Solving the Unique Math Puzzle: Finding a Two-Digit Number Given Its Digit Sum and Reverse Condition

In mathematics, many problems are solved by breaking down the given information into manageable parts and solving step by step. This article explores the process of finding a two-digit number when the sum of its digits and the condition relating the original number with the reversed number are given.

Introduction to the Problem

Consider the problem of finding a two-digit number whose digits sum up to 14, and when reversed, the new number is 23 less than twice the original number. This type of problem is not only interesting but also provides a clear example of how to use algebraic methods and logical reasoning.

Setting Up the Equations

To solve this problem, let's denote the two-digit number as XY, where X is the tens digit and Y is the units digit. We are given two conditions:

The sum of the digits is 14: (X Y 14) The reversed number is 23 less than twice the original number: (10Y X 2(1 Y) - 23)

Step 1: Simplifying the Second Condition

The second condition can be simplified as follows:

[(10Y X) 2 2Y - 23]

Subtracting (2Y X) from both sides, we get:

[8Y - 19X -23]

Step 2: Solving the System of Equations

We now have a system of two equations:

[begin{align*}X Y 14 8Y - 19X -23end{align*}]

We can solve this system by expressing (Y) in terms of (X) from the first equation:

[Y 14 - X]

Substituting (Y 14 - X) into the second equation:

[8(14 - X) - 19X -23]

Expanding and simplifying:

[112 - 8X - 19X -23] [-27X -135] [X 5]

Substituting (X 5) back into the equation (Y 14 - X):

[Y 14 - 5 9]

Therefore, the original number is 59.

Verification

Let's verify the solution:

The sum of the digits: (5 9 14) Reversed number: 95 Twice the original number minus 23: (2(59) - 23 118 - 23 95)

Both conditions are satisfied, confirming that the original number is indeed 59.

Conclusion

This problem illustrates how to solve a two-digit number problem using algebraic methods to set up and solve a system of equations. By systematically breaking down the problem, we can solve it step by step, ensuring that both conditions are met.

Keywords: Two-digit number, digit sum, reverse number