Question

In Exercises 17-22, write a mathematical model for the number problem, and solve the problem. Find two consecutive numbers whose sum is 525 .

Short answer

The two consecutive numbers whose sum is 525 are 262 and 263

Step-by-step solution

Defining Variables

Denote the first number as \( n \) and the second consecutive number as \( n + 1 \). These are our variables that represent the consecutive numbers in the problem.

Setting up the Equation

The problem states that the sum of these two numbers is 525. Therefore, we can set up the equation as \( n + (n + 1) = 525 \).

Simplifying the Equation

Consolidate the like terms on the left-hand side of the equation to simplify it. This results in \( 2n + 1 = 525 \).

Further Simplification & Solving for n

Subtract 1 from both sides of the equation, leading to \( 2n = 524 \). Dividing each side by 2 gives \( n = 262 \).

Find the Second Number

Now, compute the second consecutive number by adding one to the first number, \( n + 1 = 262 + 1 = 263 \).

Key concepts

Algebraic Equations

Understanding algebraic equations is crucial for solving many types of problems, including finding consecutive numbers. These equations are mathematical statements that include unknown values, known as variables, along with numbers and operations. In our example, the problem asks to find two consecutive numbers whose sum is 525.

To start, we define a variable, let's say \( n \), to represent the first number. The next consecutive number is simply one more than the first, so it would be \( n + 1 \). This understanding allows us to construct an equation expressing the sum: \( n + (n + 1) = 525 \). When you see equations like this, remember that they are meant to find the value of the variable that makes the statement true.

Through a series of simplifications—combining like terms and isolating the variable—we break down the original problem into a more manageable form, eventually solving for \( n \) to find the first number. It's important to perform the same operation on both sides of the equation to maintain its balance until we isolate the variable to find its value.

Mathematical Modeling

Mathematical modeling involves translating real-world scenarios into mathematical language to analyze and solve problems. The process can take many forms, but often involves identifying patterns, relationships, and constructor equations that reflect the situation.

In the context of our consecutive numbers problem, we create a model by defining variables to represent the numbers we're looking to find. The relationship between these numbers is clear—they're consecutive, so one is the successor of the other. To reflect this relationship, we set up an equation that models the sum of these two numbers as provided by the problem statement.

Mathematical modeling is a powerful tool that goes beyond textbook exercises. It's used across many fields, such as economics, engineering, and science, to describe complex systems and predict behavior. When engaging in mathematical modeling, it's important to consider all aspects of the problem, to choose appropriate variables, and to form an accurate representation of the problem that can be manipulated mathematically.

Problem Solving

Strategy and Execution

Problem solving is a broad skill that involves approaching a challenge with systematic thinking and executing a strategy to reach a solution. When faced with a math problem, start by understanding what is being asked. Then, select appropriate mathematical tools and techniques to address the problem.

In our example, the problem solving process started with defining the problem—finding two consecutive numbers that sum to a specific value. The next step was to select appropriate tools, namely algebraic equations, and execute a strategy to solve for our unknowns.

Verification

Problem solving doesn't end with finding an answer; verification is an essential step. After solving for the consecutive numbers, it's wise to check if the numbers indeed satisfy the original condition—their sum being 525. This reassures that the solution is correct and that no mistakes were made in the process.

Developing strong problem-solving skills involves practicing a variety of problems, reflecting on the methods used, and learning from each attempt. With continuous practice, you'll find that the ability to navigate complex problems becomes second nature.