Solving Mathematical Puzzles Involving Digit Reversal: A Comprehensive Guide

Making sense of mathematical puzzles that involve digit reversal can be both intriguing and challenging. Whether you are a student, a teacher, or simply someone who enjoys solving puzzles, this article aims to guide you through the process step-by-step. We will explore a specific problem where the sum of the digits of a two-digit number is 10 and the number formed by reversing the digits is 18 less than the original number. By the end of this article, you will be equipped with the skills to solve similar problems with ease.

Understanding the Problem

The given problem states that the sum of the digits of a two-digit number is 10. The number formed by reversing the digits is 18 less than the original number. Our goal is to find the original number.

Step-by-Step Solution

Let’s break down the solution methodically:

Represent the two-digit number as (10a b), where (a) is the tens digit and (b) is the units digit. The first condition given is that the sum of the digits is 10. This can be written as: (a b 10) The second condition is that the number formed by reversing the digits is 18 less than the original number. This can be expressed as: (10b a 10a b - 18)

Solving the Equations

Let's start by simplifying the second equation:

Begin by rearranging the terms: (10b a 10a b - 18) (10b - b 10a - a - 18) (9b - 9a -18)

Dividing the entire equation by 9:

(b - a -2)

We now have a system of equations:

(a b 10) (b - a -2)

Solving the second equation for (b):

(b a - 2)

Substitute (b a - 2) into the first equation:

(a (a - 2) 10) (2a - 2 10)

Adding 2 to both sides:

(2a 12)

Dividing by 2:

(a 6)

Substitute (a 6) back into (b a - 2):

(b 6 - 2 4)

Therefore, the original number is:

(10a b 10(6) 4 60 4 64)

To verify:

The sum of the digits is (6 4 10), correct. The reversed number is 46 and 64 - 46 18, correct.

Hence, the original number is 64.

Additional Examples

Let's examine a few more examples to reinforce the concept:

Example 1

The problem states that the sum of the digits is 14 and the number formed by reversing the digits is 18 less than the original number.

Let the number be 1y. From the problem statement, we deduce: (xy 14) (10y - 1x 21y - 23)

Solving the equations:

(10y - y 21y - 23 - 8x) (112 - 8x 19x - 23) (27x 135) (x 5) and (y 9)

The number is 59. Verification:

95 2 * 59 - 23 118 - 23 95, correct.

Example 2

The number is 56, and when the digits are reversed, you get 65, which is 9 more than 56.

Let the two-digit number be (xy). From the problem statement, we have: (x y 11) (10y x 65) (10y x - (1 y) 9)

Solving the equations:

(9x - 9y 18) (9x - y 18) (x - y 2)

Solving for (x) and (y):

(x 9) and (y 7)

Therefore, the original number is 97. Verification:

(9 7 11), correct. (79 - 56 23 - 14 9), correct.

Conclusion

Solving problems involving digit reversal requires a methodical approach using algebraic equations. By breaking down the problem into manageable parts and systematically solving the equations, you can find the correct solution. Whether you are tackling a simple or complex problem, the process remains the same. With practice and a clear understanding of the principles involved, you can confidently solve a wide range of mathematical puzzles.