Solving the Cost Problem Using Linear Equations

This problem involves finding the costs of individual items (pens and notebooks) based on given total costs for different combinations. We can solve this using a system of linear equations.

Define Variables

• Let the cost of one pen be denoted by $p$.
• Let the cost of one notebook be denoted by $n$.

Formulate Equations

From the information given in the question, we can set up two equations:

• "The cost of 2 pens and 5 notebooks is ₹187": $2p + 5n = 187 \quad (1)$
• "The cost of 8 notebooks exceeds the cost of 4 pens by 40": This means the cost of 8 notebooks is ₹40 more than the cost of 4 pens. $8n = 4p + 40 \quad (2)$

Simplify and Solve the System of Equations

First, let's simplify Equation (2):

Divide the entire equation by 4:

Now, we can express $p$ in terms of $n$: $p = 2n - 10$

Substitute this expression for $p$ into Equation (1):

Distribute the 2:

Combine the terms with $n$: $9n - 20 = 187$

Add 20 to both sides to isolate the term with $n$: $9n = 187 + 20$ $9n = 207$

Solve for $n$ by dividing by 9:

So, the cost of one notebook ($n$) is ₹23.

Calculate the Cost of a Pen

Now substitute the value of $n$ back into the expression for $p$ (derived from Equation 2):

The cost of one pen ($p$) is ₹36.

Calculate the Final Cost

The question asks for the cost of 8 pens and 7 notebooks. We can calculate this using the costs we found:

Cost = (Number of pens $\times$ Cost per pen) + (Number of notebooks $\times$ Cost per notebook)

Cost = $8p + 7n$

Substitute the values $p=36$ and $n=23$:

Calculate the cost of the pens:

Calculate the cost of the notebooks:

Add the two costs together:

Therefore, the cost of 8 pens and 7 notebooks is ₹449.