Solving Proportional Relationships Through Algebra

Proportional relationships are fundamental in mathematics, often appearing in algebraic equations and ratio problems. In this article, we will explore a problem involving two numbers in a specific ratio and how to solve for those numbers using algebraic methods.

Understanding the Problem

Let's consider a scenario where two numbers are in the ratio of 2:5. If we add 4 to each number, the new ratio becomes 4:9. Our goal is to find the original numbers.

Step-by-Step Solution

Assumption: Let the two numbers be 2x and 5x where x is a common multiplier.

Using Algebraic Equations

According to the problem, if we add 4 to each of the numbers, the new ratio becomes 4:9. This can be expressed as:

[ frac{2x 4}{5x 4} frac{4}{9} ]

Now, we can cross-multiply to eliminate the fraction:

[ 9(2x 4) 4(5x 4) ]

Expanding both sides gives:

[ 18x 36 2 16 ]

Isolating x on one side:

[ 18x 36 - 2 16 ]

[ -2x -20 ]

[ x 10 ]

Substituting x back to find the two numbers:

[ 2x 2(10) 20 ]

[ 5x 5(10) 50 ]

The two numbers are hence 20 and 50.

Verification

Addition of 4 to each number results in:

First number: 20 4 24

Second number: 50 4 54

Checking the ratio:

[ frac{24}{54} frac{4}{9} ]

This confirms our solution. The numbers are indeed 20 and 50.

Alternative Solution Methods

Using a Constant of Proportionality:

Let k denote the constant of proportionality. The problem can be expressed as:

[ frac{2k 4}{5k 4} frac{4}{9} ]

Cross-multiplying to eliminate the fraction:

[ 9(2k 4) 4(5k 4) ]

Expanding both sides:

[ 18k 36 20k 16 ]

Isolating k on one side:

[ 18k 36 - 20k 16 ]

[ -2k -20 ]

[ k 10 ]

Using the value of k to find the numbers:

[ 2k 2(10) 20 ]

[ 5k 5(10) 50 ]

Hence, the numbers are 20 and 50.

Conclusion

This problem demonstrates the importance of algebraic methods in solving proportional relationships and ratio problems. Understanding how to apply algebraic equations and cross-multiplication can significantly simplify and solve these types of problems.