Problem 31 - Algebra A Combined Function

Question

The left and right page numbers of an open book are two consecutive integers whose sum is 469 . Find these page numbers.

Step-by-Step Solution

Answer

The page numbers are 234 and 235.

Step 1: Understand the Problem Statement

We are given that the left and right page numbers of an open book are two consecutive integers. Their sum equals 469. We need to find these two page numbers.

Step 2: Define the Consecutive Integers

Let the left page number be denoted by the integer \(x\). Consequently, the right page number will be the next consecutive integer, which is \(x + 1\).

Step 3: Set Up the Equation

The problem states that the sum of these two consecutive integers is 469. We can represent this situation with an equation: \[ x + (x + 1) = 469 \]

Step 4: Solve the Equation

Begin simplifying the equation: \[ x + x + 1 = 469 \]Combine like terms:\[ 2x + 1 = 469 \]Subtract 1 from both sides: \[ 2x = 468 \]Divide both sides by 2:\[ x = 234 \]

Step 5: Determine the Consecutive Integers

Since \(x = 234\), this means the left page number is 234. Therefore, the right page number, being \(x + 1\), is 235.

Key Concepts

Summing Numbers

In mathematics, one of the fundamental concepts is summing numbers, which is simply the process of adding them together to find their combined total. This exercise involves adding two consecutive integers to find a specific sum.
Consecutive integers are numbers that follow each other in order. For example, 1 and 2 are consecutive, as are 234 and 235. In this problem, the sum of these consecutive numbers is provided, which in this case is 469.
When you're asked to sum numbers in a problem, you are essentially calculating their total value. Here, the task is to identify two consecutive integers such that their sum equals 469.

Algebraic Equation

Algebraic equations are mathematical sentences that show the equality between two expressions. In this exercise, we use an algebraic equation to solve for two unknown integers. Here's how it works:
First, you assign a variable, say \(x\), to one of the numbers. The other consecutive number can then be represented as \(x + 1\). This allows you to create an equation that represents the problem.
For example:

• Let the first page number be \(x\).
• Then, the next consecutive page number will be \(x + 1\).
• The equation formed is \(x + (x + 1) = 469\).

The equation sets up the relationship specified in the problem statement. Solving this equation enables us to determine the actual values of the two consecutive integers.

Problem-Solving Steps

When approaching any mathematical problem, effective problem-solving steps are crucial. For this problem, we break it down systematically:
1. **Understand the Problem:** Begin by reading and comprehending what is being asked. Identify that you need two consecutive integers that sum to 469.
2. **Set Up Variables:** Identify what you need to find and represent the numbers using a variable. We used \(x\) for the first integer.
3. **Formulate an Equation:** Use the relationship provided in the problem (sum equals 469) to set up an equation: \(2x + 1 = 469\).
4. **Solve the Equation:** Solve for \(x\) using algebraic operations like addition, subtraction, and division, until you find its value: \(x = 234\).
5. **Conclude with Results:** Finally, apply the solution to find that the two page numbers are 234 and 235.
Following these organized steps ensures you accurately solve problems while building a strong foundation in algebraic thinking.