Problem 57 - Precalculus: Mathematical for Calculus

Question

The sum of the digits of a two-digit number is \(7 .\) When the digits are reversed, the number is increased by \(27 .\) Find the number.

Step-by-Step Solution

Answer

The number is 25.

Step 1: Define Variables

Let the two-digit number be represented as \(10x + y\), where \(x\) is the tens digit and \(y\) is the units digit.

Step 2: Set Up Equations from the Problem

From the problem, we have two conditions. First, the sum of the digits is 7, so \(x + y = 7\). Second, reversing the digits increases the number by 27, so the reversed number is \(10y + x = 10x + y + 27\).

Step 3: Solve the Second Equation

Rearrange the equation \(10y + x = 10x + y + 27\) to get \(10y - y = 10x - x + 27\), simplifying to \(9y = 9x + 27\). Divide both sides by 9 to get \(y = x + 3\).

Step 4: Substitute into the First Equation

Substitute \(y = x + 3\) into the first equation \(x + y = 7\): \(x + (x + 3) = 7\). Simplify to get \(2x + 3 = 7\).

Step 5: Solve for x

From \(2x + 3 = 7\), subtract 3 from both sides to get \(2x = 4\). Divide both sides by 2 to find \(x = 2\).

Step 6: Find y Using x

Substitute \(x = 2\) back into the equation \(y = x + 3\), yielding \(y = 2 + 3 = 5\).

Step 7: Verify the Solution

The original number is \(10x + y = 10(2) + 5 = 25\). When the digits are reversed, the number is \(10y + x = 10(5) + 2 = 52\). The increase is indeed \(25 + 27 = 52\), so the solution is verified.

Key Concepts

Digit Reversal Problems

In digit reversal problems, you're tasked with working out a mystery number by manipulating the positions of its digits. This can set off a delightful chase of algebraic detective work! In this case, the challenge was to find a two-digit number where reversing the digits results in a number that is 27 greater than the original. By breaking this down, we understand that reversing our number translates to swapping the positions of the tens and units digits.
This specific concept is crucial because reversing digits forms the backbone of countless number riddles and puzzles:

• It enables exploration through logical reasoning.
• Tests your ability to translate written problems into mathematical equations.
• Improves your ability to think abstractly about numbers.

When solving digit reversal problems, always double-check that the new reversed number satisfies the problem's conditions—here, an increase by 27.

Algebraic Equations

Algebraic equations form the core framework that allows us to solve digit-related puzzles. In this problem, we represented the original two-digit number as \(10x + y\). Here, \(x\) and \(y\) represent the tens and units digits, respectively. Creating equations based on the problem:1. The sum of the digits is given: \(x + y = 7\).2. The reversed digits increase the number by 27: \(10y + x = 10x + y + 27\).To solve these equations, we rearranged terms to make them easier to handle. Rearranging involves isolating variables and simplifying where possible. For example, by simplifying the second equation, we find \(y = x + 3\).
This breakdown into simpler equations showcases the power of algebra in dissecting complicated problems into manageable parts:

• Transforms word problems into solvable mathematical expressions.
• Encourages detailed and critical thinking to isolate each variable.
• It serves as practice for recognizing patterns and logical sequences.

Problem-Solving Strategies

Approaching two-digit algebraic problems requires a toolbox of strategies that provide clear guidance. Let's break down the process:- **Define Variables:** Identifying each digit with a variable such as \(x\) and \(y\) helps establish a mathematical framework to start from. It forms the backbone on which equations are built.- **Set Up Equations:** Drawing each equation directly from the problem statements keeps you anchored to the conditions and always aligned with the intended solution.- **Simplify and Solve:** Simplification is key! Breaking down equations like \(9y = 9x + 27\) to \(y = x + 3\) reduces complexity and paves the way for quick solutions.- **Verify the Solution:** Verification isn't just a formality—it's essential. Ensure the reversed digit case satisfies the given conditions, cementing confidence in your solution.
By systematically following these steps, you sharpen your problem-solving skills, ensuring you tackle not just number puzzles, but a broad range of challenges more effectively.