The Difference Between a Two-Digit Number and the Number Obtained by Interchanging the Digits

In mathematics, understanding the relationship between a two-digit number and its digit interchange can provide insights into basic arithmetic operations and algebraic problem-solving. This article will explore a specific problem where the difference between a two-digit number and the number obtained by interchanging its digits is 9. We will solve for the digits of the number and verify the solution through examples.

Problem Statement

Consider a two-digit number ab, where a is the tens digit and b is the units digit. When the digits are interchanged, the new number becomes b a.

According to the problem, the difference between the original number and the number obtained by interchanging the digits is 9. This can be expressed mathematically as:

10a b - (10b a) 9

Solving the Problem Mathematically

Let's simplify and solve the equation:

Start with the given equation:

10a b - 10b - a 9

Simplify the equation by combining like terms:

9a - 9b 9

Divide the entire equation by 9:

a - b 1

This equation tells us that the difference between the tens digit and the units digit is 1. Therefore, the difference between the two digits of the number is 1.

Numerical Examples

Let's verify the solution with a few numerical examples:

21 - 12 9 32 - 23 9 43 - 34 9 54 - 45 9 65 - 56 9 76 - 67 9 87 - 78 9 98 - 89 9

Each example confirms that the difference between the original number and the number with interchanged digits is always 9.

Additional Examples

Let's consider another scenario where the sum of the digits is 15 and the difference between the product and the difference of the digits is 56. We will determine the original number and its digits.

Problem Revisited with Additional Constraints

Let the tens digit be a and the ones digit be b. The given conditions are:

10a b - 10b - a 9 a b 15 ab 56

From the first condition, we have:

9a - 9b 9 or a - b 1

From the second condition:

a b 15

Solving the System of Equations

Let's solve these equations simultaneously:

a - b 1 a b 15

Adding these two equations:

2a 16 or a 8

Substituting a 8 into the second equation:

8 b 15 or b 7

Thus, the digits are a 8 and b 7. The product of the digits is:

8 × 7 56

Conclusion

In summary, the difference between the two digits of a two-digit number, where the number and the number with interchanged digits have a 9 unit difference, is 1. With additional constraints on the sum and product of the digits, the solution can be uniquely determined as 8 and 7, resulting in the product 56.