Solving Ratios and Proportions in Mathematical Problems

To effectively solve mathematical problems involving ratios and proportions, it's crucial to understand the relationships and operations that govern these concepts. In this article, we will explore a specific problem involving two given ratios and determine the value of a complex ratio that combines further operations. Understanding these concepts is vital for students and professionals alike, as they form the foundation of advanced mathematical studies and real-world applications.

Understanding the Problem

We are given the following ratios:

x : y 3 : 5 x : 2 5 : 7

The goal is to find the value of y - z : y * z. This problem requires a step-by-step approach to understand and manipulate the given ratios to reach the desired solution.

Step-by-Step Solution

Step 1: Express y and z in terms of x

From the first ratio x : y 3 : 5, we can express y in terms of x:

y frac{5}{3}x

From the second ratio x : 2 5 : 7, we can express x in terms of 2 and 7:

frac{x}{2} frac{5}{7} Rightarrow x frac{10}{7}

Step 2: Calculate y and z using the value of x

Using the value of x, we substitute it into the equation for y:

y frac{5}{3} cdot frac{10}{7} frac{50}{21}

In the second ratio, we solve for z by using the value of x:

x frac{10}{7} Rightarrow frac{x}{2} frac{5}{7} Rightarrow z 2 cdot frac{5}{7} frac{10}{7}

Step 3: Calculate y - z and y * z to find the ratio y - z : y * z

To find y - z, we calculate:

y - z frac{50}{21} - frac{10}{7} frac{50}{21} - frac{30}{21} frac{20}{21}

To find y * z, we calculate:

y * z frac{50}{21} * frac{10}{7} frac{500}{147}

Now, we can find the ratio y - z : y * z:

frac{y - z}{y * z} frac{frac{20}{21}}{frac{500}{147}} frac{20}{21} * frac{147}{500} frac{28}{50} frac{1}{4}

Alternative Approach

Another approach involves directly using the ratios given, but this involves the same steps:

The first ratio x : y 3 : 5 can be rewritten as 15 : 25, and the second ratio x : z 5 : 7 can be rewritten as 25 : 35. Thus, x : y : z 3 : 5 : 7. To find y - z : y * z, we calculate:

y - z : y * z (5 - 7) : 5 * 7 -2 : 12 -1 : 6

Conclusion

By carefully manipulating the given ratios and performing the required operations, we can find the value of complex ratios that involve multiple variables. The step-by-step approach ensures accuracy and helps in understanding the underlying mathematical principles.