Solving Digit-Related Mathematical Problems: A Detailed Guide

Number puzzles and digit-related problems are a beloved pursuit in recreational mathematics. These problems not only challenge our logical thinking but also enhance our problem-solving skills. In this article, we will explore a fascinating digit-related problem and walk through a step-by-step solution process that can be useful for solving similar problems.

Understanding the Problem

We are given a two-digit number whose digits add up to 13. When the digits are interchanged, the new number decreases by 45. Our goal is to find this number.

Step-by-Step Solution

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

The sum of the digits is 13: a b 13 When the digits are interchanged, the new number is 45 less than the original number: 10b a 10a b - 45

To simplify the second equation, we can rearrange it as follows:

10b a 10a b - 45

Rearranging gives:

10b - b - 10a -45

Simplifying further:

9b - 9a -45

Dividing the entire equation by 9:

b - a -5 or b a - 5

Now, we have a system of equations:

a b 13 b a - 5

Substituting the second equation into the first:

a (a - 5) 13

Simplifying:

2a - 5 13

Adding 5 to both sides:

2a 18

Dividing by 2:

a 9

Substituting a 9 back into the equation for b:

b 9 - 5 4

Thus, the digits are a 9 and b 4. Therefore, the two-digit number is:

10a b 109 4 90 4 94

Verification

To verify:

The sum of the digits: 9 4 13 – Correct. Interchanging the digits gives 49, and 94 - 49 45 – Correct.

Hence, the two-digit number is 94.

Additional Example

Let's consider another problem: Finding a two-digit number such that the sum of its digits is 12 and when the digits are interchanged, the number decreases by 36.

Let us assume that the ones place digit of the number be t and the digit of tens place be o. Then the number will be 10o t.

The sum of the digits is 12: t o 12. When the digits are interchanged, the number is 36 less than the original number: 10t o 10o t - 36.

Rearranging the second equation:

10t o 10o t - 36

Rearranging gives:

10t - t - 10o -36

Simplifying further:

9t - 10o -36

From the first equation, we know o 12 - t. Substituting this into the second equation:

9t - 10(12 - t) -36

Simplifying:

9t - 120 10t -36

Merging terms:

19t - 120 -36

Adding 120 to both sides:

19t 84

Dividing by 19:

t 8

Substituting t 8 back into the first equation:

o 12 - 8 4

Thus, the digits are t 8 and o 4. Therefore, the two-digit number is 10o t 104 8 84.

Conclusion

By studying and solving these types of digit-related problems, we can improve our understanding of number relationships and develop a stronger problem-solving mindset. In this article, we have demonstrated how to solve a two-digit number problem using logical reasoning and algebraic methods. These skills are invaluable not only in recreational mathematics but also in various real-world applications where numerical analysis is required.

Key Takeaways

Digit-related problems often involve setting up and solving systems of linear equations. Logical thinking and algebraic manipulation are essential tools for solving such problems. Verification is a critical step in confirming the correctness of the solution.

Feel free to explore more problems and practice solving them to strengthen your skills further.