Problem 25 - Introductory and Intermediate Algebra for College Students 4th

Question

Use the fact that page numbers on facing pages of a book are consecutive integers. The sum of the page numbers on the facing pages of a book is \(629 .\) What are the page numbers?

Step-by-Step Solution

Answer

The page numbers on the facing pages of the book are 314 and 315.

Step 1: Interpret the problem and denote the two numbers

Let's say the page number on the left page is \(x\). Because page numbers on facing pages of a book are consecutive integers, the page number on the right page will be \(x+1\).

Step 2: Set up the equation

According to the problem, the sum of the page numbers on the facing pages is 629. So, we can set up the equation as: \(x + (x + 1) = 629\).

Step 3: Solve the equation

Combine like terms to simplify the equation: \(2x + 1 = 629\). Then to isolate \(2x\), we subtract 1 from both sides of our equation: \(2x = 629 - 1 = 628\). Finally, we solve for \(x\), by dividing both sides of the equation by 2: \(x = 628 / 2 = 314\).

Step 4: Find the consecutive number

The next page (right side) would be \(x+1\), so we just need to add 1 to our previous result: \(314 + 1= 315\).

Key Concepts

Page Numbers

In books, page numbers are typically arranged in a sequence. Particularly for a pair of facing pages, these numbers are consecutive integers. This means that if one page has a number, the next page has a number that is one more than the previous one. So, for example, pages marked with 50 and 51 are consecutive. Understanding this concept is crucial when dealing with problems related to page numbers, because it helps us discover relationships between them. Facing pages in a book always follow this consecutive pattern, and knowing this can help you set up mathematical models to solve related problems.

Equations

An equation is a mathematical statement that asserts the equality of two expressions. In the context of consecutive page numbers, we often use equations to represent their relationships. For instance, if we let the page number on the left be denoted by \(x\), the page on the right will be \(x+1\) because it is the next consecutive number.In our problem, the equation \[x + (x + 1) = 629\]is set up to represent the sum of these two consecutive page numbers. An equation like this succinctly summarizes the entire relationship in a mathematical form, allowing us to perform operations and subsequently solve for unknown variables.

Solving Equations

Solving equations involves finding the value(s) of the variable(s) that make the equation true. This can involve various techniques such as simplifying expressions, combining like terms, and performing operations such as addition, subtraction, multiplication, or division on both sides of the equation.In our example, after writing the equation \[2x + 1 = 629\]based on the relationship between the consecutive pages, we need to simplify and isolate the variable \(x\). We first subtract 1 from each side, resulting in \[2x = 628\]Then, to find \(x\), we divide both sides by 2, giving us \[x = 314\]This tells us that the first page number is 314. Finally, since the goal is to also find the next consecutive number, we add 1 to 314, resulting in 315. This kind of step-by-step problem-solving is fundamental for solving equations effectively and confirms our understanding of the relationship between numbers.