Solving the Cost of Pens and Pencils Word Problem

This problem involves finding the original costs of pens and pencils given two different scenarios and then calculating the cost of a specific combination.

Step 1: Define Variables

Let the original cost of one pen be represented by '$p$' and the original cost of one pencil be represented by '$c$'.

Step 2: Formulate Equations from the Given Information

We are given two pieces of information that can be translated into linear equations:

• The cost of 7 pens and 17 pencils is ₹92. This gives us the first equation:

  $ 7p + 17c = 92 \quad (1) $

• If the cost of a pen decreases by ₹0.50 (new cost is $p - 0.50$) and the cost of a pencil increases by ₹8 (new cost is $c + 8$), then the cost of 8 pens and 14 pencils is ₹186. This gives us the second equation:

  $ 8(p - 0.50) + 14(c + 8) = 186 $

  Let's simplify this equation:

  $ 8p - 4 + 14c + 112 = 186 $

  $ 8p + 14c + 108 = 186 $

  $ 8p + 14c = 186 - 108 $

  $ 8p + 14c = 78 $

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

  $ 4p + 7c = 39 \quad (2) $

Step 3: Solve the System of Linear Equations

Now we have a system of two linear equations with two variables:

1. $7p + 17c = 92$
2. $4p + 7c = 39$

We can use the method of elimination or substitution. Let's use elimination.

Multiply equation (1) by 4 and equation (2) by 7 to make the coefficients of '$p$' the same:

Multiplying (1) by 4:

$ 4 \times (7p + 17c = 92) \implies 28p + 68c = 368 \quad (3) $

Multiplying (2) by 7:

$ 7 \times (4p + 7c = 39) \implies 28p + 49c = 273 \quad (4) $

Now, subtract equation (4) from equation (3):

$ (28p + 68c) - (28p + 49c) = 368 - 273 $

$ 19c = 95 $

Solve for '$c$':

$ c = \frac{95}{19} $

$ c = 5 $

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

Now substitute the value of '$c$' (which is 5) back into one of the original equations (let's use equation 2) to find '$p$':

$ 4p + 7c = 39 $

$ 4p + 7(5) = 39 $

$ 4p + 35 = 39 $

$ 4p = 39 - 35 $

$ 4p = 4 $

$ p = \frac{4}{4} $

$ p = 1 $

So, the original cost of a pen is ₹1.

Step 4: Calculate the Required Cost

The question asks for the original cost of 9 pens and 6 pencils.

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

Cost = $9p + 6c$

Substitute the values we found for '$p$' and '$c$':

Cost = $9(1) + 6(5)$

Cost = $9 + 30$

Cost = $39$

Conclusion

The original cost of 9 pens and 6 pencils is ₹39.