Problem Analysis: Cost Calculation

This problem involves finding the original costs of pens and pencils based on information from two different scenarios. We need to set up and solve a system of linear equations to determine these costs and then calculate the price for a specific combination.

Setting Up Algebraic Equations

Let $p$ represent the original cost of one pen in Rupees (₹) and $c$ represent the original cost of one pencil in Rupees (₹).

Scenario 1: The cost of 19 pens and 13 pencils is ₹254.

This can be written as the equation:

$19p + 13c = 254 \quad (1)$

Scenario 2: The cost of a pen decreases by ₹6 (new cost $p-6$) and the cost of a pencil increases by ₹9 (new cost $c+9$). The cost of 14 pens and 2 pencils under these new conditions is ₹106.

This translates to the equation:

$14(p-6) + 2(c+9) = 106$

Solving the System of Equations

First, let's simplify the second equation:

$14p - 84 + 2c + 18 = 106$

Combine the constant terms:

$14p + 2c - 66 = 106$

Add 66 to both sides:

$14p + 2c = 106 + 66$

$14p + 2c = 172$

Divide the entire equation by 2 for simplification:

$7p + c = 86 \quad (2)$

Now, we can express $c$ in terms of $p$ using equation (2):

$c = 86 - 7p$

Substitute this expression for $c$ into equation (1):

$19p + 13(86 - 7p) = 254$

Distribute the 13:

$19p + 1118 - 91p = 254$

Combine the terms with $p$:

$1118 - 72p = 254$

Rearrange the equation to solve for $p$:

$1118 - 254 = 72p$

$864 = 72p$

Divide by 72 to find the value of $p$:

$p = \frac{864}{72}$

$p = 12$

So, the original cost of one pen is ₹12.

Now, substitute the value of $p$ back into the expression for $c$ (from equation 2):

$c = 86 - 7p$

$c = 86 - 7(12)$

$c = 86 - 84$

$c = 2$

Therefore, the original cost of one pencil is ₹2.

Calculating Original Cost

The question asks for the original cost of 12 pens and 15 pencils.

Original Cost = (Number of pens × Original cost per pen) + (Number of pencils × Original cost per pencil)

Original Cost = $12p + 15c$

Substitute the values $p=12$ and $c=2$:

Original Cost = $12(12) + 15(2)$

Original Cost = $144 + 30$

Original Cost = $174$

Final Answer Calculation

The original cost of 12 pens and 15 pencils is ₹174.