Solving Proportional Relationships: When 15% of x is Three Times 10% of y

In various mathematical and real-world scenarios, understanding and solving proportional relationships is essential. This article delves into the problem of finding the ratio between two variables, x and y, when 15% of x is three times the 10% of y. The process involves algebraic manipulation, understanding percentage calculations, and interpreting ratios. Let's explore this concept step-by-step.

Understanding the Problem

The given problem states that 15% of x is equal to three times 10% of y. Mathematically, this can be expressed as:

0.15x 3 × 0.10y

Step-by-Step Solution

Step 1: Simplify the Equation

We start by simplifying the equation to make it easier to manipulate:

0.15x 0.30y

Step 2: Isolate the Variables

To find the ratio of x to y, we need to isolate the variables on both sides of the equation. We can do this by dividing both sides by y and 0.15:

Divide both sides by 0.15:

(frac{0.15x}{0.15} frac{0.30y}{0.15})

(x 2y)

Step 3: Express the Ratio

Now, to express this as a ratio, we have:

x : y 2 : 1

Explanation of the Solution

The final answer, x : y 2 : 1, indicates that for every unit of y, there are two units of x. This relationship can be confirmed by the following steps:

1. Start with the given equation: 0.15x 0.30y

2. Simplify to: x 2y

3. Express as a ratio: x : y 2 : 1

Additional Examples and Applications

This concept can be extended to a variety of real-world applications such as financial analysis, scientific measurements, and everyday problem-solving. For instance, if in a business scenario, 15% of the total sales (x) from one product equals three times 10% of the total sales (y) from another product, the ratio of the sales from the first product to the second would be 2:1.

Key Takeaways

Proportional Relationships: Understanding the relationship between two variables is crucial in many areas. Percentage Calculations: Knowing how to manipulate percentages helps in solving a wide range of problems. Ratio and Proportion: Expressing the relationship between variables in a ratio simplifies problem-solving.

By practicing and applying these concepts, you can enhance your problem-solving skills in mathematics and beyond.