A person bought some articles at the rate of 5 per rupee and the same number at the rate of 4 per rupee. He mixed both the types and sold at the rate of 9 for 2 rupees. In this business he suffered a loss of 3. The total number of articles bought by him was

Options

A. A) 1090
B. B) 1080
C. C) 540
D. D) 545

Answer

To solve the problem step by step, let's break it down:

### Step 1: Understand the Cost Price (CP)
The person bought articles at two different rates:
- **Rate 1**: 5 articles for 1 rupee
- **Rate 2**: 4 articles for 1 rupee

Let the number of articles bought at each rate be \( x \).

#### Calculation of Total Cost Price:
1. For articles bought at the rate of 5 per rupee:
- Cost for \( x \) articles = \( \frac{x}{5} \) rupees
2. For articles bought at the rate of 4 per rupee:
- Cost for \( x \) articles = \( \frac{x}{4} \) rupees

Total Cost Price (CP) = \( \frac{x}{5} + \frac{x}{4} \)

### Step 2: Find a Common Denominator
To add the fractions, we need a common denominator:
- The LCM of 5 and 4 is 20.

Rewriting the fractions:
- \( \frac{x}{5} = \frac{4x}{20} \)
- \( \frac{x}{4} = \frac{5x}{20} \)

Now, adding them:
\[ \text{Total CP} = \frac{4x + 5x}{20} = \frac{9x}{20} \]

### Step 3: Understand the Selling Price (SP)
The person sells the articles at the rate of 9 articles for 2 rupees.

#### Calculation of Total Selling Price:
Total number of articles = \( 2x \) (since he bought \( x \) articles at each rate).

Selling Price (SP) for \( 2x \) articles:
- For 9 articles, the price is 2 rupees.
- Therefore, for \( 2x \) articles, the price is:
\[ \text{SP} = \frac{2}{9} \times 2x = \frac{4x}{9} \]

### Step 4: Calculate the Loss
According to the problem, the person suffered a loss of 3 rupees.
The formula for loss is:
\[ \text{Loss} = \text{CP} - \text{SP} \]
Given that the loss is 3 rupees:
\[ \frac{9x}{20} - \frac{4x}{9} = 3 \]

### Step 5: Solve the Equation
To solve the equation, we first find a common denominator for the fractions:
- The LCM of 20 and 9 is 180.

Rewriting the fractions:
- \( \frac{9x}{20} = \frac{81x}{180} \)
- \( \frac{4x}{9} = \frac{80x}{180} \)

Now substituting back into the equation:
\[ \frac{81x}{180} - \frac{80x}{180} = 3 \]
This simplifies to:
\[ \frac{x}{180} = 3 \]

### Step 6: Solve for \( x \)
Multiplying both sides by 180:
\[ x = 3 \times 180 = 540 \]

### Step 7: Total Number of Articles
Since he bought \( x \) articles at each rate, the total number of articles is:
\[ \text{Total Articles} = 2x = 2 \times 540 = 1080 \]

### Final Answer
The total number of articles bought by him was **1080**.

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