Problem 14 - 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? Draw a diagram of the cover.

Step-by-Step Solution

Answer

The photos should each be \(2 \frac{1}{4}\) inches wide.

Step 1: Convert all measurements to the same unit

The total cover width and the margins are provided in inches and fractions of an inch. First, it's important to convert all measurements to the same unit for consistency. Here, all measurements are converted to fractions of an inch. The total width of the cover is \(6 \frac{1}{2}\) inches which is equal to \(\frac{13}{2}\) inches.

Step 2: Sum up the spaces taken by margins and between photos

Sum the left and right margins, and the space between the photos. So, the total space taken up by the margins and between the photos equals \(2 \times \frac{3}{4}\) inches (margins) + \(\frac{1}{2}\) inch (space between photos) = \(\frac{3}{2}\) inches + \(\frac{1}{2}\) inch = 2 inches.

Step 3: Calculate the width for the photos

Subtract the total space taken by the margins and between the photos from the total cover width to get the total space for the photos. So, the total available space for the photos equals \(\frac{13}{2}\) inches - 2 inches = \(\frac{9}{2}\) inches. Since there will be two photos across the width, the width of each photo is \(\frac{9}{2}\) inches divided by 2, which gives us \(\frac{9}{4}\) inches or \(2 \frac{1}{4}\) inches.

Key Concepts

Measurement Conversion

When working with different measurements, it's crucial to convert everything into a consistent format. This eases calculations and reduces errors. Initially, when you have a mixed number like \(6 \frac{1}{2}\) inches, you should convert it to an improper fraction to simplify arithmetic. Here, \(6 \frac{1}{2}\) becomes \(\frac{13}{2}\) inches.
It's helpful to practice going back and forth between mixed numbers and improper fractions, as each has its own benefits. Mixed numbers are easier to read and visualize, while improper fractions make arithmetic operations a breeze.
Remember, when converting fractions, all dimensions should match the eventual format required in your solution. Consistency is key!

Algebraic Reasoning

Algebraic reasoning involves identifying the variables and relationships between different quantities in a problem. In this exercise, the problem is finding out the width of each photo by using given measurements. You need to first determine what needs to be subtracted from the total width, such as the margins and space between photos.

Using algebraic reasoning:

• Identify total given width: \(\frac{13}{2}\) inches.
• Understand the contribution of each margin: \(2 \times \frac{3}{4}\) inches.
• Account for space between photos: \(\frac{1}{2}\) inch.

Now by adding these constraints together, \(\frac{3}{2} + \frac{1}{2} = 2\) inches, you can calculate the remaining width for the photos. This type of reasoning makes it easy to break down complex problems into smaller, more manageable parts.

Problem-Solving in Geometry

In geometric problems, especially those involving spaces and arrangements like in designing covers, visualization is crucial. Drawing diagrams helps illustrate the problem and ensures a clear understanding of all elements involved.
To solve the given exercise, visualize the cover by sketching a rectangle. Label its entire width as \(6 \frac{1}{2}\) inches. Then mark the left and right margins of \(\frac{3}{4}\) inch each and the gap between two photos as \(\frac{1}{2}\) inch.
This visual setup confirms space allocation and allows for easier manipulation and calculation of remaining areas. Therefore, by showing this, you can see how each photo fit within the available space calculated earlier, thus ensuring geometric reasonings align closely with algebraic calculations.