Solving a Two-Digit Number Puzzle: Methods and Insights

Let's explore a number puzzle where the sum of the digits of a two-digit number is 7, and when the digits are reversed, the new number is 25 less than twice the original number. We'll solve this puzzle using both logical reasoning and algebraic methods.

Understanding the Problem

Given a two-digit number, let the tens digit be represented as ( x ) and the units digit as ( y ). The following conditions apply:

The sum of the digits is 7: ( x y 7 ) When the digits are reversed, the new number is 25 less than twice the original number: ( 10y x 2(1 y) - 25 )

Solving the Puzzle Logically

First, let's try to solve this puzzle by considering the table given:

Original Number Sumu of Digits Reversed Number Twice the Original Number 25 Less Than Twice the Original Number 25 - 12 13 1 3 4 31 - 13 18 13 × 2 26 26 - 25 1

Given the constraints and the logical process, we can deduce the original number by systematically checking each pair of digits that sum to 7. The pair 1 and 4 is a good candidate since it's the only pair that fits the second condition.

Solving the Puzzle Algebraically

We can also solve this puzzle using algebraic equations. Let's write down the equations:

The sum of the digits: ( x y 7 ) Reversing the digits and the given condition: ( 10y x 2(1 y) - 25 )

Rewrite the second equation for clarity:

10y x 2 2y - 25

Rearrange the terms:

10y x - 2 - 2y -25

Simplify:

8y - 19x -25

Now, we have two equations:

1. ( x y 7 )

2. ( 8y - 19x -25 )

Solve the first equation for ( y ):

( y 7 - x )

Substitute ( y ) into the second equation:

( 8(7 - x) - 19x -25 )

Expand and combine like terms:

56 - 8x - 19x -25

56 - 27x -25

Subtract 56 from both sides:

-27x -81

Divide by -27:

x 3

Substitute ( x 3 ) back into the first equation:

3 y 7

y 4

So the original number is ( 1 y 10(3) 4 34 ).

Conclusion

The original number that satisfies the given conditions is 34. This was found using both logical reasoning and algebraic methods. The algebraic solution provides a systematic approach to solving similar puzzles.