Problem 18 - Fundamentals of Mathematics

Question

The sum of three consecutive integers is \(48 .\) What are they?

Step-by-Step Solution

Answer

The integers are 15, 16, and 17.

Step 1: Define Variables

Let's define the three consecutive integers. If we denote the first integer by \(x\), then the next two consecutive integers can be expressed as \(x+1\) and \(x+2\).

Step 2: Set Up the Equation

The problem states that the sum of these three integers is 48. Therefore, we can write the equation: \(x + (x+1) + (x+2) = 48\).

Step 3: Simplify the Equation

We simplify the equation by combining like terms. This gives us: \(3x + 3 = 48\).

Step 4: Solve for x

Subtract 3 from both sides of the equation to isolate the term with \(x\). This gives us \(3x = 45\). Then, divide both sides by 3 to solve for \(x\): \(x = 15\).

Step 5: Find the Consecutive Integers

Using the value of \(x\) we found, the first integer is 15. The second integer is \(15 + 1 = 16\), and the third integer is \(15 + 2 = 17\).

Key Concepts

Equation Solving

In the context of finding consecutive integers whose sum is a known value, the process of equation solving plays a crucial role. The first step is to define variables clearly. Here, we chose to define the first integer as \(x\). This is the foundation for setting up the mathematical equation. By identifying subsequent integers as expressions of \(x\), namely \(x+1\) and \(x+2\), we simplify the problem into terms we can solve more easily.

The next step involves setting up an equation that represents the problem statement. In this exercise, the sum of the integers must equal 48. Thus, the mathematical representation becomes:\[ x + (x+1) + (x+2) = 48 \](When variables are understood in terms of a single unknown, the complexity of the problem is reduced to a manageable linear equation.) Simplifying this equation by combining like terms: \(3x+3 = 48\), allows us to start isolating the variable. Solving such linear equations typically involves strategically adding, subtracting, multiplying, or dividing both sides until \(x\) is isolated. This methodological approach simplifies potentially complex problems into small, manageable steps that can lead to a solution with clarity and consistency.

Integer Equations

Integer equations involve dealing specifically with whole numbers, which do not include fractions or decimals. In our example of consecutive numbers whose sum is 48, these whole numbers are the integers of our equation. The integers are characterized by their distinct nature wherein gaps (without intermediary decimal or fractional values) lie between them. This very quality makes problems involving consecutive integers straightforward due to the predictable pattern of incrementation, such as \(n, n+1, n+2\), where 'n' is the first integer.

By focusing on integer properties, these problems stress the importance of understanding equations through the lens of integer arithmetic. These integer characteristics simplify solving equations since they narrow down possibilities (in integers plus increments, rather than an infinite spectrum of possible values that includes fractions or decimals).

• Whole numbers only
• Predictable patterns
• Easy to set up and solve

Integer problems not only hone your skills in solving equations but also in recognizing patterns, making them invaluable in reinforcing foundational mathematics.

Mathematical Problem Solving

At the heart of exercises like finding consecutive integers is mathematical problem solving, an essential skill set that allows tackling a wide range of math problems. This skill involves not just solving equations, but interpreting a problem correctly and knowing which principles to apply. It starts with understanding the problem statement—determining what is given, what needs to be found, and under what conditions. This is then followed by logically organizing your thoughts and expressing the problem in mathematical terms.

Breaking down the problem into smaller, systematic steps facilitates the solving process. This also includes reviewing each solution step for accuracy, choosing the proper arithmetic operations to isolate and solve for variables, and double-checking calculations. Mathematical problem solving further encompasses the practice of verifying solutions. For example, you will ensure the integers solved initially do indeed sum up to the given total, as done by substituting back into the equation: \(15 + 16 + 17 = 48\). This confirms that each step was executed correctly.

• Problem interpretation
• Logical organization
• Verification

By integrating these strategic steps, problem-solving becomes not only possible but also engaging and rewarding.