Solving for Numbers in a Given Ratio and Difference

In mathematics, problems involving ratios and differences are common and can often be solved with a bit of algebraic manipulation. This article focuses on a specific problem: two numbers are in the ratio 7:9 and their difference is 10. The objective is to find their sum and specifically, the greater number.

Problem Statement:

Consider two numbers that are in the ratio of 7:9 and their difference is 10. Find the greater number and their sum.

Solution Methods:

There are several methods to solve this problem. Below, we will illustrate a few of these methods.

Method 1: Using Algebraic Equations

Let's denote the smaller number as x times 7 and the larger number as x times 9. According to the given conditions:

Let the first number be 7x Let the second number be 9x The difference between the numbers is 10, so we have 9x - 7x 10

Let's solve the equation step by step:

9x - 7x 10 2x 10 x 5 The greater number is 9x 9 * 5 45

Method 2: Using a Constant Multiplier

Another approach is to assume a constant multiplier x, and use the given ratio and difference:

Let the first number be 7x Let the second number be 9x So, 9x - 7x 10 2x 10 x 5 The greater number is 9x 9 * 5 45

Method 3: Direct Steps

For a more straightforward method, check out the following steps:

Let the first number be 7x Let the second number be 9x The difference is 9x - 7x 2x 10 x 10 / 2 5 The greater number is 9x 9 * 5 45

Summary:

The greater number is 45. Adding this to the smaller number (35), their sum is 80.

Conclusion:

The problem can be solved using algebraic manipulation or a simple method by setting up the ratio and solving for the variables. The greater number is 45, and their sum is 80.