Mathematical Operations to Convert Ratios: Subtracting a Constant to Change 3:4 to 2:3

Let us delve into a problem that involves manipulating ratios to transform one given ratio into another. Specifically, we need to determine the value to be subtracted from both terms in the ratio 3:4 so that the new ratio becomes 2:3. This article will guide you through the mathematical steps involved in finding this value.

Understanding the Problem

The given ratio is 3:4, and we want to transform it into the ratio 2:3 by subtracting the same whole number from each term.

Setting Up the Equations

To solve this, we represent the numbers in the ratio as 3x and 4x, where x is a positive constant. We use the property of ratios and set up the following equation:

(frac{3x - k}{4x - k} frac{2}{3})

Here, k is the constant to be subtracted. Cross-multiplying to eliminate the fractions gives us:

(3(3x - k) 2(4x - k))

Expanding and rearranging terms, we get:

(9x - 3k 8x - 2k Rightarrow 9x - 8x -2k 3k Rightarrow x k Rightarrow k x)

This means the value to be subtracted is equal to x, the original constant.

Practical Examples and Solutions

Example 1: Let's solve another way. Consider the fractions 3/4 and 2/3. We need to find the difference that when subtracted from 3/4, it results in 2/3.

(3/4 - 2/3 9/12 - 8/12 1/12)

This means 1/12 needs to be subtracted from 3/4 to get 2/3.

Verification:

Verifying the result:

(3/4 - 1/12 (9-1)/12 8/12 2/3)

This confirms our solution.

Generalization

Let n be the number to be subtracted. Then:

(frac{3-n}{4-n} frac{2}{3})

Cross-multiplying gives us:

(3(3-n) 2(4-n) Rightarrow 9-3n 8-2n Rightarrow 9-8 3n-2n Rightarrow n1)

This confirms that the value to be subtracted is 1.

Conclusion

In summary, to change the ratio 3:4 to 2:3, the same constant value (equal to x in our initial representation) must be subtracted from both terms. This constant is the simplest form of the original ratio. In numerical form, the value to be subtracted is 1, as shown in the specific examples and more generalized algebraic solutions.

Key Takeaways

To change a given ratio, identify the constant value to be subtracted from both terms. Use the cross-multiplication and algebraic manipulation method to find this value. The value of x (or n) can be determined through simple algebra and verification steps.

Further Reading and Resources

For more on ratio operations and advanced algebraic techniques, refer to the following resources:

Math is Fun: Ratios Brightstorm: Ratios and Proportions

Understanding these mathematical operations is crucial in various fields including mathematics, engineering, and economics.