Solving Mathematical Puzzles: Two-Digit Number with Digit Sum and Reversal

In mathematics, puzzles that involve finding a two-digit number based on certain constraints can be intriguing. In this article, we will explore a specific type of puzzle that involves the digit sum and digit reversal of a two-digit number. We will solve various puzzles step-by-step, providing clear explanations and solutions.

Problem 1: The Sum of Digits is 6, and the Reversed Number Decreases by 29

Let's start with a puzzle where the sum of the digits of a two-digit number is 6, and the number decreases by 29 when reversed.

Given:

The sum of the digits is 6. The reversed number is decreased by 29.

Step 1: Let the two-digit number be ab, where a is the tens digit and b is the units digit.

So, the number can be represented as 10a b, and the reversed number as 10b a.

Equation 1: a b 6

Equation 2: 10b a 10a b - 29

Subtract 10a b from both sides of Equation 2:

9b - 9a -29

Divide both sides by 9:

b - a -3

Now we have:

a b 6 b - a -3

Add the two equations:

2b 3

b 3 3 6 - 3 3

b 3

Substitute b 3 into the first equation:

a 3 6

a 6 - 3 3

The two-digit number is 33. However, this does not fit the problem conditions. Let's solve it again with the correct solution:

Step 2: Correct Solution

b 12 - a (from original condition)

10a b 10b a - 29

10a (12 - a) 10(12 - a) a - 29

9a 12 120 - 9a - 29

18a 87

a 87 / 18 4.83 8 (since we are dealing with integers)

b 12 - 8 4

The two-digit number is 84.

Problem 2: Reversal and Subtraction by 54

Let the two-digit number be xy, where x is the tens digit and y is the units digit.

Given:

The sum of digits is 8. The reversed number is decreased by 54.

Step 1: Let the two-digit number be 1 y, and the reversed number be 10y x.

Equation 1: x y 8

Equation 2: 10y x 1 y - 54

Subtract 1 y from both sides:

9y - 9x -54

Divide both sides by 9:

y - x -6

Now we have:

x y 8 y - x -6

Add the two equations:

2y 2

y 1

Substitute y 1 into the first equation:

x 1 8

x 7

The two-digit number is 71.

Problem 3: Reversal and Subtraction by 27

Given:

The number is represented as ab. Reverse the digits to get 10b a. Reverse decreases by 27.

Step 1: Let the two-digit number be 10a b, and the reversed number be 10b a.

Equation 1: a b 7

Equation 2: 10b a 10a b - 27

Subtract 10a b from both sides:

9b - 9a -27

Divide both sides by 9:

b - a -3

Now we have:

a b 7 b - a -3

Add the two equations:

2b 4

b 2

Substitute b 2 into the first equation:

a 2 7

a 5

The two-digit number is 52.

Problem 4: Reversal and Subtraction by 36

Given:

x is one of the digits. The initial number can be represented as 112 - x - 1012 - x - x 36.

Step 1: Let the two-digit number be 1x, and the reversed number be 10 - 1.

Equation 1: 1x - 10 1 36

101x - 10 36

18x 144

x 144 / 18 8

The larger digit is 8.

The numbers are 84 and 48.

Problem 5: Reversal with Digit Substitution

Given:

The original number is represented as 1y. The reversed number is 10yx. The sum of digits is 8. The reversed number is decreased by 54.

Step 1: Let the two-digit number be 1 y.

Equation 1: x y 8

Equation 2: 10y x - (1 y) 54

10y x - 1 - y 54

9y - 9x 54

Divide both sides by 9:

y - x 6

Now we have:

x y 8 y - x 6

Add the two equations:

2y 14

y 7

Substitute y 7 into the first equation:

x 7 8

x 1

The two-digit number is 71.

Note: There was a mistake in the original solution where y was mistakenly substituted as 1, leading to an incorrect solution. The correct solution is 71.

Conclusion

In these problems, we utilized the fact that the sum of the digits and the relationship between the original number and the reversed number can be used to form equations and solve for the digits. By carefully setting up and solving these equations, we can determine the original two-digit number in each case.