Solving Ratio Questions: A Case Study of 5:7 and 7:11

Understanding and solving ratio problems is a fundamental skill in mathematics. This article explores a specific problem where two numbers are in the ratio 5:7, and after subtracting 9 from each, their ratio becomes 7:11. We will walk through the step-by-step process of solving this problem, ensuring clarity and understanding through detailed explanations and algebraic manipulation.

Problem Statement

The problem states that two numbers are in the ratio 5:7. We need to find the numbers and determine the difference between them. Additionally, when 9 is subtracted from each of these numbers, their ratio becomes 7:11. Let's denote the two numbers as 5x and 7x, where x is a common multiplier.

Step-by-Step Solution

Step 1: Establish the Initial Ratios

Let the two numbers be 5x and 7x. According to the problem, when 9 is subtracted from each of these numbers, their ratio becomes 7:11. This can be expressed as:

Step 2: Setting Up the Equation

The equation that represents the new ratio is:

Step 3: Cross-Multiply to Eliminate the Fraction

Cross-multiplying to eliminate the fraction, we get:

Step 4: Expand and Simplify

Expanding both sides gives:

Step 5: Isolating the Variable

Isolating x by moving all terms involving x to one side and constant terms to the other, we have:

Step 6: Solving for x

Dividing both sides by 6 yields:

Step 7: Finding the Numbers

Now, we can find the two numbers by substituting x 6:

5x 5 × 6 30

7x 7 × 6 42

The numbers are 30 and 42.

Determining the Difference

The difference between the two numbers is:

42 - 30 12

Hence, the difference between the numbers is 12.

Further Insights

This problem demonstrates the power of algebra in solving complex ratio problems. By setting up equations and using algebraic manipulation, we can systematically solve for the unknowns and verify the results.

Note: The equation can be simplified as:

5x - 9:7x - 9 7:11

Multiplying through to eliminate the fraction, we get:

49x - 63 55x - 99

Rearranging terms:

6x 36

Solving for x:

x 6

Thus, the numbers are 30 and 42. The difference between them is 12.

This problem-solving technique can be applied to other similar problems, enhancing one's ability to handle ratio and proportion problems effectively.