Arun and Prabhat have some books with them. Once Prabhat said to Arun that if Arun gives 3 books to Prabhat then Arun will have only `1/2` of the books that Prabhat will have with him. Then Arun asked Prabhat that if Prabhat gives only two books to Arun, then Prabhat will have as many books as Arun will have. The total number of books that Arun and Prabhat have with them is :

Options

A. A)25
B. B)56
C. C)30
D. D)can't be determined

Answer

To solve the problem step by step, we will define the number of books Arun has as \( a \) and the number of books Prabhat has as \( p \).

### Step 1: Set up the equations based on the conditions given in the problem.

1. **First Condition**: If Arun gives 3 books to Prabhat, then Arun will have half the number of books that Prabhat will have after receiving those 3 books.
- After Arun gives 3 books: Arun will have \( a - 3 \) books.
- Prabhat will have \( p + 3 \) books.
- The equation based on the condition is:
\[ a - 3 = \frac{1}{2}(p + 3) \]

2. **Second Condition**: If Prabhat gives 2 books to Arun, then Prabhat will have as many books as Arun will have after receiving those 2 books.
- After Prabhat gives 2 books: Arun will have \( a + 2 \) books.
- Prabhat will have \( p - 2 \) books.
- The equation based on this condition is:
\[ p - 2 = a + 2 \]

### Step 2: Solve the equations.

1. **From the first equation**:
\[ a - 3 = \frac{1}{2}(p + 3) \]
Multiply both sides by 2 to eliminate the fraction:
\[ 2(a - 3) = p + 3 \]
Simplifying gives:
\[ 2a - 6 = p + 3 \implies p = 2a - 9 \quad \text{(Equation 1)} \]

2. **From the second equation**:
\[ p - 2 = a + 2 \]
Rearranging gives:
\[ p = a + 4 \quad \text{(Equation 2)} \]

### Step 3: Substitute Equation 1 into Equation 2.

Substituting \( p = 2a - 9 \) from Equation 1 into Equation 2:
\[ 2a - 9 = a + 4 \]
Now, solve for \( a \):
\[ 2a - a = 4 + 9 \]
\[ a = 13 \]

### Step 4: Find \( p \) using the value of \( a \).

Now substitute \( a = 13 \) back into either equation to find \( p \). Using Equation 2:
\[ p = a + 4 = 13 + 4 = 17 \]

### Step 5: Calculate the total number of books.

The total number of books that Arun and Prabhat have is:
\[ a + p = 13 + 17 = 30 \]

### Final Answer:
The total number of books that Arun and Prabhat have is **30**.

---