Solving for the Sum of Numbers Given Their Ratio and the Sum of Squares

In mathematics, solving problems related to ratios and the sum of squares can be quite intriguing. This article explores a specific problem where we need to determine the sum of three numbers given their ratio and the sum of their squares. Let's dive into the solution step by step.

Problem Statement

The problem states that the ratio of three numbers is 3:4:5, and the sum of the squares of these numbers is 1250. We need to find the sum of the numbers.

Step-by-Step Solution

Step 1: Express the numbers in terms of a common factor.

Let the three numbers be 3x, 4x, and 5x, where x is a common factor. This is based on the given ratio 3:4:5.

Step 2: Write the equation for the sum of the squares.

The sum of the squares of the numbers is given as 1250. Therefore, we can write the equation as follows:

[3x^2 4x^2 5x^2 1250]

Simplifying the left side of the equation:

[9x^2 16x^2 25x^2 1250]

[5^2 1250]

Step 3: Solve for x2.

Divide both sides of the equation by 50 to find x2:

[x^2 frac{1250}{50} 25]

Step 4: Find the value of x.

Take the square root of both sides to find the value of x:

[x sqrt{25} 5]

Step 5: Determine the actual numbers.

Now that we have the value of x, we can determine the actual numbers:

First number: [3x 3 times 5 15]

Second number: [4x 4 times 5 20]

Third number: [5x 5 times 5 25]

Step 6: Calculate the sum of the numbers.

[15 20 25 60]

Therefore, the sum of the numbers is 60.

Additional Considerations

It is important to note that while we found x to be 5, the common factor x can also be -5. However, since we are typically concerned with positive numbers in most mathematical contexts, we consider the positive value of x. If we had used -5, the numbers would have been -15, -20, and -25, and their sum would be -60.

Conclusion

This problem illustrates the process of solving for unknowns in a ratio using the sum of squares. Understanding such problems enhances analytical and problem-solving skills in mathematics. Whether the sum of the numbers is 60 or -60, the methodology remains the same.

Keywords

ratio of numbers, sum of squares, problem solving in mathematics