Exploring Number Pairs with a Given Ratio and Difference

When faced with a problem that asks you to find two numbers with a specific ratio and a certain difference, the solution can be both straightforward and intriguing. In this article, we will explore a method to solve such problems, focusing on the ratio of 9:11 and a difference of 6, aside from the most obvious pair, 27 and 33.

Understanding the Problem

The problem states that the ratio of two numbers is 9:11, and their difference is 6. One possible solution is 27 and 33, but the question specifies that this pair cannot be the correct answer. We need to find another pair of numbers that satisfy these conditions.

Solving the Problem: Algebraic Approach

Let's denote the two numbers by 9x and 11x, where x is a common factor. According to the problem, the difference between these two numbers is 6. We can set up the following equation:

11x - 9x 6

Simplifying the left side of the equation:

2x 6

Solving for x:

x 6 / 2 3

Substituting the value of x back into the expressions for the two numbers:

9x 9 × 3 27

11x 11 × 3 33

This confirms that 27 and 33 are indeed a solution, but the problem specifies that they cannot be the answer. So, we need to look for another solution. One alternative solution is to consider negative numbers, as the original problem does not restrict them to positive integers.

Exploring Additional Solutions

Another possible solution is to use negative numbers. Since the ratio 9:11 still holds, we can use -27 and -33. Let's verify:

The difference between -33 and -27 is:

-33 - (-27) -33 27 -6

The ratio of -27 to -33 is:

-27 / -33 27 / 33 9 / 11

Therefore, -27 and -33 also satisfy the conditions given in the problem.

Logical Approach

To find the solution logically, we can start with the basic ratio of 9 and 11. We can double or triple these numbers to find a pair that matches the difference of 6:

9 × 2 18, and 11 × 2 22. The difference is 4. 9 × 3 27, and 11 × 3 33. The difference is 6.

This confirms that 27 and 33 (with a difference of 6) and -27 and -33 (which also have a difference of 6) are valid solutions.

Simplifying the Process

To make the process of finding such numbers easier, we can use algebraic substitution:

Let's denote the two numbers as 9x and 11x. The difference is given by:

11x - 9x 6

Dividing both sides by 2:

2x 6

Thus, x 3.

Substituting the value of x back in:
9x 9 × 3 27
11x 11 × 3 33

Therefore, the numbers are 27 and 33, or their negative counterparts -27 and -33.

Conclusion

In conclusion, we have explored how to find number pairs with a given ratio and a specific difference, beyond the most obvious solution. By using algebraic substitution and logical reasoning, we have found that -27 and -33 are another valid pair of numbers that satisfy the conditions of the problem. This method can be applied to similar problems, making it a useful skill for solving ratio and difference problems.