Solving the Ratio of Two Numbers Problem Efficiently

## Introduction to the Ratio of Two Numbers Problem

The problem of finding two numbers given their ratio and sum is a classic example used to teach students about ratio and algebraic equations. A common scenario involves two numbers with a given ratio, and the sum of these numbers. This article will explore an efficient approach to solve such problems, providing a clear algebraic solution. We will introduce the problem, break down the process, and offer a step-by-step solution. By the end, you will be confident in tackling similar ratio problems.

The Problem Statement

Given that the ratio of two numbers is 5:2, and their sum is 49, we need to find these two numbers. The goal is to understand and apply the steps required to find the values of these numbers.

Step-by-Step Solution

### Step 1: Assign Variables to the Numbers

Let's denote the two numbers as 5x and 2x, where x is a common factor.

- Explanation: The ratio 5:2 means one number is 5 times x, and the other is 2 times x.

Step 2: Express the Sum of the Numbers

The sum of the two numbers is given as 49. Therefore, we can write the equation:

5x 2x 49

Simplify the equation:

7x 49

Step 3: Solve for x

To find x, divide both sides of the equation by 7:

x 49 / 7

x 7

Step 4: Calculate the Two Numbers

Now that we have the value of x, we can find the two numbers:

First number (5x): 5 * 7 35

Second number (2x): 2 * 7 14

To summarize, the two numbers are 35 and 14.

Conclusion

In conclusion, using the ratio and the sum, we efficiently determined the two numbers. This method is applicable to a wide range of similar problems, where you are given a ratio and the sum to find the individual numbers. By breaking down the problem into manageable steps, we can easily solve for the unknowns, ensuring a clear and thorough understanding.

### Summary of Key Steps

Assign variables to the numbers (5x and 2x). Express the sum of the numbers (7x 49). Solve for the common factor (x 49 / 7). Calculate the two numbers (5x and 2x).

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