A bought some oranges at the rate of 4 for Rs 1. I kept `(1)/(5)` with my self and sold remaining at the rate of 3 for Rs 1 and earn profit of Rs 1. Find the number of oranges I bought ?

Answer

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

### Step 1: Define the Variables
Let the total number of oranges bought be \( x \).

### Step 2: Calculate the Cost Price
A bought the oranges at the rate of 4 for Rs 1. Therefore, the cost price (CP) of \( x \) oranges can be calculated as follows:
\[ \text{Cost Price (CP)} = \frac{x}{4} \text{ (since 4 oranges cost Rs 1)} \]

### Step 3: Determine the Number of Oranges Kept and Sold
A kept \( \frac{1}{5} \) of the oranges with himself. Thus, the number of oranges kept is:
\[ \text{Oranges kept} = \frac{x}{5} \]
The remaining oranges sold will be:
\[ \text{Oranges sold} = x - \frac{x}{5} = \frac{5x - x}{5} = \frac{4x}{5} \]

### Step 4: Calculate the Selling Price
A sold the remaining oranges at the rate of 3 for Rs 1. Therefore, the selling price (SP) of the oranges sold can be calculated as follows:
\[ \text{Selling Price (SP)} = \frac{\text{Oranges sold}}{3} = \frac{\frac{4x}{5}}{3} = \frac{4x}{15} \]

### Step 5: Set Up the Profit Equation
According to the problem, A earned a profit of Rs 1. The profit can be defined as:
\[ \text{Profit} = \text{Selling Price} - \text{Cost Price} \]
Substituting the values we have:
\[ 1 = \frac{4x}{15} - \frac{x}{4} \]

### Step 6: Solve the Profit Equation
To solve the equation, we first find a common denominator for the fractions. The least common multiple of 15 and 4 is 60. Thus, we rewrite the equation:
\[ 1 = \frac{4x \cdot 4}{60} - \frac{x \cdot 15}{60} \]
This simplifies to:
\[ 1 = \frac{16x - 15x}{60} \]
\[ 1 = \frac{x}{60} \]
Multiplying both sides by 60 gives:
\[ x = 60 \]

### Conclusion
The total number of oranges A bought is \( 60 \).