Solving the Cost Problem: Pens and Pencils

This problem involves finding the original costs of pens and pencils given two different scenarios involving their prices and total costs. We can solve this using a system of linear equations.

Setting Up the Equations

First, let's define the variables:

• Let \( p \) be the original cost of one pen (in ₹).
• Let \( c \) be the original cost of one pencil (in ₹).

Now, let's translate the information given in the problem into equations:

Scenario 1: The cost of 7 pens and 13 pencils is ₹128.

This gives us our first equation:

$7p + 13c = 128$

Scenario 2: The cost of a pen decreases by ₹7, and the cost of a pencil increases by ₹2. The new cost of 14 pens and 10 pencils is ₹98.

The new cost of a pen is \( p - 7 \).

The new cost of a pencil is \( c + 2 \).

The equation for this scenario is:

$14(p - 7) + 10(c + 2) = 98$

Let's simplify the second equation:

$14p - 98 + 10c + 20 = 98$

$14p + 10c - 78 = 98$

$14p + 10c = 98 + 78$

$14p + 10c = 176$

We can simplify this further by dividing the entire equation by 2:

$7p + 5c = 88$

Solving for Pen and Pencil Costs

Now we have a system of two linear equations:

We can solve this system using the elimination method. Notice that the coefficient of \( p \) is the same (7) in both equations. Let's subtract Equation 2 from Equation 1:

$(7p + 13c) - (7p + 5c) = 128 - 88$

$7p + 13c - 7p - 5c = 40$

$8c = 40$

Now, solve for \( c \):

$c = \frac{40}{8}$

$c = 5$

So, the original cost of a pencil is ₹5.

Next, substitute the value of \( c \) (which is 5) back into either Equation 1 or Equation 2 to find \( p \). Let's use the simplified Equation 2:

$7p + 5c = 88$

$7p + 5(5) = 88$

$7p + 25 = 88$

Subtract 25 from both sides:

$7p = 88 - 25$

$7p = 63$

Now, solve for \( p \):

$p = \frac{63}{7}$

$p = 9$

So, the original cost of a pen is ₹9.

Calculating the Original Cost

The question asks for the original cost of 9 pens and 7 pencils.

We need to calculate: $9p + 7c$

Substitute the values we found for \( p \) and \( c \):

$9(9) + 7(5)$

$81 + 35$

$116$

Therefore, the original cost of 9 pens and 7 pencils is ₹116.