Question

You are designing a newspaper page with three photos. The page is \(13 \frac{1}{4}\) inches wide with 1 inch margins on both sides. You need to allow \(\frac{3}{4}\) inch between photographs. How wide should you make the photos if they are of equal size? Sketch a diagram of the newspaper page.

Short answer

The width of each photo should be \(3 \frac{1}{4}\) inches.

Step-by-step solution

Determine Total Available Space

First, we need to calculate the total space available for photos on the page. The total width of the page is \(13 \frac{1}{4}\) inches and we have 1-inch margins on both side. This makes a total of 2 inches of margins. This means the space available for the photos (and the spaces in between) is \(13 \frac{1}{4}\) inches - 2 inches = \(11 \frac{1}{4}\) inches.

Determine Space Occupied by Spaces Between Photos

We need to calculate the total amount of space taken by the spaces between photos. We have 2 spaces, each of \(\frac{3}{4}\) inch. So total space occupied by the spaces between the photos is 2 * \(\frac{3}{4}\) inches = \(1 \frac{1}{2}\) inches.

Calculate Space Available for Photos

Finally, we calculate the total space available for the photos. The space for photos is given by the total available space - the space taken by spaces between photos. We found that the total available space is \(11 \frac{1}{4}\) inches and the total space consumed by the spaces is \(1 \frac{1}{2}\) inches. This means the space available for photos is \(11 \frac{1}{4}\) inches - \(1 \frac{1}{2}\) inches = \(9 \frac{3}{4}\) inches.

Determine Width of Each Photo

Since all the photos are of the same size, the width of each photo will be the total space available for photos divided by the number of photos which is 3 in this case. We have \(9 \frac{3}{4}\) inches of space available for photos. Hence the width of each photo will be \(9 \frac{3}{4}\) inches / 3 = \(3 \frac{1}{4}\) inches.

Key concepts

Linear Equations

Linear equations are fundamental tools in algebra that represent the relationship between two variables, typically on a two-dimensional plane. These equations can be used to solve for unknown quantities and are characterized by each term being either a constant or the product of a constant and a single variable.

In the context of the newspaper page design, the problem translates to a linear equation when we want to define the width of each photo. We express the total width equation as the sum of the widths of the three photos plus the spaces between them. This simple equation can be written as follows:
\[3x + 2y = W\]
where \(x\) is the width of each photo, \(y\) is the width of the space between the photos, and \(W\) is the total width available for the photos and spaces. We can solve this equation for \(x\), which represents the width needed for each of the three photos.

Fraction Operations

Operations with fractions are a crucial part of algebra, particularly when dealing with real-life problems. In our exercise, fraction operations allow us to subtract the margins and spaces from the total width to find the width available for the photos.

Here's how to perform these operations in our scenario:

• Subtract the margins (2 inches in total) from the page width to calculate the available space.
• Multiply the number of spaces (2) by the width of each space (\(\frac{3}{4}\) inch) to find the total space occupied by the spaces.
• Finally, subtract the space occupied by the spaces from the available space to find the total width that can be allocated to the photos.

Understanding how to work with fractions is essential as they frequently appear in algebraic problems, and mastering these operations is key for accurate calculations.

Problem-Solving Strategies

Problem-solving strategies in algebra involve a step-by-step approach to break down complex problems into manageable parts. The newspaper page design task showcases this strategy effectively.

The steps followed to reach a solution are:

• Defining the problem and understanding the requirements.
• Breaking down the problem into smaller, calculable parts (like margins, spaces between photos, and photo widths).
• Using algebraic methods (like equations and operations with fractions) to systematically solve each part.
• Combining the solutions of the individual parts to solve the overall problem.

Such strategies not only provide structure to the problem-solving process but also make it easier to identify and correct errors along the way.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations that represent a specific value. In our newspaper page example, the space each photo will take (\(3 \frac{1}{4}\) inches) can be seen as an algebraic expression. It is important to understand the role of each component within the expression:

• Numbers and fractions represent known quantities, like the width of the page or the margins.
• Variables stand in for the unknowns that we need to solve for, such as the width of each photo.
• Mathematical operations (add, subtract, multiply, divide) tell us how these quantities interact with each other.

Recognizing and manipulating algebraic expressions is critical in solving algebra problems because it allows one to abstract and simplify complex scenarios into workable mathematics.