Question

I bought 5 pens, 7 pencils and 4 erasers. Rajan bought 6 pens, 8 erasers and 14 pencils for an amount which is half more than what I had paid. What percent of the total amount paid by me was paid for the pens?
(a) 37.5%
(b) 62.5%
(c) 50%
(d) None of these

Answer 1

Hint: We start solving the problem by assigning the variables for total amount, cost pen, pencil and eraser. We then write our first equation using the condition that we bought 5 pens, 7 pencils and 4 erasers. We then write our second equation using the condition that Rajan bought 6 pens, 8 erasers and 14 pencils for the amount which is half more than we paid. Using these two equations we find the cost of a pen in terms of total cost. We then find the percentage of the amount we paid for pens using the percentage of ‘y’ in ‘x’ = $\dfrac{y}{x}\times 100$.

Complete step by step answer:
We have bought 5 pens, 7 pencils and 4 erasers and Rajan bought 6 pens, 8 erasers and 14 pencils. He paid an amount which is half more than what I had paid. We need to find the percent of total amount paid for pens by us.
Let us assume the total amount paid by us is ‘A’ and the cost of pens, pencils and erasers be ‘p’, ‘s’, ‘e’.
We have bought 5 pens, 7 pencils and 4 erasers by paying the amount ‘A’.
So, we get $5p+7s+4e=A$---(1).
According to the problem, Rajan paid half more than what we paid. So, the amount paid by Rajan is $\left( 1+\dfrac{1}{2} \right)A=\dfrac{3}{2}A$ and Rajan bought 6 pens, 8 erasers and 14 pencils by paying the amount $\dfrac{3}{2}A$.
So, we get $6p+14s+8e=\dfrac{3}{2}A$.
$\Rightarrow 10p+14s+8e-4p=\dfrac{3}{2}A$.
$\Rightarrow 2\left( 5p+7s+4e \right)-4p=\dfrac{3}{2}A$.
From equation (1) we get,
$\Rightarrow 2A-4p=\dfrac{3}{2}A$.
$\Rightarrow 2A-\dfrac{3}{2}A=4p$.
$\Rightarrow \dfrac{A}{2}=4p$.
$\Rightarrow p=\dfrac{A}{2\times 4}$.
$\Rightarrow p=\dfrac{A}{8}$.
So, the cost of each pen is $\dfrac{A}{8}$. Let us find the cost of 5 pens.
So, we have cost 5 pens as $5p=5\left( \dfrac{A}{8} \right)$.
$5p=\dfrac{5A}{8}$. But we have paid a total amount of A.
Let us find the percentage of $\dfrac{5A}{8}$ in A. We know that the percentage of ‘y’ in ‘x’ is calculated as $\dfrac{y}{x}\times 100$.
So, we get a percentage of $\dfrac{5A}{8}$ in A as $\dfrac{\dfrac{5A}{8}}{A}\times 100$.
$\Rightarrow \dfrac{5}{8}\times 100=5\times 12.5=62.5%$.
So, we have paid 62.5% of the total amount paid for pens.
The correct option for the given problem is (b).

Note:
Whenever we get this type of problem, we first need to start by assigning the variables for the unknowns. This makes us proceed through the problem and makes calculations easy. We should allocate random values for costs of pen, pencil and eraser as the actual amount paid by us is not given in the problem. We can also find the cost of pencil and eraser using the cost of the pen we just found.