Solving Algebraic Puzzles: A Case of Pencils

Algebraic puzzles are fun and challenging ways to explore the beauty and utility of mathematics. Today, let's dive into a simple yet insightful puzzle that revolves around the distribution of pencils between two individuals, Ali and Aslam. How many pencils does each person have initially, and how does the problem evolve to meet a specific condition? Here’s the problem in detail:

Problem Statement

Ali has two times the number of pencils as Aslam has. If 10 more pencils are added to the total pencils they already had, then it would make a total of 100 pencils. How many pencils did Ali have initially?

A Simple Algebraic Analysis

Let's denote the number of pencils Aslam has as p. Consequently, Ali has 2p pencils. Initially, the total number of pencils is p 2p 3p. To meet the condition of the problem, we add 10 more pencils to the total, thus:

[2p p 10 100]

Simplifying the equation, we get:

[3p 10 100]

Subtracting 10 from both sides:

[3p 90]

Dividing by 3:

[p 30]

So, Aslam initially had 30 pencils. Since Ali had twice as many as Aslam, we can calculate Ali's pencils as follows:

[2p 2 times 30 60]

Therefore, Ali initially had 60 pencils.

A Verification and a Simplified Solution

Let's verify the solution by adding 10 to the total number of pencils:

[30 60 10 100]

This confirms our solution. Here is another confirmation using a different but equivalent method:

[2 times 100 - 10 60]

Let's use a more algebraic approach to summarize:

[x text{number of pencils Aslam has}] [2x text{number of pencils Ali has}] [2x x 10 100] [3x 10 100] [3x 90] [x 30] [2 times 30 60]

Thus, Aslam has 30 pencils, and Ali has 60 pencils initially.

Conclusion

Mathematics is not just about complex equations. Simple algebraic puzzles like this one can be fun and engaging, teaching us valuable problem-solving skills. Whether you're a student or a casual observer, these puzzles offer a delightful challenge and can deepen your understanding of how algebra works in real-life scenarios. If you're interested in solving more such puzzles or learning more math concepts, there are countless resources available online and in print.