Solving a Two-Digit Number Puzzle: A Comprehensive Guide

Today, we explore the realm of mathematical puzzles and learn how to solve a particular challenge involving two-digit numbers. The problem at hand is as follows:

Problem Statement

Given a two-digit number, where the sum of its digits is 9, and the number formed by reversing the digits is 45 more than the original number, what is the original number?

Understanding the Problem

Let's denote the two-digit number as 10a b, where a is the tens digit and b is the units digit. We are given two conditions:

a b 9 10b a is 45 more than 10a b

By restating the second condition, we have:

10b a 10a b 45

Creating a Solvable Equation

Let's simplify the second equation to find a relationship between a and b. First, rearrange the terms:

10b a - 10a - b 45

This simplifies to:

9b - 9a 45

Next, divide the whole equation by 9:

b - a 5

Simultaneously Solving the Equations

Now, we have two simultaneous equations:

a b 9 b - a 5

From the second equation, we can express b in terms of a.

b a 5

Substitute this into the first equation:

a (a 5) 9

This simplifies to:

2a 5 9

Subtract 5 from both sides:

2a 4

Divide by 2:

a 2

Now, substitute a 2 back into the equation b a 5 to find b:

b 2 5 7

Conclusion

The original two-digit number is:

10a b 10(2) 7 27

Therefore, the original number is 27.

Additional Insights

Another approach to solving this problem involves a different set of numbers, such as 36 and 63. These numbers satisfy the conditions that the reversed number is 27 more than the original number and that the sum of the digits is 9.

Using similar steps, we can solve for x and y in a form of xy 9:

9x - 9y 27

This simplifies to:

x - y 3

Solving these equations, we find:

x 6 and y 3

Hence, the original number is 36.

Mathematical Insight

Understanding the nature of digit reversal and the conditions provided can simplify the problem significantly. The key is decomposing the problem into manageable parts and solving the resulting equations.

Key Takeaways

Use the properties of digits and their relationships to form equations. Solve the equations simultaneously to find the values of individual digits. Verify the solution by checking the conditions provided.

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