Solving Number Puzzles: A Step-by-Step Guide Using Algebra

Number puzzles and riddles are not only entertaining but also a great way to hone your mathematical skills. In this article, we will walk through the process of solving various number puzzles using algebraic equations. We will focus on a particular example where a two-digit number is 27 less than the number formed by reversing the digits, and the sum of the digits is 15. We will elaborate on how to set up equations, solve for variables, and verify our solution.

Solving for a Two-Digit Number

Let the two-digit number be represented as (10a b), where (a) is the tens digit and (b) is the units digit.

According to the problem, we have two equations:

The number is 27 less than the number formed by reversing the digits: The sum of the digits is 15:

Step 1: Set Up the Equations

The number is 27 less than the number formed by reversing the digits:

(10a b 10b a - 27)

The sum of the digits is 15:

(a b 15)

Step 2: Solve for One of the Variables

Starting with the first equation:

(10a b 10b a - 27)

(10a - a b - 10b -27)

(9a - 9b -27)

(a - b -3)

From this equation, we can express (a) in terms of (b):

(a b - 3)

Step 3: Substitute and Solve

Substitute (a b - 3) into the second equation:

(b - 3 b 15)

(2b - 3 15)

(2b 18)

(b 9)

Now, substitute (b 9) back into the equation (a b - 3):

(a 9 - 3 6)

Thus, the digits are (a 6) and (b 9).

Therefore, the two-digit number is:

(10a b 10 times 6 9 69)

Step 4: Verify the Solution

Verify that the number formed by reversing the digits is 96, and check if 69 is indeed 27 less than 96:

(96 - 69 27)

Check the sum of the digits:

(6 9 15)

The solution satisfies both conditions, confirming that the two-digit number is 69.

Additional Examples and Verification

In the additional examples provided, we solve similar puzzles with different steps and checks:

Example 1

Let the number be (1y). (xy 14) or (y 14 - x). (8y 112 - 8x) (Equation 1). 10yx 2 1y - 23, which simplifies to (10y x 22y - 23) (Equation 2). Equate Equation 1 and Equation 2:

(112 - 8x 19x - 23)

(27x 135)

(x 5)

(y 9)

The number is 59. Verification: (95 2 times 59 - 23 118 - 23 95). Correct.

Example 2

Let the original number be represented by (XY), where (X) is the Most Significant Digit (MSD). (X Y 14) ……. (Equation 1) (10Y X - 23 21 Y) (10Y X - 23 20 2Y) (10Y X - 23 20 2Y) Multiplying the first equation by 8: (8X 8Y 112) ……… (Equation 3) Subtracting Equation 3 from Equation 2: ((19X - 8Y) - (8X 8Y) 23 - 112) Simplifying: (27X 135) (X 5) and (Y 9)

The original number is 59. Verification: (2(59) - 23 118 - 23 95). Correct.

Example 3

(t o 14) (10o t 210t o - 23) (10o t 20t 2o - 23) (t 14 - o) (10o 14 - o 20 14 - o - 23) (9o 14 280 - 20o - 23) (9o 14 257 - 18o) (27o 243) (o 9) (t 5)

The original number is 59. Verification: (2(59) - 23 118 - 23 95). Correct.

By using algebraic equations, we can systematically solve these puzzles and ensure the accuracy of our answers. This method not only helps in finding the correct answer but also in verifying the solution.

Conclusion

Number puzzles can be effectively solved using algebraic equations. The process involves setting up equations based on the given conditions, solving for variables step by step, and verifying the solution. This approach ensures accuracy and provides a clear understanding of the problem. Whether it's a two-digit number puzzle, or a more complex set of conditions, algebraic methods offer a reliable and efficient solution. Happy puzzling, and keep practicing to improve your mathematical skills!