Solving a Ratio Problem: Exploring the Concept and Method

In mathematics, particularly in algebra, solving problems involving ratios is a common task. Ratios provide a way to compare the quantities of different items or numbers. In this article, we will explore a specific problem where two numbers are in a given ratio, and the problem statement involves changing these numbers in a certain way to find new values and ratios. This common type of problem helps in understanding algebraic manipulation and the concept of variables.

Problem

Given two numbers in the ratio of 4:3. If 4 is added to the first number and 5 is subtracted from the second number, the new ratio becomes 4:1. The challenge is to determine the original numbers.

Original Problem: Two numbers are in the ratio of 4:3. If 4 is added to the first and 5 is subtracted from the second, then the new ratio becomes 4:1. What are the numbers?

Solution

To solve this problem, we can use algebraic methods. Let's define the two numbers as 4x and 3x.

Step-by-Step Solution

Define Variables: Let the first number be 4x and the second number be 3x. Set Up the Equation: According to the problem, if 4 is added to the first number and 5 is subtracted from the second number, the new ratio becomes 4:1. This can be written as:

4/1 (4x 4) / (3x - 5)

Simplify the Equation: Cross-multiply to solve for x.

4(3x - 5) 1(4x 4)

Solve for x:

12x - 20 4x 4

12x - 4x 24 20

8x 24

x 24 / 8 3

Determine the Original Numbers: Using the value of x, we can find the original numbers.

4x 4 × 3 12

3x 3 × 3 9

Conclusion

The original numbers are 12 and 9. This solution demonstrates the use of algebraic techniques to solve ratio problems. Such problems are not only useful in mathematics but also in real-life scenarios where proportions need to be maintained or adjusted.

Additional Resources

For further learning, you can explore similar problems and solutions by visiting websites or educational platforms that specialize in algebra and mathematical problem solving.

Keywords: ratio problem, algebraic solution, mathematical problem solving