Solving a Mathematical Puzzle: Half a Number Multiplied by a Constant Equals Its Square

Mathematics often presents fascinating puzzles that challenge our understanding and problem-solving skills. One such intriguing puzzle involves finding a number such that half of the number, when multiplied by a certain constant, equals the square of the number. This question not only tests our algebraic skills but also helps in understanding the properties of square numbers. Let's explore this mathematical puzzle in detail and see how we can solve it using a step-by-step approach.

The Problem Statement

Let us denote the unknown number as x. According to the problem, half of this number multiplied by 10 equals its square. In mathematical terms, this can be expressed as:

(frac{x}{2} times 10 x^2)

Step-by-Step Solution

First, we simplify the left side of the equation:

(frac{x}{2} times 10 5x)

This gives us the equation:

(5x x^2)

To solve this equation, we rearrange it to bring all terms to one side:

(x^2 - 5x 0)

We then factor out the common term, x:

(x(x - 5) 0)

This results in two solutions:

(x 0) or (x 5)

While x 0 is a valid numerical solution, it often does not make practical sense in the context of many real-world problems. Therefore, the number that satisfies the given condition is:

x 5

Alternative Approaches

Let's examine another approach using variables:

Let y (frac{x}{2}). According to the problem statement, y times 10 should equal x^2. Substituting y back into the equation, we get:

10y x^2

Recall that y (frac{x}{2}), so:

10 (frac{x}{2}) x^2

Which simplifies to:

5x x^2

Rearranging gives:

x^2 - 5x 0

Factoring out x again:

x(x - 5) 0

Hence, the solutions are the same:

x 0 or x 5

Given the context, x 5 is the meaningful solution.

Conclusion

This mathematical puzzle demonstrates the power of algebraic manipulation and the importance of recognizing the properties of square numbers. The solution, x 5, satisfies the original problem statement where half of 5 (which is 2.5) multiplied by 10 gives 25, which is the square of 5.

Further Exploration

Understanding such puzzles and equations can enhance our problem-solving skills and deepen our comprehension of mathematical concepts. Consider exploring more such problems to sharpen your skills and appreciate the beauty of mathematics.