Solving the Cost of Pens and Pencils Problem

This problem asks us to find the total cost of 8 pens and 3 pencils, given the total cost of 32 pens and 12 pencils.

Understanding the Given Information

We are given the following information:

• The cost of 32 pens and 12 pencils is Rs. 790.

We need to find the cost of 8 pens and 3 pencils.

Finding the Relationship

Let the cost of one pen be \( P \) and the cost of one pencil be \( L \).

The given information can be written as an equation:

\( 32P + 12L = 790 \)

We are asked to find the value of \( 8P + 3L \).

Let's look at the coefficients of \( P \) and \( L \) in the expression we need to find (8 and 3) and compare them to the coefficients in the given equation (32 and 12). We can observe a relationship:

• The number of pens changes from 32 to 8. \( 32 \div 4 = 8 \).
• The number of pencils changes from 12 to 3. \( 12 \div 4 = 3 \).

This indicates that the combination of items we are interested in (8 pens and 3 pencils) is exactly one-fourth (\( \frac{1}{4} \)) of the combination given (32 pens and 12 pencils).

Calculating the Total Cost

Since the number of both pens and pencils is scaled down by the same factor (divided by 4), the total cost will also be scaled down by the same factor.

So, the cost of 8 pens and 3 pencils will be one-fourth of the cost of 32 pens and 12 pencils.

Cost of 8 pens and 3 pencils \( = \frac{\text{Cost of 32 pens and 12 pencils}}{4} \)

Cost of 8 pens and 3 pencils \( = \frac{790}{4} \)

Now, let's perform the division:

\( 790 \div 4 \)

\( 790 \div 4 = (700 + 90) \div 4 \)

\( = 700 \div 4 + 90 \div 4 \)

\( = 175 + 22.5 \)

\( = 197.5 \)

Alternatively, using long division:

The result of the division is 197.5.

So, the total cost of 8 pens and 3 pencils together is Rs. 197.5.

Summary of the Solution

Given: Cost of 32 pens + 12 pencils = Rs. 790

To find: Cost of 8 pens + 3 pencils

Observe that \( 8 = 32 \div 4 \) and \( 3 = 12 \div 4 \).

Therefore, Cost of 8 pens + 3 pencils \( = \frac{\text{Cost of 32 pens + 12 pencils}}{4} = \frac{790}{4} = 197.5 \).

The total cost of 8 pens and 3 pencils together is Rs. 197.5.

Revision Table: Key Concepts

Additional Information: Linear Equations

This problem could also be viewed in terms of linear equations, although the direct proportionality makes it simpler.

Let \(P\) be the cost of one pen and \(L\) be the cost of one pencil.

The given information translates to a linear equation in two variables:

\( 32P + 12L = 790 \quad \text{(Equation 1)} \)

We need to find the value of the expression \( 8P + 3L \).

Notice that if we divide Equation 1 by 4 on both sides, we get:

\( \frac{32P + 12L}{4} = \frac{790}{4} \)

\( \frac{32P}{4} + \frac{12L}{4} = \frac{790}{4} \)

\( 8P + 3L = 197.5 \)

This confirms that the value of the expression \( 8P + 3L \) is indeed 197.5.

In general, if you have an expression \( aX + bY = C \) and you need to find the value of \( k aX + k bY \), it will be \( k C \). This applies here with \( X=P \), \( Y=L \), \( a=32 \), \( b=12 \), \( C=790 \), and \( k=1/4 \).