Solving for the Sum of Two Numbers Given Their Proportional Relationship and Difference

In mathematics, we often need to solve problems involving the proportional relationship between two numbers and their difference. This type of problem can be approached in various ways, from basic algebraic manipulation to more complex mathematical models. Below, we will solve a specific example involving a ratio of 10:7 and a difference of 105 between the two numbers.

Algebraic Approach

The problem statement is as follows: The ratio of two numbers is 10:7 and their difference is 105. What is the sum of these numbers?

Step 1: Setting Up the Equations

We can set up a proportion to represent the relationship between the two numbers. Let's denote the two numbers as A and B with A being the larger number.

The ratio of the two numbers is given as 10:7, which can be written as:

A:B 10:7

Alternatively, we can represent this relationship using variables:

Let A 10k and B 7k, where k is a constant.

Step 2: Using the Given Difference

The difference between the two numbers is given as 105:

A - B 105

Substituting the values of A and B from the ratio, we get:

10k - 7k 105

Now, we can solve for k by simplifying the equation:

Step 3: Finding the Value of (k)

10k - 7k 105

3k 105

k 105 / 3

k 35

Step 4: Calculating the Values of A and B

Now that we have the value of k, we can find the actual values of A and B by substituting k 35 back into the proportional relationship:

A 10k 10 * 35 350

B 7k 7 * 35 245

Step 5: Calculating the Sum of the Numbers

The sum of the two numbers is:

A B 350 245 595

Conclusion

By solving the problem using the algebraic representation and the given difference, we have determined that the sum of the two numbers with a ratio of 10:7 and a difference of 105 is 595.

Alternative Methods

There are also alternative methods to solve this problem, such as the following:

Using the direct proportion formula to set up the equations and solve for the constants.

Solving the problem using a more geometric or visual approach, such as drawing a diagram.

Key Points

1. Understanding the given ratio and how to represent it algebraically.

2. Utilizing the difference to find the value of the constant in the proportional relationship.

3. Calculating the individual values of the numbers from the proportional constants.

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Ratio of numbers - The relationship between two quantities expressed as a comparison of their values.

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Difference between numbers - The numerical difference between two values, used to find the constants in proportional relationships.

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