Understanding the Relationship Between A, B, and C

When the ratio A:B:C is given as 2:3:4, it means that the values of A, B, and C maintain these specific proportions. Let's dive into the detailed steps to find the ratios A/B, B/C, and A/C, and see how they interrelate.

Step 1: Expressing Ratios as a Function of a Common Variable

To simplify the process, let's assign a common variable 'x' such that: A 2x B 3x C 4x

Step 2: Calculating A/B, B/C, and A/C

We can now calculate the individual ratios more easily. A/B 2x / 3x 2/3 B/C 3x / 4x 3/4 A/C 2x / 4x 1/2 Now, let's see these ratios together: A/B : B/C : A/C 2/3 : 3/4 : 1/2.

Step 3: Finding the Least Common Multiple (LCM)

To convert these fractions into a common format, we need to find the LCM of the denominators (3, 4, and 2). The LCM of 3, 4, and 2 is 12. Let's express each fraction with the denominator 12. A/B 2/3 8/12 B/C 3/4 9/12 A/C 1/2 6/12 Thus, A/B : B/C : A/C 8/12 : 9/12 : 6/12, which simplifies to 8 : 9 : 6.

Conclusion

Therefore, the ratios A/B : B/C : A/C, when A:B:C is 2:3:4, are 8:9:6. This method of expressing the ratios as a function of a common variable and using the least common multiple (LCM) is a fundamental approach in solving such problems. Understanding and practicing these techniques can significantly enhance your problem-solving skills in algebra and geometry.

Ensure you practice similar problems to reinforce your understanding of how to manipulate ratios and proportions effectively. This knowledge is crucial for various applications in mathematics and related fields.