Problem 15 - Algebra 1

Question

Pover DESIGN In Exercises \(14-16\), you want the cover of a sports media guide to show two photos across its width. The cover is \(6 \frac{1}{2}\) inches wide, and the left and right margins are each \(\frac{3}{4}\) inch. The space between the photos is \(\frac{1}{2}\) inch. How wide should you make the photos? \- Write an equation to model the problem. Use your diagram to help.

Step-by-Step Solution

Answer

Each photo should be \(2 \frac{3}{4}\) inches wide.

Step 1: Identify the given measures

The total width of the cover is \(6 \frac{1}{2}\) inches, the left and right margins are each \(\frac{3}{4}\) inch, and the space between the photos is \(\frac{1}{2}\) inch.

Step 2: Transform the width

First, convert \(6 \frac{1}{2}\) to an improper fraction to facilitate the calculation. Arriving at the result of \(13/2\) inches for the total width.

Step 3: Formulate the equation

The equation should demonstrate that the total width of the cover ( \(13/2\) inches) is equal to the sum of the space occupied by the two photos, the margins, and the space between the photos. This can be represented as: width = 2*photo width + left margin + right margin + space between photos. If we let x represent the width of the photos, the equation becomes: \(13/2\) = 2x + \(\frac{3}{4}\) + \(\frac{3}{4}\) + \(\frac{1}{2}\).

Step 4: Simplify the equation

Adding up the margins and space between photos: \(\frac{3}{4}\) + \(\frac{3}{4}\) + \(\frac{1}{2}\) = \(\frac{2}{2}\). Substituting in the equation, there is: \(13/2\) = 2x + 2

Step 5: Solve for x (photo width)

The equation can be rearranged to solve for x by subtracting 2 from both sides of the equation which gives: \(11/2\) = 2x. To isolate x, divide both sides by 2, so x = \(\frac{11}{4}\) or \(2\frac{3}{4}\) inches.

Key Concepts

Problem-Solving

In problem-solving, we first need to understand the situation entirely. This means identifying all elements given in the problem and setting them up to find the missing piece. In this exercise, we are tasked with finding how wide each photo on the cover should be.
To approach this:

• Identify what information you have: You know the total width of the cover, the margins, and the space between the photos.
• Understand what you need to find: The width of each photo.
• Create a plan: Use this information to form an equation that can help find the unknown.

By organizing the data and relationships, forming an equation becomes practical, and you can move towards solving it.

Fractions

Fractions are central in this exercise as measurements are given in mixed numbers and fractions.
When dealing with fractions:

• Transform mixed numbers into improper fractions to make calculations straightforward.
• Keep track of all fractions and their simplifications.

For instance, converting the mixed number height of the cover, \(6 \frac{1}{2}\), becomes \(\frac{13}{2}\), making it easier to incorporate into the equation.
Additionally, adding fractions like the margin (\(\frac{3}{4}\)) and the space between the photos (\(\frac{1}{2}\)) requires finding a common denominator, which is crucial for accurate and simplified results.

Geometry

Understanding basic geometric concepts helps in visualizing the problem. Geometry allows us to comprehend how the elements fit together on the cover.
Key aspects to note are:

• The cover's total width includes margins and spaces, which are geometric properties representing real-world constraints.
• Sketching a diagram can oftentimes clarify spatial relationships and help in forming the equation.

Visual tools like diagrams not only make the understanding better but also help reinforce the logical thinking required to solve the problem.

Modeling Equations

Modeling equations involve creating a mathematical representation of a real-world problem, which is what we have done here.
The process follows these steps:

• Identify every component contributing to the total width: two photos, two margins, and the gap between photos.
• Create a mathematical equation using these components.
• Represent the unknown values with variables—in this case, the width of each photo is \(x\).
• Use simple algebra to solve the equation, i.e., finding \(x\) in the equation \(\frac{11}{2} = 2x\).

This technique breaks down complex measurements into approachable calculations, helping solve everyday space and design-related queries.