Problem 74 - Intermediate Algebra

Question

Digits Problem. The sum of the digits of a two-digit number is 10. If we interchange the digits, then the new number formed is 54 less than the original. Find the original number.

Step-by-Step Solution

Answer

The original number is 82.

Step 1: Defining the Variables

Let the two-digit number be represented as \(10x + y\), where \(x\) is the tens digit and \(y\) is the units digit. The problem states that the sum of the digits is 10, giving us the equation \(x + y = 10\).

Step 2: Formulating the Second Equation

If we interchange the digits, the new number formed can be represented as \(10y + x\). The problem states that this new number is 54 less than the original, which provides the equation \((10y + x) + 54 = (10x + y)\).

Step 3: Simplifying the Second Equation

Distribute and rearrange the terms of the equation \((10y + x) + 54 = (10x + y)\). This simplifies to \(10y + x + 54 = 10x + y\). Simplify further to obtain the equation \(9y + 54 = 9x\).

Step 4: Solving the System of Equations

From the simplified equation \(9y + 54 = 9x\), divide every term by 9 to get \(y + 6 = x\). You now have a system of equations: \(x + y = 10\) and \(y + 6 = x\).

Step 5: Substituting and Solving

Substitute \(y + 6 = x\) into the equation \(x + y = 10\). This gives \((y + 6) + y = 10\), which simplifies to \(2y + 6 = 10\). Solve this equation: \(2y = 4\), so \(y = 2\).

Step 6: Finding the Value of x

With \(y = 2\), substitute back into \(y + 6 = x\) to find \(x\). This gives \(2 + 6 = 8\), so \(x = 8\).

Step 7: Determine the Original Number

The original number is formed using \(x = 8\) and \(y = 2\), so the number is \(10x + y = 10(8) + 2 = 82\).

Key Concepts

Systems of Equations

Systems of equations are a fundamental concept in algebra. A system of equations consists of two or more equations with the same set of variables. The purpose is to find the values of these variables that satisfy all the equations at the same time. In our problem, we used two equations to model the constraints given:

• The sum of the digits is 10: \( x + y = 10 \).
• The number with interchanged digits is 54 less than the original: \( (10y + x) + 54 = (10x + y) \).

These equations describe a two-digit number problem, where \( x \) and \( y \) represent the tens and units digits, respectively. To solve this system:
1. Express one of the variables in terms of the other using one equation.
2. Substitute this expression into the other equation to find the values of the variables.
3. Ensure to check the solutions with both initial equations to confirm correctness.

Word Problems

Word problems are mathematical exercises where the problem is presented in a narrative format using words instead of equations. The challenge is to extract mathematical expressions from the text, which requires careful reading and critical thinking. In our digits problem, the scenario involves understanding and translating:

• The relationship between digits when summed: the sum is 10.
• The transformation when digits are swapped: the rearranged number is 54 less.

This process involves identifying the quantities involved (digits of the number) and the relationships between them (sum and difference conditions). Effective strategies include:
- Highlight key information and relationships as you read.
- Define variables clearly to represent unknowns.
- Convert descriptive sentences into algebraic expressions.
This skill helps simplify and solve real-life scenarios through mathematics, often seen in entrance exams and standardized tests.

Digit Problems

Digit problems are a unique type of word problem that focus specifically on numbers and their digits. These problems typically involve:

• Setting relationships between the digits of a number.
• Manipulating the number by changing its digits.

For example, consider our exercise. Here, we are dealing with two-digit numbers, where we denote the number as \( 10x + y \) from its digits. The first equation from such a problem could be the sum of the digits. The second usually involves forming a new number through some operation on the digits, like reversing them and comparing values.When solving digit problems:
- Always outline what each variable represents.
- Carefully analyze how changing digits affect the number.
- Derive relationships accordingly, ensuring each part corresponds to the problem's requirements.Digit problems help sharpen logical thinking and algebraic manipulation skills as students apply math concepts to solve seemingly simple, yet intricate puzzles.