Solving Mathematical Problems Involving Ratios and Differences

Mathematics, especially when dealing with ratios and differences, can sometimes be challenging. In this article, we will explore a specific problem and break down the solution step by step. This understanding is crucial for anyone looking to enhance their problem-solving skills or for those preparing for mathematics assessments.

Problem Statement

In the given problem, we are presented with two numbers whose ratio is 3:8 and whose difference is 115. The objective is to determine both numbers, with a specific focus on finding the larger number.

Solution

Step 1: Define the Problem Mathematically

Let us denote the two numbers by a and b. The given ratio a/b 3/8 can be represented as:

a 3b/8

Additionally, the difference between the two numbers is given as:

a - b 115

Step 2: Substitute and Solve the Equations

Substituting the expression for a into the difference equation:

3b/8 - b 115

Multiplying the entire equation by 8 to clear the denominator:

3b - 8b 920

Combining like terms:

-5b 920

Dividing both sides by -5:

b -920/5 -184

Step 3: Find the Value of the Larger Number

Substituting b -184 into the difference equation:

a - (-184) 115

Simplifying:

a 184 115

Therefore:

a -184 115 -69

The larger number is thus -184.

Conclusion

In this mathematical problem, we successfully illustrated the steps to determine the numbers given a ratio and a difference. While the solution involves negative numbers in this specific case, understanding the problem-solving methodology is more critical. This type of problem-solving is not only applicable to the given scenario but also enhances problem-solving skills in various mathematical contexts.

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