A man bought a number of oranges at 5 for a rupee and an equal number at 4 for a rupee. He mixed the two and sold at 9 for 2. If the loss is Rs 3 then find the number of total or anges.

Options

A. a)2160
B. b)540
C. c)1080
D. d)270

Answer

To solve the problem step by step, we will break it down into manageable parts.

### Step 1: Determine the cost price of oranges bought at different rates.
- The man buys oranges at two different rates:
- First, he buys oranges at the rate of 5 for 1 rupee. Therefore, the cost of 1 orange is \( \frac{1}{5} \) rupees.
- Let the number of oranges bought at this rate be \( x \). The total cost for these oranges is \( \frac{x}{5} \) rupees.

- Next, he buys the same number of oranges \( x \) at the rate of 4 for 1 rupee. Therefore, the cost of 1 orange is \( \frac{1}{4} \) rupees.
- The total cost for these oranges is \( \frac{x}{4} \) rupees.

### Step 2: Calculate the total cost price (CP) of all oranges.
- The total cost price for all the oranges is the sum of the costs from both purchases:
\[ CP = \frac{x}{5} + \frac{x}{4} \]
- To add these fractions, we need a common denominator, which is 20:
\[ CP = \frac{4x}{20} + \frac{5x}{20} = \frac{9x}{20} \text{ rupees} \]

### Step 3: Determine the selling price (SP) of the mixed oranges.
- The man mixes the two batches of oranges, resulting in a total of \( 2x \) oranges.
- He sells them at the rate of 9 for 2 rupees, which means the selling price for 1 orange is:
\[ SP = \frac{2}{9} \text{ rupees} \]
- Therefore, the total selling price for \( 2x \) oranges is:
\[ SP = 2x \times \frac{2}{9} = \frac{4x}{9} \text{ rupees} \]

### Step 4: Set up the equation using the loss information.
- We are given that the loss incurred is 3 rupees. The loss can be expressed as:
\[ \text{Loss} = CP - SP \]
- Substituting the values we found:
\[ 3 = \frac{9x}{20} - \frac{4x}{9} \]

### Step 5: Solve the equation for \( x \).
- To solve for \( x \), we first need to eliminate the fractions by finding a common denominator. The least common multiple of 20 and 9 is 180:
\[ 3 = \frac{9x \times 9}{180} - \frac{4x \times 20}{180} \]
\[ 3 = \frac{81x}{180} - \frac{80x}{180} \]
\[ 3 = \frac{x}{180} \]
- Multiplying both sides by 180 gives:
\[ x = 540 \]

### Step 6: Calculate the total number of oranges.
- Since the man bought an equal number of oranges at both rates, the total number of oranges is:
\[ \text{Total oranges} = x + x = 2x = 2 \times 540 = 1080 \]

### Final Answer:
The total number of oranges is **1080**.

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