Problem 41 - Intermediate Algebra

Question

The sum of the digits of a three-digit number is 8. Twice the hundreds digit plus the tens digit is equal to the ones digit. If the digits of the number are reversed, the new number is 82 more than twice the original number. What is the three-digit number?

Step-by-Step Solution

Answer

The three-digit number is 215.

Step 1: Define Variables

Let the three-digit number be represented as \(100a + 10b + c\), where \(a\), \(b\), and \(c\) are the hundreds, tens, and units digits respectively. These digits must satisfy \(1 \leq a \leq 9\) and \(0 \leq b, c \leq 9\).

Step 2: Sum of the Digits

We are given that the sum of the three digits is 8. This can be written as the equation: \[ a + b + c = 8 \]

Step 3: Relationship Among Digits

Twice the hundreds digit plus the tens digit equals the ones digit. This provides the equation: \[ 2a + b = c \]

Step 4: Reverse Number Condition

When the digits are reversed, the new number \(100c + 10b + a\) is 82 more than twice the original number. This gives: \[ 100c + 10b + a = 2(100a + 10b + c) + 82 \] Simplifying this equation, we have: \[ 100c + 10b + a = 200a + 20b + 2c + 82 \] Rearranging, we get: \[ 98c - 199a - 10b = 82 \]

Step 5: Solve the Equations

We have three equations now: \(a + b + c = 8\), \(2a + b = c\), and \(98c - 199a - 10b = 82\). From \(a + b + c = 8\), substitute \(c = 2a + b\) from the second equation to get \(a + b + 2a + b = 8\), which simplifies to \(3a + 2b = 8\).Substituting \(c = 2a + b\) into \(98c - 199a - 10b = 82\) yields:\[ 98(2a + b) - 199a - 10b = 82 \]\[ 196a + 98b - 199a - 10b = 82 \]\[ -3a + 88b = 82 \]Divide through by 2:\[ -3a + 44b = 41 \]

Step 6: Solve the System of Equations

Now solve the two linear equations: \(3a + 2b = 8\) and \(-3a + 44b = 41\).Add the two equations to eliminate \(a\):\[ 3a + 2b - 3a + 44b = 8 + 41 \]\[ 46b = 49 \]Thus, \(b = \frac{49}{46}\). Since \(b\) must be an integer, check the nearest integers. If \(b = 1\), then \(3a + 2(1) = 8\) gives \(3a = 6\), so \(a = 2\).Substituting \(a = 2\) and \(b = 1\) into \(c = 2a + b\), we find \(c = 4 + 1 = 5\).

Step 7: Verify the Solution

Check if 215 is correct:- Sum of digits: \(2 + 1 + 5 = 8\).- Twice the hundreds plus tens equals ones: \(2(2) + 1 = 5\).- Reversed number is 512. Is 512 = 2 * 215 + 82? Calculate: 2*215 = 430, add 82 = 512.All conditions are satisfied.

Key Concepts

Digit Problems

Digit problems can be fun and challenging puzzles that often appear in mathematical exercises. These problems generally require breaking down multi-digit numbers to individual digits and analyzing the relationships between them.

• Digit problems commonly focus on sums, multiplying, reversing, or rearranging digits.
• Understanding different ways digits can combine to meet specified requirements is crucial.

To tackle these problems effectively: 1. Understand the given conditions clearly. 2. Define variables for each digit, often starting from the leftmost digit such as hundreds for three-digit numbers. 3. Use equations to represent problem conditions. For instance, if the problem states that the sum of digits is a certain number, create an equation to represent that.
Digit problems also serve as great practice for reinforcing concepts like place value and operation balancing, making them a useful tool in enhancing problem-solving skills.

Three-Digit Numbers

A three-digit number is any number ranging from 100 to 999. Each three-digit number is composed of hundreds, tens, and units (ones) digits. Understanding their structure and properties is key to solving digit problems involving these numbers.

• The hundreds digit (0a1) ranges from 1 to 9 (since a three-digit number cannot start with 0).
• The tens (0b1) and units (cd) digits each range from 0 to 9.

In our example problem: - The number could be expressed in a mathematical form, such as 00a + 10b + c. - The reversal of the number, such as 00c + 10b + a, changes the order of digits, altering the number's value and potentially meeting other specified conditions.
Understanding the numerical value of a number in terms of its digits is crucial as it aids in forming equations that help solve complex digit problems effectively.

Equation Solving

Solving equations is a vital skill in mathematics, especially in problems involving numbers and digits. The goal is to find the values of variables that satisfy given conditions, represented as mathematical equations.

• First, break down each part of the problem to form basic equations representing the given conditions.
• Use substitution or elimination methods when dealing with simultaneous equations.

For instance, in the provided problem: - Start with translating word statements to equations, like the sum of digits and relationships among digits. - Use the technique of substitution, replacing one variable with another expression derived from an equation, which simplifies the problem. - Finally, solve these simplified equations to find exact values of the unknown digits.
Understanding how to manipulate and solve equations not only helps with digit problems but also enhances your overall mathematical reasoning and problem-solving capabilities. This is especially useful when dealing with real-world applications where systems of equations often arise.