Shelf A has `4/5` of the number of books that shelf B has. If 25% of the books in A are transferred to B and then 25% of the books from B are transferred to A, then the percentage of the total num ber of books that A will have is

Options

A. 25
B. 50
C. 75
D. 1

Answer

To solve the problem step by step, we will first define the number of books on each shelf and then perform the transfers as described in the question.

### Step 1: Define the number of books on each shelf
Let the number of books on shelf B be represented as \( B \). According to the problem, shelf A has \( \frac{4}{5} \) of the number of books that shelf B has. Therefore, we can express the number of books on shelf A as:
\[ A = \frac{4}{5}B \]

### Step 2: Assign values to A and B
To make calculations easier, we can assign specific values to \( A \) and \( B \). Let's assume:
\[ B = 500 \quad \text{(this is an arbitrary choice for easier calculations)} \]
Then, using the relationship from Step 1:
\[ A = \frac{4}{5} \times 500 = 400 \]

### Step 3: Calculate 25% of books in A transferred to B
Next, we need to calculate 25% of the books in shelf A that will be transferred to shelf B.
\[ \text{Books transferred from A to B} = 25\% \text{ of } A = \frac{25}{100} \times 400 = 100 \]
After transferring 100 books from A to B:
- Books in A: \( 400 - 100 = 300 \)
- Books in B: \( 500 + 100 = 600 \)

### Step 4: Calculate 25% of books in B transferred to A
Now, we will calculate 25% of the books in shelf B that will be transferred back to shelf A.
\[ \text{Books transferred from B to A} = 25\% \text{ of } B = \frac{25}{100} \times 600 = 150 \]
After transferring 150 books from B to A:
- Books in A: \( 300 + 150 = 450 \)
- Books in B: \( 600 - 150 = 450 \)

### Step 5: Calculate the percentage of total books that A will have
Now, we need to find the total number of books and the percentage of books that shelf A has.
\[ \text{Total books} = A + B = 450 + 450 = 900 \]
The percentage of total books that shelf A has is calculated as follows:
\[ \text{Percentage of books in A} = \left( \frac{A}{\text{Total books}} \right) \times 100 = \left( \frac{450}{900} \right) \times 100 = 50\% \]

### Final Answer
The percentage of the total number of books that shelf A will have is \( 50\% \).

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