Discovering the Value of x in Ratios and Ratios After Modification
Understanding the relationship between two ratios and how they change upon the addition of a constant to their terms is a fascinating mathematical puzzle. In this article, we will explore how the value of the constant x can be found when a given ratio is transformed through the addition of x.
Understanding Ratios and Their Transformation
Consider the initial ratio of a:b which is given as 4:9. This ratio can also be represented as:
1. Initial Ratio Representation
a/b 4/9
2. Ratio After Modification
When a constant x is added to each term of the original ratio, the new ratio becomes 5:11. Hence, the new terms can be represented as:
a' / b' 5/11
Expressed in terms of the original ratio, this can be further broken down as:
(a x)/(b x) 5/11
As we can see, the difference between the terms remains constant. This is due to the fact that the difference b - a does not change when x is added to each term. Therefore, we can write:
b - (a x) b - a
This implies that the difference remains the same, allowing us to reduce the ratios to a common scale. Let’s do this step by step:
3. Scaling the Ratios
The initial ratio 4:9 can be scaled as follows:
4:9 24:54 (by multiplying each term by 6)
Similarly, the new ratio 5:11 can be scaled to match the same difference:
5:11 25:55
Matching the difference, we get 54 - 24 30 and 55 - 25 30, so:
a' 24 x and b' 54 x
Therefore, x can be calculated by the difference:
25 - 24 11 - 5
Since the difference is 6, we have:
4. Solving for x
The solution involves setting up an equation based on the original and new ratios:
9a - 4b 0 and 11(a x) - 5(b x) 0
Simplifying the second equation, we get:
11a 11x - 5b - 5x 0
Combining like terms, we have:
11a - 5b 6x 0
Substituting 9a 4b, we get:
11a - 5(9a/4) 6x 0
11a - 45a/4 6x 0
11a/4 - 45a/4 6x 0
-34a/4 6x 0
6x 34a/4
6x 17a/2
Solving for x in terms of a, we get:
x (17a/2) / 6 17a/12
However, by simplifying the original problem, we find:
6x 45 - 44
6x 1
x 1/6
Hence, the value of x is 1/6.
Conclusion
The process of finding x in such problems involves scaling the original and new ratios to a common difference, applying algebraic manipulations to derive the relationship, and solving for the constant. Understanding these steps is crucial for solving similar problems involving ratio adjustments.
Understanding the value of x not only enhances algebraic problem-solving skills but also provides insight into how ratios can be manipulated through constant additions. This knowledge is valuable in various mathematical and real-world applications where proportional relationships are involved.