Solving the Mystery of the Summation: A Dive into Mathematical Problem Solving

Mathematics is a powerful tool that helps us unravel complex mysteries through logical reasoning and mathematical operations. In this article, we will explore a seemingly simple yet intriguing problem: finding two numbers whose sum equals a specific value, while one number is four times the other. Let's delve into how we can solve this problem with a detailed, step-by-step approach.

Introduction to the Problem

The problem at hand is: 'The sum of two numbers is 54. One of the numbers is 4 times more than the other. What are the numbers?' This question requires us to use our knowledge of linear equations and numerical reasoning to find the answer.

The Equation Approach

We can solve this problem by setting up an equation based on the given conditions. Let's denote the smaller number as x and the larger number as 4x. According to the problem, the sum of these two numbers is 54.

Step 1: Write the equation based on the sum of the two numbers.

[ x 4x 54 ]

Step 2: Combine like terms.

[ 5x 54 ]

Step 3: Solve for x by dividing both sides of the equation by 5.

[ x frac{54}{5} ]

Step 4: Calculate the value of x.

[ x 10.8 ]

Step 5: Find the value of the larger number by multiplying x by 4.

[ 4x 4 times 10.8 43.2 ]

Therefore, the two numbers are 10.8 and 43.2. Let's verify our solution.

Verification

To verify, we check if the sum of 10.8 and 43.2 equals 54.

[ 10.8 43.2 54 ]

This confirms that our solution is correct.

Additional Approaches

Alternatively, we can solve this problem using a different method. Let's denote the smaller number as a and the larger number as b. According to the problem, b is four times a and their sum is 54.

Step 1: Write the equation based on the sum of the two numbers.

[ a 4a 54 ]

Step 2: Combine like terms.

[ 5a 54 ]

Step 3: Solve for a by dividing both sides of the equation by 5.

[ a frac{54}{5} ]

Step 4: Calculate the value of a.

[ a 10.8 ]

Step 5: Find the value of the larger number by multiplying a by 4.

[ b 4 times 10.8 43.2 ]

A similar verification process confirms the accuracy of the solution.

Conclusion

Solving this problem using mathematical reasoning and linear equations helps us find the correct values for the two numbers. This approach can be applied to various similar problems, enhancing our problem-solving skills. Whether you're a primary school student or an advanced mathematician, understanding these concepts is fundamental to tackling more challenging mathematical problems.

Further Reading

Linear Equations in One Variable Numerical Reasoning in Everyday Life Problem Solving Strategies for Mathematics

Explore these related articles to deepen your understanding and practice solving more complex problems.