A person buys a certain number of oranges at ₹84 a dozen and two times of it at ₹72 a dozen. He sells all the oranges at ₹90 a dozen and makes a profit of ₹1680. How many oranges does he buy?

Options

A. 1584
B. 1530
C. 1440
D. 1476

Answer

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

### Step 1: Understand the Purchases
The person buys oranges in two batches:
1. The first batch is bought at ₹84 per dozen.
2. The second batch is bought at ₹72 per dozen, and he buys twice the quantity of the first batch.

Let the number of dozens in the first batch be \( x \). Therefore, the number of dozens in the second batch is \( 2x \).

### Step 2: Calculate the Cost of Purchases
- Cost of the first batch:
\[ \text{Cost}_1 = x \times 84 \]
- Cost of the second batch:
\[ \text{Cost}_2 = 2x \times 72 = 144x \]

### Step 3: Total Cost
The total cost for both batches is:
\[ \text{Total Cost} = \text{Cost}_1 + \text{Cost}_2 = 84x + 144x = 228x \]

### Step 4: Selling Price
The person sells all the oranges at ₹90 per dozen. The total number of dozens is:
\[ \text{Total dozens} = x + 2x = 3x \]
Thus, the total selling price is:
\[ \text{Total Selling Price} = 3x \times 90 = 270x \]

### Step 5: Calculate Profit
The profit made is given as ₹1680. Profit can be calculated as:
\[ \text{Profit} = \text{Total Selling Price} - \text{Total Cost} \]
Substituting the values we have:
\[ 1680 = 270x - 228x \]
This simplifies to:
\[ 1680 = 42x \]

### Step 6: Solve for \( x \)
To find \( x \):
\[ x = \frac{1680}{42} = 40 \]

### Step 7: Calculate Total Oranges
Since \( x \) represents the number of dozens in the first batch, the total number of dozens of oranges is:
\[ \text{Total dozens} = 3x = 3 \times 40 = 120 \]
To find the total number of oranges, we convert dozens to individual oranges:
\[ \text{Total oranges} = 120 \times 12 = 1440 \]

### Final Answer
The total number of oranges bought is **1440**.

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