Problem 13 - Algebra and Trigonometry

Question

Number Problem Find three consecutive integers whose sum is \(156 .\)

Step-by-Step Solution

Answer

The integers are 51, 52, and 53.

Step 1: Define the Variables

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

Step 2: Set Up the Equation

Since we know that the sum of these three integers is 156, we can write the equation:\[ x + (x + 1) + (x + 2) = 156 \]

Step 3: Simplify the Equation

Combine like terms in the equation:\[ 3x + 3 = 156 \]

Step 4: Solve for x

First, subtract 3 from both sides of the equation to isolate the terms involving \( x \):\[ 3x = 153 \]Then, divide both sides by 3 to solve for \( x \):\[ x = 51 \]

Step 5: Find the Integers

Now that we have \( x = 51 \), the three consecutive integers are:- First integer: \( x = 51 \)- Second integer: \( x + 1 = 52 \)- Third integer: \( x + 2 = 53 \)

Key Concepts

Integer Equations

Integer equations are mathematical sentences that involve integers, which are whole numbers including negative numbers but not fractions or decimals. These equations can be quite simple or complex depending on the number of unknowns and operations involved. Our focus here is on equations involving consecutive integers.

Consecutive integers are numbers that follow each other in order without any gaps. For example, 51, 52, and 53 are consecutive integers. To solve problems involving these integers, you often set up equations where the integers are represented in terms of a variable, usually called \( x \).

In the exercise given, we defined the first integer as \( x \), and the next ones as \( x + 1 \) and \( x + 2 \). The challenge was then to find these integers whose sum is 156. By forming and solving an equation, we can easily find the value of \( x \) and thus determine the actual integers.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that together describe a particular value or set of values. In our exercise, the expression \( x + (x + 1) + (x + 2) \) is an algebraic expression that represents the sum of three consecutive integers.

Creating algebraic expressions is a fundamental skill in algebra. These expressions can be manipulated to find solutions to equations. The key steps here involve using operations such as addition, subtraction, and sometimes division and multiplication to simplify expressions.

One thing to note is the importance of combining like terms. In our example, \( x + (x + 1) + (x + 2) \) can be simplified to \( 3x + 3 \). Recognizing and simplifying expressions this way is critical in mathematical problem-solving, as it makes equations easier to manage and solve.

Mathematical Problem-Solving

Mathematical problem-solving is the approach you use to find solutions to math-related questions or problems. It often starts with understanding the problem and defining variables, followed by setting up equations, solving them, and interpreting the results.

In the context of consecutive integer problems, it's essential to:

• Recognize the pattern among the integers.
• Define variables that represent these unknowns.
• Set up an equation based on given conditions (like the sum of the integers).
• Simplify and solve the equation systematically.

In our exercise, each step was carefully laid out to ensure a clear path to finding the answer. Solving for \( x \) using simple arithmetic operations like subtraction and division is a straightforward part of this process.
Understanding each of these strategies can greatly enhance your proficiency in solving integer equation problems, helping you to effectively tackle more complex mathematical challenges in the future.