Solving Mathematical Puzzles: A Step-by-Step Guide

Mathematical puzzles can be both fun and challenging. One such puzzle is the 'pencil problem,' which involves determining how many pencils a boy purchased given certain conditions. In this article, we will explore the solution in several ways, from basic algebra to more advanced methods. We will also go through the solution step-by-step, providing detailed explanations and insights.

The Pencil Problem: A Basic Approach

The problem can be stated as follows: A boy bought some pencils for 60. If he had paid 1 less for each pencil, he could have bought 5 more pencils. The question is to determine how many pencils he originally bought.

Let's begin by solving the problem using simple algebra. We will define our variables and set up the equations to find the solution.

Algebraic Solution

Define the variables:

n the number of pencils the boy bought. p the original price per pencil (which is 1).

Given information:

The total cost is 60. If the price of each pencil was 1 less, the boy could have bought 5 more pencils.

Set up the equations:

Original total cost: x * p 60 New total cost: (x 5) * (p - 1) 60

Since the total cost is the same in both cases, set the two equations equal to each other:

x * p (x 5) * (p - 1)

Solve the equation:

x 15

Therefore, the boy bought 15 pencils.

Alternative Solution Methods

There are several other methods to solve the pencil problem. Here, we will explore three additional approaches: integer factorization, trial and error, and a more advanced algebraic method.

Integer Factorization

Define the variables:

n1 the original number of pencils. n2 the number of pencils he would have bought if the price was reduced by 1. p1 the original price per pencil (which is 1). p2 the new price per pencil (which is 1 - 1 0).

Given total cost: n1 * p1 n2 * p2 60

Set up the ratio:

n2 / n1 p1 / p2

Manipulate the ratio to include the additional pencils:

(n2 - n1) / n1 (p1 - p2) / p2

Simplify the equation:

(n2 - n1) / n1 * n2 (p1 - p2) / p2 * n2

Substitute the known values:

5 / n1 * n2 1 / 60

Factorize and find integer solutions:

n1 * n2 300 300 15 * 20

The integer pair that satisfies the equation is:

n1 15 n2 20

Trial and Error

Start with a reasonable guess and check if it satisfies the given conditions:

n 15 p1 4 (since 60 / 15 4)

Check the second condition:

(60 / (4 - 1)) - (60 / 4) 20 - 15 5

Verify the solution:

n1 15

Advanced Algebraic Method

Define the variables:

x the original price per pencil (which is 1).

Given the conditions:

The boy paid 60 for x pencils. If the price was reduced by 1, he would buy 5 more pencils.

Set up the equations:

60 / x - 1 - 60 / x 5

Factor and solve the quadratic equation:

6 - 60 5x^2 - 5x 5x^2 - 5x - 60 x^2 - x - 12 5x^2 - 5x - 60 0

Solve for x:

x 4

Verify the solution:

60 / 4 15

Therefore, the boy originally bought 15 pencils.

Conclusion

Through various methods, we have successfully solved the pencil problem and confirmed that the boy originally bought 15 pencils. This problem demonstrates the power of algebraic and logical reasoning in solving mathematical puzzles. Whether you are a student or an enthusiast, these methods can help you tackle similar problems with confidence.