Solving the Cost of Pens and Pencils: An Algebraic Puzzle

In this article, we explore a classic algebraic puzzle that involves determining the cost of pens and pencils. The problem is presented with an interesting twist, making it a perfect exercise for students and enthusiasts of mathematics.

The Puzzle

Given the following conditions:

3 pens cost the same as 5 pencils. The cost of 2 pens and 4 pencils is Rs. 22.

We are asked to find the price of 10 pens.

Step-by-Step Solution

First, let's denote the cost of one pen as P and the cost of one pencil as C.

Step 1: Express the Relationships

From the problem, we have two key pieces of information:

1. The cost of 3 pens is equal to the cost of 5 pencils:

3P 5C

2. The cost of 2 pens and 4 pencils is Rs. 22:

2P 4C 22

Step 2: Solve for C

From the first equation, we can solve for C:

C (3/5)P

Step 3: Substitute and Simplify

Now, substitute C from the first equation into the second equation:

2P 4((3/5)P) 22

2P (12/5)P 22

To combine the terms, convert 2P to a fraction with a denominator of 5:

(10/5)P (12/5)P 22

This simplifies to:

(22/5)P 22

Step 4: Solve for P

Multiply both sides by 5:

22P 110

Now, divide by 22:

P 5

Step 5: Find the Cost of One Pencil

Substitute P 5 back into the first equation:

3(5) 5C

This simplifies to:

15 5C rarr; C 3

Step 6: Calculate the Price of 10 Pens

Since the cost of one pen is Rs. 5, the cost of 10 pens is:

10P 10(5) Rs. 50

Verification

Let's verify the solution:

3 pens 5 pencils; 3(5) 15 5 pencils (from above) 3(3) 9 2 pens 4 pencils 2(5) 4(3) 10 12 22 10 pens 10(5) 50

The solution is consistent with the given conditions, confirming that the cost of 10 pens is Rs. 50.

Additional Insights

This algebraic puzzle is a great way to practice problem-solving skills. It demonstrates the importance of setting up equations and using substitution to simplify complex problems. By understanding and applying these algebraic techniques, you can solve a wide range of real-world problems that involve cost calculations and relationships between different quantities.