Solving Pens and Cost Proportion Problems

This problem involves understanding the relationship between the number of pens and their cost. We are given the cost of a certain number of pens and asked to find how many pens can be bought for a different amount of money.

Understanding the Problem

We are told that 192 pens cost Rs. 10. We need to find out how many pens can be purchased with Rs. 5.

We can assume that the cost of the pens is directly proportional to the number of pens. This means if the cost decreases, the number of pens that can be bought for that cost will also decrease by the same proportion, assuming the price per pen remains constant.

Step-by-Step Solution

Here's how we can solve this pens cost calculation problem:

1. Identify the initial relationship: 192 pens cost Rs. 10.
2. Identify the new cost: Rs. 5.
3. Compare the new cost to the initial cost. The new cost (Rs. 5) is exactly half of the initial cost (Rs. 10).
4. Since the number of pens is directly proportional to the cost, if the cost is halved, the number of pens that can be bought must also be halved.
5. Calculate half of the initial number of pens: \( \frac{192}{2} \)
6. \( \frac{192}{2} = 96 \)

So, for Rs. 5, you can buy 96 pens.

Verification

Let's verify this using the unit cost method. Though not necessary for this simple problem, it confirms the proportionality.

• Cost of 192 pens = Rs. 10
• Cost of 1 pen = \( \frac{\text{Rs. } 10}{192} \)
• Number of pens for Rs. 5 = \( \frac{\text{Total Cost}}{\text{Cost per pen}} \)
• Number of pens = \( \frac{5}{\frac{10}{192}} = 5 \times \frac{192}{10} = \frac{5 \times 192}{10} = \frac{960}{10} = 96 \)

Both methods give the same result, 96 pens.

Revision Table: Pens Cost Problem

The cost is halved (from Rs. 10 to Rs. 5), so the quantity is also halved (from 192 to 96).

Additional Information: Direct Proportion

In a direct proportion, two quantities increase or decrease at the same rate. If quantity A is directly proportional to quantity B, we can write this as \( A \propto B \). This means that the ratio \( \frac{A}{B} \) is constant, or \( A = kB \), where \( k \) is the constant of proportionality.

In this pens cost problem, the number of pens (\( N \)) is directly proportional to the cost (\( C \)). So, \( N \propto C \), or \( \frac{N}{C} = k \).

We have two scenarios:

• Scenario 1: \( N_1 = 192 \), \( C_1 = 10 \). So, \( \frac{192}{10} = k \).
• Scenario 2: \( N_2 = ? \), \( C_2 = 5 \). So, \( \frac{N_2}{5} = k \).

Since \( k \) is the same in both cases:

\( \frac{N_2}{5} = \frac{192}{10} \)

To find \( N_2 \), we can cross-multiply or simply multiply both sides by 5:

\( N_2 = \frac{192}{10} \times 5 \)

\( N_2 = \frac{192 \times 5}{10} \)

\( N_2 = \frac{960}{10} \)

\( N_2 = 96 \)

This confirms the result using the direct proportion formula.