Solving for Two Numbers Given Their Sum and Multiplicative Relationship

Mathematics often involves solving real-world problems through equations. One such problem involves finding two numbers where their sum is given, and one number is a multiple of the other. Let's explore how to find these numbers using a specific example.

Problem Statement

The sum of two numbers is 100. The larger number is 3 times the smaller number. What are the two numbers?

Solution Steps

To solve this problem, we can use basic algebraic equations. Let's define the smaller number as x. According to the problem, the larger number is 3 times the smaller number, so we denote it as 3x.

Step 1: Set Up the Equation

Given that the sum of the two numbers is 100, we can write the following equation:

x 3x 100

Step 2: Combine Like Terms

On the left side of the equation, we can combine the like terms:

4x 100

Step 3: Solve for x

To find the value of x, we divide both sides of the equation by 4:

x frac{100}{4} 25

So, the smaller number is 25.

Step 4: Find the Larger Number

The larger number is 3 times the smaller number, which gives us:

3x 3 times 25 75

Conclusion

Therefore, the two numbers are:

Verification

To verify the solution, we check if the sum of these two numbers is indeed 100:

25 75 100 (Verification passes)

Additionally, we can check if the larger number is 3 times the smaller number:

75 3 times 25 (Verification passes)

End of Solution

Following these steps, we find that the two numbers are 25 and 75. This solution demonstrates a straightforward approach to solving problems where the sum of two numbers and their multiplicative relationship is given.