Problem 40 - Prealgebra

Question

NUMBER SENSE Three times the sum of three consecutive integers \(x\) \(x+1,\) and \(x+2,\) is 72. What are the integers?

Step-by-Step Solution

Answer

The integers are 7, 8, and 9.

Step 1: Identify the integers

Let's represent the three consecutive integers as \( x \), \( x+1 \), and \( x+2 \).

Step 2: Set up the equation

The problem states that three times the sum of these integers is 72. First, find the sum of these integers, which is \( x + (x + 1) + (x + 2) = 3x + 3 \). Then, set up the equation for three times the sum: \( 3(3x + 3) = 72 \).

Step 3: Simplify the equation

Simplify the equation by distributing the 3: \( 9x + 9 = 72 \).

Step 4: Solve for \( x \)

Subtract 9 from both sides of the equation to isolate terms with \( x \) on one side: \( 9x = 63 \). Then, divide both sides by 9 to solve for \( x \): \( x = 7 \).

Step 5: Determine the three consecutive integers

Substitute \( x = 7 \) into the expressions for the integers: \( x = 7 \), \( x+1 = 8 \), \( x+2 = 9 \). Therefore, the consecutive integers are 7, 8, and 9.

Key Concepts

Number Sense

In mathematics, number sense refers to a basic understanding of numbers and their relationships. When dealing with consecutive integers, we're talking about numbers that follow one right after another without any gaps. For example, 4, 5, and 6 are consecutive integers. This idea helps us recognize patterns and make sense of mathematical problems.

To deepen number sense, consider the properties of integers. Each consecutive integer can be represented by a simple algebraic expression. This representation helps us develop equations effortlessly when solving problems. In the exercise, we used the terms \( x \), \( x+1 \), and \( x+2 \). Understanding this pattern simplifies arithmetic calculations and enhances logical thinking skills.

Developing number sense is crucial as it improves problem-solving abilities and allows for quick reasoning in contexts involving numerical data. It enables students to check their work instinctively and make accurate estimations.

Equations

Equations are mathematical statements that assert the equality of two expressions. They play a vital role in mathematics as they allow us to solve problems by finding the values of unknown variables.

In the given exercise, setting up the equation involves expressing mathematical conditions using algebra. We initially summed the consecutive integers: \( x + (x+1) + (x+2)\) to obtain \( 3x + 3 \). The problem specified three times this sum, leading us to formulate \(3(3x + 3) = 72\).

Through these steps, we standardize the problem-making process. Simplifying the equation by distributing numbers and isolating the variable, we reveal the solution. It's akin to unraveling layers until we get to the heart of the problem.

When solving such equations, remember:

• Distribute coefficients as needed
• Combine like terms for simplicity
• Keep balancing both sides to maintain equality
• Systematically isolate the variable
• Verify the results for accuracy

These practices are essential to mastering equations.

Problem Solving

Problem solving is one of the key skills developed through mathematics. It involves not only understanding the problem but also devising a way to tackle it. In our exercise, recognizing and defining the problem were foundational steps.

Firstly, understanding the requirement helps. Here, it was "three times the sum of three consecutive integers equals 72". This guided the entire process.

Next, we designed a plan by setting up an equation based on the problem description. The steps involved logical procedures: identifying integers, summing them, and equating to the given total. As we worked through the equation, simplifying and solving it revealed the solution, which are the integers 7, 8, and 9.

Problem solving tips:

• Clearly understand what the problem is asking
• Organize knowns and unknowns efficiently
• Follow a structured procedure (like solving equations)
• Look for patterns or connections
• Review the process to confirm the solution is correct

Effective problem-solving skills boost analytical thinking, a valuable tool extending beyond mathematics.