Vijay had some bananas and he divided them into two lots A and B. He sold the first lot at the rate of RS. 2 for 3 bananas and the second lot at the rate of Rs 1 per banana and got a total of Rs. 400. If he had sold the first lot at the rate of Rs. 1 per banana and the second lot at the rate of Rs.4 for 5 bananas , his total collection would have been Rs 460. Find the total number of bananas he had.

Options

A. `500`
B. `600`
C. `700`
D. `800`

Explanation

To solve the problem step by step, we will first define the variables and then set up the equations based on the information given in the question.

### Step 1: Define the Variables
Let:
- \( X \) = Number of bananas in lot A
- \( Y \) = Number of bananas in lot B

### Step 2: Set Up the First Equation
According to the first scenario, Vijay sold:
- Lot A at the rate of Rs. 2 for 3 bananas, which means the price per banana is \( \frac{2}{3} \).
- Lot B at the rate of Rs. 1 per banana.

The total earning from both lots is Rs. 400. Therefore, we can write the first equation as:
\[ \frac{2}{3}X + Y = 400 \]
To eliminate the fraction, we can multiply the entire equation by 3:
\[ 2X + 3Y = 1200 \quad \text{(Equation 1)} \]

### Step 3: Set Up the Second Equation
In the second scenario, Vijay sold:
- Lot A at the rate of Rs. 1 per banana.
- Lot B at the rate of Rs. 4 for 5 bananas, which means the price per banana is \( \frac{4}{5} \).

The total earning from both lots in this case is Rs. 460. Therefore, we can write the second equation as:
\[ X + \frac{4}{5}Y = 460 \]
Again, to eliminate the fraction, we can multiply the entire equation by 5:
\[ 5X + 4Y = 2300 \quad \text{(Equation 2)} \]

### Step 4: Solve the System of Equations
Now we have the following system of equations:
1. \( 2X + 3Y = 1200 \)
2. \( 5X + 4Y = 2300 \)

We will solve these equations using the elimination method. First, we can multiply Equation 1 by 5 and Equation 2 by 2 to align the coefficients of \( X \):
\[ 10X + 15Y = 6000 \quad \text{(Equation 3)} \]
\[ 10X + 8Y = 4600 \quad \text{(Equation 4)} \]

### Step 5: Subtract the Equations
Now, we will subtract Equation 4 from Equation 3:
\[ (10X + 15Y) - (10X + 8Y) = 6000 - 4600 \]
This simplifies to:
\[ 7Y = 1400 \]
Dividing both sides by 7 gives:
\[ Y = 200 \]

### Step 6: Substitute to Find \( X \)
Now, substitute \( Y = 200 \) back into Equation 1 to find \( X \):
\[ 2X + 3(200) = 1200 \]
This simplifies to:
\[ 2X + 600 = 1200 \]
Subtracting 600 from both sides gives:
\[ 2X = 600 \]
Dividing both sides by 2 gives:
\[ X = 300 \]

### Step 7: Calculate Total Bananas
Now that we have both \( X \) and \( Y \):
\[ X + Y = 300 + 200 = 500 \]

### Final Answer
The total number of bananas Vijay had is **500**.

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