Solving Ratio and Difference Problems in Mathematics

Making your content appealing and SEO-friendly is crucial in today's digital landscape. The given problem involves two numbers in a specific ratio and a given difference, which can be solved using algebraic equations. Let's explore this problem and its solution in detail.

Understanding the Problem

Two numbers have a ratio of 12:5, and their difference is 49. We need to find these two numbers. This involves setting up an equation based on the given information and solving it step-by-step.

Setting Up the Equation

Let's denote the two numbers as 12x and 5x, where x is a common multiplier. Since the difference between the two numbers is 49, we can set up the following equation:

Step-by-Step Solution

Start with the given ratio and difference:

12x - 5x 49

Simplify the equation:

7x 49

Divide both sides by 7:

x 7

Find the two numbers using the value of x:

The first number is:

12x 12 * 7 84

The second number is:

5x 5 * 7 35

The two numbers are 84 and 35.

Visual Representation of the Solution

Here is a visual representation of the solution:

35
84

Alternative Solutions

There are alternative methods to solve this problem, as shown below:

Method 1

Let the Number be in the ratio 12x : 5x. The difference is:

12x - 5x 7x 49

Solve for x:

x 49 / 7 7

Therefore, the numbers are:

12x 84
5x 35

Method 2

The difference is 7, and 49 is 7 times 7. Therefore, the two numbers are:

7 * 12 84
7 * 5 35

Method 3

The difference is represented as:

12x - 5x 7x 49

Solve for x:

x 49 / 7 7

Therefore, the numbers are:

12x 84
5x 35

Conclusion

The two numbers in the ratio 12:5, given that their difference is 49, are 84 and 35. Understanding and applying the common multiplier method is key to solving such problems. This method can be applied to other similar problems involving ratios and differences.