Anushka and Vijay had 60 postcards together. After giving 8 postcards to Anushka, Vijay have twice as many postcards as Anushka have. How many postcards did Anushka have at first?

Options

A. 18
B. 36
C. 24
D. 12

Answer

To solve the problem step by step, let's define the variables and set up the equations based on the information given.

### Step 1: Define the Variables
Let:
- \( X \) = the number of postcards Anushka had initially
- \( Y \) = the number of postcards Vijay had initially

### Step 2: Set Up the First Equation
According to the problem, Anushka and Vijay together had 60 postcards. Therefore, we can write the first equation as:
\[ X + Y = 60 \]

### Step 3: Understand the Postcard Transfer
After Vijay gives 8 postcards to Anushka:
- Anushka will have \( X + 8 \) postcards.
- Vijay will have \( Y - 8 \) postcards.

### Step 4: Set Up the Second Equation
The problem states that after giving away the postcards, Vijay has twice as many postcards as Anushka. This gives us the second equation:
\[ Y - 8 = 2(X + 8) \]

### Step 5: Simplify the Second Equation
Now, let's simplify the second equation:
1. Expand the right side:
\[ Y - 8 = 2X + 16 \]
2. Rearranging gives:
\[ Y - 2X = 16 + 8 \]
\[ Y - 2X = 24 \]

### Step 6: Solve the System of Equations
Now we have two equations:
1. \( X + Y = 60 \) (Equation 1)
2. \( Y - 2X = 24 \) (Equation 2)

We can solve these equations simultaneously. From Equation 1, we can express \( Y \) in terms of \( X \):
\[ Y = 60 - X \]

### Step 7: Substitute \( Y \) in Equation 2
Now substitute \( Y \) in Equation 2:
\[ (60 - X) - 2X = 24 \]
This simplifies to:
\[ 60 - 3X = 24 \]

### Step 8: Solve for \( X \)
Now, isolate \( X \):
1. Subtract 60 from both sides:
\[ -3X = 24 - 60 \]
\[ -3X = -36 \]
2. Divide by -3:
\[ X = 12 \]

### Step 9: Find \( Y \)
Now that we have \( X \), we can find \( Y \):
\[ Y = 60 - X = 60 - 12 = 48 \]

### Conclusion
Thus, Anushka had **12 postcards** initially.