
- Type 2: \( 16 \) rupees per kg

### Step 2: Selling Price
The selling price of the mixture is \( 17.8 \) rupees per kg.

### Step 3: Set Up the Allegation
We will use the allegation method to find the ratio in which the two types of rice should be mixed. The formula for allegation is as follows:

\[ \text{Price of Type 1} - \text{Mean Price} \quad \text{and} \quad \text{Mean Price} - \text{Price of Type 2} \]

Calculating the differences:
- Difference between Type 1 and Mean Price:
\[ 20 - 17.8 = 2.2 \]

- Difference between Mean Price and Type 2:
\[ 17.8 - 16 = 1.8 \]

### Step 4: Set Up the Ratio
Using the differences calculated, we can set up the ratio:
\[ \frac{2.2}{1.8} = \frac{22}{18} = \frac{11}{9} \]

This means the ratio of Type 1 rice to Type 2 rice is \( 11:9 \).

### Step 5: Relate to Given Ratio
The shopkeeper mixes the rice in the ratio \( n : (n + 1) \). For the shopkeeper to incur a loss, we need:
\[ \frac{n}{n + 1} < \frac{11}{20} \]

### Step 6: Solve the Inequality
Cross-multiplying gives:
\[ 20n < 11(n + 1) \]
Expanding the right side:
\[ 20n < 11n + 11 \]
Rearranging gives:
\[ 20n - 11n < 11 \]
\[ 9n < 11 \]
Dividing both sides by 9:
\[ n < \frac{11}{9} \approx 1.22 \]

### Step 7: Find Minimum Integer Value of \( n \)
Since \( n \) must be an integer, the maximum integer value for \( n \) that satisfies this condition is \( n = 1 \).

### Step 8: Check for Loss Condition
To ensure the shopkeeper is in loss, we need to check the condition:
\[ \frac{n}{n + 1} > \frac{9}{11} \]

Cross-multiplying gives:
\[ 11n > 9(n + 1) \]
Expanding gives:
\[ 11n > 9n + 9 \]
Rearranging gives:
\[ 11n - 9n > 9 \]
\[ 2n > 9 \]
Dividing both sides by 2:
\[ n > 4.5 \]

### Step 9: Conclusion
The minimum integer value of \( n \) that satisfies this condition is \( n = 5 \).

Thus, the answer is:
\[ \boxed{5} \]