Question

A pen costs 7 times as that of a pencil. If Raju buys 3 pencils and 2 pens and pays Rs 34, what is the cost of a pencil and a pen?

Answer 1

Hint: We will first assume the cost of one pen and one pencil to be any variable. Then we will obtain the first relation between these variables from the given condition. We will find the cost of 3 pencils and the cost of 2 pens. Then we will find the sum of the cost of 3 pencils and the cost of 2 pens and we will equate the obtained sum with the given sum. From there, we will get the value of the variable and hence the cost of a pencil and a pen.

Complete step-by-step answer:
Let the cost of one pen be \[x\] and the cost of one pencil be \[y\].
As it is given that the cost of a pen is 7 times that of a pencil.
Therefore,
\[x = 7y\] ………. \[\left( 1 \right)\]
Now, we will find the cost of 3 pencils. We know that the cost of one pencil is equal to \[y\]. So the cost of 3 pencils will be equal to \[3y\] and again we will find the cost of 2 pens. We know that the cost of one pen is equal to \[x\]. So the cost of 2 pens will be equal to \[2x\].
Raju pays Rs 34 for 3 pencils and 2 pens.
Therefore, we get
\[2x + 3y = 34\] …….. \[\left( 2 \right)\]
Now, we will substitute the value of \[x\] obtained in equation \[\left( 1 \right)\] in equation \[\left( 2 \right)\].
 \[ \Rightarrow 2 \times 7y + 3y = 34\]
On multiplying the terms, we get
\[ \Rightarrow 14y + 3y = 34\]
On adding the like terms, we get
\[ \Rightarrow 17y = 34\]
On dividing both sides by 17, we get
\[\begin{array}{l} \Rightarrow \frac{{17y}}{{17}} = \frac{{34}}{{17}}\\ \Rightarrow y = 2\end{array}\]
Now, we will substitute the value of \[y\] in equation \[\left( 1 \right)\].
\[\begin{array}{l}x = 7 \times 2\\ \Rightarrow x = 14\end{array}\]
Hence, the cost of one pen is equal to Rs 14 and cost of one pencil is equal to Rs 2.

Note: To solve this problem, we first assume the value that we need to calculate to be any variable and form the required number of the linear equations including the variable. A linear equation is an equation that has the highest degree of two and has only one solution. To find the value of the given number of variables, we need the same number of equations including the variables to get the value of all the variables.