Problem 79 - Elementary Algebra

Question

The sum of three consecutive integers is 60 . Find the integers.

Step-by-Step Solution

Answer

The integers are 19, 20, and 21.

Step 1: Understanding Consecutive Integers

Consecutive integers are numbers that follow each other in order, without any numbers missing in between. If we denote the first integer as \( x \), then the next two consecutive integers are \( x+1 \) and \( x+2 \).

Step 2: Setting Up the Equation

Since we have the sum of these three consecutive integers equal to 60, we can set up an equation: \( x + (x+1) + (x+2) = 60 \).

Step 3: Simplifying the Equation

Combine the like terms in the equation: \( 3x + 3 = 60 \). This simplifies the expression to make it easier to solve.

Step 4: Solving for the First Integer

To solve for \( x \), subtract 3 from both sides of the equation: \( 3x = 57 \). Then, divide both sides by 3 to find \( x \): \( x = 19 \).

Step 5: Finding All Three Integers

The first integer is \( 19 \). Since they must be consecutive, the next integers are \( x+1 = 20 \) and \( x+2 = 21 \). Thus, the three consecutive integers are 19, 20, and 21.

Key Concepts

Consecutive Integers

Consecutive integers are numbers that appear right after each other in order. To understand this better, think of the sequence of natural numbers: 1, 2, 3, 4, 5, and so on. Each number comes immediately after the last one without skipping any numbers. For example:

• The numbers 4, 5, and 6 are consecutive integers because they follow one after the another without any gap.
• Numbers like 7, 8, and 9 are also consecutive.

When dealing with problems involving consecutive integers, it's common to express them with algebraic expressions, such as:

• Let the first integer be denoted by \( x \).
• The next consecutive integer would be \( x+1 \).
• The integer after that is \( x+2 \).

This pattern helps in setting up equations that allow us to solve various math problems involving sequences of numbers.

Mathematical Problem Solving

Mathematical problem solving is a systematic approach to finding solutions to problems. It often involves understanding the problem, setting up equations, and solving them. Let's break it down:

• Understanding the Problem: First, we must interpret what the problem is asking. For instance, if we are to find the sum of three consecutive integers, we need to know what consecutive integers are.
• Setting Up an Equation: Once we understand the problem, we express it in terms of mathematical equations or expressions. Continuing with our example, we'd set up an equation that represents the relationship between the three integers and their sum.
• Simplifying and Solving: It's crucial to simplify the equation by combining like terms and performing basic arithmetic operations. After simplification, we solve the equation to find the values of the unknown variables.

Effective mathematical problem solving involves a clear understanding of basic math concepts and careful step-by-step execution.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. They are a fundamental part of algebra and are very useful for solving equations.In the context of solving equations involving consecutive integers, algebraic expressions represent unknown values and their relationships. For instance, let's consider the expression \( x, x+1, x+2 \) for three consecutive integers:

• Here, \( x \) stands for the first integer.
• \( x+1 \) represents the next integer in the sequence.
• \( x+2 \) is the third integer.

Using algebraic expressions makes it easier to visualize and solve problems, especially when those problems involve operations like addition and are part of an equation, such as finding their sum.