Problem 73 - Precalculus Mathematics 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: Setup Equations

Let the tens digit be \( x \) and the units digit be \( y \). The number can be represented as \( 10x + y \). Since the sum of the digits is 7, we can write the equation: \( x + y = 7 \).

Step 2: Reverse Digits

When the digits are reversed, the new number is \( 10y + x \). We know that reversing the digits increases the number by 27, giving us the equation: \( 10y + x = 10x + y + 27 \).

Step 3: Simplify Second Equation

Rearrange and simplify the second equation: \( 10y + x = 10x + y + 27 \). Subtract \( x \) and \( y \) from both sides to obtain: \( 9y = 9x + 27 \).

Step 4: Solve for One Variable

Divide the entire equation by 9 to simplify: \( y = x + 3 \).

Step 5: Substitute and Solve

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

Step 6: Calculate Variable Values

Solve for \( x \) by subtracting 3 from both sides: \( 2x = 4 \). Divide by 2, getting \( x = 2 \). Substitute back to find \( y \): \( y = x + 3 = 2 + 3 = 5 \).

Step 7: Find the Original Number

The tens digit is 2 and the unit digit is 5. Therefore, the original number is \( 10x + y = 25 \).

Key Concepts

Linear Equations

Linear equations are foundational in algebra, providing a straightforward way to represent relationships between quantities. In this exercise, the core idea is to use linear equations to describe the relationships between the digits of a two-digit number and how they interact when manipulated.

A linear equation in one variable has the general form \( ax + b = c \), where \( x \) represents the unknown variable that we want to solve for, and \( a \), \( b \), and \( c \) are constants. However, more complex problems might require simultaneous equations, as seen in this exercise.

• The first equation \( x + y = 7 \) represents the sum of the digits.
• The second equation is derived from rearranging the digits: \( 10y + x = 10x + y + 27 \).

These equations are used to find values of \( x \) and \( y \), showing how linear equations can model real-world situations in algebra.

Digit Sums

The sum of digits can provide important insights into number properties. In this exercise, the condition \( x + y = 7 \) shows a direct relationship between the digits of the original number.

This concept is not only helpful in number theory but also vital in solving puzzles and problems where the sum gives constraints on possible values.

• Understanding digit sums can help verify calculations and estimate values quickly.
• Digit sums can simplify complex numbers, making them more manageable.

Digit sums often help us better understand the structural properties of numbers, especially when determining all possible combinations leading to a specific result.

Equation Solving

Equation solving is a critical skill in algebra that helps find unknown values that satisfy given mathematical statements. This process involves several steps, which can include setting up equations, simplification, and substitution.

In our exercise, solving started with two key equations representing different conditions of the same number. We substituted and rearranged these into simpler forms until the unknowns were isolated.

• Start with constructing equations based on the problem description.
• Simplify equations by combining like terms and isolating variable terms.
• Use substitution to replace variables and further simplify the equations.

Ultimately, by applying these methods orderly, the desired solution is achieved, demonstrating how logical steps can efficiently lead to solving complex equations.

Number Properties

Number properties are the inherent characteristics and rules numbers follow, vital for solving many mathematical problems. In our exercise, these properties assist in confirming that calculations are correct.

The most relevant number properties used here include place value understanding and number rearrangement techniques, which lead to new numbers.

• Place value is crucial in forming numbers, as seen in the translation from digits \( x \) and \( y \) to actual number representation \( 10x + y \).
• When digits switch places, properties of addition and equality help verify the correctness of the equations.

Recognizing and utilizing these properties can streamline problem-solving processes, ensuring solutions adhere to the foundational rules governing numbers.