Solving Numeric Puzzles: Finding the Two-Digit Number

Many times, numerical problems are presented with specific clues that help us identify a particular number. These can be quite challenging yet rewarding, as they test both logical thinking and mathematical skills. This article delves into a specific type of numerical puzzle involving two-digit numbers with a clever twist.

Problem Statement

A two-digit number has a unique property: the digit in the tens' place is three times that of the ones' place. When the digits are reversed, the new number is 36 less than the original number. What is the two-digit number?

Solution Approach

To solve this problem, we will use algebra to represent the two-digit number and then solve a system of equations derived from the given conditions. Let's denote the two-digit number as 10tu200b u, where t is the digit in the tens place and u is the digit in the ones place.

Step-by-Step Solution

Condition 1: The digit in the tens' place is three times that of the ones' place.

The first condition tells us that t is three times u.

Mathematically, this can be represented as:

t 3u

Condition 2: Reversing the digits results in a new number that is 36 less than the original number.

When the digits are reversed, the number becomes 10u t. According to the problem, this new number is 36 less than the original number.

Mathematically, this can be represented as:

10u t 10t u - 36

Simplifying the Equations

The second equation can be simplified step by step:

Rearrange the equation to group like terms:

10u - u - 10t -36

9u - 9t -36

Divide the entire equation by 9:

u - t -4

Which can be rewritten as:

t u 4

Solving the System of Equations

We now have two equations:

t 3u t u 4

Setting these equations equal to each other gives:

3u u 4

Subtract u from both sides:

2u 4

Divide by 2:

u 2

Substituting u 2 back into one of the original equations to find t:

t 3(2) 6

Conclusion

Thus, the original two-digit number is:

10t u 10(6) 2 60 2 62

To verify, the number reversed is 26, and the difference between 62 and 26 is indeed 36, satisfying the problem's condition.

Therefore, the number is 62.

Alternative Solutions Using Reasoning

Alternatively, we can analyze the problem using logical reasoning. By assessing the possible two-digit numbers where the tens digit is three times the units digit, the only candidates are: 21, 42, 63, and 84.

Among these, only 84 satisfies the second condition: when reversed, the difference is exactly 36:

84 - 48 36

Conclusion

The two-digit number that fulfills both conditions is 84.