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

### Concept & Formula This problem utilizes systems of linear equations. Even though there are three variables and only two equations, we can solve for a grouped variable by observing proportional relationships (scaling) within the coefficients. $$ \text{Percentage} = \frac{\text{Part}}{\text{Whole}} \times 100 $$ ### Step-by-Step Solution * **Given:** * Let the price of a pen be $p$, a pencil be $c$, and an eraser be $e$. * My total amount ($M$) = $5p + 7c + 4e$. * Rajan's total amount ($R$) = $6p + 14c + 8e$. * Rajan paid "half more" than I paid, meaning $R = M + 0.5M = 1.5M$. * **Calculation / Deduction:** * Equate Rajan's cost to $1.5 \times$ My cost: * $6p + 14c + 8e = 1.5(5p + 7c + 4e)$ * Expand the right side: * $6p + 14c + 8e = 7.5p + 10.5c + 6e$ * Rearrange the terms to group the variables. Let's move $p$ to one side and $c, e$ to the other: * $14c - 10.5c + 8e - 6e = 7.5p - 6p$ * $3.5c + 2e = 1.5p$ * Multiply the entire equation by $2$ to clear the decimals: * $7c + 4e = 3p$ * We need the percentage of my total amount that went toward pens. * My total amount $M = 5p + (7c + 4e)$. * Substitute the block $(7c + 4e)$ with $3p$: * $M = 5p + 3p = 8p$. * Amount spent on pens = $5p$. * Percentage = $\frac{5p}{8p} \times 100 = \frac{5}{8} \times 100$. * $\frac{5}{8} \times 100 = 62.5\%$. ### Exam Strategy & Shortcut Look closely at the coefficients. Notice that Rajan bought $14$ pencils and $8$ erasers, which is exactly double my $7$ pencils and $4$ erasers. This strong pattern signals that you should group pencils and erasers together into a single unit. Once you deduce $7c + 4e = 3p$, you instantly know your total bill was equivalent to buying $8$ pens ($5+3$). $5$ out of $8$ is standard fraction knowledge: $\frac{1}{8} = 12.5\%$, so $\frac{5}{8} = 62.5\%$. ### Common Pitfall Students often waste time trying to find the individual prices of $p$, $c$, and $e$. It is mathematically impossible to find unique individual values here because there are $3$ variables but only $1$ comparative equation. You must substitute blocks of variables. ### Final Answer **Therefore, the correct answer is 62.5%.**