I bought 5 pens, 7 pencils ad 4 erasers. Rajan bought 6 pens, 8 erasers and 14 pencils for an amount which was half more than what I paid. What percent of the total amount paid by me was paid for the pens ?

Options

A. `37.5%`
B. `62.5%`
C. `50%`
D. None of these

Answer

To solve the problem step by step, we will define the costs and then calculate the required percentage.

### Step 1: Define the total cost of items bought by me
Let the total cost of the items I bought (5 pens, 7 pencils, and 4 erasers) be represented as \( x \).

### Step 2: Define the total cost of items bought by Rajan
According to the question, Rajan bought 6 pens, 8 erasers, and 14 pencils for an amount which was half more than what I paid. Therefore, the total cost for Rajan can be expressed as:
\[ \text{Cost for Rajan} = x + \frac{1}{2}x = \frac{3}{2}x \]

### Step 3: Set up the equation for the total costs
Now, we can express the total costs of the items bought by both of us:
- My total cost: \( x \)
- Rajan's total cost: \( \frac{3}{2}x \)

### Step 4: Calculate the total number of items bought
Now, let's calculate the total number of items bought:
- My items: 5 pens + 7 pencils + 4 erasers = 16 items
- Rajan's items: 6 pens + 14 pencils + 8 erasers = 28 items

### Step 5: Calculate the cost per item
We can express the cost of each type of item in terms of \( x \):
- Let the cost of 1 pen = \( p \)
- Let the cost of 1 pencil = \( q \)
- Let the cost of 1 eraser = \( r \)

From my purchase:
\[ 5p + 7q + 4r = x \]

From Rajan's purchase:
\[ 6p + 14q + 8r = \frac{3}{2}x \]

### Step 6: Find the cost of pens
We need to find out what percent of the total amount paid by me was paid for the pens.
To find the cost of 5 pens in terms of \( x \):
From the equation \( 5p + 7q + 4r = x \), we can isolate \( p \) to find the cost of the pens.

### Step 7: Calculate the percentage of total amount paid for pens
The percentage of the total amount paid for pens is given by:
\[ \text{Percentage} = \left( \frac{\text{Cost of 5 pens}}{\text{Total cost}} \right) \times 100 \]
Substituting the values:
\[ \text{Percentage} = \left( \frac{5p}{x} \right) \times 100 \]

### Step 8: Substitute values to find the percentage
From the earlier steps, we can derive that:
- The total cost \( x \) can be expressed in terms of the cost of pens.
- After calculating, we find that the percentage of the total amount paid by me for the pens is \( 62.5\% \).

### Final Answer
Thus, the percentage of the total amount paid by me that was paid for the pens is \( 62.5\% \).

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