Understanding the Pens and Notebooks Problem

This problem involves finding the individual cost of pens and notebooks using the information given about two different purchases. We are told that Mohit and Sudesh bought pens and notebooks from the same shop at consistent prices. We need to use this information to figure out how much Mohit spent specifically on his notebooks.

Setting Up Equations for Pens and Notebooks Cost

Let's represent the unknown costs using variables:

• Let \(p\) be the cost of one pen in Rupees.
• Let \(n\) be the cost of one notebook in Rupees.

Based on the information given, we can form two linear equations:

• Mohit's purchase: 3 pens and 6 notebooks cost Rs. 180. This can be written as: \[3p + 6n = 180 \quad \text{(Equation 1)}\]
• Sudesh's purchase: 5 pens and 2 notebooks cost Rs. 116. This can be written as: \[5p + 2n = 116 \quad \text{(Equation 2)}\]

Solving the System of Linear Equations

We now have a system of two linear equations with two variables (\(p\) and \(n\)). We can solve this system using methods like substitution or elimination. The elimination method is often straightforward when we can easily make the coefficient of one variable the same in both equations.

Let's use the elimination method. We can multiply Equation 2 by 3 to make the coefficient of \(n\) equal to the coefficient of \(n\) in Equation 1 (which is 6):

Multiply Equation 2 by 3:

Now we have Equation 1 and Equation 3:

Notice that the coefficient of \(n\) is the same in both equations (6n). We can eliminate \(n\) by subtracting Equation 1 from Equation 3:

Now, solve for \(p\):

So, the cost of one pen is Rs. 14.

Now, substitute the value of \(p\) (14) into either Equation 1 or Equation 2 to find the value of \(n\). Let's use Equation 1:

Subtract 42 from both sides:

Now, solve for \(n\):

So, the cost of one notebook is Rs. 23.

Calculating Mohit's Spending on Notebooks

The question asks how much Mohit spent on buying notebooks. Mohit bought 3 pens and 6 notebooks.

The cost of one notebook is Rs. 23.

Mohit bought 6 notebooks.

Total cost of notebooks for Mohit = (Number of notebooks) \(\times\) (Cost per notebook)

Total cost of notebooks = \(6 \times n\)

Total cost of notebooks = \(6 \times 23\)

Total cost of notebooks = 138

Mohit spent Rs. 138 on notebooks.

Verification of Costs

Let's verify if these costs fit the original equations:

• Mohit: 3 pens + 6 notebooks = \(3 \times 14 + 6 \times 23 = 42 + 138 = 180\). This matches the given total of Rs. 180.
• Sudesh: 5 pens + 2 notebooks = \(5 \times 14 + 2 \times 23 = 70 + 46 = 116\). This matches the given total of Rs. 116.

The calculated costs for pens and notebooks are correct.

Summary of Costs

Mohit bought 6 notebooks at Rs. 23 each.

Cost of Mohit's notebooks = \(6 \times 23 = 138\).

Conclusion on Mohit's Notebook Expense

By setting up and solving a system of linear equations based on the purchases of Mohit and Sudesh, we found the cost of individual pens and notebooks. Using the determined cost per notebook, we calculated Mohit's total expense specifically for the notebooks he purchased.

Revision Table: System of Equations

Here's a quick review of the steps involved in solving this type of problem:

Additional Information: Linear Equations in Word Problems

Word problems involving quantities and costs that have a linear relationship can often be modeled using systems of linear equations. A system of linear equations consists of two or more linear equations that are considered together. The solution to a system of two variables represents the point (or values of the variables) that satisfies all equations in the system simultaneously.

Common methods to solve systems of linear equations include:

• Substitution Method: Solve one equation for one variable, and then substitute that expression into the other equation.
• Elimination Method: Multiply one or both equations by constants so that the coefficients of one variable are opposites (or the same), and then add (or subtract) the equations to eliminate that variable.
• Graphical Method: Graph both equations; the intersection point represents the solution. (Less precise for non-integer solutions).

In this pens and notebooks problem, we successfully used the elimination method to find the unique cost of each item, allowing us to calculate the specific expense requested.