A man ordered 10 physics books and some chemistry books. The price of chemistry book is twice the price of a Physics book. While preparing the bill. the clerk interchanged the number of physics and chemistry books by mistake.
which decreased the bill by `12 (1)/(2) % ` . The ratio of the number of physics books to the number of chemistry books in the original order is

Options

A. `3 : 5`
B. `3 : 4`
C. `4 : 5`
D. `2 : 3`

Answer

To solve the problem step by step, we will denote the price of a physics book as \( k \) and the number of chemistry books as \( x \).

### Step 1: Understand the Initial Setup
- The man ordered 10 physics books and \( x \) chemistry books.
- The price of a chemistry book is twice that of a physics book, so the price of a chemistry book is \( 2k \).

### Step 2: Calculate the Original Bill
The total cost of the original order can be calculated as:
\[ \text{Total Cost} = (\text{Number of Physics Books} \times \text{Price of Physics Book}) + (\text{Number of Chemistry Books} \times \text{Price of Chemistry Book}) \]
\[ \text{Total Cost} = (10 \times k) + (x \times 2k) = 10k + 2xk = k(10 + 2x) \]

### Step 3: Calculate the Bill After Interchanging Books
When the clerk interchanged the number of books, he charged for \( x \) physics books and 10 chemistry books. The new total cost becomes:
\[ \text{New Total Cost} = (x \times k) + (10 \times 2k) = xk + 20k = k(x + 20) \]

### Step 4: Set Up the Equation Based on the Decrease in Bill
The problem states that the bill decreased by \( 12.5\% \) (which is \( \frac{1}{8} \)). Therefore, we can express this as:
\[ \text{New Total Cost} = \text{Original Total Cost} - \frac{1}{8} \times \text{Original Total Cost} \]
This simplifies to:
\[ \text{New Total Cost} = \frac{7}{8} \times \text{Original Total Cost} \]
Substituting the expressions for the total costs:
\[ k(x + 20) = \frac{7}{8} \times k(10 + 2x) \]

### Step 5: Simplify the Equation
We can cancel \( k \) from both sides (assuming \( k \neq 0 \)):
\[ x + 20 = \frac{7}{8}(10 + 2x) \]
Now, multiply both sides by 8 to eliminate the fraction:
\[ 8(x + 20) = 7(10 + 2x) \]
Expanding both sides:
\[ 8x + 160 = 70 + 14x \]

### Step 6: Rearranging the Equation
Rearranging gives:
\[ 8x - 14x = 70 - 160 \]
\[ -6x = -90 \]
Dividing both sides by -6:
\[ x = 15 \]

### Step 7: Find the Ratio of Physics to Chemistry Books
Now that we know the number of chemistry books \( x = 15 \) and the number of physics books is 10, we can find the ratio:
\[ \text{Ratio} = \frac{\text{Number of Physics Books}}{\text{Number of Chemistry Books}} = \frac{10}{15} = \frac{2}{3} \]

### Final Answer
The ratio of the number of physics books to the number of chemistry books in the original order is \( \frac{2}{3} \).

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